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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Binary-Report Peer Prediction for Real-Valued Signal Spaces</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/2503.16280v1/"/></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/2503.16280v1#S1" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><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/2503.16280v1#S1.SS1" title="In 1 Introduction ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Motivating example: Output Agreement (OA)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S1.SS2" title="In 1 Introduction ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S1.SS3" title="In 1 Introduction ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Related Work</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Output Agreement</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/2503.16280v1#S2.SS1" title="In 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS2" title="In 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Equilibrium Characterization</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/2503.16280v1#S2.SS2.SSS0.Px1" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Results.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS3" title="In 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Dynamics</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="In 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>A Gaussian Model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Dasgupta-Ghosh</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/2503.16280v1#S3.SS1" title="In 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Equilibrium Characterization</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/2503.16280v1#S3.SS1.SSS0.Px1" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Equilibrium results.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.SS2" title="In 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Dynamics</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.SS3" title="In 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Gaussian Model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Determinant-based Mutual Information (DMI) Mechanism</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/2503.16280v1#S4.SS1" title="In 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Equilibrium Characterization</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/2503.16280v1#S4.SS1.SSS0.Px1" title="In 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Equilibrium results.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4.SS1.SSS0.Px2" title="In 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Beyond consistent threshold strategies.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Robust Bayesian Truth Serum</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/2503.16280v1#S5.SS1" title="In 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Equilibrium Characterization</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.SS2" title="In 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Gaussian Model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Experiments</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/2503.16280v1#S6.SS1" title="In 6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Skewness</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6.SS2" title="In 6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Multimodality</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Discussion</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/2503.16280v1#S7.SS0.SSS0.Px1" title="In 7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Adding flexibility.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7.SS0.SSS0.Px2" title="In 7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Beyond binary reports.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7.SS0.SSS0.Px3" title="In 7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Effort.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7.SS0.SSS0.Px4" title="In 7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title">Further broadening the model.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A1" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Output Agreement</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/2503.16280v1#A1.SS1" title="In Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.1 </span>Equilibrium Characterization Generalization</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A1.SS2" title="In Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.2 </span>Dynamics Generalization</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Dasgupta-Ghosh</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/2503.16280v1#A2.SS1" title="In Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Omitted Proofs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2.SS2" title="In Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.2 </span>Equilibrium Characterization Generalization</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2.SS3" title="In Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.3 </span>Dynamics Generalization</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A3" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Omitted Proofs for DMI</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/2503.16280v1#A3.SS1" title="In Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.1 </span>Proof of Theorem <span class="ltx_text ltx_ref_tag">9</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A4" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Omitted Proofs for RBTS</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/2503.16280v1#A4.SS1" title="In Appendix D Omitted Proofs for RBTS ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1 </span>Gaussian Model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A5" title="In Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">E </span>Omitted Experiments</span></a></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">Binary-Report Peer Prediction for Real-Valued Signal Spaces</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Rafael Frongillo <br class="ltx_break"/>University of Colorado Boulder <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id1.1.id1">raf@colorado.edu</span> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ian Kash <br class="ltx_break"/>University of Illinois Chicago <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id2.1.id1">iankash@uic.edu</span> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Mary Monroe <br class="ltx_break"/>University of Colorado Boulder <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id3.1.id1">mary.monroe@colorado.edu</span> </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id4.id1">Theoretical guarantees about peer prediction mechanisms typically rely on the discreteness of the signal and report space. However, we posit that a discrete signal model is not realistic: in practice, agents observe richer information and map their signals to a discrete report. In this paper, we formalize a model with real-valued signals and binary reports. We study a natural class of symmetric strategies where agents map their information to a binary value according to a single real-valued threshold. We characterize equilibria for several well-known peer prediction mechanisms which are known to be truthful under the binary report model. In general, even when every threshold would correspond to a truthful equilibrium in the binary signal model, only certain thresholds remain equilibria in our model. Furthermore, by studying the dynamics of this threshold, we find that some of these equilibria are unstable. These results suggest important limitations for the deployment of existing peer prediction mechanisms in practice.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.3">Eliciting high-quality information from agents without verification is a well-studied problem, motivated by settings such as peer grading or data labeling where the ground truth is unknown, subjective, or hard to acquire. <em class="ltx_emph ltx_font_italic" id="S1.p1.3.1">Peer prediction</em> mechanisms aim to incentivize truthful behavior of agents by paying them based on how their reports correlate with other submissions, without the need for ground truth. Agents are typically modeled as receiving a signal, for example <math alttext="H" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_H</annotation></semantics></math> (“high”) or <math alttext="L" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_L</annotation></semantics></math> (“low”) for the quality of an essay, from a fixed joint probability distribution <math alttext="P" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mi id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_P</annotation></semantics></math> unknown to the mechanism. Upon seeing their signal, agents update their beliefs and, under the right conditions, are incentivized to truthfully report the signal they receive.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.3">In practice, however, the signal space of information that agents receive is much richer than the small number of categories typically studied in the literature. An essay contains far more information than a simple <math alttext="H" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_H</annotation></semantics></math> or <math alttext="L" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_L</annotation></semantics></math> impression. As a result, one would expect the same agent to form a range of posterior beliefs based on these more nuanced impressions. What are the implications of this more nuanced signal space?<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>One implication which we do not focus on is the well-known issue of spurious correlation, such as adopting a strategy of conditioning one’s report on the first word of the essay; see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7" title="7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a>.</span></span></span> Existing results often rely on a clear separation in the joint distribution <math alttext="P" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">P</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_P</annotation></semantics></math> to control the incentives, making it unclear how one would expect these mechanisms to perform in practice.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.5">To address this question, we introduce and study a model of binary-report peer prediction for real-valued signal spaces. In this model, agents receive a real number as their signal and then based on it select a report of <math alttext="H" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_H</annotation></semantics></math> or <math alttext="L" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_L</annotation></semantics></math>. We focus on the natural class of threshold strategies, where an agent reports <math alttext="H" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_H</annotation></semantics></math> if and only if their signal exceeds a particular threshold. For example, an agent may report <math alttext="H" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_H</annotation></semantics></math> if they deem the quality of an essay to be above 0.7, and <math alttext="L" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mi id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><ci id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">italic_L</annotation></semantics></math> otherwise.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">We generally find that mechanisms which are truthful under the original binary signal model fail to yield correct incentives under the more nuanced model with real-valued signals. Intuitively, agents with signals near the threshold may have an incentive to misreport, as the separation that enforces truthfulness breaks down. We give necessary and sufficient conditions for a threshold to be an equilibrium, i.e., not suffer from this issue, in a variety of peer prediction mechanisms. Furthermore we study <em class="ltx_emph ltx_font_italic" id="S1.p4.1.1">dynamics</em> arising if a small fraction of agents with the greatest incentive to do so change their report, causing the threshold to slowly shift. We then distinguish <em class="ltx_emph ltx_font_italic" id="S1.p4.1.2">stable</em> equilibria under our dynamics as reasonable to arise naturally, from unstable equilibria.</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>Motivating example: Output Agreement (OA)</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.8">Consider Output Agreement (OA), which provides a reward of 1 if an agent’s report agrees with a peer and 0 otherwise <cite class="ltx_cite ltx_citemacro_citep">[Von Ahn and Dabbish, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib17" title="">2004</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib18" title="">2008</a>]</cite>. In our more nuanced, real-valued signal model, each agent <math alttext="i" 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" xref="S1.SS1.p1.1.m1.1.1.cmml">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">italic_i</annotation></semantics></math> recieves a signal we model as a random variable <math alttext="X_{i}=Z+Z_{i}" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.1"><semantics id="S1.SS1.p1.2.m2.1a"><mrow id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml"><msub id="S1.SS1.p1.2.m2.1.1.2" xref="S1.SS1.p1.2.m2.1.1.2.cmml"><mi id="S1.SS1.p1.2.m2.1.1.2.2" xref="S1.SS1.p1.2.m2.1.1.2.2.cmml">X</mi><mi id="S1.SS1.p1.2.m2.1.1.2.3" xref="S1.SS1.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p1.2.m2.1.1.1" xref="S1.SS1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S1.SS1.p1.2.m2.1.1.3" xref="S1.SS1.p1.2.m2.1.1.3.cmml"><mi id="S1.SS1.p1.2.m2.1.1.3.2" xref="S1.SS1.p1.2.m2.1.1.3.2.cmml">Z</mi><mo id="S1.SS1.p1.2.m2.1.1.3.1" xref="S1.SS1.p1.2.m2.1.1.3.1.cmml">+</mo><msub id="S1.SS1.p1.2.m2.1.1.3.3" xref="S1.SS1.p1.2.m2.1.1.3.3.cmml"><mi id="S1.SS1.p1.2.m2.1.1.3.3.2" xref="S1.SS1.p1.2.m2.1.1.3.3.2.cmml">Z</mi><mi id="S1.SS1.p1.2.m2.1.1.3.3.3" xref="S1.SS1.p1.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.1b"><apply id="S1.SS1.p1.2.m2.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1"><eq id="S1.SS1.p1.2.m2.1.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1.1"></eq><apply id="S1.SS1.p1.2.m2.1.1.2.cmml" xref="S1.SS1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p1.2.m2.1.1.2.1.cmml" xref="S1.SS1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S1.SS1.p1.2.m2.1.1.2.2.cmml" xref="S1.SS1.p1.2.m2.1.1.2.2">𝑋</ci><ci id="S1.SS1.p1.2.m2.1.1.2.3.cmml" xref="S1.SS1.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S1.SS1.p1.2.m2.1.1.3.cmml" xref="S1.SS1.p1.2.m2.1.1.3"><plus id="S1.SS1.p1.2.m2.1.1.3.1.cmml" xref="S1.SS1.p1.2.m2.1.1.3.1"></plus><ci id="S1.SS1.p1.2.m2.1.1.3.2.cmml" xref="S1.SS1.p1.2.m2.1.1.3.2">𝑍</ci><apply id="S1.SS1.p1.2.m2.1.1.3.3.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p1.2.m2.1.1.3.3.1.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.SS1.p1.2.m2.1.1.3.3.2.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3.2">𝑍</ci><ci id="S1.SS1.p1.2.m2.1.1.3.3.3.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.1c">X_{i}=Z+Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_Z + italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="Z" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.1"><semantics id="S1.SS1.p1.3.m3.1a"><mi id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.1b"><ci id="S1.SS1.p1.3.m3.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.1d">italic_Z</annotation></semantics></math> and <math alttext="Z_{i}" class="ltx_Math" display="inline" id="S1.SS1.p1.4.m4.1"><semantics id="S1.SS1.p1.4.m4.1a"><msub id="S1.SS1.p1.4.m4.1.1" xref="S1.SS1.p1.4.m4.1.1.cmml"><mi id="S1.SS1.p1.4.m4.1.1.2" xref="S1.SS1.p1.4.m4.1.1.2.cmml">Z</mi><mi id="S1.SS1.p1.4.m4.1.1.3" xref="S1.SS1.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.4.m4.1b"><apply id="S1.SS1.p1.4.m4.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.4.m4.1.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S1.SS1.p1.4.m4.1.1.2.cmml" xref="S1.SS1.p1.4.m4.1.1.2">𝑍</ci><ci id="S1.SS1.p1.4.m4.1.1.3.cmml" xref="S1.SS1.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.4.m4.1c">Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.4.m4.1d">italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are both unit normals with <math alttext="Z" class="ltx_Math" display="inline" id="S1.SS1.p1.5.m5.1"><semantics id="S1.SS1.p1.5.m5.1a"><mi id="S1.SS1.p1.5.m5.1.1" xref="S1.SS1.p1.5.m5.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.5.m5.1b"><ci id="S1.SS1.p1.5.m5.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.5.m5.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.5.m5.1d">italic_Z</annotation></semantics></math> shared by all agents and <math alttext="Z_{i}" class="ltx_Math" display="inline" id="S1.SS1.p1.6.m6.1"><semantics id="S1.SS1.p1.6.m6.1a"><msub id="S1.SS1.p1.6.m6.1.1" xref="S1.SS1.p1.6.m6.1.1.cmml"><mi id="S1.SS1.p1.6.m6.1.1.2" xref="S1.SS1.p1.6.m6.1.1.2.cmml">Z</mi><mi id="S1.SS1.p1.6.m6.1.1.3" xref="S1.SS1.p1.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.6.m6.1b"><apply id="S1.SS1.p1.6.m6.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.6.m6.1.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="S1.SS1.p1.6.m6.1.1.2.cmml" xref="S1.SS1.p1.6.m6.1.1.2">𝑍</ci><ci id="S1.SS1.p1.6.m6.1.1.3.cmml" xref="S1.SS1.p1.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.6.m6.1c">Z_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.6.m6.1d">italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> unique to agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p1.7.m7.1"><semantics id="S1.SS1.p1.7.m7.1a"><mi id="S1.SS1.p1.7.m7.1.1" xref="S1.SS1.p1.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.7.m7.1b"><ci id="S1.SS1.p1.7.m7.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.7.m7.1d">italic_i</annotation></semantics></math>. In a peer grading setting the signals <math alttext="X_{i}" class="ltx_Math" display="inline" id="S1.SS1.p1.8.m8.1"><semantics id="S1.SS1.p1.8.m8.1a"><msub id="S1.SS1.p1.8.m8.1.1" xref="S1.SS1.p1.8.m8.1.1.cmml"><mi id="S1.SS1.p1.8.m8.1.1.2" xref="S1.SS1.p1.8.m8.1.1.2.cmml">X</mi><mi id="S1.SS1.p1.8.m8.1.1.3" xref="S1.SS1.p1.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.8.m8.1b"><apply id="S1.SS1.p1.8.m8.1.1.cmml" xref="S1.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.8.m8.1.1.1.cmml" xref="S1.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S1.SS1.p1.8.m8.1.1.2.cmml" xref="S1.SS1.p1.8.m8.1.1.2">𝑋</ci><ci id="S1.SS1.p1.8.m8.1.1.3.cmml" xref="S1.SS1.p1.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.8.m8.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.8.m8.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> might capture a fine-grained assessment of the quality of an assignment.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.6">In contrast, the traditional signal structure for OA is usually taken to be finite, so take the simplest binary case where the set of signals is <math alttext="\{H,L\}" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.2"><semantics id="S1.SS1.p2.1.m1.2a"><mrow id="S1.SS1.p2.1.m1.2.3.2" xref="S1.SS1.p2.1.m1.2.3.1.cmml"><mo id="S1.SS1.p2.1.m1.2.3.2.1" stretchy="false" xref="S1.SS1.p2.1.m1.2.3.1.cmml">{</mo><mi id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml">H</mi><mo id="S1.SS1.p2.1.m1.2.3.2.2" xref="S1.SS1.p2.1.m1.2.3.1.cmml">,</mo><mi id="S1.SS1.p2.1.m1.2.2" xref="S1.SS1.p2.1.m1.2.2.cmml">L</mi><mo id="S1.SS1.p2.1.m1.2.3.2.3" stretchy="false" xref="S1.SS1.p2.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.2b"><set id="S1.SS1.p2.1.m1.2.3.1.cmml" xref="S1.SS1.p2.1.m1.2.3.2"><ci id="S1.SS1.p2.1.m1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1">𝐻</ci><ci id="S1.SS1.p2.1.m1.2.2.cmml" xref="S1.SS1.p2.1.m1.2.2">𝐿</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.2c">\{H,L\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.2d">{ italic_H , italic_L }</annotation></semantics></math>. The mechanism designer has some intended mapping between the real-valued and binary signal spaces. Given the interpretation of <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.1"><semantics id="S1.SS1.p2.2.m2.1a"><mi id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.1b"><ci id="S1.SS1.p2.2.m2.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p2.3.m3.1"><semantics id="S1.SS1.p2.3.m3.1a"><mi id="S1.SS1.p2.3.m3.1.1" xref="S1.SS1.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.3.m3.1b"><ci id="S1.SS1.p2.3.m3.1.1.cmml" xref="S1.SS1.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.3.m3.1d">italic_L</annotation></semantics></math> as “high” and “low”, it is natural for this mapping to be determined by a real-valued threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p2.4.m4.1"><semantics id="S1.SS1.p2.4.m4.1a"><mi id="S1.SS1.p2.4.m4.1.1" xref="S1.SS1.p2.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.4.m4.1b"><ci id="S1.SS1.p2.4.m4.1.1.cmml" xref="S1.SS1.p2.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.4.m4.1d">italic_τ</annotation></semantics></math>, with all values above the threshold corresponding to <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p2.5.m5.1"><semantics id="S1.SS1.p2.5.m5.1a"><mi id="S1.SS1.p2.5.m5.1.1" xref="S1.SS1.p2.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.5.m5.1b"><ci id="S1.SS1.p2.5.m5.1.1.cmml" xref="S1.SS1.p2.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.5.m5.1d">italic_H</annotation></semantics></math> and all below it corresponding to <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p2.6.m6.1"><semantics id="S1.SS1.p2.6.m6.1a"><mi id="S1.SS1.p2.6.m6.1.1" xref="S1.SS1.p2.6.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.6.m6.1b"><ci id="S1.SS1.p2.6.m6.1.1.cmml" xref="S1.SS1.p2.6.m6.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.6.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.6.m6.1d">italic_L</annotation></semantics></math>. In peer grading this naturally corresponds to an intution that a “good-enough” assignment might be graded as satisfactory while a poor one would be deemed unsatisfactory.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.9">Let us suppose first that the mechanism designer announces a desired threshold of <math alttext="\tau=0" class="ltx_Math" display="inline" id="S1.SS1.p3.1.m1.1"><semantics id="S1.SS1.p3.1.m1.1a"><mrow id="S1.SS1.p3.1.m1.1.1" xref="S1.SS1.p3.1.m1.1.1.cmml"><mi id="S1.SS1.p3.1.m1.1.1.2" xref="S1.SS1.p3.1.m1.1.1.2.cmml">τ</mi><mo id="S1.SS1.p3.1.m1.1.1.1" xref="S1.SS1.p3.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS1.p3.1.m1.1.1.3" xref="S1.SS1.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.1.m1.1b"><apply id="S1.SS1.p3.1.m1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1"><eq id="S1.SS1.p3.1.m1.1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1.1"></eq><ci id="S1.SS1.p3.1.m1.1.1.2.cmml" xref="S1.SS1.p3.1.m1.1.1.2">𝜏</ci><cn id="S1.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.SS1.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.1.m1.1d">italic_τ = 0</annotation></semantics></math>. In the binary signal model with this threshold and distribution (i.e. agents only receive a signal of <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.1"><semantics id="S1.SS1.p3.2.m2.1a"><mi id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.1b"><ci id="S1.SS1.p3.2.m2.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.1d">italic_H</annotation></semantics></math> or <math alttext="L" 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">L</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">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.3.m3.1d">italic_L</annotation></semantics></math> as determined by <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p3.4.m4.1"><semantics id="S1.SS1.p3.4.m4.1a"><mi id="S1.SS1.p3.4.m4.1.1" xref="S1.SS1.p3.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.4.m4.1b"><ci id="S1.SS1.p3.4.m4.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.4.m4.1d">italic_τ</annotation></semantics></math>), OA is truthful because agent <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p3.5.m5.1"><semantics id="S1.SS1.p3.5.m5.1a"><mi id="S1.SS1.p3.5.m5.1.1" xref="S1.SS1.p3.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.5.m5.1b"><ci id="S1.SS1.p3.5.m5.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.5.m5.1d">italic_i</annotation></semantics></math> receiving <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p3.6.m6.1"><semantics id="S1.SS1.p3.6.m6.1a"><mi id="S1.SS1.p3.6.m6.1.1" xref="S1.SS1.p3.6.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.6.m6.1b"><ci id="S1.SS1.p3.6.m6.1.1.cmml" xref="S1.SS1.p3.6.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.6.m6.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.6.m6.1d">italic_H</annotation></semantics></math> makes it more likely that all other agents received <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p3.7.m7.1"><semantics id="S1.SS1.p3.7.m7.1a"><mi id="S1.SS1.p3.7.m7.1.1" xref="S1.SS1.p3.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.7.m7.1b"><ci id="S1.SS1.p3.7.m7.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.7.m7.1d">italic_H</annotation></semantics></math> than <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p3.8.m8.1"><semantics id="S1.SS1.p3.8.m8.1a"><mi id="S1.SS1.p3.8.m8.1.1" xref="S1.SS1.p3.8.m8.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.8.m8.1b"><ci id="S1.SS1.p3.8.m8.1.1.cmml" xref="S1.SS1.p3.8.m8.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.8.m8.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.8.m8.1d">italic_L</annotation></semantics></math> and vice versa (<math alttext="\Pr[H|H],\Pr[L|L]>0.5" class="ltx_Math" display="inline" id="S1.SS1.p3.9.m9.4"><semantics id="S1.SS1.p3.9.m9.4a"><mrow 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xref="S1.SS1.p3.9.m9.3.3.1.1.1.1.1.1.3">𝐻</ci></apply></apply><apply id="S1.SS1.p3.9.m9.4.4.2.2.2.2.cmml" xref="S1.SS1.p3.9.m9.4.4.2.2.2.1"><ci id="S1.SS1.p3.9.m9.2.2.cmml" xref="S1.SS1.p3.9.m9.2.2">Pr</ci><apply id="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.cmml" xref="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1"><csymbol cd="latexml" id="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.1.cmml" xref="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.1">conditional</csymbol><ci id="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.2.cmml" xref="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.2">𝐿</ci><ci id="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.3.cmml" xref="S1.SS1.p3.9.m9.4.4.2.2.2.1.1.1.3">𝐿</ci></apply></apply></list><cn id="S1.SS1.p3.9.m9.4.4.4.cmml" type="float" xref="S1.SS1.p3.9.m9.4.4.4">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.9.m9.4c">\Pr[H|H],\Pr[L|L]>0.5</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.9.m9.4d">roman_Pr [ italic_H | italic_H ] , roman_Pr [ italic_L | italic_L ] > 0.5</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.5">In the real-valued case, we can no longer speak of OA as truthful because it is no longer a direct revelation mechanism, but we can ask whether reporting as intended by <math alttext="\tau" 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">τ</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">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.1.m1.1d">italic_τ</annotation></semantics></math> is a (Bayes-Nash) equilibrium. It turns out <math alttext="\tau=0" class="ltx_Math" display="inline" id="S1.SS1.p4.2.m2.1"><semantics id="S1.SS1.p4.2.m2.1a"><mrow id="S1.SS1.p4.2.m2.1.1" xref="S1.SS1.p4.2.m2.1.1.cmml"><mi id="S1.SS1.p4.2.m2.1.1.2" xref="S1.SS1.p4.2.m2.1.1.2.cmml">τ</mi><mo id="S1.SS1.p4.2.m2.1.1.1" xref="S1.SS1.p4.2.m2.1.1.1.cmml">=</mo><mn id="S1.SS1.p4.2.m2.1.1.3" xref="S1.SS1.p4.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.2.m2.1b"><apply id="S1.SS1.p4.2.m2.1.1.cmml" xref="S1.SS1.p4.2.m2.1.1"><eq id="S1.SS1.p4.2.m2.1.1.1.cmml" xref="S1.SS1.p4.2.m2.1.1.1"></eq><ci id="S1.SS1.p4.2.m2.1.1.2.cmml" xref="S1.SS1.p4.2.m2.1.1.2">𝜏</ci><cn id="S1.SS1.p4.2.m2.1.1.3.cmml" type="integer" xref="S1.SS1.p4.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.2.m2.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.2.m2.1d">italic_τ = 0</annotation></semantics></math> is such an equilibrium. For example, if <math alttext="x_{i}=0.11" class="ltx_Math" display="inline" id="S1.SS1.p4.3.m3.1"><semantics id="S1.SS1.p4.3.m3.1a"><mrow id="S1.SS1.p4.3.m3.1.1" xref="S1.SS1.p4.3.m3.1.1.cmml"><msub id="S1.SS1.p4.3.m3.1.1.2" xref="S1.SS1.p4.3.m3.1.1.2.cmml"><mi id="S1.SS1.p4.3.m3.1.1.2.2" xref="S1.SS1.p4.3.m3.1.1.2.2.cmml">x</mi><mi id="S1.SS1.p4.3.m3.1.1.2.3" xref="S1.SS1.p4.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p4.3.m3.1.1.1" xref="S1.SS1.p4.3.m3.1.1.1.cmml">=</mo><mn id="S1.SS1.p4.3.m3.1.1.3" xref="S1.SS1.p4.3.m3.1.1.3.cmml">0.11</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.3.m3.1b"><apply id="S1.SS1.p4.3.m3.1.1.cmml" xref="S1.SS1.p4.3.m3.1.1"><eq id="S1.SS1.p4.3.m3.1.1.1.cmml" xref="S1.SS1.p4.3.m3.1.1.1"></eq><apply id="S1.SS1.p4.3.m3.1.1.2.cmml" xref="S1.SS1.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p4.3.m3.1.1.2.1.cmml" xref="S1.SS1.p4.3.m3.1.1.2">subscript</csymbol><ci id="S1.SS1.p4.3.m3.1.1.2.2.cmml" xref="S1.SS1.p4.3.m3.1.1.2.2">𝑥</ci><ci id="S1.SS1.p4.3.m3.1.1.2.3.cmml" xref="S1.SS1.p4.3.m3.1.1.2.3">𝑖</ci></apply><cn id="S1.SS1.p4.3.m3.1.1.3.cmml" type="float" xref="S1.SS1.p4.3.m3.1.1.3">0.11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.3.m3.1c">x_{i}=0.11</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0.11</annotation></semantics></math>, we have <math alttext="\Pr[X_{j}>0|x_{i}]\approx 0.52>0.5" class="ltx_Math" display="inline" id="S1.SS1.p4.4.m4.2"><semantics id="S1.SS1.p4.4.m4.2a"><mrow id="S1.SS1.p4.4.m4.2.2" xref="S1.SS1.p4.4.m4.2.2.cmml"><mrow id="S1.SS1.p4.4.m4.2.2.1.1" xref="S1.SS1.p4.4.m4.2.2.1.2.cmml"><mi id="S1.SS1.p4.4.m4.1.1" xref="S1.SS1.p4.4.m4.1.1.cmml">Pr</mi><mo id="S1.SS1.p4.4.m4.2.2.1.1a" xref="S1.SS1.p4.4.m4.2.2.1.2.cmml">⁡</mo><mrow id="S1.SS1.p4.4.m4.2.2.1.1.1" xref="S1.SS1.p4.4.m4.2.2.1.2.cmml"><mo id="S1.SS1.p4.4.m4.2.2.1.1.1.2" stretchy="false" xref="S1.SS1.p4.4.m4.2.2.1.2.cmml">[</mo><mrow id="S1.SS1.p4.4.m4.2.2.1.1.1.1" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.cmml"><msub id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.cmml"><mi id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.2" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.2.cmml">X</mi><mi id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.3" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.3.cmml">j</mi></msub><mo id="S1.SS1.p4.4.m4.2.2.1.1.1.1.1" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.1.cmml">></mo><mrow id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.cmml"><mn id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.2" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.2.cmml">0</mn><mo fence="false" id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.1" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.1.cmml">|</mo><msub id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.cmml"><mi id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.2" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.2.cmml">x</mi><mi id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.3" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><mo id="S1.SS1.p4.4.m4.2.2.1.1.1.3" stretchy="false" xref="S1.SS1.p4.4.m4.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S1.SS1.p4.4.m4.2.2.3" xref="S1.SS1.p4.4.m4.2.2.3.cmml">≈</mo><mn id="S1.SS1.p4.4.m4.2.2.4" xref="S1.SS1.p4.4.m4.2.2.4.cmml">0.52</mn><mo id="S1.SS1.p4.4.m4.2.2.5" xref="S1.SS1.p4.4.m4.2.2.5.cmml">></mo><mn id="S1.SS1.p4.4.m4.2.2.6" xref="S1.SS1.p4.4.m4.2.2.6.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.4.m4.2b"><apply id="S1.SS1.p4.4.m4.2.2.cmml" xref="S1.SS1.p4.4.m4.2.2"><and id="S1.SS1.p4.4.m4.2.2a.cmml" xref="S1.SS1.p4.4.m4.2.2"></and><apply id="S1.SS1.p4.4.m4.2.2b.cmml" xref="S1.SS1.p4.4.m4.2.2"><approx id="S1.SS1.p4.4.m4.2.2.3.cmml" xref="S1.SS1.p4.4.m4.2.2.3"></approx><apply id="S1.SS1.p4.4.m4.2.2.1.2.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1"><ci id="S1.SS1.p4.4.m4.1.1.cmml" xref="S1.SS1.p4.4.m4.1.1">Pr</ci><apply id="S1.SS1.p4.4.m4.2.2.1.1.1.1.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1"><gt id="S1.SS1.p4.4.m4.2.2.1.1.1.1.1.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.1"></gt><apply id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.1.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2">subscript</csymbol><ci id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.2.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.2">𝑋</ci><ci id="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.3.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.2.3">𝑗</ci></apply><apply id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3"><csymbol cd="latexml" id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.1.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.1">conditional</csymbol><cn id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.2.cmml" type="integer" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.2">0</cn><apply id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.1.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.2.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.2">𝑥</ci><ci id="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.3.cmml" xref="S1.SS1.p4.4.m4.2.2.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply></apply><cn id="S1.SS1.p4.4.m4.2.2.4.cmml" type="float" xref="S1.SS1.p4.4.m4.2.2.4">0.52</cn></apply><apply id="S1.SS1.p4.4.m4.2.2c.cmml" xref="S1.SS1.p4.4.m4.2.2"><gt id="S1.SS1.p4.4.m4.2.2.5.cmml" xref="S1.SS1.p4.4.m4.2.2.5"></gt><share href="https://arxiv.org/html/2503.16280v1#S1.SS1.p4.4.m4.2.2.4.cmml" id="S1.SS1.p4.4.m4.2.2d.cmml" xref="S1.SS1.p4.4.m4.2.2"></share><cn id="S1.SS1.p4.4.m4.2.2.6.cmml" type="float" xref="S1.SS1.p4.4.m4.2.2.6">0.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.4.m4.2c">\Pr[X_{j}>0|x_{i}]\approx 0.52>0.5</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.4.m4.2d">roman_Pr [ italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > 0 | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] ≈ 0.52 > 0.5</annotation></semantics></math>, so reporting <math alttext="H" 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">H</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">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.5.m5.1d">italic_H</annotation></semantics></math> as intended is indeed the best response.</p> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.14">Now suppose the desired threshold were <math alttext="\tau=0.1" class="ltx_Math" display="inline" id="S1.SS1.p5.1.m1.1"><semantics id="S1.SS1.p5.1.m1.1a"><mrow id="S1.SS1.p5.1.m1.1.1" xref="S1.SS1.p5.1.m1.1.1.cmml"><mi id="S1.SS1.p5.1.m1.1.1.2" xref="S1.SS1.p5.1.m1.1.1.2.cmml">τ</mi><mo id="S1.SS1.p5.1.m1.1.1.1" xref="S1.SS1.p5.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS1.p5.1.m1.1.1.3" xref="S1.SS1.p5.1.m1.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.1.m1.1b"><apply id="S1.SS1.p5.1.m1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1"><eq id="S1.SS1.p5.1.m1.1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1.1"></eq><ci id="S1.SS1.p5.1.m1.1.1.2.cmml" xref="S1.SS1.p5.1.m1.1.1.2">𝜏</ci><cn id="S1.SS1.p5.1.m1.1.1.3.cmml" type="float" xref="S1.SS1.p5.1.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.1.m1.1c">\tau=0.1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.1.m1.1d">italic_τ = 0.1</annotation></semantics></math>. In the binary signal model, <math alttext="\Pr[H|H]=\Pr[X_{j}>0.1|X_{i}>0.1]\approx 0.64" class="ltx_Math" display="inline" id="S1.SS1.p5.2.m2.4"><semantics id="S1.SS1.p5.2.m2.4a"><mrow id="S1.SS1.p5.2.m2.4.4" xref="S1.SS1.p5.2.m2.4.4.cmml"><mrow id="S1.SS1.p5.2.m2.3.3.1.1" xref="S1.SS1.p5.2.m2.3.3.1.2.cmml"><mi id="S1.SS1.p5.2.m2.1.1" xref="S1.SS1.p5.2.m2.1.1.cmml">Pr</mi><mo id="S1.SS1.p5.2.m2.3.3.1.1a" xref="S1.SS1.p5.2.m2.3.3.1.2.cmml">⁡</mo><mrow id="S1.SS1.p5.2.m2.3.3.1.1.1" xref="S1.SS1.p5.2.m2.3.3.1.2.cmml"><mo id="S1.SS1.p5.2.m2.3.3.1.1.1.2" stretchy="false" xref="S1.SS1.p5.2.m2.3.3.1.2.cmml">[</mo><mrow id="S1.SS1.p5.2.m2.3.3.1.1.1.1" xref="S1.SS1.p5.2.m2.3.3.1.1.1.1.cmml"><mi id="S1.SS1.p5.2.m2.3.3.1.1.1.1.2" xref="S1.SS1.p5.2.m2.3.3.1.1.1.1.2.cmml">H</mi><mo fence="false" id="S1.SS1.p5.2.m2.3.3.1.1.1.1.1" xref="S1.SS1.p5.2.m2.3.3.1.1.1.1.1.cmml">|</mo><mi id="S1.SS1.p5.2.m2.3.3.1.1.1.1.3" xref="S1.SS1.p5.2.m2.3.3.1.1.1.1.3.cmml">H</mi></mrow><mo id="S1.SS1.p5.2.m2.3.3.1.1.1.3" stretchy="false" xref="S1.SS1.p5.2.m2.3.3.1.2.cmml">]</mo></mrow></mrow><mo id="S1.SS1.p5.2.m2.4.4.4" xref="S1.SS1.p5.2.m2.4.4.4.cmml">=</mo><mrow id="S1.SS1.p5.2.m2.4.4.2.1" xref="S1.SS1.p5.2.m2.4.4.2.2.cmml"><mi id="S1.SS1.p5.2.m2.2.2" xref="S1.SS1.p5.2.m2.2.2.cmml">Pr</mi><mo id="S1.SS1.p5.2.m2.4.4.2.1a" xref="S1.SS1.p5.2.m2.4.4.2.2.cmml">⁡</mo><mrow id="S1.SS1.p5.2.m2.4.4.2.1.1" xref="S1.SS1.p5.2.m2.4.4.2.2.cmml"><mo id="S1.SS1.p5.2.m2.4.4.2.1.1.2" stretchy="false" xref="S1.SS1.p5.2.m2.4.4.2.2.cmml">[</mo><mrow id="S1.SS1.p5.2.m2.4.4.2.1.1.1" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.cmml"><msub id="S1.SS1.p5.2.m2.4.4.2.1.1.1.3" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.3.cmml"><mi id="S1.SS1.p5.2.m2.4.4.2.1.1.1.3.2" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.3.2.cmml">X</mi><mi id="S1.SS1.p5.2.m2.4.4.2.1.1.1.3.3" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.3.3.cmml">j</mi></msub><mo id="S1.SS1.p5.2.m2.4.4.2.1.1.1.2" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.2.cmml">></mo><mrow 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id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.4" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.4.cmml">0.1</mn></mrow></mrow><mo id="S1.SS1.p5.2.m2.4.4.2.1.1.3" stretchy="false" xref="S1.SS1.p5.2.m2.4.4.2.2.cmml">]</mo></mrow></mrow><mo id="S1.SS1.p5.2.m2.4.4.5" xref="S1.SS1.p5.2.m2.4.4.5.cmml">≈</mo><mn id="S1.SS1.p5.2.m2.4.4.6" xref="S1.SS1.p5.2.m2.4.4.6.cmml">0.64</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.2.m2.4b"><apply id="S1.SS1.p5.2.m2.4.4.cmml" xref="S1.SS1.p5.2.m2.4.4"><and id="S1.SS1.p5.2.m2.4.4a.cmml" xref="S1.SS1.p5.2.m2.4.4"></and><apply id="S1.SS1.p5.2.m2.4.4b.cmml" xref="S1.SS1.p5.2.m2.4.4"><eq id="S1.SS1.p5.2.m2.4.4.4.cmml" xref="S1.SS1.p5.2.m2.4.4.4"></eq><apply id="S1.SS1.p5.2.m2.3.3.1.2.cmml" xref="S1.SS1.p5.2.m2.3.3.1.1"><ci id="S1.SS1.p5.2.m2.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1">Pr</ci><apply id="S1.SS1.p5.2.m2.3.3.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.3.3.1.1.1.1"><csymbol cd="latexml" id="S1.SS1.p5.2.m2.3.3.1.1.1.1.1.cmml" 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xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1"><times id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.2.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.2"></times><cn id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.3.cmml" type="float" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.3">0.1</cn><apply id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.2.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1"><csymbol cd="latexml" id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.2.1.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.2">ket</csymbol><apply id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.2.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.2">𝑋</ci><ci id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.3.cmml" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.4.cmml" type="float" xref="S1.SS1.p5.2.m2.4.4.2.1.1.1.1.4">0.1</cn></apply></apply></apply></apply><apply id="S1.SS1.p5.2.m2.4.4c.cmml" xref="S1.SS1.p5.2.m2.4.4"><approx id="S1.SS1.p5.2.m2.4.4.5.cmml" xref="S1.SS1.p5.2.m2.4.4.5"></approx><share href="https://arxiv.org/html/2503.16280v1#S1.SS1.p5.2.m2.4.4.2.cmml" id="S1.SS1.p5.2.m2.4.4d.cmml" xref="S1.SS1.p5.2.m2.4.4"></share><cn id="S1.SS1.p5.2.m2.4.4.6.cmml" type="float" xref="S1.SS1.p5.2.m2.4.4.6">0.64</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.2.m2.4c">\Pr[H|H]=\Pr[X_{j}>0.1|X_{i}>0.1]\approx 0.64</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.2.m2.4d">roman_Pr [ italic_H | italic_H ] = roman_Pr [ italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > 0.1 | italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0.1 ] ≈ 0.64</annotation></semantics></math> and <math alttext="\Pr[L|L]\approx 0.69" class="ltx_Math" display="inline" id="S1.SS1.p5.3.m3.2"><semantics id="S1.SS1.p5.3.m3.2a"><mrow id="S1.SS1.p5.3.m3.2.2" xref="S1.SS1.p5.3.m3.2.2.cmml"><mrow id="S1.SS1.p5.3.m3.2.2.1.1" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml"><mi id="S1.SS1.p5.3.m3.1.1" xref="S1.SS1.p5.3.m3.1.1.cmml">Pr</mi><mo id="S1.SS1.p5.3.m3.2.2.1.1a" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml">⁡</mo><mrow id="S1.SS1.p5.3.m3.2.2.1.1.1" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml"><mo id="S1.SS1.p5.3.m3.2.2.1.1.1.2" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml">[</mo><mrow id="S1.SS1.p5.3.m3.2.2.1.1.1.1" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml"><mi id="S1.SS1.p5.3.m3.2.2.1.1.1.1.2" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.2.cmml">L</mi><mo fence="false" id="S1.SS1.p5.3.m3.2.2.1.1.1.1.1" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.1.cmml">|</mo><mi id="S1.SS1.p5.3.m3.2.2.1.1.1.1.3" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.3.cmml">L</mi></mrow><mo id="S1.SS1.p5.3.m3.2.2.1.1.1.3" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S1.SS1.p5.3.m3.2.2.2" xref="S1.SS1.p5.3.m3.2.2.2.cmml">≈</mo><mn id="S1.SS1.p5.3.m3.2.2.3" xref="S1.SS1.p5.3.m3.2.2.3.cmml">0.69</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.3.m3.2b"><apply id="S1.SS1.p5.3.m3.2.2.cmml" xref="S1.SS1.p5.3.m3.2.2"><approx id="S1.SS1.p5.3.m3.2.2.2.cmml" xref="S1.SS1.p5.3.m3.2.2.2"></approx><apply id="S1.SS1.p5.3.m3.2.2.1.2.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1"><ci id="S1.SS1.p5.3.m3.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1">Pr</ci><apply id="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1"><csymbol cd="latexml" id="S1.SS1.p5.3.m3.2.2.1.1.1.1.1.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.1">conditional</csymbol><ci id="S1.SS1.p5.3.m3.2.2.1.1.1.1.2.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.2">𝐿</ci><ci id="S1.SS1.p5.3.m3.2.2.1.1.1.1.3.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.3">𝐿</ci></apply></apply><cn id="S1.SS1.p5.3.m3.2.2.3.cmml" type="float" xref="S1.SS1.p5.3.m3.2.2.3">0.69</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.3.m3.2c">\Pr[L|L]\approx 0.69</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.3.m3.2d">roman_Pr [ italic_L | italic_L ] ≈ 0.69</annotation></semantics></math> are both greater than <math alttext="0.5" class="ltx_Math" display="inline" id="S1.SS1.p5.4.m4.1"><semantics id="S1.SS1.p5.4.m4.1a"><mn id="S1.SS1.p5.4.m4.1.1" xref="S1.SS1.p5.4.m4.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.4.m4.1b"><cn id="S1.SS1.p5.4.m4.1.1.cmml" type="float" xref="S1.SS1.p5.4.m4.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.4.m4.1c">0.5</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.4.m4.1d">0.5</annotation></semantics></math>, so OA remains truthful. Now in the real-valued model consider an agent receiving a signal of <math alttext="x_{i}=0.11" class="ltx_Math" display="inline" id="S1.SS1.p5.5.m5.1"><semantics id="S1.SS1.p5.5.m5.1a"><mrow id="S1.SS1.p5.5.m5.1.1" xref="S1.SS1.p5.5.m5.1.1.cmml"><msub id="S1.SS1.p5.5.m5.1.1.2" xref="S1.SS1.p5.5.m5.1.1.2.cmml"><mi id="S1.SS1.p5.5.m5.1.1.2.2" xref="S1.SS1.p5.5.m5.1.1.2.2.cmml">x</mi><mi id="S1.SS1.p5.5.m5.1.1.2.3" xref="S1.SS1.p5.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p5.5.m5.1.1.1" xref="S1.SS1.p5.5.m5.1.1.1.cmml">=</mo><mn id="S1.SS1.p5.5.m5.1.1.3" xref="S1.SS1.p5.5.m5.1.1.3.cmml">0.11</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.5.m5.1b"><apply id="S1.SS1.p5.5.m5.1.1.cmml" xref="S1.SS1.p5.5.m5.1.1"><eq id="S1.SS1.p5.5.m5.1.1.1.cmml" xref="S1.SS1.p5.5.m5.1.1.1"></eq><apply id="S1.SS1.p5.5.m5.1.1.2.cmml" xref="S1.SS1.p5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p5.5.m5.1.1.2.1.cmml" xref="S1.SS1.p5.5.m5.1.1.2">subscript</csymbol><ci id="S1.SS1.p5.5.m5.1.1.2.2.cmml" xref="S1.SS1.p5.5.m5.1.1.2.2">𝑥</ci><ci id="S1.SS1.p5.5.m5.1.1.2.3.cmml" xref="S1.SS1.p5.5.m5.1.1.2.3">𝑖</ci></apply><cn id="S1.SS1.p5.5.m5.1.1.3.cmml" type="float" xref="S1.SS1.p5.5.m5.1.1.3">0.11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.5.m5.1c">x_{i}=0.11</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0.11</annotation></semantics></math>. As <math alttext="x_{i}" class="ltx_Math" display="inline" id="S1.SS1.p5.6.m6.1"><semantics id="S1.SS1.p5.6.m6.1a"><msub id="S1.SS1.p5.6.m6.1.1" xref="S1.SS1.p5.6.m6.1.1.cmml"><mi id="S1.SS1.p5.6.m6.1.1.2" xref="S1.SS1.p5.6.m6.1.1.2.cmml">x</mi><mi id="S1.SS1.p5.6.m6.1.1.3" xref="S1.SS1.p5.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.6.m6.1b"><apply id="S1.SS1.p5.6.m6.1.1.cmml" xref="S1.SS1.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS1.p5.6.m6.1.1.1.cmml" xref="S1.SS1.p5.6.m6.1.1">subscript</csymbol><ci id="S1.SS1.p5.6.m6.1.1.2.cmml" xref="S1.SS1.p5.6.m6.1.1.2">𝑥</ci><ci id="S1.SS1.p5.6.m6.1.1.3.cmml" xref="S1.SS1.p5.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.6.m6.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is just above the threshold, the intended report is <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p5.7.m7.1"><semantics id="S1.SS1.p5.7.m7.1a"><mi id="S1.SS1.p5.7.m7.1.1" xref="S1.SS1.p5.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.7.m7.1b"><ci id="S1.SS1.p5.7.m7.1.1.cmml" xref="S1.SS1.p5.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.7.m7.1d">italic_H</annotation></semantics></math>. But <math alttext="\Pr[X_{j}>0.1|x_{i}]\approx 0.48<0.5" class="ltx_Math" display="inline" id="S1.SS1.p5.8.m8.2"><semantics id="S1.SS1.p5.8.m8.2a"><mrow id="S1.SS1.p5.8.m8.2.2" xref="S1.SS1.p5.8.m8.2.2.cmml"><mrow id="S1.SS1.p5.8.m8.2.2.1.1" xref="S1.SS1.p5.8.m8.2.2.1.2.cmml"><mi id="S1.SS1.p5.8.m8.1.1" xref="S1.SS1.p5.8.m8.1.1.cmml">Pr</mi><mo id="S1.SS1.p5.8.m8.2.2.1.1a" xref="S1.SS1.p5.8.m8.2.2.1.2.cmml">⁡</mo><mrow id="S1.SS1.p5.8.m8.2.2.1.1.1" xref="S1.SS1.p5.8.m8.2.2.1.2.cmml"><mo id="S1.SS1.p5.8.m8.2.2.1.1.1.2" stretchy="false" xref="S1.SS1.p5.8.m8.2.2.1.2.cmml">[</mo><mrow id="S1.SS1.p5.8.m8.2.2.1.1.1.1" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.cmml"><msub id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.cmml"><mi id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.2" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.2.cmml">X</mi><mi id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.3" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.3.cmml">j</mi></msub><mo id="S1.SS1.p5.8.m8.2.2.1.1.1.1.1" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.1.cmml">></mo><mrow id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.cmml"><mn id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.2" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.2.cmml">0.1</mn><mo fence="false" id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.1" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.1.cmml">|</mo><msub id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.cmml"><mi id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.2" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.2.cmml">x</mi><mi id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.3" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><mo id="S1.SS1.p5.8.m8.2.2.1.1.1.3" stretchy="false" xref="S1.SS1.p5.8.m8.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S1.SS1.p5.8.m8.2.2.3" xref="S1.SS1.p5.8.m8.2.2.3.cmml">≈</mo><mn id="S1.SS1.p5.8.m8.2.2.4" xref="S1.SS1.p5.8.m8.2.2.4.cmml">0.48</mn><mo id="S1.SS1.p5.8.m8.2.2.5" xref="S1.SS1.p5.8.m8.2.2.5.cmml"><</mo><mn id="S1.SS1.p5.8.m8.2.2.6" xref="S1.SS1.p5.8.m8.2.2.6.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.8.m8.2b"><apply id="S1.SS1.p5.8.m8.2.2.cmml" xref="S1.SS1.p5.8.m8.2.2"><and id="S1.SS1.p5.8.m8.2.2a.cmml" xref="S1.SS1.p5.8.m8.2.2"></and><apply id="S1.SS1.p5.8.m8.2.2b.cmml" xref="S1.SS1.p5.8.m8.2.2"><approx id="S1.SS1.p5.8.m8.2.2.3.cmml" xref="S1.SS1.p5.8.m8.2.2.3"></approx><apply id="S1.SS1.p5.8.m8.2.2.1.2.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1"><ci id="S1.SS1.p5.8.m8.1.1.cmml" xref="S1.SS1.p5.8.m8.1.1">Pr</ci><apply id="S1.SS1.p5.8.m8.2.2.1.1.1.1.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1"><gt id="S1.SS1.p5.8.m8.2.2.1.1.1.1.1.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.1"></gt><apply id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.1.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2">subscript</csymbol><ci id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.2.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.2">𝑋</ci><ci id="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.3.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.2.3">𝑗</ci></apply><apply id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3"><csymbol cd="latexml" id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.1.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.1">conditional</csymbol><cn id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.2.cmml" type="float" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.2">0.1</cn><apply id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.1.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.2.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.2">𝑥</ci><ci id="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.3.cmml" xref="S1.SS1.p5.8.m8.2.2.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply></apply><cn id="S1.SS1.p5.8.m8.2.2.4.cmml" type="float" xref="S1.SS1.p5.8.m8.2.2.4">0.48</cn></apply><apply id="S1.SS1.p5.8.m8.2.2c.cmml" xref="S1.SS1.p5.8.m8.2.2"><lt id="S1.SS1.p5.8.m8.2.2.5.cmml" xref="S1.SS1.p5.8.m8.2.2.5"></lt><share href="https://arxiv.org/html/2503.16280v1#S1.SS1.p5.8.m8.2.2.4.cmml" id="S1.SS1.p5.8.m8.2.2d.cmml" xref="S1.SS1.p5.8.m8.2.2"></share><cn id="S1.SS1.p5.8.m8.2.2.6.cmml" type="float" xref="S1.SS1.p5.8.m8.2.2.6">0.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.8.m8.2c">\Pr[X_{j}>0.1|x_{i}]\approx 0.48<0.5</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.8.m8.2d">roman_Pr [ italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT > 0.1 | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] ≈ 0.48 < 0.5</annotation></semantics></math>. Thus the best response is <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p5.9.m9.1"><semantics id="S1.SS1.p5.9.m9.1a"><mi id="S1.SS1.p5.9.m9.1.1" xref="S1.SS1.p5.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.9.m9.1b"><ci id="S1.SS1.p5.9.m9.1.1.cmml" xref="S1.SS1.p5.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.9.m9.1d">italic_L</annotation></semantics></math>, so reporting according to <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p5.10.m10.1"><semantics id="S1.SS1.p5.10.m10.1a"><mi id="S1.SS1.p5.10.m10.1.1" xref="S1.SS1.p5.10.m10.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.10.m10.1b"><ci id="S1.SS1.p5.10.m10.1.1.cmml" xref="S1.SS1.p5.10.m10.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.10.m10.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.10.m10.1d">italic_τ</annotation></semantics></math> is <span class="ltx_text ltx_font_bold" id="S1.SS1.p5.14.1">not</span> an equilibrium. The above analysis will give the same qualitative result for any finite nonzero <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p5.11.m11.1"><semantics id="S1.SS1.p5.11.m11.1a"><mi id="S1.SS1.p5.11.m11.1.1" xref="S1.SS1.p5.11.m11.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.11.m11.1b"><ci id="S1.SS1.p5.11.m11.1.1.cmml" xref="S1.SS1.p5.11.m11.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.11.m11.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.11.m11.1d">italic_τ</annotation></semantics></math>, intuitively because with a positive threshold any agent receiving a signal slightly above the threshold will think that the average agent will have a signal below it and vice versa for negative thresholds. Thus, the <em class="ltx_emph ltx_font_italic" id="S1.SS1.p5.14.2">only</em> finite initial threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p5.12.m12.1"><semantics id="S1.SS1.p5.12.m12.1a"><mi id="S1.SS1.p5.12.m12.1.1" xref="S1.SS1.p5.12.m12.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.12.m12.1b"><ci id="S1.SS1.p5.12.m12.1.1.cmml" xref="S1.SS1.p5.12.m12.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.12.m12.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.12.m12.1d">italic_τ</annotation></semantics></math> for which reporting according to <math alttext="\tau" class="ltx_Math" display="inline" id="S1.SS1.p5.13.m13.1"><semantics id="S1.SS1.p5.13.m13.1a"><mi id="S1.SS1.p5.13.m13.1.1" xref="S1.SS1.p5.13.m13.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.13.m13.1b"><ci id="S1.SS1.p5.13.m13.1.1.cmml" xref="S1.SS1.p5.13.m13.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.13.m13.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.13.m13.1d">italic_τ</annotation></semantics></math> is an equilibrium is <math alttext="\tau=0" class="ltx_Math" display="inline" id="S1.SS1.p5.14.m14.1"><semantics id="S1.SS1.p5.14.m14.1a"><mrow id="S1.SS1.p5.14.m14.1.1" xref="S1.SS1.p5.14.m14.1.1.cmml"><mi id="S1.SS1.p5.14.m14.1.1.2" xref="S1.SS1.p5.14.m14.1.1.2.cmml">τ</mi><mo id="S1.SS1.p5.14.m14.1.1.1" xref="S1.SS1.p5.14.m14.1.1.1.cmml">=</mo><mn id="S1.SS1.p5.14.m14.1.1.3" xref="S1.SS1.p5.14.m14.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.14.m14.1b"><apply id="S1.SS1.p5.14.m14.1.1.cmml" xref="S1.SS1.p5.14.m14.1.1"><eq id="S1.SS1.p5.14.m14.1.1.1.cmml" xref="S1.SS1.p5.14.m14.1.1.1"></eq><ci id="S1.SS1.p5.14.m14.1.1.2.cmml" xref="S1.SS1.p5.14.m14.1.1.2">𝜏</ci><cn id="S1.SS1.p5.14.m14.1.1.3.cmml" type="integer" xref="S1.SS1.p5.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.14.m14.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.14.m14.1d">italic_τ = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.15">Given that <math alttext="\tau=0.1" class="ltx_Math" display="inline" id="S1.SS1.p6.1.m1.1"><semantics id="S1.SS1.p6.1.m1.1a"><mrow 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">τ</mi><mo id="S1.SS1.p6.1.m1.1.1.1" xref="S1.SS1.p6.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS1.p6.1.m1.1.1.3" xref="S1.SS1.p6.1.m1.1.1.3.cmml">0.1</mn></mrow><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"><eq id="S1.SS1.p6.1.m1.1.1.1.cmml" xref="S1.SS1.p6.1.m1.1.1.1"></eq><ci id="S1.SS1.p6.1.m1.1.1.2.cmml" xref="S1.SS1.p6.1.m1.1.1.2">𝜏</ci><cn id="S1.SS1.p6.1.m1.1.1.3.cmml" type="float" xref="S1.SS1.p6.1.m1.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.1.m1.1c">\tau=0.1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.1.m1.1d">italic_τ = 0.1</annotation></semantics></math> is not an equilibrium, what should we expect to happen if the mechanism designer announces it? We saw that agents with signals <math alttext="x_{i}" class="ltx_Math" display="inline" id="S1.SS1.p6.2.m2.1"><semantics id="S1.SS1.p6.2.m2.1a"><msub id="S1.SS1.p6.2.m2.1.1" xref="S1.SS1.p6.2.m2.1.1.cmml"><mi id="S1.SS1.p6.2.m2.1.1.2" xref="S1.SS1.p6.2.m2.1.1.2.cmml">x</mi><mi id="S1.SS1.p6.2.m2.1.1.3" xref="S1.SS1.p6.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.2.m2.1b"><apply id="S1.SS1.p6.2.m2.1.1.cmml" xref="S1.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p6.2.m2.1.1.1.cmml" xref="S1.SS1.p6.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p6.2.m2.1.1.2.cmml" xref="S1.SS1.p6.2.m2.1.1.2">𝑥</ci><ci id="S1.SS1.p6.2.m2.1.1.3.cmml" xref="S1.SS1.p6.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.2.m2.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> slightly above <math alttext="0.1" class="ltx_Math" display="inline" id="S1.SS1.p6.3.m3.1"><semantics id="S1.SS1.p6.3.m3.1a"><mn id="S1.SS1.p6.3.m3.1.1" xref="S1.SS1.p6.3.m3.1.1.cmml">0.1</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.3.m3.1b"><cn id="S1.SS1.p6.3.m3.1.1.cmml" type="float" xref="S1.SS1.p6.3.m3.1.1">0.1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.3.m3.1c">0.1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.3.m3.1d">0.1</annotation></semantics></math> have an incentive to report <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p6.4.m4.1"><semantics id="S1.SS1.p6.4.m4.1a"><mi id="S1.SS1.p6.4.m4.1.1" xref="S1.SS1.p6.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.4.m4.1b"><ci id="S1.SS1.p6.4.m4.1.1.cmml" xref="S1.SS1.p6.4.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.4.m4.1d">italic_L</annotation></semantics></math> instead of <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p6.5.m5.1"><semantics id="S1.SS1.p6.5.m5.1a"><mi id="S1.SS1.p6.5.m5.1.1" xref="S1.SS1.p6.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.5.m5.1b"><ci id="S1.SS1.p6.5.m5.1.1.cmml" xref="S1.SS1.p6.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.5.m5.1d">italic_H</annotation></semantics></math>. This incentive weakens as <math alttext="x_{i}" class="ltx_Math" display="inline" id="S1.SS1.p6.6.m6.1"><semantics id="S1.SS1.p6.6.m6.1a"><msub id="S1.SS1.p6.6.m6.1.1" xref="S1.SS1.p6.6.m6.1.1.cmml"><mi id="S1.SS1.p6.6.m6.1.1.2" xref="S1.SS1.p6.6.m6.1.1.2.cmml">x</mi><mi id="S1.SS1.p6.6.m6.1.1.3" xref="S1.SS1.p6.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.6.m6.1b"><apply id="S1.SS1.p6.6.m6.1.1.cmml" xref="S1.SS1.p6.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS1.p6.6.m6.1.1.1.cmml" xref="S1.SS1.p6.6.m6.1.1">subscript</csymbol><ci id="S1.SS1.p6.6.m6.1.1.2.cmml" xref="S1.SS1.p6.6.m6.1.1.2">𝑥</ci><ci id="S1.SS1.p6.6.m6.1.1.3.cmml" xref="S1.SS1.p6.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.6.m6.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> grows, so it is natural to expect only a small fraction of agents to deviate from the intended strategy, specifically those with signals closest to but above <math alttext="0.1" class="ltx_Math" display="inline" id="S1.SS1.p6.7.m7.1"><semantics id="S1.SS1.p6.7.m7.1a"><mn id="S1.SS1.p6.7.m7.1.1" xref="S1.SS1.p6.7.m7.1.1.cmml">0.1</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.7.m7.1b"><cn id="S1.SS1.p6.7.m7.1.1.cmml" type="float" xref="S1.SS1.p6.7.m7.1.1">0.1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.7.m7.1c">0.1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.7.m7.1d">0.1</annotation></semantics></math>. But effectively, this raises our threshold from <math alttext="\tau_{0}=0.1" class="ltx_Math" display="inline" id="S1.SS1.p6.8.m8.1"><semantics id="S1.SS1.p6.8.m8.1a"><mrow id="S1.SS1.p6.8.m8.1.1" xref="S1.SS1.p6.8.m8.1.1.cmml"><msub id="S1.SS1.p6.8.m8.1.1.2" xref="S1.SS1.p6.8.m8.1.1.2.cmml"><mi id="S1.SS1.p6.8.m8.1.1.2.2" xref="S1.SS1.p6.8.m8.1.1.2.2.cmml">τ</mi><mn id="S1.SS1.p6.8.m8.1.1.2.3" xref="S1.SS1.p6.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S1.SS1.p6.8.m8.1.1.1" xref="S1.SS1.p6.8.m8.1.1.1.cmml">=</mo><mn id="S1.SS1.p6.8.m8.1.1.3" xref="S1.SS1.p6.8.m8.1.1.3.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.8.m8.1b"><apply id="S1.SS1.p6.8.m8.1.1.cmml" xref="S1.SS1.p6.8.m8.1.1"><eq id="S1.SS1.p6.8.m8.1.1.1.cmml" xref="S1.SS1.p6.8.m8.1.1.1"></eq><apply id="S1.SS1.p6.8.m8.1.1.2.cmml" xref="S1.SS1.p6.8.m8.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p6.8.m8.1.1.2.1.cmml" xref="S1.SS1.p6.8.m8.1.1.2">subscript</csymbol><ci id="S1.SS1.p6.8.m8.1.1.2.2.cmml" xref="S1.SS1.p6.8.m8.1.1.2.2">𝜏</ci><cn id="S1.SS1.p6.8.m8.1.1.2.3.cmml" type="integer" xref="S1.SS1.p6.8.m8.1.1.2.3">0</cn></apply><cn id="S1.SS1.p6.8.m8.1.1.3.cmml" type="float" xref="S1.SS1.p6.8.m8.1.1.3">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.8.m8.1c">\tau_{0}=0.1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.8.m8.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0.1</annotation></semantics></math> to some higher value, say <math alttext="\tau_{1}=0.13" class="ltx_Math" display="inline" id="S1.SS1.p6.9.m9.1"><semantics id="S1.SS1.p6.9.m9.1a"><mrow id="S1.SS1.p6.9.m9.1.1" xref="S1.SS1.p6.9.m9.1.1.cmml"><msub id="S1.SS1.p6.9.m9.1.1.2" xref="S1.SS1.p6.9.m9.1.1.2.cmml"><mi id="S1.SS1.p6.9.m9.1.1.2.2" xref="S1.SS1.p6.9.m9.1.1.2.2.cmml">τ</mi><mn id="S1.SS1.p6.9.m9.1.1.2.3" xref="S1.SS1.p6.9.m9.1.1.2.3.cmml">1</mn></msub><mo id="S1.SS1.p6.9.m9.1.1.1" xref="S1.SS1.p6.9.m9.1.1.1.cmml">=</mo><mn id="S1.SS1.p6.9.m9.1.1.3" xref="S1.SS1.p6.9.m9.1.1.3.cmml">0.13</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.9.m9.1b"><apply id="S1.SS1.p6.9.m9.1.1.cmml" xref="S1.SS1.p6.9.m9.1.1"><eq id="S1.SS1.p6.9.m9.1.1.1.cmml" xref="S1.SS1.p6.9.m9.1.1.1"></eq><apply id="S1.SS1.p6.9.m9.1.1.2.cmml" xref="S1.SS1.p6.9.m9.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p6.9.m9.1.1.2.1.cmml" xref="S1.SS1.p6.9.m9.1.1.2">subscript</csymbol><ci id="S1.SS1.p6.9.m9.1.1.2.2.cmml" xref="S1.SS1.p6.9.m9.1.1.2.2">𝜏</ci><cn id="S1.SS1.p6.9.m9.1.1.2.3.cmml" type="integer" xref="S1.SS1.p6.9.m9.1.1.2.3">1</cn></apply><cn id="S1.SS1.p6.9.m9.1.1.3.cmml" type="float" xref="S1.SS1.p6.9.m9.1.1.3">0.13</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.9.m9.1c">\tau_{1}=0.13</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.9.m9.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.13</annotation></semantics></math>. Since the same argument applies to our new threshold, in future tasks further deviations could lead to <math alttext="\tau_{2}=0.16" class="ltx_Math" display="inline" id="S1.SS1.p6.10.m10.1"><semantics id="S1.SS1.p6.10.m10.1a"><mrow id="S1.SS1.p6.10.m10.1.1" xref="S1.SS1.p6.10.m10.1.1.cmml"><msub id="S1.SS1.p6.10.m10.1.1.2" xref="S1.SS1.p6.10.m10.1.1.2.cmml"><mi id="S1.SS1.p6.10.m10.1.1.2.2" xref="S1.SS1.p6.10.m10.1.1.2.2.cmml">τ</mi><mn id="S1.SS1.p6.10.m10.1.1.2.3" xref="S1.SS1.p6.10.m10.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS1.p6.10.m10.1.1.1" xref="S1.SS1.p6.10.m10.1.1.1.cmml">=</mo><mn id="S1.SS1.p6.10.m10.1.1.3" xref="S1.SS1.p6.10.m10.1.1.3.cmml">0.16</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.10.m10.1b"><apply id="S1.SS1.p6.10.m10.1.1.cmml" xref="S1.SS1.p6.10.m10.1.1"><eq id="S1.SS1.p6.10.m10.1.1.1.cmml" xref="S1.SS1.p6.10.m10.1.1.1"></eq><apply id="S1.SS1.p6.10.m10.1.1.2.cmml" xref="S1.SS1.p6.10.m10.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p6.10.m10.1.1.2.1.cmml" xref="S1.SS1.p6.10.m10.1.1.2">subscript</csymbol><ci id="S1.SS1.p6.10.m10.1.1.2.2.cmml" xref="S1.SS1.p6.10.m10.1.1.2.2">𝜏</ci><cn id="S1.SS1.p6.10.m10.1.1.2.3.cmml" type="integer" xref="S1.SS1.p6.10.m10.1.1.2.3">2</cn></apply><cn id="S1.SS1.p6.10.m10.1.1.3.cmml" type="float" xref="S1.SS1.p6.10.m10.1.1.3">0.16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.10.m10.1c">\tau_{2}=0.16</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.10.m10.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.16</annotation></semantics></math>. Thus, we should expect that, over time, the system will converge toward the “trival” equilibrium at <math alttext="\tau=\infty" class="ltx_Math" display="inline" id="S1.SS1.p6.11.m11.1"><semantics id="S1.SS1.p6.11.m11.1a"><mrow id="S1.SS1.p6.11.m11.1.1" xref="S1.SS1.p6.11.m11.1.1.cmml"><mi id="S1.SS1.p6.11.m11.1.1.2" xref="S1.SS1.p6.11.m11.1.1.2.cmml">τ</mi><mo id="S1.SS1.p6.11.m11.1.1.1" xref="S1.SS1.p6.11.m11.1.1.1.cmml">=</mo><mi id="S1.SS1.p6.11.m11.1.1.3" mathvariant="normal" xref="S1.SS1.p6.11.m11.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.11.m11.1b"><apply id="S1.SS1.p6.11.m11.1.1.cmml" xref="S1.SS1.p6.11.m11.1.1"><eq id="S1.SS1.p6.11.m11.1.1.1.cmml" xref="S1.SS1.p6.11.m11.1.1.1"></eq><ci id="S1.SS1.p6.11.m11.1.1.2.cmml" xref="S1.SS1.p6.11.m11.1.1.2">𝜏</ci><infinity id="S1.SS1.p6.11.m11.1.1.3.cmml" xref="S1.SS1.p6.11.m11.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.11.m11.1c">\tau=\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.11.m11.1d">italic_τ = ∞</annotation></semantics></math> where agents report <math alttext="L" class="ltx_Math" display="inline" id="S1.SS1.p6.12.m12.1"><semantics id="S1.SS1.p6.12.m12.1a"><mi id="S1.SS1.p6.12.m12.1.1" xref="S1.SS1.p6.12.m12.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.12.m12.1b"><ci id="S1.SS1.p6.12.m12.1.1.cmml" xref="S1.SS1.p6.12.m12.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.12.m12.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.12.m12.1d">italic_L</annotation></semantics></math> regardless of their signal. The same logic shows that announcing <math alttext="\tau<0" class="ltx_Math" display="inline" id="S1.SS1.p6.13.m13.1"><semantics id="S1.SS1.p6.13.m13.1a"><mrow id="S1.SS1.p6.13.m13.1.1" xref="S1.SS1.p6.13.m13.1.1.cmml"><mi id="S1.SS1.p6.13.m13.1.1.2" xref="S1.SS1.p6.13.m13.1.1.2.cmml">τ</mi><mo id="S1.SS1.p6.13.m13.1.1.1" xref="S1.SS1.p6.13.m13.1.1.1.cmml"><</mo><mn id="S1.SS1.p6.13.m13.1.1.3" xref="S1.SS1.p6.13.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.13.m13.1b"><apply id="S1.SS1.p6.13.m13.1.1.cmml" xref="S1.SS1.p6.13.m13.1.1"><lt id="S1.SS1.p6.13.m13.1.1.1.cmml" xref="S1.SS1.p6.13.m13.1.1.1"></lt><ci id="S1.SS1.p6.13.m13.1.1.2.cmml" xref="S1.SS1.p6.13.m13.1.1.2">𝜏</ci><cn id="S1.SS1.p6.13.m13.1.1.3.cmml" type="integer" xref="S1.SS1.p6.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.13.m13.1c">\tau<0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.13.m13.1d">italic_τ < 0</annotation></semantics></math> will lead toward the trivial “always <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.p6.14.m14.1"><semantics id="S1.SS1.p6.14.m14.1a"><mi id="S1.SS1.p6.14.m14.1.1" xref="S1.SS1.p6.14.m14.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.14.m14.1b"><ci id="S1.SS1.p6.14.m14.1.1.cmml" xref="S1.SS1.p6.14.m14.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.14.m14.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.14.m14.1d">italic_H</annotation></semantics></math>” equilibrium at <math alttext="\tau=-\infty" class="ltx_Math" display="inline" id="S1.SS1.p6.15.m15.1"><semantics id="S1.SS1.p6.15.m15.1a"><mrow id="S1.SS1.p6.15.m15.1.1" xref="S1.SS1.p6.15.m15.1.1.cmml"><mi id="S1.SS1.p6.15.m15.1.1.2" xref="S1.SS1.p6.15.m15.1.1.2.cmml">τ</mi><mo id="S1.SS1.p6.15.m15.1.1.1" xref="S1.SS1.p6.15.m15.1.1.1.cmml">=</mo><mrow id="S1.SS1.p6.15.m15.1.1.3" xref="S1.SS1.p6.15.m15.1.1.3.cmml"><mo id="S1.SS1.p6.15.m15.1.1.3a" xref="S1.SS1.p6.15.m15.1.1.3.cmml">−</mo><mi id="S1.SS1.p6.15.m15.1.1.3.2" mathvariant="normal" xref="S1.SS1.p6.15.m15.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.15.m15.1b"><apply id="S1.SS1.p6.15.m15.1.1.cmml" xref="S1.SS1.p6.15.m15.1.1"><eq id="S1.SS1.p6.15.m15.1.1.1.cmml" xref="S1.SS1.p6.15.m15.1.1.1"></eq><ci id="S1.SS1.p6.15.m15.1.1.2.cmml" xref="S1.SS1.p6.15.m15.1.1.2">𝜏</ci><apply id="S1.SS1.p6.15.m15.1.1.3.cmml" xref="S1.SS1.p6.15.m15.1.1.3"><minus id="S1.SS1.p6.15.m15.1.1.3.1.cmml" xref="S1.SS1.p6.15.m15.1.1.3"></minus><infinity id="S1.SS1.p6.15.m15.1.1.3.2.cmml" xref="S1.SS1.p6.15.m15.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.15.m15.1c">\tau=-\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.15.m15.1d">italic_τ = - ∞</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p7"> <p class="ltx_p" id="S1.SS1.p7.1">From a dynamical systems perspective, the above shows that the <math alttext="\tau=0" class="ltx_Math" display="inline" id="S1.SS1.p7.1.m1.1"><semantics id="S1.SS1.p7.1.m1.1a"><mrow id="S1.SS1.p7.1.m1.1.1" xref="S1.SS1.p7.1.m1.1.1.cmml"><mi id="S1.SS1.p7.1.m1.1.1.2" xref="S1.SS1.p7.1.m1.1.1.2.cmml">τ</mi><mo id="S1.SS1.p7.1.m1.1.1.1" xref="S1.SS1.p7.1.m1.1.1.1.cmml">=</mo><mn id="S1.SS1.p7.1.m1.1.1.3" xref="S1.SS1.p7.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.1.m1.1b"><apply id="S1.SS1.p7.1.m1.1.1.cmml" xref="S1.SS1.p7.1.m1.1.1"><eq id="S1.SS1.p7.1.m1.1.1.1.cmml" xref="S1.SS1.p7.1.m1.1.1.1"></eq><ci id="S1.SS1.p7.1.m1.1.1.2.cmml" xref="S1.SS1.p7.1.m1.1.1.2">𝜏</ci><cn id="S1.SS1.p7.1.m1.1.1.3.cmml" type="integer" xref="S1.SS1.p7.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.1.m1.1d">italic_τ = 0</annotation></semantics></math> equilibrium is <span class="ltx_text ltx_font_italic" id="S1.SS1.p7.1.1">unstable</span> and can only be realized by starting exactly at it. Since there are no other finite equilibria, any variation or small error in the model will inexorably lead to a threshold that tends toward infinity. Given the inherent noisiness of many peer prediction applications, we argue similar to <cite class="ltx_cite ltx_citemacro_citet">Shnayder et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib14" title="">2016</a>]</cite> that <span class="ltx_text ltx_font_italic" id="S1.SS1.p7.1.2">the only reasonable equilibra in practice are stable ones</span>. Stability is thus a crucial equilibrium refinement for peer prediction.</p> </div> <div class="ltx_para" id="S1.SS1.p8"> <p class="ltx_p" id="S1.SS1.p8.1">To summarize, the standard analysis of OA in the binary signal model concludes that the mechanism has desirable incentive properties as long as agents believe others are more likely to share their signal than have the opposite. In contrast, by introducing our more nuanced real-valued signal model, we have seen that the only reasonable outcome to expect from OA is a “trivial” equilibrium where no information is gained. Thus, at least in this example, <span class="ltx_text ltx_font_italic" id="S1.SS1.p8.1.1">this shift in perspective takes OA from useful to entirely useless</span>.</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>Results</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.1">We explore the generality of our observation that moving to a real-valued signal model that refines the standard binary one substantially changes the incentives and behavior of peer prediction mechanisms. We begin by formally analyzing the Output Agreement (OA) mechanism discussed above. It is well-known in the binary signal setting that uninformative equilibria exist where all agents submit the same report. In our model, we find that uninformative equilibria not only exist, but remain stable under dynamics. That is, at least for some natural distributions, we would expect any initial threshold to drift toward <math alttext="\pm\infty" class="ltx_Math" display="inline" id="S1.SS2.p1.1.m1.1"><semantics id="S1.SS2.p1.1.m1.1a"><mrow id="S1.SS2.p1.1.m1.1.1" xref="S1.SS2.p1.1.m1.1.1.cmml"><mo id="S1.SS2.p1.1.m1.1.1a" xref="S1.SS2.p1.1.m1.1.1.cmml">±</mo><mi id="S1.SS2.p1.1.m1.1.1.2" mathvariant="normal" xref="S1.SS2.p1.1.m1.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.1.m1.1b"><apply id="S1.SS2.p1.1.m1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1"><csymbol cd="latexml" id="S1.SS2.p1.1.m1.1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1">plus-or-minus</csymbol><infinity id="S1.SS2.p1.1.m1.1.1.2.cmml" xref="S1.SS2.p1.1.m1.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.1.m1.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.1.m1.1d">± ∞</annotation></semantics></math> over time, eventually leading to blind agreement.</p> </div> <div class="ltx_para" id="S1.SS2.p2"> <p class="ltx_p" id="S1.SS2.p2.2">In the binary signal setting, a multi-task mechanism proposed by <cite class="ltx_cite ltx_citemacro_citet">Dasgupta and Ghosh [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib4" title="">2013</a>]</cite> (DG) fixes the uninformative equilibria problem in OA. The DG mechanism adds a penalty corresponding to the empirical frequency of agents’ reports across other tasks, effectively discouraging blind agreement and leading to strictly worse payments in uninformative equilibria. Under our richer signal model, we provide necessary and sufficient conditions for a threshold to be an equilibrium. Under some conditions like monotonicity, these equilibria are stable, while the uniformative equilibria are unstable. We conclude that DG is potentially useful but may only allow limited control over the equilibrium threshold, i.e., what <math alttext="H" 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">H</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">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.1.m1.1d">italic_H</annotation></semantics></math> and <math alttext="L" 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">L</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">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.2.m2.1d">italic_L</annotation></semantics></math> end up meaning.</p> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.1">We perform similar analyses for two other more complex mechanisms, Determininant Mutual Information (DMI) <cite class="ltx_cite ltx_citemacro_citep">[Kong, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib9" title="">2020</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib10" title="">2024</a>]</cite> and the Robust Bayesian Truth Serum (RBTS) <cite class="ltx_cite ltx_citemacro_citep">[Witkowski and Parkes, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib19" title="">2012</a>]</cite>. Under DMI, an interesting consequence of our model is the benefit of coordinating reports across tasks. For all four mechanisms we study we augment our general analysis with a worked example of the Gaussian case and numerical illustrations of more complex distributions that exhibit skewness and multimodality.</p> </div> <div class="ltx_para" id="S1.SS2.p4"> <p class="ltx_p" id="S1.SS2.p4.2">In general, our results show that realistic behavior under common peer prediction mechanisms departs significantly from the traditional discrete-signal theory. With a richer signal space, agents will essentially redefine the meaning of the binary <math alttext="H" class="ltx_Math" display="inline" id="S1.SS2.p4.1.m1.1"><semantics id="S1.SS2.p4.1.m1.1a"><mi id="S1.SS2.p4.1.m1.1.1" xref="S1.SS2.p4.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.1.m1.1b"><ci id="S1.SS2.p4.1.m1.1.1.cmml" xref="S1.SS2.p4.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.1.m1.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S1.SS2.p4.2.m2.1"><semantics id="S1.SS2.p4.2.m2.1a"><mi id="S1.SS2.p4.2.m2.1.1" xref="S1.SS2.p4.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p4.2.m2.1b"><ci id="S1.SS2.p4.2.m2.1.1.cmml" xref="S1.SS2.p4.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p4.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p4.2.m2.1d">italic_L</annotation></semantics></math> signals based on community norms. This paper therefore serves as a launching point for uncovering how behavior changes under peer prediction in the real world. In particular, our results will help practitioners better understand the implications of their choice of mechanism.</p> </div> </section> <section class="ltx_subsection" id="S1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.3 </span>Related Work</h3> <div class="ltx_para" id="S1.SS3.p1"> <p class="ltx_p" id="S1.SS3.p1.1">For a general overview of peer prediction, we direct the reader to <cite class="ltx_cite ltx_citemacro_citet">Faltings [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib6" title="">2023</a>], Lehmann [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib11" title="">2024</a>], Frongillo and Waggoner [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib7" title="">2024</a>]</cite>. Most relevant to our work is the distinction between peer prediction mechanisms which are <span class="ltx_text ltx_font_italic" id="S1.SS3.p1.1.1">minimal</span>, in that agents only report their signal (as in OA), and those where additional information about agent beliefs is reported (e.g. the agent’s posterior belief about the report of another agent as in the Bayesian Truth Serum <cite class="ltx_cite ltx_citemacro_citep">[Prelec, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib13" title="">2004</a>]</cite> and its variants). Another important dimension is whether the mechanism is for a single task or a larger <em class="ltx_emph ltx_font_italic" id="S1.SS3.p1.1.2">mult-task</em> collection where reports from unrelated tasks can be compared to improve incentives. We examine two multi-task mechanisms (DG and DMI) and a non-minimal one (RBTS). While more robust than OA, our results show that neither of these structures avoids the issues our work raises.</p> </div> <div class="ltx_para" id="S1.SS3.p2"> <p class="ltx_p" id="S1.SS3.p2.4">To our knowledge, our approach of assuming agents have a fundamentally richer signal than they are asked to reveal is novel. The idea using dynamical stability for equilibrium refinement in peer prediction was introduced by <cite class="ltx_cite ltx_citemacro_citet">Shnayder et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib14" title="">2016</a>]</cite>. That work looks at a population of agents playing strategies from a discrete set, such as <math alttext="\{" class="ltx_Math" display="inline" id="S1.SS3.p2.1.m1.1"><semantics id="S1.SS3.p2.1.m1.1a"><mo id="S1.SS3.p2.1.m1.1.1" stretchy="false" xref="S1.SS3.p2.1.m1.1.1.cmml">{</mo><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.1.m1.1b"><ci id="S1.SS3.p2.1.m1.1.1.cmml" xref="S1.SS3.p2.1.m1.1.1">{</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.1.m1.1c">\{</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.1.m1.1d">{</annotation></semantics></math>truthful, always <math alttext="H" class="ltx_Math" display="inline" id="S1.SS3.p2.2.m2.1"><semantics id="S1.SS3.p2.2.m2.1a"><mi id="S1.SS3.p2.2.m2.1.1" xref="S1.SS3.p2.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.2.m2.1b"><ci id="S1.SS3.p2.2.m2.1.1.cmml" xref="S1.SS3.p2.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.2.m2.1d">italic_H</annotation></semantics></math>, and always <math alttext="L" class="ltx_Math" display="inline" id="S1.SS3.p2.3.m3.1"><semantics id="S1.SS3.p2.3.m3.1a"><mi id="S1.SS3.p2.3.m3.1.1" xref="S1.SS3.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.3.m3.1b"><ci id="S1.SS3.p2.3.m3.1.1.cmml" xref="S1.SS3.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.3.m3.1d">italic_L</annotation></semantics></math> <math alttext="\}" class="ltx_Math" display="inline" id="S1.SS3.p2.4.m4.1"><semantics id="S1.SS3.p2.4.m4.1a"><mo id="S1.SS3.p2.4.m4.1.1" stretchy="false" xref="S1.SS3.p2.4.m4.1.1.cmml">}</mo><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.4.m4.1b"><ci id="S1.SS3.p2.4.m4.1.1.cmml" xref="S1.SS3.p2.4.m4.1.1">}</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.4.m4.1c">\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.4.m4.1d">}</annotation></semantics></math>. Using replicator dynamics, they measure the stability of equilibria by their basin of attraction, the volume of initial conditions leading to that equilbrium. At a very high level, their conclusions bear similarity to ours: useful equilibria are less robust for OA, and multi-task mechanisms like DG are more robust.</p> </div> <div class="ltx_para" id="S1.SS3.p3"> <p class="ltx_p" id="S1.SS3.p3.1">Looking closer, two key differences between our work and <cite class="ltx_cite ltx_citemacro_citet">Shnayder et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib14" title="">2016</a>]</cite> are the signal model and strategic model. For the signal model, they consider a discrete signal space matching the report space, as usual. For the strategic model, they consider agents learning whether to switch to another strategy in a discrete set, whereas we allow agents to choose any threshold strategy. As a result, our results differ substantially in some cases: for OA, the truthful equilibrium is unstable even in a technical sense (measure zero basin of attraction), and for DG, “truthfulness” as given by the mechanism designer’s desired threshold need not be an equilibrium.</p> </div> <div class="ltx_para" id="S1.SS3.p4"> <p class="ltx_p" id="S1.SS3.p4.1">We note that the model and tools in our paper also bear some resemblance to past work in the epistemic democracy literature. Specifically, <cite class="ltx_cite ltx_citemacro_citet">Duggan and Martinelli [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib5" title="">2001</a>]</cite> and <cite class="ltx_cite ltx_citemacro_citet">Meirowitz [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib12" title="">2002</a>]</cite> study a common-value voting setting, where there is a ground truth “correct” alternative that all agents prefer, and they each receive signals about which alternative that is. As in our paper, the alternative space is binary while the signal space is continuous. The authors similarly identify threshold equilibria, where agents map their signals to a vote for either alternative according to a cutoff point, and prove that under some conditions on the continuity and monotonicity of the prior densities over signals in each world state, there is a unique symmetric Bayes-Nash equilibrium characterized by a threshold. We note that though the tools are similar (equilibria are characterized by evaluating conditional probabilities at the threshold), the setting and goals of the papers are quite different. (The authors aim to show that the “right” alternative is chosen with high probability over the limit of the number of agents.)</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>Output Agreement</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.4">We begin by analyzing the popular peer prediction mechanism Output Agreement (OA). Output Agreement is a minimal, single task mechanism: each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_i</annotation></semantics></math> submits a report for a task and receives a positive payment if their report matches that of another agent <math alttext="j" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_j</annotation></semantics></math> on the same task. In the binary signal model, truthful reporting forms a correlated equilibrium under some assumptions about the information structure. 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xref="S2.p1.4.m4.4.4.2.1.1.1.3.cmml">L</mi></mrow><mo id="S2.p1.4.m4.4.4.2.1.1.3" stretchy="false" xref="S2.p1.4.m4.4.4.2.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.4b"><apply id="S2.p1.4.m4.4.4.cmml" xref="S2.p1.4.m4.4.4"><gt id="S2.p1.4.m4.4.4.3.cmml" xref="S2.p1.4.m4.4.4.3"></gt><apply id="S2.p1.4.m4.3.3.1.2.cmml" xref="S2.p1.4.m4.3.3.1.1"><ci id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1">Pr</ci><apply id="S2.p1.4.m4.3.3.1.1.1.1.cmml" xref="S2.p1.4.m4.3.3.1.1.1.1"><csymbol cd="latexml" id="S2.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S2.p1.4.m4.3.3.1.1.1.1.1">conditional</csymbol><ci id="S2.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S2.p1.4.m4.3.3.1.1.1.1.2">𝐿</ci><ci id="S2.p1.4.m4.3.3.1.1.1.1.3.cmml" xref="S2.p1.4.m4.3.3.1.1.1.1.3">𝐿</ci></apply></apply><apply id="S2.p1.4.m4.4.4.2.2.cmml" xref="S2.p1.4.m4.4.4.2.1"><ci id="S2.p1.4.m4.2.2.cmml" xref="S2.p1.4.m4.2.2">Pr</ci><apply id="S2.p1.4.m4.4.4.2.1.1.1.cmml" xref="S2.p1.4.m4.4.4.2.1.1.1"><csymbol 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However, <em class="ltx_emph ltx_font_italic" id="S2.p1.4.2">uninformative</em> equilibria where agents misreport their information also exist. Specifically, agents can coordinate to all submit the same report and each receive a maximum payment. Nonetheless, Output Agreement remains popular because of its simplicity. We thus aim to characterize OA under our richer signal space model to help practitioners understand how behavior occurs in the real world.</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>Model</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.8">We begin by introducing the model that, with minor variations to accommodate the form of different peer prediction mechanisms, will be used througout the paper. There are <math alttext="n" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_n</annotation></semantics></math> agents; each agent <math alttext="i" 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">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_i</annotation></semantics></math> receives a signal <math alttext="X_{i}\in\mathbb{R}" 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.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><msub id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2.2" xref="S2.SS1.p1.3.m3.1.1.2.2.cmml">X</mi><mi id="S2.SS1.p1.3.m3.1.1.2.3" xref="S2.SS1.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p1.3.m3.1.1.1" xref="S2.SS1.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><in id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1.1"></in><apply id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.2.1.cmml" xref="S2.SS1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.2.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2.2">𝑋</ci><ci id="S2.SS1.p1.3.m3.1.1.2.3.cmml" xref="S2.SS1.p1.3.m3.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">X_{i}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> representing information gained about a shared task. The signals are drawn from a joint distribution <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">𝒟</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">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">caligraphic_D</annotation></semantics></math> that is symmetric in the sense that each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mi id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><ci id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation 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xref="S2.SS1.p1.6.m6.3.3.1.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.3b"><apply id="S2.SS1.p1.6.m6.3.3.cmml" xref="S2.SS1.p1.6.m6.3.3"><eq id="S2.SS1.p1.6.m6.3.3.2.cmml" xref="S2.SS1.p1.6.m6.3.3.2"></eq><apply id="S2.SS1.p1.6.m6.3.3.3.cmml" xref="S2.SS1.p1.6.m6.3.3.3"><times id="S2.SS1.p1.6.m6.3.3.3.1.cmml" xref="S2.SS1.p1.6.m6.3.3.3.1"></times><ci id="S2.SS1.p1.6.m6.3.3.3.2.cmml" xref="S2.SS1.p1.6.m6.3.3.3.2">𝐹</ci><ci id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1">𝑥</ci></apply><apply id="S2.SS1.p1.6.m6.3.3.1.2.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1"><ci id="S2.SS1.p1.6.m6.2.2.cmml" xref="S2.SS1.p1.6.m6.2.2">Pr</ci><apply id="S2.SS1.p1.6.m6.3.3.1.1.1.1.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1"><eq id="S2.SS1.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.1"></eq><apply id="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.1.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.2.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.2">𝑋</ci><ci id="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.3.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p1.6.m6.3.3.1.1.1.1.3.cmml" xref="S2.SS1.p1.6.m6.3.3.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.3c">F(x)=\Pr[X_{i}=x]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.3d">italic_F ( italic_x ) = roman_Pr [ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x ]</annotation></semantics></math>, and (2) the same posterior distribution <math alttext="\beta(x)=\Pr[X^{\prime}=\cdot\mid X_{i}=x]" class="ltx_math_unparsed" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mrow id="S2.SS1.p1.7.m7.1b"><mi id="S2.SS1.p1.7.m7.1.2">β</mi><mrow id="S2.SS1.p1.7.m7.1.3"><mo id="S2.SS1.p1.7.m7.1.3.1" stretchy="false">(</mo><mi id="S2.SS1.p1.7.m7.1.1">x</mi><mo id="S2.SS1.p1.7.m7.1.3.2" stretchy="false">)</mo></mrow><mo id="S2.SS1.p1.7.m7.1.4">=</mo><mi id="S2.SS1.p1.7.m7.1.5">Pr</mi><mrow id="S2.SS1.p1.7.m7.1.6"><mo id="S2.SS1.p1.7.m7.1.6.1" stretchy="false">[</mo><msup id="S2.SS1.p1.7.m7.1.6.2"><mi id="S2.SS1.p1.7.m7.1.6.2.2">X</mi><mo id="S2.SS1.p1.7.m7.1.6.2.3">′</mo></msup><mo id="S2.SS1.p1.7.m7.1.6.3" rspace="0em">=</mo><mo id="S2.SS1.p1.7.m7.1.6.4" lspace="0em" rspace="0em">⋅</mo><mo id="S2.SS1.p1.7.m7.1.6.5" lspace="0em" rspace="0.167em">∣</mo><msub id="S2.SS1.p1.7.m7.1.6.6"><mi id="S2.SS1.p1.7.m7.1.6.6.2">X</mi><mi id="S2.SS1.p1.7.m7.1.6.6.3">i</mi></msub><mo id="S2.SS1.p1.7.m7.1.6.7">=</mo><mi id="S2.SS1.p1.7.m7.1.6.8">x</mi><mo id="S2.SS1.p1.7.m7.1.6.9" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">\beta(x)=\Pr[X^{\prime}=\cdot\mid X_{i}=x]</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_β ( italic_x ) = roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ⋅ ∣ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_x ]</annotation></semantics></math>. Identical marginals are realistic when the signals are exchangeable, e.g. when they have the same conditional distribution over some latent variable <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mi id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><ci id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">italic_θ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.7">Let the report space be <math alttext="\mathcal{R}=\{L,H\}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.2"><semantics id="S2.SS1.p2.1.m1.2a"><mrow id="S2.SS1.p2.1.m1.2.3" xref="S2.SS1.p2.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.2.3.2" xref="S2.SS1.p2.1.m1.2.3.2.cmml">ℛ</mi><mo id="S2.SS1.p2.1.m1.2.3.1" xref="S2.SS1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS1.p2.1.m1.2.3.3.2" xref="S2.SS1.p2.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p2.1.m1.2.3.3.1.cmml">{</mo><mi id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">L</mi><mo id="S2.SS1.p2.1.m1.2.3.3.2.2" xref="S2.SS1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.1.m1.2.2" xref="S2.SS1.p2.1.m1.2.2.cmml">H</mi><mo id="S2.SS1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS1.p2.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.2b"><apply id="S2.SS1.p2.1.m1.2.3.cmml" xref="S2.SS1.p2.1.m1.2.3"><eq id="S2.SS1.p2.1.m1.2.3.1.cmml" xref="S2.SS1.p2.1.m1.2.3.1"></eq><ci id="S2.SS1.p2.1.m1.2.3.2.cmml" xref="S2.SS1.p2.1.m1.2.3.2">ℛ</ci><set id="S2.SS1.p2.1.m1.2.3.3.1.cmml" xref="S2.SS1.p2.1.m1.2.3.3.2"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝐿</ci><ci id="S2.SS1.p2.1.m1.2.2.cmml" xref="S2.SS1.p2.1.m1.2.2">𝐻</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.2c">\mathcal{R}=\{L,H\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.2d">caligraphic_R = { italic_L , italic_H }</annotation></semantics></math>. Each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_i</annotation></semantics></math> calculates a report <math alttext="r_{i}\in\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mrow id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><msub id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2.2" xref="S2.SS1.p2.3.m3.1.1.2.2.cmml">r</mi><mi id="S2.SS1.p2.3.m3.1.1.2.3" xref="S2.SS1.p2.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.3.m3.1.1.1" xref="S2.SS1.p2.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">ℛ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><in id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1.1"></in><apply id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.2.1.cmml" xref="S2.SS1.p2.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2.2">𝑟</ci><ci id="S2.SS1.p2.3.m3.1.1.2.3.cmml" xref="S2.SS1.p2.3.m3.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p2.3.m3.1.1.3.cmml" xref="S2.SS1.p2.3.m3.1.1.3">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">r_{i}\in\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_R</annotation></semantics></math> for the task according to a deterministic strategy <math alttext="\sigma_{i}:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><msub id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2.2" xref="S2.SS1.p2.4.m4.1.1.2.2.cmml">σ</mi><mi id="S2.SS1.p2.4.m4.1.1.2.3" xref="S2.SS1.p2.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.4.m4.1.1.1.cmml">:</mo><mrow id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml"><mi id="S2.SS1.p2.4.m4.1.1.3.2" xref="S2.SS1.p2.4.m4.1.1.3.2.cmml">ℝ</mi><mo id="S2.SS1.p2.4.m4.1.1.3.1" stretchy="false" xref="S2.SS1.p2.4.m4.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.1.1.3.3" xref="S2.SS1.p2.4.m4.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><ci id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1.1">:</ci><apply id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.1.2.1.cmml" xref="S2.SS1.p2.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.1.2.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2.2">𝜎</ci><ci id="S2.SS1.p2.4.m4.1.1.2.3.cmml" xref="S2.SS1.p2.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S2.SS1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3"><ci id="S2.SS1.p2.4.m4.1.1.3.1.cmml" xref="S2.SS1.p2.4.m4.1.1.3.1">→</ci><ci id="S2.SS1.p2.4.m4.1.1.3.2.cmml" xref="S2.SS1.p2.4.m4.1.1.3.2">ℝ</ci><ci id="S2.SS1.p2.4.m4.1.1.3.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">\sigma_{i}:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : blackboard_R → caligraphic_R</annotation></semantics></math> mapping from signal to report space. The Output Agreement mechanism pays each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><ci id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_i</annotation></semantics></math> an amount <math alttext="M_{\textrm{OA}}(r_{i},r_{j})" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.2"><semantics id="S2.SS1.p2.6.m6.2a"><mrow id="S2.SS1.p2.6.m6.2.2" xref="S2.SS1.p2.6.m6.2.2.cmml"><msub id="S2.SS1.p2.6.m6.2.2.4" xref="S2.SS1.p2.6.m6.2.2.4.cmml"><mi id="S2.SS1.p2.6.m6.2.2.4.2" xref="S2.SS1.p2.6.m6.2.2.4.2.cmml">M</mi><mtext id="S2.SS1.p2.6.m6.2.2.4.3" xref="S2.SS1.p2.6.m6.2.2.4.3a.cmml">OA</mtext></msub><mo 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id="S2.SS1.p2.6.m6.2.2.2.2.2.cmml" xref="S2.SS1.p2.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.2.2.2.2.2.1.cmml" xref="S2.SS1.p2.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.6.m6.2.2.2.2.2.2.cmml" xref="S2.SS1.p2.6.m6.2.2.2.2.2.2">𝑟</ci><ci id="S2.SS1.p2.6.m6.2.2.2.2.2.3.cmml" xref="S2.SS1.p2.6.m6.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.2c">M_{\textrm{OA}}(r_{i},r_{j})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.2d">italic_M start_POSTSUBSCRIPT OA end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> as a function of reports <math alttext="r_{i},r_{j}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.2"><semantics id="S2.SS1.p2.7.m7.2a"><mrow id="S2.SS1.p2.7.m7.2.2.2" xref="S2.SS1.p2.7.m7.2.2.3.cmml"><msub id="S2.SS1.p2.7.m7.1.1.1.1" 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cd="ambiguous" id="S2.Ex1.m1.1.1.1.1.3.1.1.1.2.1.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.2">subscript</csymbol><ci id="S2.Ex1.m1.1.1.1.1.3.1.1.1.2.2.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.2.2">𝑟</ci><ci id="S2.Ex1.m1.1.1.1.1.3.1.1.1.2.3.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.2.3">𝑖</ci></apply><apply id="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.1.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.3">subscript</csymbol><ci id="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.2.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.2">𝑟</ci><ci id="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.3.cmml" xref="S2.Ex1.m1.1.1.1.1.3.1.1.1.3.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.1c">M_{\textrm{OA}}(r_{i},r_{j})=\mathbf{1}[r_{i}=r_{j}].</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.1d">italic_M start_POSTSUBSCRIPT OA end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = bold_1 [ italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.3">Thus the (ex-interim) expected utility for playing strategy <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><msub id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">σ</mi><mi id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝜎</ci><ci id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_j</annotation></semantics></math> plays according to <math alttext="\sigma_{j}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><msub id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml"><mi id="S2.SS1.p3.3.m3.1.1.2" xref="S2.SS1.p3.3.m3.1.1.2.cmml">σ</mi><mi id="S2.SS1.p3.3.m3.1.1.3" xref="S2.SS1.p3.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.2.cmml" xref="S2.SS1.p3.3.m3.1.1.2">𝜎</ci><ci id="S2.SS1.p3.3.m3.1.1.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">\sigma_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is</p> <table class="ltx_equation ltx_eqn_table" 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italic_i end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.6">We consider a natural class of strategies, <em class="ltx_emph ltx_font_italic" id="S2.SS1.p4.6.1">threshold strategies</em>, where agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mi id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_i</annotation></semantics></math> reports <math alttext="H" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mi id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><ci id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_H</annotation></semantics></math> if and only if their signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><mi id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><ci id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_x</annotation></semantics></math> satisfies <math alttext="x\geq\tau" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.1"><semantics id="S2.SS1.p4.4.m4.1a"><mrow id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml"><mi id="S2.SS1.p4.4.m4.1.1.2" xref="S2.SS1.p4.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS1.p4.4.m4.1.1.1" xref="S2.SS1.p4.4.m4.1.1.1.cmml">≥</mo><mi id="S2.SS1.p4.4.m4.1.1.3" xref="S2.SS1.p4.4.m4.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.1b"><apply id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1"><geq id="S2.SS1.p4.4.m4.1.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1.1"></geq><ci id="S2.SS1.p4.4.m4.1.1.2.cmml" xref="S2.SS1.p4.4.m4.1.1.2">𝑥</ci><ci id="S2.SS1.p4.4.m4.1.1.3.cmml" xref="S2.SS1.p4.4.m4.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.1c">x\geq\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.1d">italic_x ≥ italic_τ</annotation></semantics></math> for some fixed threshold <math alttext="\tau\in\mathbb{R}\cup\{\pm\infty\}" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.1"><semantics id="S2.SS1.p4.5.m5.1a"><mrow id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml"><mi id="S2.SS1.p4.5.m5.1.1.3" xref="S2.SS1.p4.5.m5.1.1.3.cmml">τ</mi><mo id="S2.SS1.p4.5.m5.1.1.2" xref="S2.SS1.p4.5.m5.1.1.2.cmml">∈</mo><mrow id="S2.SS1.p4.5.m5.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.cmml"><mi id="S2.SS1.p4.5.m5.1.1.1.3" xref="S2.SS1.p4.5.m5.1.1.1.3.cmml">ℝ</mi><mo id="S2.SS1.p4.5.m5.1.1.1.2" xref="S2.SS1.p4.5.m5.1.1.1.2.cmml">∪</mo><mrow id="S2.SS1.p4.5.m5.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.2.cmml"><mo id="S2.SS1.p4.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.5.m5.1.1.1.1.2.cmml">{</mo><mrow id="S2.SS1.p4.5.m5.1.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p4.5.m5.1.1.1.1.1.1a" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.cmml">±</mo><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.2" mathvariant="normal" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.2.cmml">∞</mi></mrow><mo id="S2.SS1.p4.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.5.m5.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.1b"><apply id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1"><in id="S2.SS1.p4.5.m5.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.2"></in><ci id="S2.SS1.p4.5.m5.1.1.3.cmml" xref="S2.SS1.p4.5.m5.1.1.3">𝜏</ci><apply id="S2.SS1.p4.5.m5.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1"><union id="S2.SS1.p4.5.m5.1.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.1.2"></union><ci id="S2.SS1.p4.5.m5.1.1.1.3.cmml" xref="S2.SS1.p4.5.m5.1.1.1.3">ℝ</ci><set id="S2.SS1.p4.5.m5.1.1.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.1.1.1"><apply id="S2.SS1.p4.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1">plus-or-minus</csymbol><infinity id="S2.SS1.p4.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.2"></infinity></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.1c">\tau\in\mathbb{R}\cup\{\pm\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.1d">italic_τ ∈ blackboard_R ∪ { ± ∞ }</annotation></semantics></math>. That is, for some <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m6.1"><semantics id="S2.SS1.p4.6.m6.1a"><mi id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m6.1b"><ci id="S2.SS1.p4.6.m6.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m6.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m6.1d">italic_τ</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\tau}(x)=\begin{cases}L,&x\leq\tau\\ H,&x>\tau.\end{cases}" class="ltx_Math" display="block" id="S2.Ex2.m1.5"><semantics id="S2.Ex2.m1.5a"><mrow id="S2.Ex2.m1.5.6" xref="S2.Ex2.m1.5.6.cmml"><mrow id="S2.Ex2.m1.5.6.2" 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xref="S2.Ex2.m1.5.6.2.1"></times><apply id="S2.Ex2.m1.5.6.2.2.cmml" xref="S2.Ex2.m1.5.6.2.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.6.2.2.1.cmml" xref="S2.Ex2.m1.5.6.2.2">superscript</csymbol><ci id="S2.Ex2.m1.5.6.2.2.2.cmml" xref="S2.Ex2.m1.5.6.2.2.2">𝜎</ci><ci id="S2.Ex2.m1.5.6.2.2.3.cmml" xref="S2.Ex2.m1.5.6.2.2.3">𝜏</ci></apply><ci id="S2.Ex2.m1.5.5.cmml" xref="S2.Ex2.m1.5.5">𝑥</ci></apply><apply id="S2.Ex2.m1.5.6.3.1.cmml" xref="S2.Ex2.m1.4.4"><csymbol cd="latexml" id="S2.Ex2.m1.5.6.3.1.1.cmml" xref="S2.Ex2.m1.4.4.5">cases</csymbol><ci id="S2.Ex2.m1.1.1.1.1.1.1.1.cmml" xref="S2.Ex2.m1.1.1.1.1.1.1.1">𝐿</ci><apply id="S2.Ex2.m1.2.2.2.2.2.1.cmml" xref="S2.Ex2.m1.2.2.2.2.2.1"><leq id="S2.Ex2.m1.2.2.2.2.2.1.1.cmml" xref="S2.Ex2.m1.2.2.2.2.2.1.1"></leq><ci id="S2.Ex2.m1.2.2.2.2.2.1.2.cmml" xref="S2.Ex2.m1.2.2.2.2.2.1.2">𝑥</ci><ci id="S2.Ex2.m1.2.2.2.2.2.1.3.cmml" xref="S2.Ex2.m1.2.2.2.2.2.1.3">𝜏</ci></apply><ci id="S2.Ex2.m1.3.3.3.3.1.1.1.cmml" xref="S2.Ex2.m1.3.3.3.3.1.1.1">𝐻</ci><apply id="S2.Ex2.m1.4.4.4.4.2.1.1.1.cmml" xref="S2.Ex2.m1.4.4.4.4.2.1.1"><gt id="S2.Ex2.m1.4.4.4.4.2.1.1.1.1.cmml" xref="S2.Ex2.m1.4.4.4.4.2.1.1.1.1"></gt><ci id="S2.Ex2.m1.4.4.4.4.2.1.1.1.2.cmml" xref="S2.Ex2.m1.4.4.4.4.2.1.1.1.2">𝑥</ci><ci id="S2.Ex2.m1.4.4.4.4.2.1.1.1.3.cmml" xref="S2.Ex2.m1.4.4.4.4.2.1.1.1.3">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.5c">\sigma^{\tau}(x)=\begin{cases}L,&x\leq\tau\\ H,&x>\tau.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.5d">italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_x ) = { start_ROW start_CELL italic_L , end_CELL start_CELL italic_x ≤ italic_τ end_CELL end_ROW start_ROW start_CELL italic_H , end_CELL start_CELL italic_x > italic_τ . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p4.9">We will often denote a strategy directly by its threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS1.p4.7.m1.1"><semantics id="S2.SS1.p4.7.m1.1a"><mi id="S2.SS1.p4.7.m1.1.1" xref="S2.SS1.p4.7.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.7.m1.1b"><ci id="S2.SS1.p4.7.m1.1.1.cmml" xref="S2.SS1.p4.7.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.7.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.7.m1.1d">italic_τ</annotation></semantics></math> when it is clear. Thresholding is a natural way to assign semantic meaning to the discrete labels <math alttext="H" class="ltx_Math" display="inline" id="S2.SS1.p4.8.m2.1"><semantics id="S2.SS1.p4.8.m2.1a"><mi id="S2.SS1.p4.8.m2.1.1" xref="S2.SS1.p4.8.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.8.m2.1b"><ci id="S2.SS1.p4.8.m2.1.1.cmml" xref="S2.SS1.p4.8.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.8.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.8.m2.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S2.SS1.p4.9.m3.1"><semantics id="S2.SS1.p4.9.m3.1a"><mi id="S2.SS1.p4.9.m3.1.1" xref="S2.SS1.p4.9.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.9.m3.1b"><ci id="S2.SS1.p4.9.m3.1.1.cmml" xref="S2.SS1.p4.9.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.9.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.9.m3.1d">italic_L</annotation></semantics></math> according to the continuous signal space. Moreover, in many settings the mechanism designer themselves may announce a threshold to establish norms that they would like agents to follow. For example, in peer grading a teacher may establish a higher threshold to encourage a high bar for good marks.</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>Equilibrium Characterization</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.2">We are interested in characterizing threshold strategies which are <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.2.1">symmetric Bayes-Nash equilibria</em>. That is, equilibria where each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mi id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">italic_i</annotation></semantics></math> commits to the same threshold strategy <math alttext="\sigma^{\tau}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><msup id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml"><mi id="S2.SS2.p1.2.m2.1.1.2" xref="S2.SS2.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S2.SS2.p1.2.m2.1.1.3" xref="S2.SS2.p1.2.m2.1.1.3.cmml">τ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><apply id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.2.m2.1.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">superscript</csymbol><ci id="S2.SS2.p1.2.m2.1.1.2.cmml" xref="S2.SS2.p1.2.m2.1.1.2">𝜎</ci><ci id="S2.SS2.p1.2.m2.1.1.3.cmml" xref="S2.SS2.p1.2.m2.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\sigma^{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT</annotation></semantics></math>. For brevity, we refer to these as threshold equilibria. Symmetric strategies are natural in our model since agents share the same ex-ante expected utilities.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmdefinition1.1.1.1">Definition 1</span></span><span class="ltx_text ltx_font_bold" id="Thmdefinition1.2.2">.</span> </h6> <div class="ltx_para" id="Thmdefinition1.p1"> <p class="ltx_p" id="Thmdefinition1.p1.1">A threshold strategy <math alttext="\sigma^{\tau}:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="Thmdefinition1.p1.1.m1.1"><semantics id="Thmdefinition1.p1.1.m1.1a"><mrow id="Thmdefinition1.p1.1.m1.1.1" xref="Thmdefinition1.p1.1.m1.1.1.cmml"><msup id="Thmdefinition1.p1.1.m1.1.1.2" xref="Thmdefinition1.p1.1.m1.1.1.2.cmml"><mi id="Thmdefinition1.p1.1.m1.1.1.2.2" xref="Thmdefinition1.p1.1.m1.1.1.2.2.cmml">σ</mi><mi id="Thmdefinition1.p1.1.m1.1.1.2.3" xref="Thmdefinition1.p1.1.m1.1.1.2.3.cmml">τ</mi></msup><mo id="Thmdefinition1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmdefinition1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="Thmdefinition1.p1.1.m1.1.1.3" xref="Thmdefinition1.p1.1.m1.1.1.3.cmml"><mi id="Thmdefinition1.p1.1.m1.1.1.3.2" xref="Thmdefinition1.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="Thmdefinition1.p1.1.m1.1.1.3.1" stretchy="false" xref="Thmdefinition1.p1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.1.m1.1.1.3.3" xref="Thmdefinition1.p1.1.m1.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.1.m1.1b"><apply id="Thmdefinition1.p1.1.m1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1"><ci id="Thmdefinition1.p1.1.m1.1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.1">:</ci><apply id="Thmdefinition1.p1.1.m1.1.1.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmdefinition1.p1.1.m1.1.1.2.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.2">superscript</csymbol><ci id="Thmdefinition1.p1.1.m1.1.1.2.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.2.2">𝜎</ci><ci id="Thmdefinition1.p1.1.m1.1.1.2.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.2.3">𝜏</ci></apply><apply id="Thmdefinition1.p1.1.m1.1.1.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3"><ci id="Thmdefinition1.p1.1.m1.1.1.3.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.1">→</ci><ci id="Thmdefinition1.p1.1.m1.1.1.3.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.2">ℝ</ci><ci id="Thmdefinition1.p1.1.m1.1.1.3.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.1.m1.1c">\sigma^{\tau}:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT : blackboard_R → caligraphic_R</annotation></semantics></math> is a <span class="ltx_text ltx_font_italic" id="Thmdefinition1.p1.1.1">threshold equilibrium</span> under OA if</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall x\in\mathbb{R},\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[% \sigma^{\tau}(x)=\sigma^{\tau}(x^{\prime})]\geq\mathop{\mathbb{E}}_{x^{\prime}% \sim\beta(x)}\mathbf{1}[\overline{\sigma^{\tau}(x)}=\sigma^{\tau}(x^{\prime})]," class="ltx_Math" display="block" id="S2.E2.m1.5"><semantics id="S2.E2.m1.5a"><mrow id="S2.E2.m1.5.5.1"><mrow id="S2.E2.m1.5.5.1.1.2" xref="S2.E2.m1.5.5.1.1.3.cmml"><mrow id="S2.E2.m1.5.5.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.1.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.1.1.2.1" rspace="0.167em" xref="S2.E2.m1.5.5.1.1.1.1.2.1.cmml">∀</mo><mi id="S2.E2.m1.5.5.1.1.1.1.2.2" xref="S2.E2.m1.5.5.1.1.1.1.2.2.cmml">x</mi></mrow><mo id="S2.E2.m1.5.5.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.1.1.1.cmml">∈</mo><mi id="S2.E2.m1.5.5.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.1.1.3.cmml">ℝ</mi></mrow><mo id="S2.E2.m1.5.5.1.1.2.3" xref="S2.E2.m1.5.5.1.1.3a.cmml">,</mo><mrow id="S2.E2.m1.5.5.1.1.2.2" xref="S2.E2.m1.5.5.1.1.2.2.cmml"><mrow id="S2.E2.m1.5.5.1.1.2.2.1" xref="S2.E2.m1.5.5.1.1.2.2.1.cmml"><munder id="S2.E2.m1.5.5.1.1.2.2.1.2" xref="S2.E2.m1.5.5.1.1.2.2.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.1.2.2" lspace="0em" movablelimits="false" rspace="0.167em" xref="S2.E2.m1.5.5.1.1.2.2.1.2.2.cmml">𝔼</mo><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><msup id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml"><mi id="S2.E2.m1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.3.2.cmml">x</mi><mo id="S2.E2.m1.1.1.1.3.3" xref="S2.E2.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">∼</mo><mrow id="S2.E2.m1.1.1.1.4" xref="S2.E2.m1.1.1.1.4.cmml"><mi id="S2.E2.m1.1.1.1.4.2" xref="S2.E2.m1.1.1.1.4.2.cmml">β</mi><mo id="S2.E2.m1.1.1.1.4.1" xref="S2.E2.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.4.3.2" xref="S2.E2.m1.1.1.1.4.cmml"><mo id="S2.E2.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.E2.m1.1.1.1.4.cmml">(</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">x</mi><mo id="S2.E2.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.E2.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.cmml"><mn id="S2.E2.m1.5.5.1.1.2.2.1.1.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.3.cmml">𝟏</mn><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.cmml"><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.cmml"><msup id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.cmml"><mi id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.2.cmml">σ</mi><mi id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.2.3.cmml">τ</mi></msup><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.3.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.cmml">(</mo><mi id="S2.E2.m1.4.4" xref="S2.E2.m1.4.4.cmml">x</mi><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.2.cmml">=</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.cmml"><msup id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3.cmml"><mi id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3.2.cmml">σ</mi><mi id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.3.3.cmml">τ</mi></msup><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.5.5.1.1.2.2.1.1.1.1.3" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.1.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="S2.E2.m1.5.5.1.1.2.2.3" rspace="0.1389em" xref="S2.E2.m1.5.5.1.1.2.2.3.cmml">≥</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.2" xref="S2.E2.m1.5.5.1.1.2.2.2.cmml"><munder id="S2.E2.m1.5.5.1.1.2.2.2.2" xref="S2.E2.m1.5.5.1.1.2.2.2.2.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.2.2.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S2.E2.m1.5.5.1.1.2.2.2.2.2.cmml">𝔼</mo><mrow id="S2.E2.m1.2.2.1" xref="S2.E2.m1.2.2.1.cmml"><msup id="S2.E2.m1.2.2.1.3" xref="S2.E2.m1.2.2.1.3.cmml"><mi id="S2.E2.m1.2.2.1.3.2" xref="S2.E2.m1.2.2.1.3.2.cmml">x</mi><mo id="S2.E2.m1.2.2.1.3.3" xref="S2.E2.m1.2.2.1.3.3.cmml">′</mo></msup><mo id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.2.cmml">∼</mo><mrow id="S2.E2.m1.2.2.1.4" xref="S2.E2.m1.2.2.1.4.cmml"><mi id="S2.E2.m1.2.2.1.4.2" xref="S2.E2.m1.2.2.1.4.2.cmml">β</mi><mo id="S2.E2.m1.2.2.1.4.1" xref="S2.E2.m1.2.2.1.4.1.cmml">⁢</mo><mrow id="S2.E2.m1.2.2.1.4.3.2" xref="S2.E2.m1.2.2.1.4.cmml"><mo id="S2.E2.m1.2.2.1.4.3.2.1" stretchy="false" xref="S2.E2.m1.2.2.1.4.cmml">(</mo><mi id="S2.E2.m1.2.2.1.1" xref="S2.E2.m1.2.2.1.1.cmml">x</mi><mo id="S2.E2.m1.2.2.1.4.3.2.2" stretchy="false" xref="S2.E2.m1.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.E2.m1.5.5.1.1.2.2.2.1" xref="S2.E2.m1.5.5.1.1.2.2.2.1.cmml"><mn id="S2.E2.m1.5.5.1.1.2.2.2.1.3" xref="S2.E2.m1.5.5.1.1.2.2.2.1.3.cmml">𝟏</mn><mo id="S2.E2.m1.5.5.1.1.2.2.2.1.2" xref="S2.E2.m1.5.5.1.1.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.2.cmml"><mo id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.2" stretchy="false" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.cmml"><mover accent="true" id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3.cmml"><mrow id="S2.E2.m1.3.3.1" xref="S2.E2.m1.3.3.1.cmml"><msup id="S2.E2.m1.3.3.1.3" xref="S2.E2.m1.3.3.1.3.cmml"><mi id="S2.E2.m1.3.3.1.3.2" xref="S2.E2.m1.3.3.1.3.2.cmml">σ</mi><mi id="S2.E2.m1.3.3.1.3.3" xref="S2.E2.m1.3.3.1.3.3.cmml">τ</mi></msup><mo id="S2.E2.m1.3.3.1.2" xref="S2.E2.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.3.3.1.4.2" xref="S2.E2.m1.3.3.1.cmml"><mo id="S2.E2.m1.3.3.1.4.2.1" stretchy="false" xref="S2.E2.m1.3.3.1.cmml">(</mo><mi id="S2.E2.m1.3.3.1.1" xref="S2.E2.m1.3.3.1.1.cmml">x</mi><mo id="S2.E2.m1.3.3.1.4.2.2" stretchy="false" xref="S2.E2.m1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.3.3.2" xref="S2.E2.m1.3.3.2.cmml">¯</mo></mover><mo id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.2" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.2.cmml">=</mo><mrow id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.1" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.1.cmml"><msup id="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.1.3" xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.1.3.cmml"><mi 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xref="S2.E2.m1.5.5.1.1.2.2.2.1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.5c">\forall x\in\mathbb{R},\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[% \sigma^{\tau}(x)=\sigma^{\tau}(x^{\prime})]\geq\mathop{\mathbb{E}}_{x^{\prime}% \sim\beta(x)}\mathbf{1}[\overline{\sigma^{\tau}(x)}=\sigma^{\tau}(x^{\prime})],</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.5d">∀ italic_x ∈ blackboard_R , blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_x ) = italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] ≥ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ over¯ start_ARG italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_x ) end_ARG = italic_σ start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="Thmdefinition1.p1.2">where <math alttext="\overline{H}=L,\overline{L}=H." class="ltx_Math" display="inline" id="Thmdefinition1.p1.2.m1.1"><semantics id="Thmdefinition1.p1.2.m1.1a"><mrow id="Thmdefinition1.p1.2.m1.1.1.1"><mrow id="Thmdefinition1.p1.2.m1.1.1.1.1.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.3.cmml"><mrow id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.cmml"><mover accent="true" id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.cmml"><mi id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.2.cmml">H</mi><mo id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.1" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.1" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.1.cmml">=</mo><mi id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.3" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="Thmdefinition1.p1.2.m1.1.1.1.1.2.3" xref="Thmdefinition1.p1.2.m1.1.1.1.1.3a.cmml">,</mo><mrow id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.cmml"><mover accent="true" id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.cmml"><mi id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.2" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.2.cmml">L</mi><mo id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.1" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.1.cmml">¯</mo></mover><mo id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.1" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.1.cmml">=</mo><mi id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.3" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.3.cmml">H</mi></mrow></mrow><mo id="Thmdefinition1.p1.2.m1.1.1.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.2.m1.1b"><apply id="Thmdefinition1.p1.2.m1.1.1.1.1.3.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmdefinition1.p1.2.m1.1.1.1.1.3a.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1"><eq id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.1"></eq><apply id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2"><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.1.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.1">¯</ci><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.2.2">𝐻</ci></apply><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.3.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.1.1.3">𝐿</ci></apply><apply id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2"><eq id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.1.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.1"></eq><apply id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2"><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.1.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.1">¯</ci><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.2.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.2.2">𝐿</ci></apply><ci id="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.3.cmml" xref="Thmdefinition1.p1.2.m1.1.1.1.1.2.2.3">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.2.m1.1c">\overline{H}=L,\overline{L}=H.</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.2.m1.1d">over¯ start_ARG italic_H end_ARG = italic_L , over¯ start_ARG italic_L end_ARG = italic_H .</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.2">In the case where the signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_x</annotation></semantics></math> the agent receives satisfies <math alttext="x\leq\tau" 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"><mi id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml">x</mi><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">≤</mo><mi id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml">τ</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"><leq id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1"></leq><ci id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2">𝑥</ci><ci id="S2.SS2.p2.2.m2.1.1.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_x ≤ italic_τ</annotation></semantics></math>. Condition (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E2" title="In Definition 1. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>) simplifies to</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[x^{\prime}\leq\tau]\geq% \mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[x^{\prime}>\tau],\mbox{% or equivalently }\Pr[X^{\prime}\leq\tau\mid X=x]\geq 1/2." class="ltx_Math" display="block" id="S2.Ex3.m1.4"><semantics id="S2.Ex3.m1.4a"><mrow id="S2.Ex3.m1.4.4.1"><mrow id="S2.Ex3.m1.4.4.1.1.2" xref="S2.Ex3.m1.4.4.1.1.3.cmml"><mrow id="S2.Ex3.m1.4.4.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.cmml"><mrow id="S2.Ex3.m1.4.4.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.cmml"><munder id="S2.Ex3.m1.4.4.1.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.2.cmml"><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.2.2" movablelimits="false" xref="S2.Ex3.m1.4.4.1.1.1.1.1.2.2.cmml">𝔼</mo><mrow id="S2.Ex3.m1.1.1.1" xref="S2.Ex3.m1.1.1.1.cmml"><msup id="S2.Ex3.m1.1.1.1.3" xref="S2.Ex3.m1.1.1.1.3.cmml"><mi id="S2.Ex3.m1.1.1.1.3.2" xref="S2.Ex3.m1.1.1.1.3.2.cmml">x</mi><mo id="S2.Ex3.m1.1.1.1.3.3" xref="S2.Ex3.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.Ex3.m1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.2.cmml">∼</mo><mrow id="S2.Ex3.m1.1.1.1.4" xref="S2.Ex3.m1.1.1.1.4.cmml"><mi id="S2.Ex3.m1.1.1.1.4.2" xref="S2.Ex3.m1.1.1.1.4.2.cmml">β</mi><mo id="S2.Ex3.m1.1.1.1.4.1" xref="S2.Ex3.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.1.1.1.4.3.2" xref="S2.Ex3.m1.1.1.1.4.cmml"><mo id="S2.Ex3.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.Ex3.m1.1.1.1.4.cmml">(</mo><mi id="S2.Ex3.m1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml">x</mi><mo id="S2.Ex3.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.Ex3.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.Ex3.m1.4.4.1.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.cmml"><mn id="S2.Ex3.m1.4.4.1.1.1.1.1.1.3" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.3.cmml">𝟏</mn><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.2.cmml"><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2.3" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">≤</mo><mi id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.1.3.cmml">τ</mi></mrow><mo id="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="S2.Ex3.m1.4.4.1.1.1.1.3" rspace="0.1389em" xref="S2.Ex3.m1.4.4.1.1.1.1.3.cmml">≥</mo><mrow id="S2.Ex3.m1.4.4.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.2.cmml"><munder id="S2.Ex3.m1.4.4.1.1.1.1.2.2" xref="S2.Ex3.m1.4.4.1.1.1.1.2.2.cmml"><mo id="S2.Ex3.m1.4.4.1.1.1.1.2.2.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S2.Ex3.m1.4.4.1.1.1.1.2.2.2.cmml">𝔼</mo><mrow id="S2.Ex3.m1.2.2.1" xref="S2.Ex3.m1.2.2.1.cmml"><msup id="S2.Ex3.m1.2.2.1.3" xref="S2.Ex3.m1.2.2.1.3.cmml"><mi id="S2.Ex3.m1.2.2.1.3.2" xref="S2.Ex3.m1.2.2.1.3.2.cmml">x</mi><mo id="S2.Ex3.m1.2.2.1.3.3" xref="S2.Ex3.m1.2.2.1.3.3.cmml">′</mo></msup><mo id="S2.Ex3.m1.2.2.1.2" xref="S2.Ex3.m1.2.2.1.2.cmml">∼</mo><mrow 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xref="S2.Ex3.m1.4.4.1.1.1.1.2.1.1.1.1.2.2">𝑥</ci><ci id="S2.Ex3.m1.4.4.1.1.1.1.2.1.1.1.1.2.3.cmml" xref="S2.Ex3.m1.4.4.1.1.1.1.2.1.1.1.1.2.3">′</ci></apply><ci id="S2.Ex3.m1.4.4.1.1.1.1.2.1.1.1.1.3.cmml" xref="S2.Ex3.m1.4.4.1.1.1.1.2.1.1.1.1.3">𝜏</ci></apply></apply></apply></apply></apply><apply id="S2.Ex3.m1.4.4.1.1.2.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2"><geq id="S2.Ex3.m1.4.4.1.1.2.2.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.2"></geq><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1"><times id="S2.Ex3.m1.4.4.1.1.2.2.1.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.2"></times><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.3a.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.3"><mtext id="S2.Ex3.m1.4.4.1.1.2.2.1.3.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.3"> or equivalently </mtext></ci><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1"><ci id="S2.Ex3.m1.3.3.cmml" xref="S2.Ex3.m1.3.3">Pr</ci><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1"><and id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1a.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1"></and><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1b.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1"><leq id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.3"></leq><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.1.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.2">𝑋</ci><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.3.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.2.3">′</ci></apply><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4"><csymbol cd="latexml" id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.1.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.1">conditional</csymbol><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.2.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.2">𝜏</ci><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.3.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1c.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1"><eq id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.5.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.4.cmml" id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1d.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1"></share><ci id="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.6.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.1.1.1.1.1.6">𝑥</ci></apply></apply></apply></apply><apply id="S2.Ex3.m1.4.4.1.1.2.2.3.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.3"><divide id="S2.Ex3.m1.4.4.1.1.2.2.3.1.cmml" xref="S2.Ex3.m1.4.4.1.1.2.2.3.1"></divide><cn id="S2.Ex3.m1.4.4.1.1.2.2.3.2.cmml" type="integer" xref="S2.Ex3.m1.4.4.1.1.2.2.3.2">1</cn><cn id="S2.Ex3.m1.4.4.1.1.2.2.3.3.cmml" type="integer" xref="S2.Ex3.m1.4.4.1.1.2.2.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.4c">\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[x^{\prime}\leq\tau]\geq% \mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[x^{\prime}>\tau],\mbox{% or equivalently }\Pr[X^{\prime}\leq\tau\mid X=x]\geq 1/2.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.4d">blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ] ≥ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ ] , or equivalently roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ] ≥ 1 / 2 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.8">Similarly if <math alttext="x>\tau" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m1.1"><semantics id="S2.SS2.p2.3.m1.1a"><mrow id="S2.SS2.p2.3.m1.1.1" xref="S2.SS2.p2.3.m1.1.1.cmml"><mi id="S2.SS2.p2.3.m1.1.1.2" xref="S2.SS2.p2.3.m1.1.1.2.cmml">x</mi><mo id="S2.SS2.p2.3.m1.1.1.1" xref="S2.SS2.p2.3.m1.1.1.1.cmml">></mo><mi id="S2.SS2.p2.3.m1.1.1.3" xref="S2.SS2.p2.3.m1.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m1.1b"><apply id="S2.SS2.p2.3.m1.1.1.cmml" xref="S2.SS2.p2.3.m1.1.1"><gt id="S2.SS2.p2.3.m1.1.1.1.cmml" xref="S2.SS2.p2.3.m1.1.1.1"></gt><ci id="S2.SS2.p2.3.m1.1.1.2.cmml" xref="S2.SS2.p2.3.m1.1.1.2">𝑥</ci><ci id="S2.SS2.p2.3.m1.1.1.3.cmml" xref="S2.SS2.p2.3.m1.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m1.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m1.1d">italic_x > italic_τ</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E2" title="In Definition 1. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>) simplifies to <math alttext="\Pr[X^{\prime}\leq\tau\mid X=x]\leq 1/2" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m2.2"><semantics id="S2.SS2.p2.4.m2.2a"><mrow id="S2.SS2.p2.4.m2.2.2" xref="S2.SS2.p2.4.m2.2.2.cmml"><mrow id="S2.SS2.p2.4.m2.2.2.1.1" xref="S2.SS2.p2.4.m2.2.2.1.2.cmml"><mi id="S2.SS2.p2.4.m2.1.1" xref="S2.SS2.p2.4.m2.1.1.cmml">Pr</mi><mo id="S2.SS2.p2.4.m2.2.2.1.1a" xref="S2.SS2.p2.4.m2.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS2.p2.4.m2.2.2.1.1.1" xref="S2.SS2.p2.4.m2.2.2.1.2.cmml"><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.p2.4.m2.2.2.1.2.cmml">[</mo><mrow id="S2.SS2.p2.4.m2.2.2.1.1.1.1" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.cmml"><msup id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.2" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.3" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.1.3" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.3.cmml">≤</mo><mrow id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.cmml"><mi id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.2" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.2.cmml">τ</mi><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.1" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.1.cmml">∣</mo><mi id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.3" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.1.5" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.5.cmml">=</mo><mi id="S2.SS2.p2.4.m2.2.2.1.1.1.1.6" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.6.cmml">x</mi></mrow><mo id="S2.SS2.p2.4.m2.2.2.1.1.1.3" stretchy="false" xref="S2.SS2.p2.4.m2.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S2.SS2.p2.4.m2.2.2.2" xref="S2.SS2.p2.4.m2.2.2.2.cmml">≤</mo><mrow id="S2.SS2.p2.4.m2.2.2.3" xref="S2.SS2.p2.4.m2.2.2.3.cmml"><mn id="S2.SS2.p2.4.m2.2.2.3.2" xref="S2.SS2.p2.4.m2.2.2.3.2.cmml">1</mn><mo id="S2.SS2.p2.4.m2.2.2.3.1" xref="S2.SS2.p2.4.m2.2.2.3.1.cmml">/</mo><mn id="S2.SS2.p2.4.m2.2.2.3.3" xref="S2.SS2.p2.4.m2.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m2.2b"><apply id="S2.SS2.p2.4.m2.2.2.cmml" xref="S2.SS2.p2.4.m2.2.2"><leq id="S2.SS2.p2.4.m2.2.2.2.cmml" xref="S2.SS2.p2.4.m2.2.2.2"></leq><apply id="S2.SS2.p2.4.m2.2.2.1.2.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1"><ci id="S2.SS2.p2.4.m2.1.1.cmml" xref="S2.SS2.p2.4.m2.1.1">Pr</ci><apply id="S2.SS2.p2.4.m2.2.2.1.1.1.1.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1"><and id="S2.SS2.p2.4.m2.2.2.1.1.1.1a.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1"></and><apply id="S2.SS2.p2.4.m2.2.2.1.1.1.1b.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1"><leq id="S2.SS2.p2.4.m2.2.2.1.1.1.1.3.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.3"></leq><apply id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.2">𝑋</ci><ci id="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.3.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.2.3">′</ci></apply><apply id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4"><csymbol cd="latexml" id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.1.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.1">conditional</csymbol><ci id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.2.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.2">𝜏</ci><ci id="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.3.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.SS2.p2.4.m2.2.2.1.1.1.1c.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1"><eq id="S2.SS2.p2.4.m2.2.2.1.1.1.1.5.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.p2.4.m2.2.2.1.1.1.1.4.cmml" id="S2.SS2.p2.4.m2.2.2.1.1.1.1d.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1"></share><ci id="S2.SS2.p2.4.m2.2.2.1.1.1.1.6.cmml" xref="S2.SS2.p2.4.m2.2.2.1.1.1.1.6">𝑥</ci></apply></apply></apply><apply id="S2.SS2.p2.4.m2.2.2.3.cmml" xref="S2.SS2.p2.4.m2.2.2.3"><divide id="S2.SS2.p2.4.m2.2.2.3.1.cmml" xref="S2.SS2.p2.4.m2.2.2.3.1"></divide><cn id="S2.SS2.p2.4.m2.2.2.3.2.cmml" type="integer" xref="S2.SS2.p2.4.m2.2.2.3.2">1</cn><cn id="S2.SS2.p2.4.m2.2.2.3.3.cmml" type="integer" xref="S2.SS2.p2.4.m2.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m2.2c">\Pr[X^{\prime}\leq\tau\mid X=x]\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m2.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ] ≤ 1 / 2</annotation></semantics></math>. Let <math alttext="P(\tau;x)=\Pr[X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m3.4"><semantics id="S2.SS2.p2.5.m3.4a"><mrow id="S2.SS2.p2.5.m3.4.4" xref="S2.SS2.p2.5.m3.4.4.cmml"><mrow id="S2.SS2.p2.5.m3.4.4.3" xref="S2.SS2.p2.5.m3.4.4.3.cmml"><mi id="S2.SS2.p2.5.m3.4.4.3.2" xref="S2.SS2.p2.5.m3.4.4.3.2.cmml">P</mi><mo id="S2.SS2.p2.5.m3.4.4.3.1" xref="S2.SS2.p2.5.m3.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.5.m3.4.4.3.3.2" xref="S2.SS2.p2.5.m3.4.4.3.3.1.cmml"><mo id="S2.SS2.p2.5.m3.4.4.3.3.2.1" stretchy="false" xref="S2.SS2.p2.5.m3.4.4.3.3.1.cmml">(</mo><mi id="S2.SS2.p2.5.m3.1.1" xref="S2.SS2.p2.5.m3.1.1.cmml">τ</mi><mo id="S2.SS2.p2.5.m3.4.4.3.3.2.2" xref="S2.SS2.p2.5.m3.4.4.3.3.1.cmml">;</mo><mi id="S2.SS2.p2.5.m3.2.2" xref="S2.SS2.p2.5.m3.2.2.cmml">x</mi><mo id="S2.SS2.p2.5.m3.4.4.3.3.2.3" stretchy="false" xref="S2.SS2.p2.5.m3.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.5.m3.4.4.2" xref="S2.SS2.p2.5.m3.4.4.2.cmml">=</mo><mrow id="S2.SS2.p2.5.m3.4.4.1.1" xref="S2.SS2.p2.5.m3.4.4.1.2.cmml"><mi id="S2.SS2.p2.5.m3.3.3" xref="S2.SS2.p2.5.m3.3.3.cmml">Pr</mi><mo id="S2.SS2.p2.5.m3.4.4.1.1a" xref="S2.SS2.p2.5.m3.4.4.1.2.cmml">⁡</mo><mrow id="S2.SS2.p2.5.m3.4.4.1.1.1" xref="S2.SS2.p2.5.m3.4.4.1.2.cmml"><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.2" stretchy="false" xref="S2.SS2.p2.5.m3.4.4.1.2.cmml">[</mo><mrow id="S2.SS2.p2.5.m3.4.4.1.1.1.1" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.cmml"><msup id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.2" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.2.cmml">X</mi><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.3" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.1.3" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.3.cmml">≤</mo><mrow id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.cmml"><mi id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.2" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.2.cmml">τ</mi><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.1" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.1.cmml">∣</mo><mi id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.3" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.1.5" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.5.cmml">=</mo><mi id="S2.SS2.p2.5.m3.4.4.1.1.1.1.6" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.6.cmml">x</mi></mrow><mo id="S2.SS2.p2.5.m3.4.4.1.1.1.3" stretchy="false" xref="S2.SS2.p2.5.m3.4.4.1.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m3.4b"><apply id="S2.SS2.p2.5.m3.4.4.cmml" xref="S2.SS2.p2.5.m3.4.4"><eq id="S2.SS2.p2.5.m3.4.4.2.cmml" xref="S2.SS2.p2.5.m3.4.4.2"></eq><apply id="S2.SS2.p2.5.m3.4.4.3.cmml" xref="S2.SS2.p2.5.m3.4.4.3"><times id="S2.SS2.p2.5.m3.4.4.3.1.cmml" xref="S2.SS2.p2.5.m3.4.4.3.1"></times><ci id="S2.SS2.p2.5.m3.4.4.3.2.cmml" xref="S2.SS2.p2.5.m3.4.4.3.2">𝑃</ci><list id="S2.SS2.p2.5.m3.4.4.3.3.1.cmml" xref="S2.SS2.p2.5.m3.4.4.3.3.2"><ci id="S2.SS2.p2.5.m3.1.1.cmml" xref="S2.SS2.p2.5.m3.1.1">𝜏</ci><ci id="S2.SS2.p2.5.m3.2.2.cmml" xref="S2.SS2.p2.5.m3.2.2">𝑥</ci></list></apply><apply id="S2.SS2.p2.5.m3.4.4.1.2.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1"><ci id="S2.SS2.p2.5.m3.3.3.cmml" xref="S2.SS2.p2.5.m3.3.3">Pr</ci><apply id="S2.SS2.p2.5.m3.4.4.1.1.1.1.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1"><and id="S2.SS2.p2.5.m3.4.4.1.1.1.1a.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1"></and><apply id="S2.SS2.p2.5.m3.4.4.1.1.1.1b.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1"><leq id="S2.SS2.p2.5.m3.4.4.1.1.1.1.3.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.3"></leq><apply id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.2">𝑋</ci><ci id="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.3.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.2.3">′</ci></apply><apply id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4"><csymbol cd="latexml" id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.1.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.1">conditional</csymbol><ci id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.2.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.2">𝜏</ci><ci id="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.3.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.SS2.p2.5.m3.4.4.1.1.1.1c.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1"><eq id="S2.SS2.p2.5.m3.4.4.1.1.1.1.5.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.p2.5.m3.4.4.1.1.1.1.4.cmml" id="S2.SS2.p2.5.m3.4.4.1.1.1.1d.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1"></share><ci id="S2.SS2.p2.5.m3.4.4.1.1.1.1.6.cmml" xref="S2.SS2.p2.5.m3.4.4.1.1.1.1.6">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m3.4c">P(\tau;x)=\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m3.4d">italic_P ( italic_τ ; italic_x ) = roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math> be the probability another agent reports <math alttext="L" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m4.1"><semantics id="S2.SS2.p2.6.m4.1a"><mi id="S2.SS2.p2.6.m4.1.1" xref="S2.SS2.p2.6.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m4.1b"><ci id="S2.SS2.p2.6.m4.1.1.cmml" xref="S2.SS2.p2.6.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m4.1d">italic_L</annotation></semantics></math> conditioned on seeing a signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m5.1"><semantics id="S2.SS2.p2.7.m5.1a"><mi id="S2.SS2.p2.7.m5.1.1" xref="S2.SS2.p2.7.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m5.1b"><ci id="S2.SS2.p2.7.m5.1.1.cmml" xref="S2.SS2.p2.7.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m5.1d">italic_x</annotation></semantics></math>, for a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m6.1"><semantics id="S2.SS2.p2.8.m6.1a"><mi id="S2.SS2.p2.8.m6.1.1" xref="S2.SS2.p2.8.m6.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m6.1b"><ci id="S2.SS2.p2.8.m6.1.1.cmml" xref="S2.SS2.p2.8.m6.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m6.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m6.1d">italic_τ</annotation></semantics></math>. As a summary, then, Condition (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E2" title="In Definition 1. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>) is equivalent to</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx1"> <tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall x\leq\tau,P(\tau;x)" class="ltx_Math" display="inline" id="S2.E3.m1.4"><semantics id="S2.E3.m1.4a"><mrow id="S2.E3.m1.4.4" xref="S2.E3.m1.4.4.cmml"><mrow id="S2.E3.m1.4.4.3" xref="S2.E3.m1.4.4.3.cmml"><mo id="S2.E3.m1.4.4.3.1" rspace="0.167em" xref="S2.E3.m1.4.4.3.1.cmml">∀</mo><mi id="S2.E3.m1.4.4.3.2" xref="S2.E3.m1.4.4.3.2.cmml">x</mi></mrow><mo id="S2.E3.m1.4.4.2" xref="S2.E3.m1.4.4.2.cmml">≤</mo><mrow id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.2.cmml"><mi id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml">τ</mi><mo id="S2.E3.m1.4.4.1.1.2" xref="S2.E3.m1.4.4.1.2.cmml">,</mo><mrow id="S2.E3.m1.4.4.1.1.1" xref="S2.E3.m1.4.4.1.1.1.cmml"><mi id="S2.E3.m1.4.4.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.2.cmml">P</mi><mo id="S2.E3.m1.4.4.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.3.2" xref="S2.E3.m1.4.4.1.1.1.3.1.cmml"><mo id="S2.E3.m1.4.4.1.1.1.3.2.1" stretchy="false" xref="S2.E3.m1.4.4.1.1.1.3.1.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">τ</mi><mo id="S2.E3.m1.4.4.1.1.1.3.2.2" xref="S2.E3.m1.4.4.1.1.1.3.1.cmml">;</mo><mi id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml">x</mi><mo id="S2.E3.m1.4.4.1.1.1.3.2.3" stretchy="false" xref="S2.E3.m1.4.4.1.1.1.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.4b"><apply id="S2.E3.m1.4.4.cmml" xref="S2.E3.m1.4.4"><leq id="S2.E3.m1.4.4.2.cmml" xref="S2.E3.m1.4.4.2"></leq><apply id="S2.E3.m1.4.4.3.cmml" xref="S2.E3.m1.4.4.3"><csymbol cd="latexml" id="S2.E3.m1.4.4.3.1.cmml" xref="S2.E3.m1.4.4.3.1">for-all</csymbol><ci id="S2.E3.m1.4.4.3.2.cmml" xref="S2.E3.m1.4.4.3.2">𝑥</ci></apply><list id="S2.E3.m1.4.4.1.2.cmml" xref="S2.E3.m1.4.4.1.1"><ci id="S2.E3.m1.3.3.cmml" xref="S2.E3.m1.3.3">𝜏</ci><apply id="S2.E3.m1.4.4.1.1.1.cmml" xref="S2.E3.m1.4.4.1.1.1"><times id="S2.E3.m1.4.4.1.1.1.1.cmml" xref="S2.E3.m1.4.4.1.1.1.1"></times><ci id="S2.E3.m1.4.4.1.1.1.2.cmml" xref="S2.E3.m1.4.4.1.1.1.2">𝑃</ci><list id="S2.E3.m1.4.4.1.1.1.3.1.cmml" xref="S2.E3.m1.4.4.1.1.1.3.2"><ci id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1">𝜏</ci><ci id="S2.E3.m1.2.2.cmml" xref="S2.E3.m1.2.2">𝑥</ci></list></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.4c">\displaystyle\forall x\leq\tau,P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.4d">∀ italic_x ≤ italic_τ , italic_P ( italic_τ ; italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq 1/2," class="ltx_Math" display="inline" id="S2.E3.m2.1"><semantics id="S2.E3.m2.1a"><mrow id="S2.E3.m2.1.1.1" xref="S2.E3.m2.1.1.1.1.cmml"><mrow id="S2.E3.m2.1.1.1.1" xref="S2.E3.m2.1.1.1.1.cmml"><mi id="S2.E3.m2.1.1.1.1.2" xref="S2.E3.m2.1.1.1.1.2.cmml"></mi><mo id="S2.E3.m2.1.1.1.1.1" xref="S2.E3.m2.1.1.1.1.1.cmml">≥</mo><mrow id="S2.E3.m2.1.1.1.1.3" xref="S2.E3.m2.1.1.1.1.3.cmml"><mn id="S2.E3.m2.1.1.1.1.3.2" xref="S2.E3.m2.1.1.1.1.3.2.cmml">1</mn><mo id="S2.E3.m2.1.1.1.1.3.1" xref="S2.E3.m2.1.1.1.1.3.1.cmml">/</mo><mn id="S2.E3.m2.1.1.1.1.3.3" xref="S2.E3.m2.1.1.1.1.3.3.cmml">2</mn></mrow></mrow><mo id="S2.E3.m2.1.1.1.2" xref="S2.E3.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m2.1b"><apply id="S2.E3.m2.1.1.1.1.cmml" xref="S2.E3.m2.1.1.1"><geq id="S2.E3.m2.1.1.1.1.1.cmml" 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ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall x>\tau,P(\tau;x)" class="ltx_Math" display="inline" id="S2.E4.m1.4"><semantics id="S2.E4.m1.4a"><mrow id="S2.E4.m1.4.4" xref="S2.E4.m1.4.4.cmml"><mrow id="S2.E4.m1.4.4.3" xref="S2.E4.m1.4.4.3.cmml"><mo id="S2.E4.m1.4.4.3.1" rspace="0.167em" xref="S2.E4.m1.4.4.3.1.cmml">∀</mo><mi id="S2.E4.m1.4.4.3.2" xref="S2.E4.m1.4.4.3.2.cmml">x</mi></mrow><mo id="S2.E4.m1.4.4.2" xref="S2.E4.m1.4.4.2.cmml">></mo><mrow id="S2.E4.m1.4.4.1.1" xref="S2.E4.m1.4.4.1.2.cmml"><mi id="S2.E4.m1.3.3" xref="S2.E4.m1.3.3.cmml">τ</mi><mo id="S2.E4.m1.4.4.1.1.2" xref="S2.E4.m1.4.4.1.2.cmml">,</mo><mrow id="S2.E4.m1.4.4.1.1.1" xref="S2.E4.m1.4.4.1.1.1.cmml"><mi id="S2.E4.m1.4.4.1.1.1.2" xref="S2.E4.m1.4.4.1.1.1.2.cmml">P</mi><mo id="S2.E4.m1.4.4.1.1.1.1" xref="S2.E4.m1.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.4.4.1.1.1.3.2" xref="S2.E4.m1.4.4.1.1.1.3.1.cmml"><mo id="S2.E4.m1.4.4.1.1.1.3.2.1" 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id="S2.E4.m1.4.4.1.1.1.2.cmml" xref="S2.E4.m1.4.4.1.1.1.2">𝑃</ci><list id="S2.E4.m1.4.4.1.1.1.3.1.cmml" xref="S2.E4.m1.4.4.1.1.1.3.2"><ci id="S2.E4.m1.1.1.cmml" xref="S2.E4.m1.1.1">𝜏</ci><ci id="S2.E4.m1.2.2.cmml" xref="S2.E4.m1.2.2">𝑥</ci></list></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.4c">\displaystyle\forall x>\tau,P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.4d">∀ italic_x > italic_τ , italic_P ( italic_τ ; italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq 1/2." class="ltx_Math" display="inline" id="S2.E4.m2.1"><semantics id="S2.E4.m2.1a"><mrow id="S2.E4.m2.1.1.1" xref="S2.E4.m2.1.1.1.1.cmml"><mrow id="S2.E4.m2.1.1.1.1" xref="S2.E4.m2.1.1.1.1.cmml"><mi id="S2.E4.m2.1.1.1.1.2" xref="S2.E4.m2.1.1.1.1.2.cmml"></mi><mo id="S2.E4.m2.1.1.1.1.1" xref="S2.E4.m2.1.1.1.1.1.cmml">≤</mo><mrow id="S2.E4.m2.1.1.1.1.3" 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id="S2.E4.m2.1c">\displaystyle\leq 1/2.</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m2.1d">≤ 1 / 2 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Results.</h4> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p1.1">Under these conditions, we first show that the thresholds <math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p1.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.3" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml"><mo id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3a" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml">±</mo><mi id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.1.m1.1b"><apply id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1"><eq id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.1"></eq><apply id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.3"></times></apply><apply id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are always equilibria under Output Agreement.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition1.1.1.1">Proposition 1</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition1.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition1.p1"> <p class="ltx_p" id="Thmproposition1.p1.1"><math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="Thmproposition1.p1.1.m1.1"><semantics id="Thmproposition1.p1.1.m1.1a"><mrow id="Thmproposition1.p1.1.m1.1.1" xref="Thmproposition1.p1.1.m1.1.1.cmml"><msup id="Thmproposition1.p1.1.m1.1.1.2" xref="Thmproposition1.p1.1.m1.1.1.2.cmml"><mi id="Thmproposition1.p1.1.m1.1.1.2.2" xref="Thmproposition1.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="Thmproposition1.p1.1.m1.1.1.2.3" xref="Thmproposition1.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition1.p1.1.m1.1.1.1" xref="Thmproposition1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmproposition1.p1.1.m1.1.1.3" xref="Thmproposition1.p1.1.m1.1.1.3.cmml"><mo id="Thmproposition1.p1.1.m1.1.1.3a" xref="Thmproposition1.p1.1.m1.1.1.3.cmml">±</mo><mi id="Thmproposition1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="Thmproposition1.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition1.p1.1.m1.1b"><apply id="Thmproposition1.p1.1.m1.1.1.cmml" xref="Thmproposition1.p1.1.m1.1.1"><eq id="Thmproposition1.p1.1.m1.1.1.1.cmml" xref="Thmproposition1.p1.1.m1.1.1.1"></eq><apply id="Thmproposition1.p1.1.m1.1.1.2.cmml" xref="Thmproposition1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition1.p1.1.m1.1.1.2.1.cmml" xref="Thmproposition1.p1.1.m1.1.1.2">superscript</csymbol><ci id="Thmproposition1.p1.1.m1.1.1.2.2.cmml" xref="Thmproposition1.p1.1.m1.1.1.2.2">𝜏</ci><times id="Thmproposition1.p1.1.m1.1.1.2.3.cmml" xref="Thmproposition1.p1.1.m1.1.1.2.3"></times></apply><apply id="Thmproposition1.p1.1.m1.1.1.3.cmml" xref="Thmproposition1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="Thmproposition1.p1.1.m1.1.1.3.1.cmml" xref="Thmproposition1.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="Thmproposition1.p1.1.m1.1.1.3.2.cmml" xref="Thmproposition1.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are both always threshold equilibria under OA.</p> </div> </div> <div class="ltx_proof" id="S2.SS2.SSS0.Px1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS2.SSS0.Px1.1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.1.p1.13">Let <math alttext="\tau^{*}=\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.2" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.3" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.1.cmml">=</mo><mi id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.1.m1.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1"><eq id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.1"></eq><apply id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.2.3"></times></apply><infinity id="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.1.m1.1c">\tau^{*}=\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ∞</annotation></semantics></math>. Then for all signals <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.2.m2.1a"><mi id="S2.SS2.SSS0.Px1.1.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px1.1.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.2.m2.1b"><ci id="S2.SS2.SSS0.Px1.1.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.2.m2.1d">italic_x</annotation></semantics></math> such that <math alttext="x\leq\tau^{*}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.3.m3.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.1.cmml">≤</mo><msup id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.2" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.3" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.3.m3.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1"><leq id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.1"></leq><ci id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.2">𝑥</ci><apply id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.3.m3.1c">x\leq\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.3.m3.1d">italic_x ≤ italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P(\tau^{*};x)=1\geq 1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2"><semantics id="S2.SS2.SSS0.Px1.1.p1.4.m4.2a"><mrow id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.cmml"><mrow id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.3" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.3.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.2" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.2.cmml">(</mo><msup id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.2.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.1.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.1.1.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.4" stretchy="false" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.3" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.3.cmml">=</mo><mn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.4" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.4.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.5" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.5.cmml">≥</mo><mrow id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.cmml"><mn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.2" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.1" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.3" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2b"><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2"><and id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2a.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2"></and><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2b.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2"><eq id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.3"></eq><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1"><times id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.2"></times><ci id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.3">𝑃</ci><list id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1"><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.1.1.1.1.3"></times></apply><ci id="S2.SS2.SSS0.Px1.1.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.1.1">𝑥</ci></list></apply><cn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.4.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.4">1</cn></apply><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2c.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2"><geq id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.5.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.5"></geq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.4.cmml" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2d.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2"></share><apply id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6"><divide id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.1"></divide><cn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.2">1</cn><cn id="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.4.m4.2.2.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2c">P(\tau^{*};x)=1\geq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.4.m4.2d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x ) = 1 ≥ 1 / 2</annotation></semantics></math> (since any <math alttext="X^{\prime}\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.5.m5.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.5.m5.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.2" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.2.cmml">X</mi><mo id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.3" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.5.m5.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1"><in id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.1"></in><apply id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.2">𝑋</ci><ci id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.2.3">′</ci></apply><ci id="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.5.m5.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.5.m5.1c">X^{\prime}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.5.m5.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="X^{\prime}\leq\tau^{*}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.6.m6.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.2" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.2.cmml">X</mi><mo id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.3" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.1.cmml">≤</mo><msup id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.2" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.3" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1"><leq id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.1"></leq><apply id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.2">𝑋</ci><ci id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.2.3">′</ci></apply><apply id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.6.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1c">X^{\prime}\leq\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.6.m6.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with probability one). Meanwhile, Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E4" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a> is vacuous since no signal <math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.7.m7.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.7.m7.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.7.m7.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1"><in id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.1"></in><ci id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.2">𝑥</ci><ci id="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.7.m7.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.7.m7.1c">x\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.7.m7.1d">italic_x ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="x>\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.8.m8.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.1.cmml">></mo><mi id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.3" mathvariant="normal" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.8.m8.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1"><gt id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.1"></gt><ci id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.2">𝑥</ci><infinity id="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.8.m8.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.8.m8.1c">x>\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.8.m8.1d">italic_x > ∞</annotation></semantics></math>. A similar argument follows for <math alttext="\tau^{*}=-\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.9.m9.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.9.m9.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.2" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.3" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.cmml"><mo id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3a" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.cmml">−</mo><mi id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.9.m9.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1"><eq id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.1"></eq><apply id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.2.3"></times></apply><apply id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3"><minus id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3"></minus><infinity id="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.9.m9.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.9.m9.1c">\tau^{*}=-\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = - ∞</annotation></semantics></math>: for all signals <math alttext="x>\tau^{*}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.10.m10.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.10.m10.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.1.cmml">></mo><msup id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.2" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.3" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.10.m10.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1"><gt id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.1"></gt><ci id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.2">𝑥</ci><apply id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.10.m10.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.10.m10.1c">x>\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.10.m10.1d">italic_x > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P(\tau^{*};x)=0\leq 1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2"><semantics id="S2.SS2.SSS0.Px1.1.p1.11.m11.2a"><mrow id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.cmml"><mrow id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.3" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.3.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.2" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.2.cmml">(</mo><msup id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.2" 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id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.1" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.3" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2b"><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2"><and id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2a.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2"></and><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2b.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2"><eq id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.3"></eq><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1"><times id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.2"></times><ci id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.3">𝑃</ci><list id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1"><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.1.1.1.1.3"></times></apply><ci id="S2.SS2.SSS0.Px1.1.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.1.1">𝑥</ci></list></apply><cn id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.4.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.4">0</cn></apply><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2c.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2"><leq id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.5.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.5"></leq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.4.cmml" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2d.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2"></share><apply id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6"><divide id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.1"></divide><cn id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.2">1</cn><cn id="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.1.p1.11.m11.2.2.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2c">P(\tau^{*};x)=0\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.11.m11.2d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x ) = 0 ≤ 1 / 2</annotation></semantics></math>, while Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E3" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a> is vacuous since no signal <math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.12.m12.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.12.m12.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.1.cmml">∈</mo><mi id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.12.m12.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1"><in id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.1"></in><ci id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.2">𝑥</ci><ci id="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.12.m12.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.12.m12.1c">x\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.12.m12.1d">italic_x ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="x<-\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.1.p1.13.m13.1"><semantics id="S2.SS2.SSS0.Px1.1.p1.13.m13.1a"><mrow id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.2" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.1" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.1.cmml"><</mo><mrow id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.cmml"><mo id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3a" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.cmml">−</mo><mi id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.1.p1.13.m13.1b"><apply id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1"><lt id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.1"></lt><ci id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.2">𝑥</ci><apply id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3"><minus id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3"></minus><infinity id="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.1.p1.13.m13.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.1.p1.13.m13.1c">x<-\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.1.p1.13.m13.1d">italic_x < - ∞</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p2.9">These thresholds exactly correspond to uninformative equilibria in the binary setting: if <math alttext="\tau^{*}=\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p2.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.2" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.3" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.1" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.1.cmml">=</mo><mi id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.3" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.1.m1.1b"><apply id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1"><eq id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.1"></eq><apply id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.2.3"></times></apply><infinity id="S2.SS2.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.1.m1.1c">\tau^{*}=\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ∞</annotation></semantics></math>, agents always report <math alttext="L" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.p2.2.m2.1a"><mi id="S2.SS2.SSS0.Px1.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.2.m2.1b"><ci id="S2.SS2.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.2.m2.1d">italic_L</annotation></semantics></math>, and if <math alttext="\tau^{*}=-\infty" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.p2.3.m3.1a"><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.cmml"><msup id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.2" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.3" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.1" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.cmml"><mo id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3a" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.cmml">−</mo><mi id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.2" mathvariant="normal" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.3.m3.1b"><apply id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1"><eq id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.1"></eq><apply id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.2">𝜏</ci><times id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.2.3"></times></apply><apply id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3"><minus id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3"></minus><infinity id="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p2.3.m3.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.3.m3.1c">\tau^{*}=-\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.3.m3.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = - ∞</annotation></semantics></math>, agents always report <math alttext="H" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.p2.4.m4.1a"><mi id="S2.SS2.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px1.p2.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.4.m4.1b"><ci id="S2.SS2.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.4.m4.1d">italic_H</annotation></semantics></math>. Under some smoothness conditions on the predictive posterior function, we can furthermore characterize all non-infinite threshold equilibria. 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id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.3"></leq><apply id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.2">𝑋</ci><ci id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.2.3">′</ci></apply><apply id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4"><csymbol cd="latexml" id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.1.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.1">conditional</csymbol><ci id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.2.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.2">𝑥</ci><ci id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.3.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1c.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1"><eq id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.5.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.4.cmml" id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1d.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1"></share><ci id="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.6.cmml" xref="S2.SS2.SSS0.Px1.p2.5.m5.5.5.1.1.1.1.6">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.5.m5.5c">G(x)=P(x;x)=\Pr[X^{\prime}\leq x\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.5.m5.5d">italic_G ( italic_x ) = italic_P ( italic_x ; italic_x ) = roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_x ∣ italic_X = italic_x ]</annotation></semantics></math> as the probability of another signal lying below <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.6.m6.1"><semantics id="S2.SS2.SSS0.Px1.p2.6.m6.1a"><mi id="S2.SS2.SSS0.Px1.p2.6.m6.1.1" xref="S2.SS2.SSS0.Px1.p2.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.6.m6.1b"><ci id="S2.SS2.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.6.m6.1d">italic_x</annotation></semantics></math>, conditioned on <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.7.m7.1"><semantics id="S2.SS2.SSS0.Px1.p2.7.m7.1a"><mi id="S2.SS2.SSS0.Px1.p2.7.m7.1.1" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.7.m7.1b"><ci id="S2.SS2.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.7.m7.1d">italic_x</annotation></semantics></math>. In Output Agreement, we find that threshold equilibria are characterized by thresholds <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.p2.8.m8.1a"><mi id="S2.SS2.SSS0.Px1.p2.8.m8.1.1" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.8.m8.1b"><ci id="S2.SS2.SSS0.Px1.p2.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.8.m8.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.8.m8.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.8.m8.1d">italic_τ</annotation></semantics></math> such that <math alttext="G(\tau)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.9.m9.1"><semantics id="S2.SS2.SSS0.Px1.p2.9.m9.1a"><mrow id="S2.SS2.SSS0.Px1.p2.9.m9.1.2" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.cmml"><mi id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.2" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.2.cmml">G</mi><mo id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.1" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.3.2" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.cmml"><mo id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p2.9.m9.1.1" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p2.9.m9.1b"><apply id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.cmml" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2"><times id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.1"></times><ci id="S2.SS2.SSS0.Px1.p2.9.m9.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.2.2">𝐺</ci><ci id="S2.SS2.SSS0.Px1.p2.9.m9.1.1.cmml" xref="S2.SS2.SSS0.Px1.p2.9.m9.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.9.m9.1c">G(\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.9.m9.1d">italic_G ( italic_τ )</annotation></semantics></math> crosses 1/2. More precisely, we give the following necessary and sufficient conditions.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem1.1.1.1">Theorem 1</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem1.p1"> <p class="ltx_p" id="Thmtheorem1.p1.10">Let finite threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem1.p1.1.m1.1"><semantics id="Thmtheorem1.p1.1.m1.1a"><mi id="Thmtheorem1.p1.1.m1.1.1" xref="Thmtheorem1.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.1.m1.1b"><ci id="Thmtheorem1.p1.1.m1.1.1.cmml" xref="Thmtheorem1.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.1.m1.1d">italic_τ</annotation></semantics></math> be given and <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem1.p1.2.m2.2"><semantics id="Thmtheorem1.p1.2.m2.2a"><mrow id="Thmtheorem1.p1.2.m2.2.3" xref="Thmtheorem1.p1.2.m2.2.3.cmml"><mi id="Thmtheorem1.p1.2.m2.2.3.2" xref="Thmtheorem1.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmtheorem1.p1.2.m2.2.3.1" xref="Thmtheorem1.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem1.p1.2.m2.2.3.3.2" xref="Thmtheorem1.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem1.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem1.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem1.p1.2.m2.1.1" xref="Thmtheorem1.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmtheorem1.p1.2.m2.2.3.3.2.2" xref="Thmtheorem1.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmtheorem1.p1.2.m2.2.2" xref="Thmtheorem1.p1.2.m2.2.2.cmml">x</mi><mo id="Thmtheorem1.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.2.m2.2b"><apply id="Thmtheorem1.p1.2.m2.2.3.cmml" xref="Thmtheorem1.p1.2.m2.2.3"><times id="Thmtheorem1.p1.2.m2.2.3.1.cmml" xref="Thmtheorem1.p1.2.m2.2.3.1"></times><ci id="Thmtheorem1.p1.2.m2.2.3.2.cmml" xref="Thmtheorem1.p1.2.m2.2.3.2">𝑃</ci><list id="Thmtheorem1.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem1.p1.2.m2.2.3.3.2"><ci id="Thmtheorem1.p1.2.m2.1.1.cmml" xref="Thmtheorem1.p1.2.m2.1.1">𝜏</ci><ci id="Thmtheorem1.p1.2.m2.2.2.cmml" xref="Thmtheorem1.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> be continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem1.p1.3.m3.1"><semantics id="Thmtheorem1.p1.3.m3.1a"><mi id="Thmtheorem1.p1.3.m3.1.1" xref="Thmtheorem1.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.3.m3.1b"><ci id="Thmtheorem1.p1.3.m3.1.1.cmml" xref="Thmtheorem1.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.3.m3.1d">italic_x</annotation></semantics></math>. If <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem1.p1.4.m4.1"><semantics id="Thmtheorem1.p1.4.m4.1a"><mi id="Thmtheorem1.p1.4.m4.1.1" xref="Thmtheorem1.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.4.m4.1b"><ci id="Thmtheorem1.p1.4.m4.1.1.cmml" xref="Thmtheorem1.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.4.m4.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium under the OA mechanism then <math alttext="G(\tau)=1/2" class="ltx_Math" display="inline" id="Thmtheorem1.p1.5.m5.1"><semantics id="Thmtheorem1.p1.5.m5.1a"><mrow id="Thmtheorem1.p1.5.m5.1.2" xref="Thmtheorem1.p1.5.m5.1.2.cmml"><mrow id="Thmtheorem1.p1.5.m5.1.2.2" xref="Thmtheorem1.p1.5.m5.1.2.2.cmml"><mi id="Thmtheorem1.p1.5.m5.1.2.2.2" xref="Thmtheorem1.p1.5.m5.1.2.2.2.cmml">G</mi><mo id="Thmtheorem1.p1.5.m5.1.2.2.1" xref="Thmtheorem1.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="Thmtheorem1.p1.5.m5.1.2.2.3.2" xref="Thmtheorem1.p1.5.m5.1.2.2.cmml"><mo id="Thmtheorem1.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="Thmtheorem1.p1.5.m5.1.2.2.cmml">(</mo><mi id="Thmtheorem1.p1.5.m5.1.1" xref="Thmtheorem1.p1.5.m5.1.1.cmml">τ</mi><mo id="Thmtheorem1.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="Thmtheorem1.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem1.p1.5.m5.1.2.1" xref="Thmtheorem1.p1.5.m5.1.2.1.cmml">=</mo><mrow id="Thmtheorem1.p1.5.m5.1.2.3" xref="Thmtheorem1.p1.5.m5.1.2.3.cmml"><mn id="Thmtheorem1.p1.5.m5.1.2.3.2" xref="Thmtheorem1.p1.5.m5.1.2.3.2.cmml">1</mn><mo id="Thmtheorem1.p1.5.m5.1.2.3.1" xref="Thmtheorem1.p1.5.m5.1.2.3.1.cmml">/</mo><mn id="Thmtheorem1.p1.5.m5.1.2.3.3" xref="Thmtheorem1.p1.5.m5.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.5.m5.1b"><apply id="Thmtheorem1.p1.5.m5.1.2.cmml" xref="Thmtheorem1.p1.5.m5.1.2"><eq id="Thmtheorem1.p1.5.m5.1.2.1.cmml" xref="Thmtheorem1.p1.5.m5.1.2.1"></eq><apply id="Thmtheorem1.p1.5.m5.1.2.2.cmml" xref="Thmtheorem1.p1.5.m5.1.2.2"><times id="Thmtheorem1.p1.5.m5.1.2.2.1.cmml" xref="Thmtheorem1.p1.5.m5.1.2.2.1"></times><ci id="Thmtheorem1.p1.5.m5.1.2.2.2.cmml" xref="Thmtheorem1.p1.5.m5.1.2.2.2">𝐺</ci><ci id="Thmtheorem1.p1.5.m5.1.1.cmml" xref="Thmtheorem1.p1.5.m5.1.1">𝜏</ci></apply><apply id="Thmtheorem1.p1.5.m5.1.2.3.cmml" xref="Thmtheorem1.p1.5.m5.1.2.3"><divide id="Thmtheorem1.p1.5.m5.1.2.3.1.cmml" xref="Thmtheorem1.p1.5.m5.1.2.3.1"></divide><cn id="Thmtheorem1.p1.5.m5.1.2.3.2.cmml" type="integer" xref="Thmtheorem1.p1.5.m5.1.2.3.2">1</cn><cn id="Thmtheorem1.p1.5.m5.1.2.3.3.cmml" type="integer" xref="Thmtheorem1.p1.5.m5.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.5.m5.1c">G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.5.m5.1d">italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math>. Conversely, if <math alttext="G(\tau)=1/2" class="ltx_Math" display="inline" id="Thmtheorem1.p1.6.m6.1"><semantics id="Thmtheorem1.p1.6.m6.1a"><mrow id="Thmtheorem1.p1.6.m6.1.2" xref="Thmtheorem1.p1.6.m6.1.2.cmml"><mrow id="Thmtheorem1.p1.6.m6.1.2.2" xref="Thmtheorem1.p1.6.m6.1.2.2.cmml"><mi id="Thmtheorem1.p1.6.m6.1.2.2.2" xref="Thmtheorem1.p1.6.m6.1.2.2.2.cmml">G</mi><mo id="Thmtheorem1.p1.6.m6.1.2.2.1" xref="Thmtheorem1.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="Thmtheorem1.p1.6.m6.1.2.2.3.2" xref="Thmtheorem1.p1.6.m6.1.2.2.cmml"><mo id="Thmtheorem1.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="Thmtheorem1.p1.6.m6.1.2.2.cmml">(</mo><mi id="Thmtheorem1.p1.6.m6.1.1" xref="Thmtheorem1.p1.6.m6.1.1.cmml">τ</mi><mo id="Thmtheorem1.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="Thmtheorem1.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem1.p1.6.m6.1.2.1" xref="Thmtheorem1.p1.6.m6.1.2.1.cmml">=</mo><mrow id="Thmtheorem1.p1.6.m6.1.2.3" xref="Thmtheorem1.p1.6.m6.1.2.3.cmml"><mn id="Thmtheorem1.p1.6.m6.1.2.3.2" xref="Thmtheorem1.p1.6.m6.1.2.3.2.cmml">1</mn><mo id="Thmtheorem1.p1.6.m6.1.2.3.1" xref="Thmtheorem1.p1.6.m6.1.2.3.1.cmml">/</mo><mn id="Thmtheorem1.p1.6.m6.1.2.3.3" xref="Thmtheorem1.p1.6.m6.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.6.m6.1b"><apply id="Thmtheorem1.p1.6.m6.1.2.cmml" xref="Thmtheorem1.p1.6.m6.1.2"><eq id="Thmtheorem1.p1.6.m6.1.2.1.cmml" xref="Thmtheorem1.p1.6.m6.1.2.1"></eq><apply id="Thmtheorem1.p1.6.m6.1.2.2.cmml" xref="Thmtheorem1.p1.6.m6.1.2.2"><times id="Thmtheorem1.p1.6.m6.1.2.2.1.cmml" xref="Thmtheorem1.p1.6.m6.1.2.2.1"></times><ci id="Thmtheorem1.p1.6.m6.1.2.2.2.cmml" xref="Thmtheorem1.p1.6.m6.1.2.2.2">𝐺</ci><ci id="Thmtheorem1.p1.6.m6.1.1.cmml" xref="Thmtheorem1.p1.6.m6.1.1">𝜏</ci></apply><apply id="Thmtheorem1.p1.6.m6.1.2.3.cmml" xref="Thmtheorem1.p1.6.m6.1.2.3"><divide id="Thmtheorem1.p1.6.m6.1.2.3.1.cmml" xref="Thmtheorem1.p1.6.m6.1.2.3.1"></divide><cn id="Thmtheorem1.p1.6.m6.1.2.3.2.cmml" type="integer" xref="Thmtheorem1.p1.6.m6.1.2.3.2">1</cn><cn id="Thmtheorem1.p1.6.m6.1.2.3.3.cmml" type="integer" xref="Thmtheorem1.p1.6.m6.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.6.m6.1c">G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.6.m6.1d">italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math> and either (a) <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem1.p1.7.m7.2"><semantics id="Thmtheorem1.p1.7.m7.2a"><mrow id="Thmtheorem1.p1.7.m7.2.3" xref="Thmtheorem1.p1.7.m7.2.3.cmml"><mi id="Thmtheorem1.p1.7.m7.2.3.2" xref="Thmtheorem1.p1.7.m7.2.3.2.cmml">P</mi><mo id="Thmtheorem1.p1.7.m7.2.3.1" xref="Thmtheorem1.p1.7.m7.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem1.p1.7.m7.2.3.3.2" xref="Thmtheorem1.p1.7.m7.2.3.3.1.cmml"><mo id="Thmtheorem1.p1.7.m7.2.3.3.2.1" stretchy="false" xref="Thmtheorem1.p1.7.m7.2.3.3.1.cmml">(</mo><mi id="Thmtheorem1.p1.7.m7.1.1" xref="Thmtheorem1.p1.7.m7.1.1.cmml">τ</mi><mo id="Thmtheorem1.p1.7.m7.2.3.3.2.2" xref="Thmtheorem1.p1.7.m7.2.3.3.1.cmml">;</mo><mi id="Thmtheorem1.p1.7.m7.2.2" xref="Thmtheorem1.p1.7.m7.2.2.cmml">x</mi><mo id="Thmtheorem1.p1.7.m7.2.3.3.2.3" stretchy="false" xref="Thmtheorem1.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.7.m7.2b"><apply id="Thmtheorem1.p1.7.m7.2.3.cmml" xref="Thmtheorem1.p1.7.m7.2.3"><times id="Thmtheorem1.p1.7.m7.2.3.1.cmml" xref="Thmtheorem1.p1.7.m7.2.3.1"></times><ci id="Thmtheorem1.p1.7.m7.2.3.2.cmml" xref="Thmtheorem1.p1.7.m7.2.3.2">𝑃</ci><list id="Thmtheorem1.p1.7.m7.2.3.3.1.cmml" xref="Thmtheorem1.p1.7.m7.2.3.3.2"><ci id="Thmtheorem1.p1.7.m7.1.1.cmml" xref="Thmtheorem1.p1.7.m7.1.1">𝜏</ci><ci id="Thmtheorem1.p1.7.m7.2.2.cmml" xref="Thmtheorem1.p1.7.m7.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.7.m7.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.7.m7.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is monotone decreasing in x or (b) <math alttext="P(\tau;x)-1/2" class="ltx_Math" display="inline" id="Thmtheorem1.p1.8.m8.2"><semantics id="Thmtheorem1.p1.8.m8.2a"><mrow id="Thmtheorem1.p1.8.m8.2.3" xref="Thmtheorem1.p1.8.m8.2.3.cmml"><mrow id="Thmtheorem1.p1.8.m8.2.3.2" xref="Thmtheorem1.p1.8.m8.2.3.2.cmml"><mi id="Thmtheorem1.p1.8.m8.2.3.2.2" xref="Thmtheorem1.p1.8.m8.2.3.2.2.cmml">P</mi><mo id="Thmtheorem1.p1.8.m8.2.3.2.1" xref="Thmtheorem1.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem1.p1.8.m8.2.3.2.3.2" xref="Thmtheorem1.p1.8.m8.2.3.2.3.1.cmml"><mo id="Thmtheorem1.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem1.p1.8.m8.2.3.2.3.1.cmml">(</mo><mi id="Thmtheorem1.p1.8.m8.1.1" xref="Thmtheorem1.p1.8.m8.1.1.cmml">τ</mi><mo id="Thmtheorem1.p1.8.m8.2.3.2.3.2.2" xref="Thmtheorem1.p1.8.m8.2.3.2.3.1.cmml">;</mo><mi id="Thmtheorem1.p1.8.m8.2.2" xref="Thmtheorem1.p1.8.m8.2.2.cmml">x</mi><mo id="Thmtheorem1.p1.8.m8.2.3.2.3.2.3" stretchy="false" xref="Thmtheorem1.p1.8.m8.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem1.p1.8.m8.2.3.1" xref="Thmtheorem1.p1.8.m8.2.3.1.cmml">−</mo><mrow id="Thmtheorem1.p1.8.m8.2.3.3" xref="Thmtheorem1.p1.8.m8.2.3.3.cmml"><mn id="Thmtheorem1.p1.8.m8.2.3.3.2" xref="Thmtheorem1.p1.8.m8.2.3.3.2.cmml">1</mn><mo id="Thmtheorem1.p1.8.m8.2.3.3.1" xref="Thmtheorem1.p1.8.m8.2.3.3.1.cmml">/</mo><mn id="Thmtheorem1.p1.8.m8.2.3.3.3" xref="Thmtheorem1.p1.8.m8.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.8.m8.2b"><apply id="Thmtheorem1.p1.8.m8.2.3.cmml" xref="Thmtheorem1.p1.8.m8.2.3"><minus id="Thmtheorem1.p1.8.m8.2.3.1.cmml" xref="Thmtheorem1.p1.8.m8.2.3.1"></minus><apply id="Thmtheorem1.p1.8.m8.2.3.2.cmml" xref="Thmtheorem1.p1.8.m8.2.3.2"><times id="Thmtheorem1.p1.8.m8.2.3.2.1.cmml" xref="Thmtheorem1.p1.8.m8.2.3.2.1"></times><ci id="Thmtheorem1.p1.8.m8.2.3.2.2.cmml" xref="Thmtheorem1.p1.8.m8.2.3.2.2">𝑃</ci><list id="Thmtheorem1.p1.8.m8.2.3.2.3.1.cmml" xref="Thmtheorem1.p1.8.m8.2.3.2.3.2"><ci id="Thmtheorem1.p1.8.m8.1.1.cmml" xref="Thmtheorem1.p1.8.m8.1.1">𝜏</ci><ci id="Thmtheorem1.p1.8.m8.2.2.cmml" xref="Thmtheorem1.p1.8.m8.2.2">𝑥</ci></list></apply><apply id="Thmtheorem1.p1.8.m8.2.3.3.cmml" xref="Thmtheorem1.p1.8.m8.2.3.3"><divide id="Thmtheorem1.p1.8.m8.2.3.3.1.cmml" xref="Thmtheorem1.p1.8.m8.2.3.3.1"></divide><cn id="Thmtheorem1.p1.8.m8.2.3.3.2.cmml" type="integer" xref="Thmtheorem1.p1.8.m8.2.3.3.2">1</cn><cn id="Thmtheorem1.p1.8.m8.2.3.3.3.cmml" type="integer" xref="Thmtheorem1.p1.8.m8.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.8.m8.2c">P(\tau;x)-1/2</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.8.m8.2d">italic_P ( italic_τ ; italic_x ) - 1 / 2</annotation></semantics></math> has a single crossing of <math alttext="0" class="ltx_Math" display="inline" id="Thmtheorem1.p1.9.m9.1"><semantics id="Thmtheorem1.p1.9.m9.1a"><mn id="Thmtheorem1.p1.9.m9.1.1" xref="Thmtheorem1.p1.9.m9.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.9.m9.1b"><cn id="Thmtheorem1.p1.9.m9.1.1.cmml" type="integer" xref="Thmtheorem1.p1.9.m9.1.1">0</cn></annotation-xml></semantics></math> from positive to negative then <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem1.p1.10.m10.1"><semantics id="Thmtheorem1.p1.10.m10.1a"><mi id="Thmtheorem1.p1.10.m10.1.1" xref="Thmtheorem1.p1.10.m10.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.10.m10.1b"><ci id="Thmtheorem1.p1.10.m10.1.1.cmml" xref="Thmtheorem1.p1.10.m10.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.10.m10.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.10.m10.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium under the OA mechanism.</p> </div> </div> <div class="ltx_proof" id="S2.SS2.SSS0.Px1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS2.SSS0.Px1.2.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.2.p1.7">For necessity, assume that <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.2.p1.1.m1.1a"><mi id="S2.SS2.SSS0.Px1.2.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px1.2.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.2.p1.1.m1.1b"><ci id="S2.SS2.SSS0.Px1.2.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.1.m1.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium, so Equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E3" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E4" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>) hold. Since <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.2.m2.2"><semantics id="S2.SS2.SSS0.Px1.2.p1.2.m2.2a"><mrow id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.1" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.2" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.2" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.2.p1.2.m2.2b"><apply id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3"><times id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.1"></times><ci id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.3.3.2"><ci id="S2.SS2.SSS0.Px1.2.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.2.p1.2.m2.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is continuous over <math alttext="x" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.2.p1.3.m3.1a"><mi id="S2.SS2.SSS0.Px1.2.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px1.2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.2.p1.3.m3.1b"><ci id="S2.SS2.SSS0.Px1.2.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.3.m3.1d">italic_x</annotation></semantics></math>, we must have <math alttext="\lim_{x\to\tau^{+}}P(\tau;x)=\lim_{x\to\tau^{-}}P(\tau;x)=1/2." class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.4.m4.5"><semantics id="S2.SS2.SSS0.Px1.2.p1.4.m4.5a"><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.cmml"><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.cmml"><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.cmml"><msub id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.2.cmml">lim</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.1.cmml">→</mo><msup id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.1.3.3.3.cmml">+</mo></msup></mrow></msub><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.2.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.3" rspace="0.1389em" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.cmml"><msub id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.2" lspace="0.1389em" rspace="0.167em" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.2.cmml">lim</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.1.cmml">→</mo><msup id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.1.3.3.3.cmml">−</mo></msup></mrow></msub><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.1" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.4.m4.3.3" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.3.3.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.4.2.3.1.cmml">;</mo><mi 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xref="S2.SS2.SSS0.Px1.2.p1.4.m4.5.5.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.4.m4.5c">\lim_{x\to\tau^{+}}P(\tau;x)=\lim_{x\to\tau^{-}}P(\tau;x)=1/2.</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.4.m4.5d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) = roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) = 1 / 2 .</annotation></semantics></math> But <math alttext="\lim_{x\to\tau^{+}}P(\tau;x)\leq 1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.5.m5.2"><semantics id="S2.SS2.SSS0.Px1.2.p1.5.m5.2a"><mrow id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.cmml"><mrow id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.cmml"><msub 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id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3"><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.1">→</ci><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.2">𝑥</ci><apply id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.2">𝜏</ci><plus id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.1.3.3.3"></plus></apply></apply></apply><apply id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2"><times id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.1"></times><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.2.2.3.2"><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.2">𝑥</ci></list></apply></apply><apply id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3"><divide id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.1"></divide><cn id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.2">1</cn><cn id="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.5.m5.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.5.m5.2c">\lim_{x\to\tau^{+}}P(\tau;x)\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.5.m5.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) ≤ 1 / 2</annotation></semantics></math> and <math alttext="\lim_{x\to\tau^{+}}P(\tau;x)\geq 1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2"><semantics id="S2.SS2.SSS0.Px1.2.p1.6.m6.2a"><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.cmml"><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.cmml"><msub id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.2.cmml">lim</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.1.cmml">→</mo><msup id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.3.cmml">+</mo></msup></mrow></msub><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.1" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.1" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.1.cmml">≥</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.cmml"><mn id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.2" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.1" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.3" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2b"><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3"><geq id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.1"></geq><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2"><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1">subscript</csymbol><limit id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.2"></limit><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3"><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.1">→</ci><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.2">𝑥</ci><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3">superscript</csymbol><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.2">𝜏</ci><plus id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.1.3.3.3"></plus></apply></apply></apply><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2"><times id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.1"></times><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.2.2.3.2"><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.2">𝑥</ci></list></apply></apply><apply id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3"><divide id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.1"></divide><cn id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.2">1</cn><cn id="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.6.m6.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2c">\lim_{x\to\tau^{+}}P(\tau;x)\geq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.6.m6.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) ≥ 1 / 2</annotation></semantics></math>, so we must have <math alttext="P(\tau;\tau)=G(\tau)=1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.2.p1.7.m7.3"><semantics id="S2.SS2.SSS0.Px1.2.p1.7.m7.3a"><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.cmml"><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.1" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.7.m7.1.1" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.2.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.2.p1.7.m7.2.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.3" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.3.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml"><mi id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.2.cmml">G</mi><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.1" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.3.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml"><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.3" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.3.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.5" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.5.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.cmml"><mn id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.2" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.1" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.3" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.2.p1.7.m7.3b"><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4"><and id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4a.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4"></and><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4b.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4"><eq id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.3"></eq><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2"><times id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.1"></times><ci id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.2.3.2"><ci id="S2.SS2.SSS0.Px1.2.p1.7.m7.1.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.2.p1.7.m7.2.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.2.2">𝜏</ci></list></apply><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4"><times id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.1"></times><ci id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.2.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.2">𝐺</ci><ci id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.3.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.3">𝜏</ci></apply></apply><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4c.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4"><eq id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.5.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.4.cmml" id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4d.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4"></share><apply id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6"><divide id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.1.cmml" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.1"></divide><cn id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.2">1</cn><cn id="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.2.p1.7.m7.3.4.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.2.p1.7.m7.3c">P(\tau;\tau)=G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.2.p1.7.m7.3d">italic_P ( italic_τ ; italic_τ ) = italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.3.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.3.p2.8">For sufficiency, assume <math alttext="G(\tau)=1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.3.p2.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.cmml"><mrow id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.2" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.2.cmml">G</mi><mo id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.1" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.1" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.1.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.cmml"><mn id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.1" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.3" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.1.m1.1b"><apply id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2"><eq id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.1"></eq><apply id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2"><times id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.1"></times><ci id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.2.2">𝐺</ci><ci id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.1">𝜏</ci></apply><apply id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3"><divide id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.1"></divide><cn id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.2">1</cn><cn id="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.1.m1.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.1.m1.1c">G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.1.m1.1d">italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math>. Equivalently <math alttext="P(\tau;\tau)-1/2=0" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.2.m2.2"><semantics id="S2.SS2.SSS0.Px1.3.p2.2.m2.2a"><mrow id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.cmml"><mrow id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.cmml"><mrow id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.1" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.2.m2.1.1" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.2.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.2.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.1" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.1.cmml">−</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.cmml"><mn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.1" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.3" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.3.cmml">2</mn></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.1" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.1.cmml">=</mo><mn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.3" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.2.m2.2b"><apply id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3"><eq id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.1"></eq><apply id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2"><minus id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.1"></minus><apply id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2"><times id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.1"></times><ci id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.2.3.2"><ci id="S2.SS2.SSS0.Px1.3.p2.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.2">𝜏</ci></list></apply><apply id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3"><divide id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.1"></divide><cn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.2">1</cn><cn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.2.3.3">2</cn></apply></apply><cn id="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.2.m2.2c">P(\tau;\tau)-1/2=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.2.m2.2d">italic_P ( italic_τ ; italic_τ ) - 1 / 2 = 0</annotation></semantics></math>, so (a) implies (b) and the single crossing is at <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.3.p2.3.m3.1a"><mi id="S2.SS2.SSS0.Px1.3.p2.3.m3.1.1" xref="S2.SS2.SSS0.Px1.3.p2.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.3.m3.1b"><ci id="S2.SS2.SSS0.Px1.3.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.3.m3.1d">italic_τ</annotation></semantics></math>. Thus assume (b) holds and take some <math alttext="x>\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.3.p2.4.m4.1a"><mrow id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.2" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.1" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.1.cmml">></mo><mi id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.3" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.4.m4.1b"><apply id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1"><gt id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.1"></gt><ci id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.2">𝑥</ci><ci id="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.4.m4.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.4.m4.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.4.m4.1d">italic_x > italic_τ</annotation></semantics></math>. By single crossing, <math alttext="P(\tau;x)\leq P(\tau;\tau)=G(\tau)=1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5"><semantics id="S2.SS2.SSS0.Px1.3.p2.5.m5.5a"><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.cmml"><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.1" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.1.1" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.1.1.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.2.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.2.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.2.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.3" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.3.cmml">≤</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.1" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.3.3" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.3.3.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.2.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.4.4" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.4.4.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.5" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.5.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.2.cmml">G</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.1" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.3.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.5" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.5.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.7" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.7.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.cmml"><mn id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.2" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.1" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.3" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5b"><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"><and id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6a.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"></and><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6b.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"><leq id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.3"></leq><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2"><times id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.1"></times><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.2.3.2"><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.1.1">𝜏</ci><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.2.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.2.2">𝑥</ci></list></apply><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4"><times id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.1"></times><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.2">𝑃</ci><list id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.3.2"><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.3.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.3.3">𝜏</ci><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.4.4.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.4.4">𝜏</ci></list></apply></apply><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6c.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"><eq id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.5.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.4.cmml" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6d.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"></share><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6"><times id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.1"></times><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.2">𝐺</ci><ci id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.5.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.5">𝜏</ci></apply></apply><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6e.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"><eq id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.7.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.7"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.6.cmml" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6f.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6"></share><apply id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8"><divide id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.1"></divide><cn id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.2">1</cn><cn id="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.5.m5.5.6.8.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5c">P(\tau;x)\leq P(\tau;\tau)=G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.5.m5.5d">italic_P ( italic_τ ; italic_x ) ≤ italic_P ( italic_τ ; italic_τ ) = italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math>. Similarly, if <math alttext="x\leq\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.6.m6.1"><semantics id="S2.SS2.SSS0.Px1.3.p2.6.m6.1a"><mrow id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.2" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.2.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.1" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.1.cmml">≤</mo><mi id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.3" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.6.m6.1b"><apply id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1"><leq id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.1"></leq><ci id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.2">𝑥</ci><ci id="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.3.p2.6.m6.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.6.m6.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.6.m6.1d">italic_x ≤ italic_τ</annotation></semantics></math>, <math alttext="P(\tau;x)\geq P(\tau;\tau)=1/2" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.7.m7.4"><semantics id="S2.SS2.SSS0.Px1.3.p2.7.m7.4a"><mrow id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.cmml"><mrow id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.cmml"><mi id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.2" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.2.cmml">P</mi><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.1" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.3.2" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.3.1.cmml"><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.2.3.1.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.3.p2.7.m7.1.1" 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xref="S2.SS2.SSS0.Px1.3.p2.7.m7.3.3.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.4.3.2.2" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.4.3.1.cmml">;</mo><mi id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.4" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.4.cmml">τ</mi><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.4.3.2.3" stretchy="false" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.5" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.5.cmml">=</mo><mrow id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.cmml"><mn id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.2" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.2.cmml">1</mn><mo id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.1" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.1.cmml">/</mo><mn id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.3" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.7.m7.4b"><apply id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.cmml" 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xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.1"></divide><cn id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.2">1</cn><cn id="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.3.p2.7.m7.4.5.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.7.m7.4c">P(\tau;x)\geq P(\tau;\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.7.m7.4d">italic_P ( italic_τ ; italic_x ) ≥ italic_P ( italic_τ ; italic_τ ) = 1 / 2</annotation></semantics></math>. This establishes Equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E3" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E4" title="In 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>), so <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.3.p2.8.m8.1"><semantics id="S2.SS2.SSS0.Px1.3.p2.8.m8.1a"><mi id="S2.SS2.SSS0.Px1.3.p2.8.m8.1.1" xref="S2.SS2.SSS0.Px1.3.p2.8.m8.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.3.p2.8.m8.1b"><ci id="S2.SS2.SSS0.Px1.3.p2.8.m8.1.1.cmml" xref="S2.SS2.SSS0.Px1.3.p2.8.m8.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.3.p2.8.m8.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.3.p2.8.m8.1d">italic_τ</annotation></semantics></math> is a threshold equilibrum. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p3.5">For intuition for the conditions, observe that the derivative of expected utility with respect to agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p3.1.m1.1a"><mi id="S2.SS2.SSS0.Px1.p3.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p3.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.1.m1.1b"><ci id="S2.SS2.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.1.m1.1d">italic_i</annotation></semantics></math>’s choice of threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.p3.2.m2.1a"><mi id="S2.SS2.SSS0.Px1.p3.2.m2.1.1" xref="S2.SS2.SSS0.Px1.p3.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.2.m2.1b"><ci id="S2.SS2.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.2.m2.1d">italic_τ</annotation></semantics></math> in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E1" title="In 2.1 Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>) is <math alttext="f(x)(2G(x)-1)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.3.m3.3"><semantics id="S2.SS2.SSS0.Px1.p3.3.m3.3a"><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.cmml"><mi id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.3" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.3.cmml">f</mi><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.4.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.cmml"><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.4.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p3.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.1.1.cmml">x</mi><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.4.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.cmml">)</mo></mrow><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.2a" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.cmml"><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.cmml"><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.cmml"><mn id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.2.cmml">2</mn><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1.cmml">⁢</mo><mi id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.3" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.3.cmml">G</mi><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1a" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.4.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.cmml"><mo id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.4.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p3.3.m3.2.2" xref="S2.SS2.SSS0.Px1.p3.3.m3.2.2.cmml">x</mi><mo 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id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.1"></minus><apply id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2"><times id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.1"></times><cn id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.2.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.2">2</cn><ci id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.2.3">𝐺</ci><ci id="S2.SS2.SSS0.Px1.p3.3.m3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p3.3.m3.2.2">𝑥</ci></apply><cn id="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.3.cmml" type="integer" xref="S2.SS2.SSS0.Px1.p3.3.m3.3.3.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.3.m3.3c">f(x)(2G(x)-1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.3.m3.3d">italic_f ( italic_x ) ( 2 italic_G ( italic_x ) - 1 )</annotation></semantics></math>, where <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.4.m4.1"><semantics id="S2.SS2.SSS0.Px1.p3.4.m4.1a"><mi id="S2.SS2.SSS0.Px1.p3.4.m4.1.1" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.4.m4.1b"><ci id="S2.SS2.SSS0.Px1.p3.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.4.m4.1d">italic_f</annotation></semantics></math> is the p.d.f. of <math alttext="F" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p3.5.m5.1"><semantics id="S2.SS2.SSS0.Px1.p3.5.m5.1a"><mi id="S2.SS2.SSS0.Px1.p3.5.m5.1.1" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p3.5.m5.1b"><ci id="S2.SS2.SSS0.Px1.p3.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px1.p3.5.m5.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p3.5.m5.1c">F</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p3.5.m5.1d">italic_F</annotation></semantics></math> and thus non-negative. Thus the necessary condition corresponds to the first order condition of optimizing the best response threshold and the sufficient conditions ensure this optimization is concave and quasiconcave respectively. However, the proof provided establishes optimality in the space of all possible strategies (which is required to be a threshold equilibrium), not merely that the chosen threshold is the optimal threshold.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p4"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p4.2">More broadly, this result shows that “truthfulness” in the sense of following the strategy prescribed by the mechanism designer is fragile in this setting. Only a sharply limited set of thresholds can be implemented in equilibrium—in some cases a single one. This means the mechanism designer’s ability to choose the “meaning” of <math alttext="H" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p4.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p4.1.m1.1a"><mi id="S2.SS2.SSS0.Px1.p4.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p4.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p4.1.m1.1b"><ci id="S2.SS2.SSS0.Px1.p4.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p4.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p4.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p4.1.m1.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p4.2.m2.1"><semantics id="S2.SS2.SSS0.Px1.p4.2.m2.1a"><mi id="S2.SS2.SSS0.Px1.p4.2.m2.1.1" xref="S2.SS2.SSS0.Px1.p4.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p4.2.m2.1b"><ci id="S2.SS2.SSS0.Px1.p4.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p4.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p4.2.m2.1d">italic_L</annotation></semantics></math> can be quite limited in practice.</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Dynamics</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">We have characterizations of both the uninformative infinite equilibria and more interesting finite equilibria. Which are more likely to occur? To answer this question, we turn to modeling the dynamics of the system. We are interested in studying how agents will behave under our model over time, as a tool for equilibrium refinement. That is, we want to know what threshold agents will land on as they continue to grade new batches of papers or submit new labels. Since it is likely that the mechanism shares empirical results before all agents equilibriate, we consider a dynamic setup where a small fraction of agents are able to best respond at each time step.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.9">Formally, consider the following discrete best response dynamic: the mechanism publishes an initial threshold <math alttext="\tau(0)" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mrow id="S2.SS3.p2.1.m1.1.2" xref="S2.SS3.p2.1.m1.1.2.cmml"><mi id="S2.SS3.p2.1.m1.1.2.2" xref="S2.SS3.p2.1.m1.1.2.2.cmml">τ</mi><mo id="S2.SS3.p2.1.m1.1.2.1" xref="S2.SS3.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.1.m1.1.2.3.2" xref="S2.SS3.p2.1.m1.1.2.cmml"><mo id="S2.SS3.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.1.m1.1.2.cmml">(</mo><mn id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">0</mn><mo id="S2.SS3.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><apply id="S2.SS3.p2.1.m1.1.2.cmml" xref="S2.SS3.p2.1.m1.1.2"><times id="S2.SS3.p2.1.m1.1.2.1.cmml" xref="S2.SS3.p2.1.m1.1.2.1"></times><ci id="S2.SS3.p2.1.m1.1.2.2.cmml" xref="S2.SS3.p2.1.m1.1.2.2">𝜏</ci><cn id="S2.SS3.p2.1.m1.1.1.cmml" type="integer" xref="S2.SS3.p2.1.m1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\tau(0)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_τ ( 0 )</annotation></semantics></math>, the “norms” with which agents should interpret their signals as <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mi id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><ci id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_L</annotation></semantics></math>. At each time step, a small, random fraction <math alttext="\Delta t" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><mrow id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml"><mi id="S2.SS3.p2.4.m4.1.1.2" mathvariant="normal" xref="S2.SS3.p2.4.m4.1.1.2.cmml">Δ</mi><mo id="S2.SS3.p2.4.m4.1.1.1" xref="S2.SS3.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p2.4.m4.1.1.3" xref="S2.SS3.p2.4.m4.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><apply id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1"><times id="S2.SS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1.1"></times><ci id="S2.SS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2">Δ</ci><ci id="S2.SS3.p2.4.m4.1.1.3.cmml" xref="S2.SS3.p2.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">roman_Δ italic_t</annotation></semantics></math> of agents are able to commit to a new best response threshold strategy <math alttext="\hat{\tau}(t)" class="ltx_Math" display="inline" id="S2.SS3.p2.5.m5.1"><semantics id="S2.SS3.p2.5.m5.1a"><mrow id="S2.SS3.p2.5.m5.1.2" xref="S2.SS3.p2.5.m5.1.2.cmml"><mover accent="true" id="S2.SS3.p2.5.m5.1.2.2" xref="S2.SS3.p2.5.m5.1.2.2.cmml"><mi id="S2.SS3.p2.5.m5.1.2.2.2" xref="S2.SS3.p2.5.m5.1.2.2.2.cmml">τ</mi><mo id="S2.SS3.p2.5.m5.1.2.2.1" xref="S2.SS3.p2.5.m5.1.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.p2.5.m5.1.2.1" xref="S2.SS3.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.5.m5.1.2.3.2" xref="S2.SS3.p2.5.m5.1.2.cmml"><mo id="S2.SS3.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.5.m5.1.2.cmml">(</mo><mi id="S2.SS3.p2.5.m5.1.1" xref="S2.SS3.p2.5.m5.1.1.cmml">t</mi><mo id="S2.SS3.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.5.m5.1b"><apply id="S2.SS3.p2.5.m5.1.2.cmml" xref="S2.SS3.p2.5.m5.1.2"><times id="S2.SS3.p2.5.m5.1.2.1.cmml" xref="S2.SS3.p2.5.m5.1.2.1"></times><apply id="S2.SS3.p2.5.m5.1.2.2.cmml" xref="S2.SS3.p2.5.m5.1.2.2"><ci id="S2.SS3.p2.5.m5.1.2.2.1.cmml" xref="S2.SS3.p2.5.m5.1.2.2.1">^</ci><ci id="S2.SS3.p2.5.m5.1.2.2.2.cmml" xref="S2.SS3.p2.5.m5.1.2.2.2">𝜏</ci></apply><ci id="S2.SS3.p2.5.m5.1.1.cmml" xref="S2.SS3.p2.5.m5.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.5.m5.1c">\hat{\tau}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.5.m5.1d">over^ start_ARG italic_τ end_ARG ( italic_t )</annotation></semantics></math> against the current threshold <math alttext="\tau(t)" class="ltx_Math" display="inline" id="S2.SS3.p2.6.m6.1"><semantics id="S2.SS3.p2.6.m6.1a"><mrow id="S2.SS3.p2.6.m6.1.2" xref="S2.SS3.p2.6.m6.1.2.cmml"><mi id="S2.SS3.p2.6.m6.1.2.2" xref="S2.SS3.p2.6.m6.1.2.2.cmml">τ</mi><mo id="S2.SS3.p2.6.m6.1.2.1" xref="S2.SS3.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.6.m6.1.2.3.2" xref="S2.SS3.p2.6.m6.1.2.cmml"><mo id="S2.SS3.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.6.m6.1.2.cmml">(</mo><mi id="S2.SS3.p2.6.m6.1.1" xref="S2.SS3.p2.6.m6.1.1.cmml">t</mi><mo id="S2.SS3.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.6.m6.1b"><apply id="S2.SS3.p2.6.m6.1.2.cmml" xref="S2.SS3.p2.6.m6.1.2"><times id="S2.SS3.p2.6.m6.1.2.1.cmml" xref="S2.SS3.p2.6.m6.1.2.1"></times><ci id="S2.SS3.p2.6.m6.1.2.2.cmml" xref="S2.SS3.p2.6.m6.1.2.2">𝜏</ci><ci id="S2.SS3.p2.6.m6.1.1.cmml" xref="S2.SS3.p2.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.6.m6.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.6.m6.1d">italic_τ ( italic_t )</annotation></semantics></math>. Meanwhile, the rest of the agents stick to the threshold they used previously, whether that be a previous best response or the initial <math alttext="\tau(0)" class="ltx_Math" display="inline" id="S2.SS3.p2.7.m7.1"><semantics id="S2.SS3.p2.7.m7.1a"><mrow id="S2.SS3.p2.7.m7.1.2" xref="S2.SS3.p2.7.m7.1.2.cmml"><mi id="S2.SS3.p2.7.m7.1.2.2" xref="S2.SS3.p2.7.m7.1.2.2.cmml">τ</mi><mo id="S2.SS3.p2.7.m7.1.2.1" xref="S2.SS3.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.7.m7.1.2.3.2" xref="S2.SS3.p2.7.m7.1.2.cmml"><mo id="S2.SS3.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.7.m7.1.2.cmml">(</mo><mn id="S2.SS3.p2.7.m7.1.1" xref="S2.SS3.p2.7.m7.1.1.cmml">0</mn><mo id="S2.SS3.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.7.m7.1b"><apply id="S2.SS3.p2.7.m7.1.2.cmml" xref="S2.SS3.p2.7.m7.1.2"><times id="S2.SS3.p2.7.m7.1.2.1.cmml" xref="S2.SS3.p2.7.m7.1.2.1"></times><ci id="S2.SS3.p2.7.m7.1.2.2.cmml" xref="S2.SS3.p2.7.m7.1.2.2">𝜏</ci><cn id="S2.SS3.p2.7.m7.1.1.cmml" type="integer" xref="S2.SS3.p2.7.m7.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.7.m7.1c">\tau(0)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.7.m7.1d">italic_τ ( 0 )</annotation></semantics></math>. Note that since <math alttext="\hat{\tau}(t)" class="ltx_Math" display="inline" id="S2.SS3.p2.8.m8.1"><semantics id="S2.SS3.p2.8.m8.1a"><mrow id="S2.SS3.p2.8.m8.1.2" xref="S2.SS3.p2.8.m8.1.2.cmml"><mover accent="true" id="S2.SS3.p2.8.m8.1.2.2" xref="S2.SS3.p2.8.m8.1.2.2.cmml"><mi id="S2.SS3.p2.8.m8.1.2.2.2" xref="S2.SS3.p2.8.m8.1.2.2.2.cmml">τ</mi><mo id="S2.SS3.p2.8.m8.1.2.2.1" xref="S2.SS3.p2.8.m8.1.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.p2.8.m8.1.2.1" xref="S2.SS3.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.8.m8.1.2.3.2" xref="S2.SS3.p2.8.m8.1.2.cmml"><mo id="S2.SS3.p2.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.8.m8.1.2.cmml">(</mo><mi id="S2.SS3.p2.8.m8.1.1" xref="S2.SS3.p2.8.m8.1.1.cmml">t</mi><mo id="S2.SS3.p2.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.8.m8.1b"><apply id="S2.SS3.p2.8.m8.1.2.cmml" xref="S2.SS3.p2.8.m8.1.2"><times id="S2.SS3.p2.8.m8.1.2.1.cmml" xref="S2.SS3.p2.8.m8.1.2.1"></times><apply id="S2.SS3.p2.8.m8.1.2.2.cmml" xref="S2.SS3.p2.8.m8.1.2.2"><ci id="S2.SS3.p2.8.m8.1.2.2.1.cmml" xref="S2.SS3.p2.8.m8.1.2.2.1">^</ci><ci id="S2.SS3.p2.8.m8.1.2.2.2.cmml" xref="S2.SS3.p2.8.m8.1.2.2.2">𝜏</ci></apply><ci id="S2.SS3.p2.8.m8.1.1.cmml" xref="S2.SS3.p2.8.m8.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.8.m8.1c">\hat{\tau}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.8.m8.1d">over^ start_ARG italic_τ end_ARG ( italic_t )</annotation></semantics></math> is a best response, its calculation depends on (1) the peer prediction mechanism’s payment scheme and (2) the current threshold <math alttext="\tau(t)" class="ltx_Math" display="inline" id="S2.SS3.p2.9.m9.1"><semantics id="S2.SS3.p2.9.m9.1a"><mrow id="S2.SS3.p2.9.m9.1.2" xref="S2.SS3.p2.9.m9.1.2.cmml"><mi id="S2.SS3.p2.9.m9.1.2.2" xref="S2.SS3.p2.9.m9.1.2.2.cmml">τ</mi><mo id="S2.SS3.p2.9.m9.1.2.1" xref="S2.SS3.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p2.9.m9.1.2.3.2" xref="S2.SS3.p2.9.m9.1.2.cmml"><mo id="S2.SS3.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S2.SS3.p2.9.m9.1.2.cmml">(</mo><mi id="S2.SS3.p2.9.m9.1.1" xref="S2.SS3.p2.9.m9.1.1.cmml">t</mi><mo id="S2.SS3.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S2.SS3.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.9.m9.1b"><apply id="S2.SS3.p2.9.m9.1.2.cmml" xref="S2.SS3.p2.9.m9.1.2"><times id="S2.SS3.p2.9.m9.1.2.1.cmml" xref="S2.SS3.p2.9.m9.1.2.1"></times><ci id="S2.SS3.p2.9.m9.1.2.2.cmml" xref="S2.SS3.p2.9.m9.1.2.2">𝜏</ci><ci id="S2.SS3.p2.9.m9.1.1.cmml" xref="S2.SS3.p2.9.m9.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.9.m9.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.9.m9.1d">italic_τ ( italic_t )</annotation></semantics></math>. Finally, the empirical frequency of reports is updated to reflect this new mixture of thresholds, and the process repeats. This dynamic corresponds to the following equation:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx2"> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tau(\Delta t)" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.3" xref="S2.Ex4.m1.1.1.3.cmml">τ</mi><mo id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex4.m1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.1.1.1.2" mathvariant="normal" xref="S2.Ex4.m1.1.1.1.1.1.2.cmml">Δ</mi><mo id="S2.Ex4.m1.1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Ex4.m1.1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S2.Ex4.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"><times id="S2.Ex4.m1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.2"></times><ci id="S2.Ex4.m1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.3">𝜏</ci><apply id="S2.Ex4.m1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1"><times id="S2.Ex4.m1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1.1"></times><ci id="S2.Ex4.m1.1.1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.1.1.2">Δ</ci><ci id="S2.Ex4.m1.1.1.1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle\tau(\Delta t)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">italic_τ ( roman_Δ italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left 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xref="S2.Ex4.m2.3.3.1.1.1.1.1.3.2.cmml">Δ</mi><mo id="S2.Ex4.m2.3.3.1.1.1.1.1.3.1" xref="S2.Ex4.m2.3.3.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex4.m2.3.3.1.1.1.1.1.3.3" xref="S2.Ex4.m2.3.3.1.1.1.1.1.3.3.cmml">t</mi></mrow></mrow><mo id="S2.Ex4.m2.3.3.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex4.m2.3.3.1.1.2" xref="S2.Ex4.m2.3.3.1.1.2.cmml">⁢</mo><mi id="S2.Ex4.m2.3.3.1.1.3" xref="S2.Ex4.m2.3.3.1.1.3.cmml">τ</mi><mo id="S2.Ex4.m2.3.3.1.1.2a" xref="S2.Ex4.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m2.3.3.1.1.4.2" xref="S2.Ex4.m2.3.3.1.1.cmml"><mo id="S2.Ex4.m2.3.3.1.1.4.2.1" stretchy="false" xref="S2.Ex4.m2.3.3.1.1.cmml">(</mo><mn id="S2.Ex4.m2.1.1" xref="S2.Ex4.m2.1.1.cmml">0</mn><mo id="S2.Ex4.m2.3.3.1.1.4.2.2" stretchy="false" xref="S2.Ex4.m2.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m2.3.3.1.2" xref="S2.Ex4.m2.3.3.1.2.cmml">+</mo><mrow id="S2.Ex4.m2.3.3.1.3" xref="S2.Ex4.m2.3.3.1.3.cmml"><mi id="S2.Ex4.m2.3.3.1.3.2" mathvariant="normal" xref="S2.Ex4.m2.3.3.1.3.2.cmml">Δ</mi><mo id="S2.Ex4.m2.3.3.1.3.1" xref="S2.Ex4.m2.3.3.1.3.1.cmml">⁢</mo><mi id="S2.Ex4.m2.3.3.1.3.3" xref="S2.Ex4.m2.3.3.1.3.3.cmml">t</mi><mo id="S2.Ex4.m2.3.3.1.3.1a" xref="S2.Ex4.m2.3.3.1.3.1.cmml">⁢</mo><mover accent="true" id="S2.Ex4.m2.3.3.1.3.4" xref="S2.Ex4.m2.3.3.1.3.4.cmml"><mi id="S2.Ex4.m2.3.3.1.3.4.2" xref="S2.Ex4.m2.3.3.1.3.4.2.cmml">τ</mi><mo id="S2.Ex4.m2.3.3.1.3.4.1" xref="S2.Ex4.m2.3.3.1.3.4.1.cmml">^</mo></mover><mo id="S2.Ex4.m2.3.3.1.3.1b" xref="S2.Ex4.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="S2.Ex4.m2.3.3.1.3.5.2" xref="S2.Ex4.m2.3.3.1.3.cmml"><mo id="S2.Ex4.m2.3.3.1.3.5.2.1" stretchy="false" xref="S2.Ex4.m2.3.3.1.3.cmml">(</mo><mn id="S2.Ex4.m2.2.2" xref="S2.Ex4.m2.2.2.cmml">0</mn><mo id="S2.Ex4.m2.3.3.1.3.5.2.2" stretchy="false" xref="S2.Ex4.m2.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m2.3b"><apply id="S2.Ex4.m2.3.3.cmml" xref="S2.Ex4.m2.3.3"><eq 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id="S2.Ex4.m2.3.3.1.1.3.cmml" xref="S2.Ex4.m2.3.3.1.1.3">𝜏</ci><cn id="S2.Ex4.m2.1.1.cmml" type="integer" xref="S2.Ex4.m2.1.1">0</cn></apply><apply id="S2.Ex4.m2.3.3.1.3.cmml" xref="S2.Ex4.m2.3.3.1.3"><times id="S2.Ex4.m2.3.3.1.3.1.cmml" xref="S2.Ex4.m2.3.3.1.3.1"></times><ci id="S2.Ex4.m2.3.3.1.3.2.cmml" xref="S2.Ex4.m2.3.3.1.3.2">Δ</ci><ci id="S2.Ex4.m2.3.3.1.3.3.cmml" xref="S2.Ex4.m2.3.3.1.3.3">𝑡</ci><apply id="S2.Ex4.m2.3.3.1.3.4.cmml" xref="S2.Ex4.m2.3.3.1.3.4"><ci id="S2.Ex4.m2.3.3.1.3.4.1.cmml" xref="S2.Ex4.m2.3.3.1.3.4.1">^</ci><ci id="S2.Ex4.m2.3.3.1.3.4.2.cmml" xref="S2.Ex4.m2.3.3.1.3.4.2">𝜏</ci></apply><cn id="S2.Ex4.m2.2.2.cmml" type="integer" xref="S2.Ex4.m2.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.3c">\displaystyle=(1-\Delta t)\tau(0)+\Delta t\hat{\tau}(0)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.3d">= ( 1 - roman_Δ italic_t ) italic_τ ( 0 ) + roman_Δ italic_t over^ start_ARG 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id="S2.Ex5.m1.1c">\displaystyle\tau(2\Delta t)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.1d">italic_τ ( 2 roman_Δ italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(1-\Delta t)\tau(\Delta t)+\Delta t\hat{\tau}(\Delta t)" class="ltx_Math" display="inline" id="S2.Ex5.m2.3"><semantics id="S2.Ex5.m2.3a"><mrow id="S2.Ex5.m2.3.3" xref="S2.Ex5.m2.3.3.cmml"><mi id="S2.Ex5.m2.3.3.5" xref="S2.Ex5.m2.3.3.5.cmml"></mi><mo id="S2.Ex5.m2.3.3.4" xref="S2.Ex5.m2.3.3.4.cmml">=</mo><mrow id="S2.Ex5.m2.3.3.3" xref="S2.Ex5.m2.3.3.3.cmml"><mrow id="S2.Ex5.m2.2.2.2.2" xref="S2.Ex5.m2.2.2.2.2.cmml"><mrow id="S2.Ex5.m2.1.1.1.1.1.1" xref="S2.Ex5.m2.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex5.m2.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m2.1.1.1.1.1.1.1" xref="S2.Ex5.m2.1.1.1.1.1.1.1.cmml"><mn id="S2.Ex5.m2.1.1.1.1.1.1.1.2" 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xref="S2.E5.m2.3.3.1.1.2.2.cmml">(</mo><mi id="S2.E5.m2.2.2" xref="S2.E5.m2.2.2.cmml">t</mi><mo id="S2.E5.m2.3.3.1.1.2.2.4.2.2" stretchy="false" xref="S2.E5.m2.3.3.1.1.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E5.m2.3.3.1.2" lspace="0em" xref="S2.E5.m2.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.m2.3b"><apply id="S2.E5.m2.3.3.1.1.cmml" xref="S2.E5.m2.3.3.1"><eq id="S2.E5.m2.3.3.1.1.3.cmml" xref="S2.E5.m2.3.3.1.1.3"></eq><csymbol cd="latexml" id="S2.E5.m2.3.3.1.1.4.cmml" xref="S2.E5.m2.3.3.1.1.4">absent</csymbol><apply id="S2.E5.m2.3.3.1.1.2.cmml" xref="S2.E5.m2.3.3.1.1.2"><plus id="S2.E5.m2.3.3.1.1.2.3.cmml" xref="S2.E5.m2.3.3.1.1.2.3"></plus><apply id="S2.E5.m2.3.3.1.1.1.1.cmml" xref="S2.E5.m2.3.3.1.1.1.1"><times id="S2.E5.m2.3.3.1.1.1.1.2.cmml" xref="S2.E5.m2.3.3.1.1.1.1.2"></times><apply id="S2.E5.m2.3.3.1.1.1.1.1.1.1.cmml" xref="S2.E5.m2.3.3.1.1.1.1.1.1"><times id="S2.E5.m2.3.3.1.1.1.1.1.1.1.1.cmml" 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xref="S2.E5.m2.3.3.1.1.2.2.1.1.1.3"><times id="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.1.cmml" xref="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.1"></times><ci id="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.2.cmml" xref="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.2">Δ</ci><ci id="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.3.cmml" xref="S2.E5.m2.3.3.1.1.2.2.1.1.1.3.3">𝑡</ci></apply></apply><ci id="S2.E5.m2.3.3.1.1.2.2.3.cmml" xref="S2.E5.m2.3.3.1.1.2.2.3">𝜏</ci><ci id="S2.E5.m2.2.2.cmml" xref="S2.E5.m2.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m2.3c">\displaystyle=(\Delta t)\hat{\tau}(t)+(1-\Delta t)\tau(t).</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m2.3d">= ( roman_Δ italic_t ) over^ start_ARG italic_τ end_ARG ( italic_t ) + ( 1 - roman_Δ italic_t ) italic_τ ( italic_t ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.3">We note that <math alttext="\hat{\tau}(k\Delta t)" class="ltx_Math" display="inline" id="S2.SS3.p3.1.m1.1"><semantics id="S2.SS3.p3.1.m1.1a"><mrow id="S2.SS3.p3.1.m1.1.1" xref="S2.SS3.p3.1.m1.1.1.cmml"><mover accent="true" id="S2.SS3.p3.1.m1.1.1.3" xref="S2.SS3.p3.1.m1.1.1.3.cmml"><mi id="S2.SS3.p3.1.m1.1.1.3.2" xref="S2.SS3.p3.1.m1.1.1.3.2.cmml">τ</mi><mo id="S2.SS3.p3.1.m1.1.1.3.1" xref="S2.SS3.p3.1.m1.1.1.3.1.cmml">^</mo></mover><mo id="S2.SS3.p3.1.m1.1.1.2" xref="S2.SS3.p3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS3.p3.1.m1.1.1.1.1" xref="S2.SS3.p3.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS3.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS3.p3.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p3.1.m1.1.1.1.1.1" xref="S2.SS3.p3.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS3.p3.1.m1.1.1.1.1.1.2" xref="S2.SS3.p3.1.m1.1.1.1.1.1.2.cmml">k</mi><mo id="S2.SS3.p3.1.m1.1.1.1.1.1.1" xref="S2.SS3.p3.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p3.1.m1.1.1.1.1.1.3" mathvariant="normal" xref="S2.SS3.p3.1.m1.1.1.1.1.1.3.cmml">Δ</mi><mo id="S2.SS3.p3.1.m1.1.1.1.1.1.1a" xref="S2.SS3.p3.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS3.p3.1.m1.1.1.1.1.1.4" xref="S2.SS3.p3.1.m1.1.1.1.1.1.4.cmml">t</mi></mrow><mo id="S2.SS3.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS3.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.1.m1.1b"><apply id="S2.SS3.p3.1.m1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1"><times id="S2.SS3.p3.1.m1.1.1.2.cmml" xref="S2.SS3.p3.1.m1.1.1.2"></times><apply id="S2.SS3.p3.1.m1.1.1.3.cmml" xref="S2.SS3.p3.1.m1.1.1.3"><ci id="S2.SS3.p3.1.m1.1.1.3.1.cmml" xref="S2.SS3.p3.1.m1.1.1.3.1">^</ci><ci id="S2.SS3.p3.1.m1.1.1.3.2.cmml" xref="S2.SS3.p3.1.m1.1.1.3.2">𝜏</ci></apply><apply id="S2.SS3.p3.1.m1.1.1.1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1.1.1"><times id="S2.SS3.p3.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS3.p3.1.m1.1.1.1.1.1.1"></times><ci id="S2.SS3.p3.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS3.p3.1.m1.1.1.1.1.1.2">𝑘</ci><ci id="S2.SS3.p3.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS3.p3.1.m1.1.1.1.1.1.3">Δ</ci><ci id="S2.SS3.p3.1.m1.1.1.1.1.1.4.cmml" xref="S2.SS3.p3.1.m1.1.1.1.1.1.4">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.1c">\hat{\tau}(k\Delta t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.1d">over^ start_ARG italic_τ end_ARG ( italic_k roman_Δ italic_t )</annotation></semantics></math> is technically a best response to the <em class="ltx_emph ltx_font_italic" id="S2.SS3.p3.3.1">mixture</em> of thresholds at the previous time step <math alttext="k-1" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><mrow id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml"><mi id="S2.SS3.p3.2.m2.1.1.2" xref="S2.SS3.p3.2.m2.1.1.2.cmml">k</mi><mo id="S2.SS3.p3.2.m2.1.1.1" xref="S2.SS3.p3.2.m2.1.1.1.cmml">−</mo><mn id="S2.SS3.p3.2.m2.1.1.3" xref="S2.SS3.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.1b"><apply id="S2.SS3.p3.2.m2.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1"><minus id="S2.SS3.p3.2.m2.1.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1.1"></minus><ci id="S2.SS3.p3.2.m2.1.1.2.cmml" xref="S2.SS3.p3.2.m2.1.1.2">𝑘</ci><cn id="S2.SS3.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS3.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.1c">k-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.1d">italic_k - 1</annotation></semantics></math>, rather than the probability distribution over thresholds. Ignoring this for the moment and taking the limit as <math alttext="\Delta t\to 0" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.1"><semantics id="S2.SS3.p3.3.m3.1a"><mrow id="S2.SS3.p3.3.m3.1.1" xref="S2.SS3.p3.3.m3.1.1.cmml"><mrow id="S2.SS3.p3.3.m3.1.1.2" xref="S2.SS3.p3.3.m3.1.1.2.cmml"><mi id="S2.SS3.p3.3.m3.1.1.2.2" mathvariant="normal" xref="S2.SS3.p3.3.m3.1.1.2.2.cmml">Δ</mi><mo id="S2.SS3.p3.3.m3.1.1.2.1" xref="S2.SS3.p3.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S2.SS3.p3.3.m3.1.1.2.3" xref="S2.SS3.p3.3.m3.1.1.2.3.cmml">t</mi></mrow><mo id="S2.SS3.p3.3.m3.1.1.1" stretchy="false" xref="S2.SS3.p3.3.m3.1.1.1.cmml">→</mo><mn id="S2.SS3.p3.3.m3.1.1.3" xref="S2.SS3.p3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.3.m3.1b"><apply id="S2.SS3.p3.3.m3.1.1.cmml" xref="S2.SS3.p3.3.m3.1.1"><ci id="S2.SS3.p3.3.m3.1.1.1.cmml" xref="S2.SS3.p3.3.m3.1.1.1">→</ci><apply id="S2.SS3.p3.3.m3.1.1.2.cmml" xref="S2.SS3.p3.3.m3.1.1.2"><times id="S2.SS3.p3.3.m3.1.1.2.1.cmml" xref="S2.SS3.p3.3.m3.1.1.2.1"></times><ci id="S2.SS3.p3.3.m3.1.1.2.2.cmml" xref="S2.SS3.p3.3.m3.1.1.2.2">Δ</ci><ci id="S2.SS3.p3.3.m3.1.1.2.3.cmml" xref="S2.SS3.p3.3.m3.1.1.2.3">𝑡</ci></apply><cn id="S2.SS3.p3.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.p3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.3.m3.1c">\Delta t\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.3.m3.1d">roman_Δ italic_t → 0</annotation></semantics></math> of the discrete system in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E5" title="In 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">5</span></a>, we end up with the following continuous best response dynamic:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <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="\dot{\tau}=\hat{\tau}-\tau." class="ltx_Math" display="block" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mrow id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><mover accent="true" id="S2.E6.m1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.2.cmml"><mi id="S2.E6.m1.1.1.1.1.2.2" xref="S2.E6.m1.1.1.1.1.2.2.cmml">τ</mi><mo id="S2.E6.m1.1.1.1.1.2.1" xref="S2.E6.m1.1.1.1.1.2.1.cmml">˙</mo></mover><mo id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E6.m1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.3.cmml"><mover accent="true" id="S2.E6.m1.1.1.1.1.3.2" xref="S2.E6.m1.1.1.1.1.3.2.cmml"><mi id="S2.E6.m1.1.1.1.1.3.2.2" xref="S2.E6.m1.1.1.1.1.3.2.2.cmml">τ</mi><mo id="S2.E6.m1.1.1.1.1.3.2.1" xref="S2.E6.m1.1.1.1.1.3.2.1.cmml">^</mo></mover><mo id="S2.E6.m1.1.1.1.1.3.1" xref="S2.E6.m1.1.1.1.1.3.1.cmml">−</mo><mi id="S2.E6.m1.1.1.1.1.3.3" xref="S2.E6.m1.1.1.1.1.3.3.cmml">τ</mi></mrow></mrow><mo id="S2.E6.m1.1.1.1.2" lspace="0em" xref="S2.E6.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.1b"><apply id="S2.E6.m1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1"><eq id="S2.E6.m1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1"></eq><apply id="S2.E6.m1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.2"><ci id="S2.E6.m1.1.1.1.1.2.1.cmml" xref="S2.E6.m1.1.1.1.1.2.1">˙</ci><ci id="S2.E6.m1.1.1.1.1.2.2.cmml" xref="S2.E6.m1.1.1.1.1.2.2">𝜏</ci></apply><apply id="S2.E6.m1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.3"><minus id="S2.E6.m1.1.1.1.1.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.1"></minus><apply id="S2.E6.m1.1.1.1.1.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2"><ci id="S2.E6.m1.1.1.1.1.3.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.1">^</ci><ci id="S2.E6.m1.1.1.1.1.3.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2">𝜏</ci></apply><ci id="S2.E6.m1.1.1.1.1.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">\dot{\tau}=\hat{\tau}-\tau.</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ .</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="S2.SS3.p3.4">While not, strictly speaking, the correct update from best response dynamics, this update rule has the appealing form of the threshold slowly drifting toward the best response. Alternatively, since agents near the threshold have the strongest incentive to change their strategy, this update rule could be intepreted as a form of quantal response. In any case, our focus in this work is on what sort of equilibria we should expect to end up at rather than the exact process the system takes to get there, so this choice of dynamics seems simple, natural, and amenable to analysis.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.2">We begin our study of OA under the dynamics described by Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E6" title="In 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>) with the following observation. A nice property of Output Agreement (which relates to our sufficient conditions from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem1" title="Theorem 1. ‣ Results. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>) is that for a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.1"><semantics id="S2.SS3.p4.1.m1.1a"><mi id="S2.SS3.p4.1.m1.1.1" xref="S2.SS3.p4.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.1.m1.1b"><ci id="S2.SS3.p4.1.m1.1.1.cmml" xref="S2.SS3.p4.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.1.m1.1d">italic_τ</annotation></semantics></math>, under decreasing monotonicity of <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S2.SS3.p4.2.m2.2"><semantics id="S2.SS3.p4.2.m2.2a"><mrow id="S2.SS3.p4.2.m2.2.3" xref="S2.SS3.p4.2.m2.2.3.cmml"><mi id="S2.SS3.p4.2.m2.2.3.2" xref="S2.SS3.p4.2.m2.2.3.2.cmml">P</mi><mo id="S2.SS3.p4.2.m2.2.3.1" xref="S2.SS3.p4.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.p4.2.m2.2.3.3.2" xref="S2.SS3.p4.2.m2.2.3.3.1.cmml"><mo id="S2.SS3.p4.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS3.p4.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS3.p4.2.m2.1.1" xref="S2.SS3.p4.2.m2.1.1.cmml">τ</mi><mo id="S2.SS3.p4.2.m2.2.3.3.2.2" xref="S2.SS3.p4.2.m2.2.3.3.1.cmml">;</mo><mi id="S2.SS3.p4.2.m2.2.2" xref="S2.SS3.p4.2.m2.2.2.cmml">x</mi><mo id="S2.SS3.p4.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS3.p4.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p4.2.m2.2b"><apply id="S2.SS3.p4.2.m2.2.3.cmml" xref="S2.SS3.p4.2.m2.2.3"><times id="S2.SS3.p4.2.m2.2.3.1.cmml" xref="S2.SS3.p4.2.m2.2.3.1"></times><ci id="S2.SS3.p4.2.m2.2.3.2.cmml" xref="S2.SS3.p4.2.m2.2.3.2">𝑃</ci><list id="S2.SS3.p4.2.m2.2.3.3.1.cmml" xref="S2.SS3.p4.2.m2.2.3.3.2"><ci id="S2.SS3.p4.2.m2.1.1.cmml" xref="S2.SS3.p4.2.m2.1.1">𝜏</ci><ci id="S2.SS3.p4.2.m2.2.2.cmml" xref="S2.SS3.p4.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p4.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p4.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> the best response over <em class="ltx_emph ltx_font_italic" id="S2.SS3.p4.2.1">all</em> strategies is always a threshold even out of equilibrium.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition2.1.1.1">Proposition 2</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition2.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition2.p1"> <p class="ltx_p" id="Thmproposition2.p1.7">Assume for a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmproposition2.p1.1.m1.1"><semantics id="Thmproposition2.p1.1.m1.1a"><mi id="Thmproposition2.p1.1.m1.1.1" xref="Thmproposition2.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.1.m1.1b"><ci id="Thmproposition2.p1.1.m1.1.1.cmml" xref="Thmproposition2.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.1.m1.1d">italic_τ</annotation></semantics></math> that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmproposition2.p1.2.m2.2"><semantics id="Thmproposition2.p1.2.m2.2a"><mrow id="Thmproposition2.p1.2.m2.2.3" xref="Thmproposition2.p1.2.m2.2.3.cmml"><mi id="Thmproposition2.p1.2.m2.2.3.2" xref="Thmproposition2.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmproposition2.p1.2.m2.2.3.1" xref="Thmproposition2.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmproposition2.p1.2.m2.2.3.3.2" xref="Thmproposition2.p1.2.m2.2.3.3.1.cmml"><mo id="Thmproposition2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmproposition2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmproposition2.p1.2.m2.1.1" xref="Thmproposition2.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmproposition2.p1.2.m2.2.3.3.2.2" xref="Thmproposition2.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmproposition2.p1.2.m2.2.2" xref="Thmproposition2.p1.2.m2.2.2.cmml">x</mi><mo id="Thmproposition2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmproposition2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.2.m2.2b"><apply id="Thmproposition2.p1.2.m2.2.3.cmml" xref="Thmproposition2.p1.2.m2.2.3"><times id="Thmproposition2.p1.2.m2.2.3.1.cmml" xref="Thmproposition2.p1.2.m2.2.3.1"></times><ci id="Thmproposition2.p1.2.m2.2.3.2.cmml" xref="Thmproposition2.p1.2.m2.2.3.2">𝑃</ci><list id="Thmproposition2.p1.2.m2.2.3.3.1.cmml" xref="Thmproposition2.p1.2.m2.2.3.3.2"><ci id="Thmproposition2.p1.2.m2.1.1.cmml" xref="Thmproposition2.p1.2.m2.1.1">𝜏</ci><ci id="Thmproposition2.p1.2.m2.2.2.cmml" xref="Thmproposition2.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmproposition2.p1.3.m3.1"><semantics id="Thmproposition2.p1.3.m3.1a"><mi id="Thmproposition2.p1.3.m3.1.1" xref="Thmproposition2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.3.m3.1b"><ci id="Thmproposition2.p1.3.m3.1.1.cmml" xref="Thmproposition2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.3.m3.1d">italic_x</annotation></semantics></math>. If all agents are playing according threshold strategy <math alttext="\tau" class="ltx_Math" display="inline" id="Thmproposition2.p1.4.m4.1"><semantics id="Thmproposition2.p1.4.m4.1a"><mi id="Thmproposition2.p1.4.m4.1.1" xref="Thmproposition2.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.4.m4.1b"><ci id="Thmproposition2.p1.4.m4.1.1.cmml" xref="Thmproposition2.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.4.m4.1d">italic_τ</annotation></semantics></math>, the unique best response of an agent across all strategies <math alttext="\sigma:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="Thmproposition2.p1.5.m5.1"><semantics id="Thmproposition2.p1.5.m5.1a"><mrow id="Thmproposition2.p1.5.m5.1.1" xref="Thmproposition2.p1.5.m5.1.1.cmml"><mi id="Thmproposition2.p1.5.m5.1.1.2" xref="Thmproposition2.p1.5.m5.1.1.2.cmml">σ</mi><mo id="Thmproposition2.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmproposition2.p1.5.m5.1.1.1.cmml">:</mo><mrow id="Thmproposition2.p1.5.m5.1.1.3" xref="Thmproposition2.p1.5.m5.1.1.3.cmml"><mi id="Thmproposition2.p1.5.m5.1.1.3.2" xref="Thmproposition2.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="Thmproposition2.p1.5.m5.1.1.3.1" stretchy="false" xref="Thmproposition2.p1.5.m5.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="Thmproposition2.p1.5.m5.1.1.3.3" xref="Thmproposition2.p1.5.m5.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.5.m5.1b"><apply id="Thmproposition2.p1.5.m5.1.1.cmml" xref="Thmproposition2.p1.5.m5.1.1"><ci id="Thmproposition2.p1.5.m5.1.1.1.cmml" xref="Thmproposition2.p1.5.m5.1.1.1">:</ci><ci id="Thmproposition2.p1.5.m5.1.1.2.cmml" xref="Thmproposition2.p1.5.m5.1.1.2">𝜎</ci><apply id="Thmproposition2.p1.5.m5.1.1.3.cmml" xref="Thmproposition2.p1.5.m5.1.1.3"><ci id="Thmproposition2.p1.5.m5.1.1.3.1.cmml" xref="Thmproposition2.p1.5.m5.1.1.3.1">→</ci><ci id="Thmproposition2.p1.5.m5.1.1.3.2.cmml" xref="Thmproposition2.p1.5.m5.1.1.3.2">ℝ</ci><ci id="Thmproposition2.p1.5.m5.1.1.3.3.cmml" xref="Thmproposition2.p1.5.m5.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.5.m5.1c">\sigma:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.5.m5.1d">italic_σ : blackboard_R → caligraphic_R</annotation></semantics></math> is to play according to threshold strategy <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="Thmproposition2.p1.6.m6.1"><semantics id="Thmproposition2.p1.6.m6.1a"><mover accent="true" id="Thmproposition2.p1.6.m6.1.1" xref="Thmproposition2.p1.6.m6.1.1.cmml"><mi id="Thmproposition2.p1.6.m6.1.1.2" xref="Thmproposition2.p1.6.m6.1.1.2.cmml">τ</mi><mo id="Thmproposition2.p1.6.m6.1.1.1" xref="Thmproposition2.p1.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.6.m6.1b"><apply id="Thmproposition2.p1.6.m6.1.1.cmml" xref="Thmproposition2.p1.6.m6.1.1"><ci id="Thmproposition2.p1.6.m6.1.1.1.cmml" xref="Thmproposition2.p1.6.m6.1.1.1">^</ci><ci id="Thmproposition2.p1.6.m6.1.1.2.cmml" xref="Thmproposition2.p1.6.m6.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.6.m6.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.6.m6.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> satisfying <math alttext="P(\tau;\hat{\tau})=1/2" class="ltx_Math" display="inline" id="Thmproposition2.p1.7.m7.2"><semantics id="Thmproposition2.p1.7.m7.2a"><mrow id="Thmproposition2.p1.7.m7.2.3" xref="Thmproposition2.p1.7.m7.2.3.cmml"><mrow id="Thmproposition2.p1.7.m7.2.3.2" xref="Thmproposition2.p1.7.m7.2.3.2.cmml"><mi id="Thmproposition2.p1.7.m7.2.3.2.2" xref="Thmproposition2.p1.7.m7.2.3.2.2.cmml">P</mi><mo id="Thmproposition2.p1.7.m7.2.3.2.1" xref="Thmproposition2.p1.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="Thmproposition2.p1.7.m7.2.3.2.3.2" xref="Thmproposition2.p1.7.m7.2.3.2.3.1.cmml"><mo id="Thmproposition2.p1.7.m7.2.3.2.3.2.1" stretchy="false" xref="Thmproposition2.p1.7.m7.2.3.2.3.1.cmml">(</mo><mi id="Thmproposition2.p1.7.m7.1.1" xref="Thmproposition2.p1.7.m7.1.1.cmml">τ</mi><mo id="Thmproposition2.p1.7.m7.2.3.2.3.2.2" xref="Thmproposition2.p1.7.m7.2.3.2.3.1.cmml">;</mo><mover accent="true" id="Thmproposition2.p1.7.m7.2.2" xref="Thmproposition2.p1.7.m7.2.2.cmml"><mi id="Thmproposition2.p1.7.m7.2.2.2" xref="Thmproposition2.p1.7.m7.2.2.2.cmml">τ</mi><mo id="Thmproposition2.p1.7.m7.2.2.1" xref="Thmproposition2.p1.7.m7.2.2.1.cmml">^</mo></mover><mo id="Thmproposition2.p1.7.m7.2.3.2.3.2.3" stretchy="false" xref="Thmproposition2.p1.7.m7.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproposition2.p1.7.m7.2.3.1" xref="Thmproposition2.p1.7.m7.2.3.1.cmml">=</mo><mrow id="Thmproposition2.p1.7.m7.2.3.3" xref="Thmproposition2.p1.7.m7.2.3.3.cmml"><mn id="Thmproposition2.p1.7.m7.2.3.3.2" xref="Thmproposition2.p1.7.m7.2.3.3.2.cmml">1</mn><mo id="Thmproposition2.p1.7.m7.2.3.3.1" xref="Thmproposition2.p1.7.m7.2.3.3.1.cmml">/</mo><mn id="Thmproposition2.p1.7.m7.2.3.3.3" xref="Thmproposition2.p1.7.m7.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition2.p1.7.m7.2b"><apply id="Thmproposition2.p1.7.m7.2.3.cmml" xref="Thmproposition2.p1.7.m7.2.3"><eq id="Thmproposition2.p1.7.m7.2.3.1.cmml" xref="Thmproposition2.p1.7.m7.2.3.1"></eq><apply id="Thmproposition2.p1.7.m7.2.3.2.cmml" xref="Thmproposition2.p1.7.m7.2.3.2"><times id="Thmproposition2.p1.7.m7.2.3.2.1.cmml" xref="Thmproposition2.p1.7.m7.2.3.2.1"></times><ci id="Thmproposition2.p1.7.m7.2.3.2.2.cmml" xref="Thmproposition2.p1.7.m7.2.3.2.2">𝑃</ci><list id="Thmproposition2.p1.7.m7.2.3.2.3.1.cmml" xref="Thmproposition2.p1.7.m7.2.3.2.3.2"><ci id="Thmproposition2.p1.7.m7.1.1.cmml" xref="Thmproposition2.p1.7.m7.1.1">𝜏</ci><apply id="Thmproposition2.p1.7.m7.2.2.cmml" xref="Thmproposition2.p1.7.m7.2.2"><ci id="Thmproposition2.p1.7.m7.2.2.1.cmml" xref="Thmproposition2.p1.7.m7.2.2.1">^</ci><ci id="Thmproposition2.p1.7.m7.2.2.2.cmml" xref="Thmproposition2.p1.7.m7.2.2.2">𝜏</ci></apply></list></apply><apply id="Thmproposition2.p1.7.m7.2.3.3.cmml" xref="Thmproposition2.p1.7.m7.2.3.3"><divide id="Thmproposition2.p1.7.m7.2.3.3.1.cmml" xref="Thmproposition2.p1.7.m7.2.3.3.1"></divide><cn id="Thmproposition2.p1.7.m7.2.3.3.2.cmml" type="integer" xref="Thmproposition2.p1.7.m7.2.3.3.2">1</cn><cn id="Thmproposition2.p1.7.m7.2.3.3.3.cmml" type="integer" xref="Thmproposition2.p1.7.m7.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition2.p1.7.m7.2c">P(\tau;\hat{\tau})=1/2</annotation><annotation encoding="application/x-llamapun" id="Thmproposition2.p1.7.m7.2d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = 1 / 2</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S2.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS3.1.p1"> <p class="ltx_p" id="S2.SS3.1.p1.14">If <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.1.p1.1.m1.1"><semantics id="S2.SS3.1.p1.1.m1.1a"><mi id="S2.SS3.1.p1.1.m1.1.1" xref="S2.SS3.1.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.1.m1.1b"><ci id="S2.SS3.1.p1.1.m1.1.1.cmml" xref="S2.SS3.1.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.1.m1.1d">italic_τ</annotation></semantics></math> is the current threshold strategy that all other agents are following, an agent who receives signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.1.p1.2.m2.1"><semantics id="S2.SS3.1.p1.2.m2.1a"><mi id="S2.SS3.1.p1.2.m2.1.1" xref="S2.SS3.1.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.2.m2.1b"><ci id="S2.SS3.1.p1.2.m2.1.1.cmml" xref="S2.SS3.1.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.2.m2.1d">italic_x</annotation></semantics></math> will best respond with <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.1.p1.3.m3.1"><semantics id="S2.SS3.1.p1.3.m3.1a"><mi id="S2.SS3.1.p1.3.m3.1.1" xref="S2.SS3.1.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.3.m3.1b"><ci id="S2.SS3.1.p1.3.m3.1.1.cmml" xref="S2.SS3.1.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.3.m3.1d">italic_H</annotation></semantics></math> if <math alttext="\Pr[X\leq\tau\mid X=x]\leq 1/2" class="ltx_Math" display="inline" id="S2.SS3.1.p1.4.m4.2"><semantics id="S2.SS3.1.p1.4.m4.2a"><mrow id="S2.SS3.1.p1.4.m4.2.2" xref="S2.SS3.1.p1.4.m4.2.2.cmml"><mrow id="S2.SS3.1.p1.4.m4.2.2.1.1" xref="S2.SS3.1.p1.4.m4.2.2.1.2.cmml"><mi id="S2.SS3.1.p1.4.m4.1.1" xref="S2.SS3.1.p1.4.m4.1.1.cmml">Pr</mi><mo id="S2.SS3.1.p1.4.m4.2.2.1.1a" xref="S2.SS3.1.p1.4.m4.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS3.1.p1.4.m4.2.2.1.1.1" xref="S2.SS3.1.p1.4.m4.2.2.1.2.cmml"><mo id="S2.SS3.1.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S2.SS3.1.p1.4.m4.2.2.1.2.cmml">[</mo><mrow id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.2" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.2.cmml">X</mi><mo id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.3" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.3.cmml">≤</mo><mrow id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.cmml"><mi id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.2" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.2.cmml">τ</mi><mo id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.1" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.1.cmml">∣</mo><mi id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.3" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.5" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.5.cmml">=</mo><mi id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.6" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.6.cmml">x</mi></mrow><mo id="S2.SS3.1.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="S2.SS3.1.p1.4.m4.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S2.SS3.1.p1.4.m4.2.2.2" xref="S2.SS3.1.p1.4.m4.2.2.2.cmml">≤</mo><mrow id="S2.SS3.1.p1.4.m4.2.2.3" xref="S2.SS3.1.p1.4.m4.2.2.3.cmml"><mn id="S2.SS3.1.p1.4.m4.2.2.3.2" xref="S2.SS3.1.p1.4.m4.2.2.3.2.cmml">1</mn><mo id="S2.SS3.1.p1.4.m4.2.2.3.1" xref="S2.SS3.1.p1.4.m4.2.2.3.1.cmml">/</mo><mn id="S2.SS3.1.p1.4.m4.2.2.3.3" xref="S2.SS3.1.p1.4.m4.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.4.m4.2b"><apply id="S2.SS3.1.p1.4.m4.2.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2"><leq id="S2.SS3.1.p1.4.m4.2.2.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2.2"></leq><apply id="S2.SS3.1.p1.4.m4.2.2.1.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1"><ci id="S2.SS3.1.p1.4.m4.1.1.cmml" xref="S2.SS3.1.p1.4.m4.1.1">Pr</ci><apply id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1"><and id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1a.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1"></and><apply id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1b.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1"><leq id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.3"></leq><ci id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.2">𝑋</ci><apply id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4"><csymbol cd="latexml" id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.1.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.1">conditional</csymbol><ci id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.2.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.2">𝜏</ci><ci id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.3.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1c.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1"><eq id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.5.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS3.1.p1.4.m4.2.2.1.1.1.1.4.cmml" id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1d.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1"></share><ci id="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.6.cmml" xref="S2.SS3.1.p1.4.m4.2.2.1.1.1.1.6">𝑥</ci></apply></apply></apply><apply id="S2.SS3.1.p1.4.m4.2.2.3.cmml" xref="S2.SS3.1.p1.4.m4.2.2.3"><divide id="S2.SS3.1.p1.4.m4.2.2.3.1.cmml" xref="S2.SS3.1.p1.4.m4.2.2.3.1"></divide><cn id="S2.SS3.1.p1.4.m4.2.2.3.2.cmml" type="integer" xref="S2.SS3.1.p1.4.m4.2.2.3.2">1</cn><cn id="S2.SS3.1.p1.4.m4.2.2.3.3.cmml" type="integer" xref="S2.SS3.1.p1.4.m4.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.4.m4.2c">\Pr[X\leq\tau\mid X=x]\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.4.m4.2d">roman_Pr [ italic_X ≤ italic_τ ∣ italic_X = italic_x ] ≤ 1 / 2</annotation></semantics></math>, and <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.1.p1.5.m5.1"><semantics id="S2.SS3.1.p1.5.m5.1a"><mi id="S2.SS3.1.p1.5.m5.1.1" xref="S2.SS3.1.p1.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.5.m5.1b"><ci id="S2.SS3.1.p1.5.m5.1.1.cmml" xref="S2.SS3.1.p1.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.5.m5.1d">italic_L</annotation></semantics></math> otherwise. Since <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S2.SS3.1.p1.6.m6.2"><semantics id="S2.SS3.1.p1.6.m6.2a"><mrow id="S2.SS3.1.p1.6.m6.2.3" xref="S2.SS3.1.p1.6.m6.2.3.cmml"><mi id="S2.SS3.1.p1.6.m6.2.3.2" xref="S2.SS3.1.p1.6.m6.2.3.2.cmml">P</mi><mo id="S2.SS3.1.p1.6.m6.2.3.1" xref="S2.SS3.1.p1.6.m6.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.1.p1.6.m6.2.3.3.2" xref="S2.SS3.1.p1.6.m6.2.3.3.1.cmml"><mo id="S2.SS3.1.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S2.SS3.1.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S2.SS3.1.p1.6.m6.1.1" xref="S2.SS3.1.p1.6.m6.1.1.cmml">τ</mi><mo id="S2.SS3.1.p1.6.m6.2.3.3.2.2" xref="S2.SS3.1.p1.6.m6.2.3.3.1.cmml">;</mo><mi id="S2.SS3.1.p1.6.m6.2.2" xref="S2.SS3.1.p1.6.m6.2.2.cmml">x</mi><mo id="S2.SS3.1.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S2.SS3.1.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.6.m6.2b"><apply id="S2.SS3.1.p1.6.m6.2.3.cmml" xref="S2.SS3.1.p1.6.m6.2.3"><times id="S2.SS3.1.p1.6.m6.2.3.1.cmml" xref="S2.SS3.1.p1.6.m6.2.3.1"></times><ci id="S2.SS3.1.p1.6.m6.2.3.2.cmml" xref="S2.SS3.1.p1.6.m6.2.3.2">𝑃</ci><list id="S2.SS3.1.p1.6.m6.2.3.3.1.cmml" xref="S2.SS3.1.p1.6.m6.2.3.3.2"><ci id="S2.SS3.1.p1.6.m6.1.1.cmml" xref="S2.SS3.1.p1.6.m6.1.1">𝜏</ci><ci id="S2.SS3.1.p1.6.m6.2.2.cmml" xref="S2.SS3.1.p1.6.m6.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.6.m6.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.6.m6.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.1.p1.7.m7.1"><semantics id="S2.SS3.1.p1.7.m7.1a"><mi id="S2.SS3.1.p1.7.m7.1.1" xref="S2.SS3.1.p1.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.7.m7.1b"><ci id="S2.SS3.1.p1.7.m7.1.1.cmml" xref="S2.SS3.1.p1.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.7.m7.1d">italic_x</annotation></semantics></math>, there will be a unique point <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.8.m8.1"><semantics id="S2.SS3.1.p1.8.m8.1a"><mover accent="true" id="S2.SS3.1.p1.8.m8.1.1" xref="S2.SS3.1.p1.8.m8.1.1.cmml"><mi id="S2.SS3.1.p1.8.m8.1.1.2" xref="S2.SS3.1.p1.8.m8.1.1.2.cmml">τ</mi><mo id="S2.SS3.1.p1.8.m8.1.1.1" xref="S2.SS3.1.p1.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.8.m8.1b"><apply id="S2.SS3.1.p1.8.m8.1.1.cmml" xref="S2.SS3.1.p1.8.m8.1.1"><ci id="S2.SS3.1.p1.8.m8.1.1.1.cmml" xref="S2.SS3.1.p1.8.m8.1.1.1">^</ci><ci id="S2.SS3.1.p1.8.m8.1.1.2.cmml" xref="S2.SS3.1.p1.8.m8.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.8.m8.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.8.m8.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> such that <math alttext="P(\tau;\hat{\tau})=1/2" class="ltx_Math" display="inline" id="S2.SS3.1.p1.9.m9.2"><semantics id="S2.SS3.1.p1.9.m9.2a"><mrow id="S2.SS3.1.p1.9.m9.2.3" xref="S2.SS3.1.p1.9.m9.2.3.cmml"><mrow id="S2.SS3.1.p1.9.m9.2.3.2" xref="S2.SS3.1.p1.9.m9.2.3.2.cmml"><mi id="S2.SS3.1.p1.9.m9.2.3.2.2" xref="S2.SS3.1.p1.9.m9.2.3.2.2.cmml">P</mi><mo id="S2.SS3.1.p1.9.m9.2.3.2.1" xref="S2.SS3.1.p1.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.1.p1.9.m9.2.3.2.3.2" xref="S2.SS3.1.p1.9.m9.2.3.2.3.1.cmml"><mo id="S2.SS3.1.p1.9.m9.2.3.2.3.2.1" stretchy="false" xref="S2.SS3.1.p1.9.m9.2.3.2.3.1.cmml">(</mo><mi id="S2.SS3.1.p1.9.m9.1.1" xref="S2.SS3.1.p1.9.m9.1.1.cmml">τ</mi><mo id="S2.SS3.1.p1.9.m9.2.3.2.3.2.2" xref="S2.SS3.1.p1.9.m9.2.3.2.3.1.cmml">;</mo><mover accent="true" id="S2.SS3.1.p1.9.m9.2.2" xref="S2.SS3.1.p1.9.m9.2.2.cmml"><mi id="S2.SS3.1.p1.9.m9.2.2.2" xref="S2.SS3.1.p1.9.m9.2.2.2.cmml">τ</mi><mo id="S2.SS3.1.p1.9.m9.2.2.1" xref="S2.SS3.1.p1.9.m9.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.1.p1.9.m9.2.3.2.3.2.3" stretchy="false" xref="S2.SS3.1.p1.9.m9.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.1.p1.9.m9.2.3.1" xref="S2.SS3.1.p1.9.m9.2.3.1.cmml">=</mo><mrow id="S2.SS3.1.p1.9.m9.2.3.3" xref="S2.SS3.1.p1.9.m9.2.3.3.cmml"><mn id="S2.SS3.1.p1.9.m9.2.3.3.2" xref="S2.SS3.1.p1.9.m9.2.3.3.2.cmml">1</mn><mo id="S2.SS3.1.p1.9.m9.2.3.3.1" xref="S2.SS3.1.p1.9.m9.2.3.3.1.cmml">/</mo><mn id="S2.SS3.1.p1.9.m9.2.3.3.3" xref="S2.SS3.1.p1.9.m9.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.9.m9.2b"><apply id="S2.SS3.1.p1.9.m9.2.3.cmml" xref="S2.SS3.1.p1.9.m9.2.3"><eq id="S2.SS3.1.p1.9.m9.2.3.1.cmml" xref="S2.SS3.1.p1.9.m9.2.3.1"></eq><apply id="S2.SS3.1.p1.9.m9.2.3.2.cmml" xref="S2.SS3.1.p1.9.m9.2.3.2"><times id="S2.SS3.1.p1.9.m9.2.3.2.1.cmml" xref="S2.SS3.1.p1.9.m9.2.3.2.1"></times><ci id="S2.SS3.1.p1.9.m9.2.3.2.2.cmml" xref="S2.SS3.1.p1.9.m9.2.3.2.2">𝑃</ci><list id="S2.SS3.1.p1.9.m9.2.3.2.3.1.cmml" xref="S2.SS3.1.p1.9.m9.2.3.2.3.2"><ci id="S2.SS3.1.p1.9.m9.1.1.cmml" xref="S2.SS3.1.p1.9.m9.1.1">𝜏</ci><apply id="S2.SS3.1.p1.9.m9.2.2.cmml" xref="S2.SS3.1.p1.9.m9.2.2"><ci id="S2.SS3.1.p1.9.m9.2.2.1.cmml" xref="S2.SS3.1.p1.9.m9.2.2.1">^</ci><ci id="S2.SS3.1.p1.9.m9.2.2.2.cmml" xref="S2.SS3.1.p1.9.m9.2.2.2">𝜏</ci></apply></list></apply><apply id="S2.SS3.1.p1.9.m9.2.3.3.cmml" xref="S2.SS3.1.p1.9.m9.2.3.3"><divide id="S2.SS3.1.p1.9.m9.2.3.3.1.cmml" xref="S2.SS3.1.p1.9.m9.2.3.3.1"></divide><cn id="S2.SS3.1.p1.9.m9.2.3.3.2.cmml" type="integer" xref="S2.SS3.1.p1.9.m9.2.3.3.2">1</cn><cn id="S2.SS3.1.p1.9.m9.2.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.9.m9.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.9.m9.2c">P(\tau;\hat{\tau})=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.9.m9.2d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = 1 / 2</annotation></semantics></math>, with <math alttext="P(\tau;x)\geq 1/2" class="ltx_Math" display="inline" id="S2.SS3.1.p1.10.m10.2"><semantics id="S2.SS3.1.p1.10.m10.2a"><mrow id="S2.SS3.1.p1.10.m10.2.3" xref="S2.SS3.1.p1.10.m10.2.3.cmml"><mrow id="S2.SS3.1.p1.10.m10.2.3.2" xref="S2.SS3.1.p1.10.m10.2.3.2.cmml"><mi id="S2.SS3.1.p1.10.m10.2.3.2.2" xref="S2.SS3.1.p1.10.m10.2.3.2.2.cmml">P</mi><mo id="S2.SS3.1.p1.10.m10.2.3.2.1" xref="S2.SS3.1.p1.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.1.p1.10.m10.2.3.2.3.2" xref="S2.SS3.1.p1.10.m10.2.3.2.3.1.cmml"><mo id="S2.SS3.1.p1.10.m10.2.3.2.3.2.1" stretchy="false" xref="S2.SS3.1.p1.10.m10.2.3.2.3.1.cmml">(</mo><mi id="S2.SS3.1.p1.10.m10.1.1" xref="S2.SS3.1.p1.10.m10.1.1.cmml">τ</mi><mo id="S2.SS3.1.p1.10.m10.2.3.2.3.2.2" xref="S2.SS3.1.p1.10.m10.2.3.2.3.1.cmml">;</mo><mi id="S2.SS3.1.p1.10.m10.2.2" xref="S2.SS3.1.p1.10.m10.2.2.cmml">x</mi><mo id="S2.SS3.1.p1.10.m10.2.3.2.3.2.3" stretchy="false" xref="S2.SS3.1.p1.10.m10.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.1.p1.10.m10.2.3.1" xref="S2.SS3.1.p1.10.m10.2.3.1.cmml">≥</mo><mrow id="S2.SS3.1.p1.10.m10.2.3.3" xref="S2.SS3.1.p1.10.m10.2.3.3.cmml"><mn id="S2.SS3.1.p1.10.m10.2.3.3.2" xref="S2.SS3.1.p1.10.m10.2.3.3.2.cmml">1</mn><mo id="S2.SS3.1.p1.10.m10.2.3.3.1" xref="S2.SS3.1.p1.10.m10.2.3.3.1.cmml">/</mo><mn id="S2.SS3.1.p1.10.m10.2.3.3.3" xref="S2.SS3.1.p1.10.m10.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.10.m10.2b"><apply id="S2.SS3.1.p1.10.m10.2.3.cmml" xref="S2.SS3.1.p1.10.m10.2.3"><geq id="S2.SS3.1.p1.10.m10.2.3.1.cmml" xref="S2.SS3.1.p1.10.m10.2.3.1"></geq><apply id="S2.SS3.1.p1.10.m10.2.3.2.cmml" xref="S2.SS3.1.p1.10.m10.2.3.2"><times id="S2.SS3.1.p1.10.m10.2.3.2.1.cmml" xref="S2.SS3.1.p1.10.m10.2.3.2.1"></times><ci id="S2.SS3.1.p1.10.m10.2.3.2.2.cmml" xref="S2.SS3.1.p1.10.m10.2.3.2.2">𝑃</ci><list id="S2.SS3.1.p1.10.m10.2.3.2.3.1.cmml" xref="S2.SS3.1.p1.10.m10.2.3.2.3.2"><ci id="S2.SS3.1.p1.10.m10.1.1.cmml" xref="S2.SS3.1.p1.10.m10.1.1">𝜏</ci><ci id="S2.SS3.1.p1.10.m10.2.2.cmml" xref="S2.SS3.1.p1.10.m10.2.2">𝑥</ci></list></apply><apply id="S2.SS3.1.p1.10.m10.2.3.3.cmml" xref="S2.SS3.1.p1.10.m10.2.3.3"><divide id="S2.SS3.1.p1.10.m10.2.3.3.1.cmml" xref="S2.SS3.1.p1.10.m10.2.3.3.1"></divide><cn id="S2.SS3.1.p1.10.m10.2.3.3.2.cmml" type="integer" xref="S2.SS3.1.p1.10.m10.2.3.3.2">1</cn><cn id="S2.SS3.1.p1.10.m10.2.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.10.m10.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.10.m10.2c">P(\tau;x)\geq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.10.m10.2d">italic_P ( italic_τ ; italic_x ) ≥ 1 / 2</annotation></semantics></math> for <math alttext="x\leq\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.11.m11.1"><semantics id="S2.SS3.1.p1.11.m11.1a"><mrow id="S2.SS3.1.p1.11.m11.1.1" xref="S2.SS3.1.p1.11.m11.1.1.cmml"><mi id="S2.SS3.1.p1.11.m11.1.1.2" xref="S2.SS3.1.p1.11.m11.1.1.2.cmml">x</mi><mo id="S2.SS3.1.p1.11.m11.1.1.1" xref="S2.SS3.1.p1.11.m11.1.1.1.cmml">≤</mo><mover accent="true" id="S2.SS3.1.p1.11.m11.1.1.3" xref="S2.SS3.1.p1.11.m11.1.1.3.cmml"><mi id="S2.SS3.1.p1.11.m11.1.1.3.2" xref="S2.SS3.1.p1.11.m11.1.1.3.2.cmml">τ</mi><mo id="S2.SS3.1.p1.11.m11.1.1.3.1" xref="S2.SS3.1.p1.11.m11.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.11.m11.1b"><apply id="S2.SS3.1.p1.11.m11.1.1.cmml" xref="S2.SS3.1.p1.11.m11.1.1"><leq id="S2.SS3.1.p1.11.m11.1.1.1.cmml" xref="S2.SS3.1.p1.11.m11.1.1.1"></leq><ci id="S2.SS3.1.p1.11.m11.1.1.2.cmml" xref="S2.SS3.1.p1.11.m11.1.1.2">𝑥</ci><apply id="S2.SS3.1.p1.11.m11.1.1.3.cmml" xref="S2.SS3.1.p1.11.m11.1.1.3"><ci id="S2.SS3.1.p1.11.m11.1.1.3.1.cmml" xref="S2.SS3.1.p1.11.m11.1.1.3.1">^</ci><ci id="S2.SS3.1.p1.11.m11.1.1.3.2.cmml" xref="S2.SS3.1.p1.11.m11.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.11.m11.1c">x\leq\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.11.m11.1d">italic_x ≤ over^ start_ARG italic_τ end_ARG</annotation></semantics></math> and <math alttext="P(\tau;x)\leq 1/2" class="ltx_Math" display="inline" id="S2.SS3.1.p1.12.m12.2"><semantics id="S2.SS3.1.p1.12.m12.2a"><mrow id="S2.SS3.1.p1.12.m12.2.3" xref="S2.SS3.1.p1.12.m12.2.3.cmml"><mrow id="S2.SS3.1.p1.12.m12.2.3.2" xref="S2.SS3.1.p1.12.m12.2.3.2.cmml"><mi id="S2.SS3.1.p1.12.m12.2.3.2.2" xref="S2.SS3.1.p1.12.m12.2.3.2.2.cmml">P</mi><mo id="S2.SS3.1.p1.12.m12.2.3.2.1" xref="S2.SS3.1.p1.12.m12.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.1.p1.12.m12.2.3.2.3.2" xref="S2.SS3.1.p1.12.m12.2.3.2.3.1.cmml"><mo id="S2.SS3.1.p1.12.m12.2.3.2.3.2.1" stretchy="false" xref="S2.SS3.1.p1.12.m12.2.3.2.3.1.cmml">(</mo><mi id="S2.SS3.1.p1.12.m12.1.1" xref="S2.SS3.1.p1.12.m12.1.1.cmml">τ</mi><mo id="S2.SS3.1.p1.12.m12.2.3.2.3.2.2" xref="S2.SS3.1.p1.12.m12.2.3.2.3.1.cmml">;</mo><mi id="S2.SS3.1.p1.12.m12.2.2" xref="S2.SS3.1.p1.12.m12.2.2.cmml">x</mi><mo id="S2.SS3.1.p1.12.m12.2.3.2.3.2.3" stretchy="false" xref="S2.SS3.1.p1.12.m12.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.1.p1.12.m12.2.3.1" xref="S2.SS3.1.p1.12.m12.2.3.1.cmml">≤</mo><mrow id="S2.SS3.1.p1.12.m12.2.3.3" xref="S2.SS3.1.p1.12.m12.2.3.3.cmml"><mn id="S2.SS3.1.p1.12.m12.2.3.3.2" xref="S2.SS3.1.p1.12.m12.2.3.3.2.cmml">1</mn><mo id="S2.SS3.1.p1.12.m12.2.3.3.1" xref="S2.SS3.1.p1.12.m12.2.3.3.1.cmml">/</mo><mn id="S2.SS3.1.p1.12.m12.2.3.3.3" xref="S2.SS3.1.p1.12.m12.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.12.m12.2b"><apply id="S2.SS3.1.p1.12.m12.2.3.cmml" xref="S2.SS3.1.p1.12.m12.2.3"><leq id="S2.SS3.1.p1.12.m12.2.3.1.cmml" xref="S2.SS3.1.p1.12.m12.2.3.1"></leq><apply id="S2.SS3.1.p1.12.m12.2.3.2.cmml" xref="S2.SS3.1.p1.12.m12.2.3.2"><times id="S2.SS3.1.p1.12.m12.2.3.2.1.cmml" xref="S2.SS3.1.p1.12.m12.2.3.2.1"></times><ci id="S2.SS3.1.p1.12.m12.2.3.2.2.cmml" xref="S2.SS3.1.p1.12.m12.2.3.2.2">𝑃</ci><list id="S2.SS3.1.p1.12.m12.2.3.2.3.1.cmml" xref="S2.SS3.1.p1.12.m12.2.3.2.3.2"><ci id="S2.SS3.1.p1.12.m12.1.1.cmml" xref="S2.SS3.1.p1.12.m12.1.1">𝜏</ci><ci id="S2.SS3.1.p1.12.m12.2.2.cmml" xref="S2.SS3.1.p1.12.m12.2.2">𝑥</ci></list></apply><apply id="S2.SS3.1.p1.12.m12.2.3.3.cmml" xref="S2.SS3.1.p1.12.m12.2.3.3"><divide id="S2.SS3.1.p1.12.m12.2.3.3.1.cmml" xref="S2.SS3.1.p1.12.m12.2.3.3.1"></divide><cn id="S2.SS3.1.p1.12.m12.2.3.3.2.cmml" type="integer" xref="S2.SS3.1.p1.12.m12.2.3.3.2">1</cn><cn id="S2.SS3.1.p1.12.m12.2.3.3.3.cmml" type="integer" xref="S2.SS3.1.p1.12.m12.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.12.m12.2c">P(\tau;x)\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.12.m12.2d">italic_P ( italic_τ ; italic_x ) ≤ 1 / 2</annotation></semantics></math> for <math alttext="x>\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.13.m13.1"><semantics id="S2.SS3.1.p1.13.m13.1a"><mrow id="S2.SS3.1.p1.13.m13.1.1" xref="S2.SS3.1.p1.13.m13.1.1.cmml"><mi id="S2.SS3.1.p1.13.m13.1.1.2" xref="S2.SS3.1.p1.13.m13.1.1.2.cmml">x</mi><mo id="S2.SS3.1.p1.13.m13.1.1.1" xref="S2.SS3.1.p1.13.m13.1.1.1.cmml">></mo><mover accent="true" id="S2.SS3.1.p1.13.m13.1.1.3" xref="S2.SS3.1.p1.13.m13.1.1.3.cmml"><mi id="S2.SS3.1.p1.13.m13.1.1.3.2" xref="S2.SS3.1.p1.13.m13.1.1.3.2.cmml">τ</mi><mo id="S2.SS3.1.p1.13.m13.1.1.3.1" xref="S2.SS3.1.p1.13.m13.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.13.m13.1b"><apply id="S2.SS3.1.p1.13.m13.1.1.cmml" xref="S2.SS3.1.p1.13.m13.1.1"><gt id="S2.SS3.1.p1.13.m13.1.1.1.cmml" xref="S2.SS3.1.p1.13.m13.1.1.1"></gt><ci id="S2.SS3.1.p1.13.m13.1.1.2.cmml" xref="S2.SS3.1.p1.13.m13.1.1.2">𝑥</ci><apply id="S2.SS3.1.p1.13.m13.1.1.3.cmml" xref="S2.SS3.1.p1.13.m13.1.1.3"><ci id="S2.SS3.1.p1.13.m13.1.1.3.1.cmml" xref="S2.SS3.1.p1.13.m13.1.1.3.1">^</ci><ci id="S2.SS3.1.p1.13.m13.1.1.3.2.cmml" xref="S2.SS3.1.p1.13.m13.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.13.m13.1c">x>\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.13.m13.1d">italic_x > over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. This threshold <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.1.p1.14.m14.1"><semantics id="S2.SS3.1.p1.14.m14.1a"><mover accent="true" id="S2.SS3.1.p1.14.m14.1.1" xref="S2.SS3.1.p1.14.m14.1.1.cmml"><mi id="S2.SS3.1.p1.14.m14.1.1.2" xref="S2.SS3.1.p1.14.m14.1.1.2.cmml">τ</mi><mo id="S2.SS3.1.p1.14.m14.1.1.1" xref="S2.SS3.1.p1.14.m14.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS3.1.p1.14.m14.1b"><apply id="S2.SS3.1.p1.14.m14.1.1.cmml" xref="S2.SS3.1.p1.14.m14.1.1"><ci id="S2.SS3.1.p1.14.m14.1.1.1.cmml" xref="S2.SS3.1.p1.14.m14.1.1.1">^</ci><ci id="S2.SS3.1.p1.14.m14.1.1.2.cmml" xref="S2.SS3.1.p1.14.m14.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.1.p1.14.m14.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.1.p1.14.m14.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> thus corresponds to the best response strategy. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS3.p5"> <p class="ltx_p" id="S2.SS3.p5.1">We can now use concepts from dynamical systems and characterize equilibria as <span class="ltx_text ltx_font_italic" id="S2.SS3.p5.1.1">stable</span> or <span class="ltx_text ltx_font_italic" id="S2.SS3.p5.1.2">unstable</span> according to properties of the function <math alttext="G" class="ltx_Math" display="inline" id="S2.SS3.p5.1.m1.1"><semantics id="S2.SS3.p5.1.m1.1a"><mi id="S2.SS3.p5.1.m1.1.1" xref="S2.SS3.p5.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.1.m1.1b"><ci id="S2.SS3.p5.1.m1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.1.m1.1d">italic_G</annotation></semantics></math>. This distinction captures the behavior of the dynamics near the equilibrium: do they drive the system toward the equilibrium (stable) or away from it (unstable)? More precisely, an equilibrium is (locally) stable if a sufficiently small perturbation yields dynamics that always return to it while it is unstable if all perturbations diverge from it.</p> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.2">We argue that only stable equilibria are reasonable to find in practice. Otherwise any small error in the choice of <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S2.SS3.p6.1.m1.1"><semantics id="S2.SS3.p6.1.m1.1a"><msup id="S2.SS3.p6.1.m1.1.1" xref="S2.SS3.p6.1.m1.1.1.cmml"><mi id="S2.SS3.p6.1.m1.1.1.2" xref="S2.SS3.p6.1.m1.1.1.2.cmml">τ</mi><mo id="S2.SS3.p6.1.m1.1.1.3" xref="S2.SS3.p6.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.1.m1.1b"><apply id="S2.SS3.p6.1.m1.1.1.cmml" xref="S2.SS3.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p6.1.m1.1.1.1.cmml" xref="S2.SS3.p6.1.m1.1.1">superscript</csymbol><ci id="S2.SS3.p6.1.m1.1.1.2.cmml" xref="S2.SS3.p6.1.m1.1.1.2">𝜏</ci><times id="S2.SS3.p6.1.m1.1.1.3.cmml" xref="S2.SS3.p6.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.1.m1.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> will fail to yield the desired equilibrium as the dynamics push away from it. Since the equilibria depend on the joint distribution <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S2.SS3.p6.2.m2.1"><semantics id="S2.SS3.p6.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p6.2.m2.1.1" xref="S2.SS3.p6.2.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p6.2.m2.1b"><ci id="S2.SS3.p6.2.m2.1.1.cmml" xref="S2.SS3.p6.2.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p6.2.m2.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p6.2.m2.1d">caligraphic_D</annotation></semantics></math>, which will typically not be perfectly known to the mechanism designer, such small errors are to be expected. As a result, we view stability as an important equilibrium refinement in the context of peer prediction.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem2.1.1.1">Theorem 2</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem2.p1"> <p class="ltx_p" id="Thmtheorem2.p1.7">Assume for all <math alttext="\tau\in\mathbb{R}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.1.m1.1"><semantics id="Thmtheorem2.p1.1.m1.1a"><mrow id="Thmtheorem2.p1.1.m1.1.1" xref="Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="Thmtheorem2.p1.1.m1.1.1.2" xref="Thmtheorem2.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmtheorem2.p1.1.m1.1.1.1" xref="Thmtheorem2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem2.p1.1.m1.1.1.3" xref="Thmtheorem2.p1.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.1.m1.1b"><apply id="Thmtheorem2.p1.1.m1.1.1.cmml" xref="Thmtheorem2.p1.1.m1.1.1"><in id="Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="Thmtheorem2.p1.1.m1.1.1.1"></in><ci id="Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="Thmtheorem2.p1.1.m1.1.1.2">𝜏</ci><ci id="Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="Thmtheorem2.p1.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.1.m1.1c">\tau\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.1.m1.1d">italic_τ ∈ blackboard_R</annotation></semantics></math> that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem2.p1.2.m2.2"><semantics id="Thmtheorem2.p1.2.m2.2a"><mrow id="Thmtheorem2.p1.2.m2.2.3" xref="Thmtheorem2.p1.2.m2.2.3.cmml"><mi id="Thmtheorem2.p1.2.m2.2.3.2" xref="Thmtheorem2.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmtheorem2.p1.2.m2.2.3.1" xref="Thmtheorem2.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem2.p1.2.m2.2.3.3.2" xref="Thmtheorem2.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem2.p1.2.m2.1.1" xref="Thmtheorem2.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmtheorem2.p1.2.m2.2.3.3.2.2" xref="Thmtheorem2.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmtheorem2.p1.2.m2.2.2" xref="Thmtheorem2.p1.2.m2.2.2.cmml">x</mi><mo id="Thmtheorem2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.2.m2.2b"><apply id="Thmtheorem2.p1.2.m2.2.3.cmml" xref="Thmtheorem2.p1.2.m2.2.3"><times id="Thmtheorem2.p1.2.m2.2.3.1.cmml" xref="Thmtheorem2.p1.2.m2.2.3.1"></times><ci id="Thmtheorem2.p1.2.m2.2.3.2.cmml" xref="Thmtheorem2.p1.2.m2.2.3.2">𝑃</ci><list id="Thmtheorem2.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem2.p1.2.m2.2.3.3.2"><ci id="Thmtheorem2.p1.2.m2.1.1.cmml" xref="Thmtheorem2.p1.2.m2.1.1">𝜏</ci><ci id="Thmtheorem2.p1.2.m2.2.2.cmml" xref="Thmtheorem2.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem2.p1.3.m3.1"><semantics id="Thmtheorem2.p1.3.m3.1a"><mi id="Thmtheorem2.p1.3.m3.1.1" xref="Thmtheorem2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.3.m3.1b"><ci id="Thmtheorem2.p1.3.m3.1.1.cmml" xref="Thmtheorem2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.3.m3.1d">italic_x</annotation></semantics></math>. Then if <math alttext="G(\tau)" class="ltx_Math" display="inline" id="Thmtheorem2.p1.4.m4.1"><semantics id="Thmtheorem2.p1.4.m4.1a"><mrow id="Thmtheorem2.p1.4.m4.1.2" xref="Thmtheorem2.p1.4.m4.1.2.cmml"><mi id="Thmtheorem2.p1.4.m4.1.2.2" xref="Thmtheorem2.p1.4.m4.1.2.2.cmml">G</mi><mo id="Thmtheorem2.p1.4.m4.1.2.1" xref="Thmtheorem2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="Thmtheorem2.p1.4.m4.1.2.3.2" xref="Thmtheorem2.p1.4.m4.1.2.cmml"><mo id="Thmtheorem2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="Thmtheorem2.p1.4.m4.1.2.cmml">(</mo><mi id="Thmtheorem2.p1.4.m4.1.1" xref="Thmtheorem2.p1.4.m4.1.1.cmml">τ</mi><mo id="Thmtheorem2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="Thmtheorem2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.4.m4.1b"><apply id="Thmtheorem2.p1.4.m4.1.2.cmml" xref="Thmtheorem2.p1.4.m4.1.2"><times id="Thmtheorem2.p1.4.m4.1.2.1.cmml" xref="Thmtheorem2.p1.4.m4.1.2.1"></times><ci id="Thmtheorem2.p1.4.m4.1.2.2.cmml" xref="Thmtheorem2.p1.4.m4.1.2.2">𝐺</ci><ci id="Thmtheorem2.p1.4.m4.1.1.cmml" xref="Thmtheorem2.p1.4.m4.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.4.m4.1c">G(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.4.m4.1d">italic_G ( italic_τ )</annotation></semantics></math> is strictly increasing at equilibrium point <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.5.m5.1"><semantics id="Thmtheorem2.p1.5.m5.1a"><msup id="Thmtheorem2.p1.5.m5.1.1" xref="Thmtheorem2.p1.5.m5.1.1.cmml"><mi id="Thmtheorem2.p1.5.m5.1.1.2" xref="Thmtheorem2.p1.5.m5.1.1.2.cmml">τ</mi><mo id="Thmtheorem2.p1.5.m5.1.1.3" xref="Thmtheorem2.p1.5.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.5.m5.1b"><apply id="Thmtheorem2.p1.5.m5.1.1.cmml" xref="Thmtheorem2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="Thmtheorem2.p1.5.m5.1.1">superscript</csymbol><ci id="Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="Thmtheorem2.p1.5.m5.1.1.2">𝜏</ci><times id="Thmtheorem2.p1.5.m5.1.1.3.cmml" xref="Thmtheorem2.p1.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.5.m5.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.5.m5.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.6.m6.1"><semantics id="Thmtheorem2.p1.6.m6.1a"><msup id="Thmtheorem2.p1.6.m6.1.1" xref="Thmtheorem2.p1.6.m6.1.1.cmml"><mi id="Thmtheorem2.p1.6.m6.1.1.2" xref="Thmtheorem2.p1.6.m6.1.1.2.cmml">τ</mi><mo id="Thmtheorem2.p1.6.m6.1.1.3" xref="Thmtheorem2.p1.6.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.6.m6.1b"><apply id="Thmtheorem2.p1.6.m6.1.1.cmml" xref="Thmtheorem2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="Thmtheorem2.p1.6.m6.1.1.1.cmml" xref="Thmtheorem2.p1.6.m6.1.1">superscript</csymbol><ci id="Thmtheorem2.p1.6.m6.1.1.2.cmml" xref="Thmtheorem2.p1.6.m6.1.1.2">𝜏</ci><times id="Thmtheorem2.p1.6.m6.1.1.3.cmml" xref="Thmtheorem2.p1.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.6.m6.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.6.m6.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is unstable, while if it is instead strictly decreasing <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem2.p1.7.m7.1"><semantics id="Thmtheorem2.p1.7.m7.1a"><msup id="Thmtheorem2.p1.7.m7.1.1" xref="Thmtheorem2.p1.7.m7.1.1.cmml"><mi id="Thmtheorem2.p1.7.m7.1.1.2" xref="Thmtheorem2.p1.7.m7.1.1.2.cmml">τ</mi><mo id="Thmtheorem2.p1.7.m7.1.1.3" xref="Thmtheorem2.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem2.p1.7.m7.1b"><apply id="Thmtheorem2.p1.7.m7.1.1.cmml" xref="Thmtheorem2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmtheorem2.p1.7.m7.1.1.1.cmml" xref="Thmtheorem2.p1.7.m7.1.1">superscript</csymbol><ci id="Thmtheorem2.p1.7.m7.1.1.2.cmml" xref="Thmtheorem2.p1.7.m7.1.1.2">𝜏</ci><times id="Thmtheorem2.p1.7.m7.1.1.3.cmml" xref="Thmtheorem2.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem2.p1.7.m7.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem2.p1.7.m7.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is stable.</p> </div> </div> <div class="ltx_proof" id="S2.SS3.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS3.2.p1"> <p class="ltx_p" id="S2.SS3.2.p1.5">Consider the dynamics at time step <math alttext="t" class="ltx_Math" display="inline" id="S2.SS3.2.p1.1.m1.1"><semantics id="S2.SS3.2.p1.1.m1.1a"><mi id="S2.SS3.2.p1.1.m1.1.1" xref="S2.SS3.2.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.1.m1.1b"><ci id="S2.SS3.2.p1.1.m1.1.1.cmml" xref="S2.SS3.2.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.1.m1.1d">italic_t</annotation></semantics></math>, with the current threshold <math alttext="\tau(t)" class="ltx_Math" display="inline" id="S2.SS3.2.p1.2.m2.1"><semantics id="S2.SS3.2.p1.2.m2.1a"><mrow id="S2.SS3.2.p1.2.m2.1.2" xref="S2.SS3.2.p1.2.m2.1.2.cmml"><mi id="S2.SS3.2.p1.2.m2.1.2.2" xref="S2.SS3.2.p1.2.m2.1.2.2.cmml">τ</mi><mo id="S2.SS3.2.p1.2.m2.1.2.1" xref="S2.SS3.2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.2.p1.2.m2.1.2.3.2" xref="S2.SS3.2.p1.2.m2.1.2.cmml"><mo id="S2.SS3.2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.2.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.2.p1.2.m2.1.1" xref="S2.SS3.2.p1.2.m2.1.1.cmml">t</mi><mo id="S2.SS3.2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.2.m2.1b"><apply id="S2.SS3.2.p1.2.m2.1.2.cmml" xref="S2.SS3.2.p1.2.m2.1.2"><times id="S2.SS3.2.p1.2.m2.1.2.1.cmml" xref="S2.SS3.2.p1.2.m2.1.2.1"></times><ci id="S2.SS3.2.p1.2.m2.1.2.2.cmml" xref="S2.SS3.2.p1.2.m2.1.2.2">𝜏</ci><ci id="S2.SS3.2.p1.2.m2.1.1.cmml" xref="S2.SS3.2.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.2.m2.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.2.m2.1d">italic_τ ( italic_t )</annotation></semantics></math>. Since we are considering a fixed time <math alttext="t" class="ltx_Math" display="inline" id="S2.SS3.2.p1.3.m3.1"><semantics id="S2.SS3.2.p1.3.m3.1a"><mi id="S2.SS3.2.p1.3.m3.1.1" xref="S2.SS3.2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.3.m3.1b"><ci id="S2.SS3.2.p1.3.m3.1.1.cmml" xref="S2.SS3.2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.3.m3.1d">italic_t</annotation></semantics></math>, we refer to <math alttext="\tau(t)" class="ltx_Math" display="inline" id="S2.SS3.2.p1.4.m4.1"><semantics id="S2.SS3.2.p1.4.m4.1a"><mrow id="S2.SS3.2.p1.4.m4.1.2" xref="S2.SS3.2.p1.4.m4.1.2.cmml"><mi id="S2.SS3.2.p1.4.m4.1.2.2" xref="S2.SS3.2.p1.4.m4.1.2.2.cmml">τ</mi><mo id="S2.SS3.2.p1.4.m4.1.2.1" xref="S2.SS3.2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.2.p1.4.m4.1.2.3.2" xref="S2.SS3.2.p1.4.m4.1.2.cmml"><mo id="S2.SS3.2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS3.2.p1.4.m4.1.2.cmml">(</mo><mi id="S2.SS3.2.p1.4.m4.1.1" xref="S2.SS3.2.p1.4.m4.1.1.cmml">t</mi><mo id="S2.SS3.2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS3.2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.4.m4.1b"><apply id="S2.SS3.2.p1.4.m4.1.2.cmml" xref="S2.SS3.2.p1.4.m4.1.2"><times id="S2.SS3.2.p1.4.m4.1.2.1.cmml" xref="S2.SS3.2.p1.4.m4.1.2.1"></times><ci id="S2.SS3.2.p1.4.m4.1.2.2.cmml" xref="S2.SS3.2.p1.4.m4.1.2.2">𝜏</ci><ci id="S2.SS3.2.p1.4.m4.1.1.cmml" xref="S2.SS3.2.p1.4.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.4.m4.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.4.m4.1d">italic_τ ( italic_t )</annotation></semantics></math> as <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.2.p1.5.m5.1"><semantics id="S2.SS3.2.p1.5.m5.1a"><mi id="S2.SS3.2.p1.5.m5.1.1" xref="S2.SS3.2.p1.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.2.p1.5.m5.1b"><ci id="S2.SS3.2.p1.5.m5.1.1.cmml" xref="S2.SS3.2.p1.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.2.p1.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.2.p1.5.m5.1d">italic_τ</annotation></semantics></math> throughout the proof.</p> </div> <div class="ltx_para" id="S2.SS3.3.p2"> <p class="ltx_p" id="S2.SS3.3.p2.11">We consider what happens when an agent receives a signal exactly at <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.3.p2.1.m1.1"><semantics id="S2.SS3.3.p2.1.m1.1a"><mi id="S2.SS3.3.p2.1.m1.1.1" xref="S2.SS3.3.p2.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.1.m1.1b"><ci id="S2.SS3.3.p2.1.m1.1.1.cmml" xref="S2.SS3.3.p2.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.1.m1.1d">italic_τ</annotation></semantics></math>. There are three cases to consider. In the first, <math alttext="G(\tau)=1/2" class="ltx_Math" display="inline" id="S2.SS3.3.p2.2.m2.1"><semantics id="S2.SS3.3.p2.2.m2.1a"><mrow id="S2.SS3.3.p2.2.m2.1.2" xref="S2.SS3.3.p2.2.m2.1.2.cmml"><mrow id="S2.SS3.3.p2.2.m2.1.2.2" xref="S2.SS3.3.p2.2.m2.1.2.2.cmml"><mi id="S2.SS3.3.p2.2.m2.1.2.2.2" xref="S2.SS3.3.p2.2.m2.1.2.2.2.cmml">G</mi><mo id="S2.SS3.3.p2.2.m2.1.2.2.1" xref="S2.SS3.3.p2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.3.p2.2.m2.1.2.2.3.2" xref="S2.SS3.3.p2.2.m2.1.2.2.cmml"><mo id="S2.SS3.3.p2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.3.p2.2.m2.1.2.2.cmml">(</mo><mi id="S2.SS3.3.p2.2.m2.1.1" xref="S2.SS3.3.p2.2.m2.1.1.cmml">τ</mi><mo id="S2.SS3.3.p2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.3.p2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.3.p2.2.m2.1.2.1" xref="S2.SS3.3.p2.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS3.3.p2.2.m2.1.2.3" xref="S2.SS3.3.p2.2.m2.1.2.3.cmml"><mn id="S2.SS3.3.p2.2.m2.1.2.3.2" xref="S2.SS3.3.p2.2.m2.1.2.3.2.cmml">1</mn><mo id="S2.SS3.3.p2.2.m2.1.2.3.1" xref="S2.SS3.3.p2.2.m2.1.2.3.1.cmml">/</mo><mn id="S2.SS3.3.p2.2.m2.1.2.3.3" xref="S2.SS3.3.p2.2.m2.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.2.m2.1b"><apply id="S2.SS3.3.p2.2.m2.1.2.cmml" xref="S2.SS3.3.p2.2.m2.1.2"><eq id="S2.SS3.3.p2.2.m2.1.2.1.cmml" xref="S2.SS3.3.p2.2.m2.1.2.1"></eq><apply id="S2.SS3.3.p2.2.m2.1.2.2.cmml" xref="S2.SS3.3.p2.2.m2.1.2.2"><times id="S2.SS3.3.p2.2.m2.1.2.2.1.cmml" xref="S2.SS3.3.p2.2.m2.1.2.2.1"></times><ci id="S2.SS3.3.p2.2.m2.1.2.2.2.cmml" xref="S2.SS3.3.p2.2.m2.1.2.2.2">𝐺</ci><ci id="S2.SS3.3.p2.2.m2.1.1.cmml" xref="S2.SS3.3.p2.2.m2.1.1">𝜏</ci></apply><apply id="S2.SS3.3.p2.2.m2.1.2.3.cmml" xref="S2.SS3.3.p2.2.m2.1.2.3"><divide id="S2.SS3.3.p2.2.m2.1.2.3.1.cmml" xref="S2.SS3.3.p2.2.m2.1.2.3.1"></divide><cn id="S2.SS3.3.p2.2.m2.1.2.3.2.cmml" type="integer" xref="S2.SS3.3.p2.2.m2.1.2.3.2">1</cn><cn id="S2.SS3.3.p2.2.m2.1.2.3.3.cmml" type="integer" xref="S2.SS3.3.p2.2.m2.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.2.m2.1c">G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.2.m2.1d">italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math>; then the expected utility of reporting <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.3.p2.3.m3.1"><semantics id="S2.SS3.3.p2.3.m3.1a"><mi id="S2.SS3.3.p2.3.m3.1.1" xref="S2.SS3.3.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.3.m3.1b"><ci id="S2.SS3.3.p2.3.m3.1.1.cmml" xref="S2.SS3.3.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.3.m3.1d">italic_L</annotation></semantics></math> is <math alttext="\Pr[X^{\prime}\leq\tau\mid X=\tau]=1/2" class="ltx_Math" display="inline" id="S2.SS3.3.p2.4.m4.2"><semantics 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xref="S2.SS3.3.p2.4.m4.2.2.1.1.1.1.6">𝜏</ci></apply></apply></apply><apply id="S2.SS3.3.p2.4.m4.2.2.3.cmml" xref="S2.SS3.3.p2.4.m4.2.2.3"><divide id="S2.SS3.3.p2.4.m4.2.2.3.1.cmml" xref="S2.SS3.3.p2.4.m4.2.2.3.1"></divide><cn id="S2.SS3.3.p2.4.m4.2.2.3.2.cmml" type="integer" xref="S2.SS3.3.p2.4.m4.2.2.3.2">1</cn><cn id="S2.SS3.3.p2.4.m4.2.2.3.3.cmml" type="integer" xref="S2.SS3.3.p2.4.m4.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.4.m4.2c">\Pr[X^{\prime}\leq\tau\mid X=\tau]=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.4.m4.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_τ ] = 1 / 2</annotation></semantics></math>, while the expected utility of reporting <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.3.p2.5.m5.1"><semantics id="S2.SS3.3.p2.5.m5.1a"><mi id="S2.SS3.3.p2.5.m5.1.1" xref="S2.SS3.3.p2.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.5.m5.1b"><ci id="S2.SS3.3.p2.5.m5.1.1.cmml" xref="S2.SS3.3.p2.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.5.m5.1d">italic_H</annotation></semantics></math> is <math alttext="\Pr[X^{\prime}>\tau\mid X=\tau]=1/2" class="ltx_Math" display="inline" id="S2.SS3.3.p2.6.m6.2"><semantics id="S2.SS3.3.p2.6.m6.2a"><mrow id="S2.SS3.3.p2.6.m6.2.2" xref="S2.SS3.3.p2.6.m6.2.2.cmml"><mrow id="S2.SS3.3.p2.6.m6.2.2.1.1" xref="S2.SS3.3.p2.6.m6.2.2.1.2.cmml"><mi id="S2.SS3.3.p2.6.m6.1.1" xref="S2.SS3.3.p2.6.m6.1.1.cmml">Pr</mi><mo id="S2.SS3.3.p2.6.m6.2.2.1.1a" xref="S2.SS3.3.p2.6.m6.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS3.3.p2.6.m6.2.2.1.1.1" xref="S2.SS3.3.p2.6.m6.2.2.1.2.cmml"><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.2" stretchy="false" xref="S2.SS3.3.p2.6.m6.2.2.1.2.cmml">[</mo><mrow id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.cmml"><msup id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.cmml"><mi id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.2" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.3" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.3" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.3.cmml">></mo><mrow id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.cmml"><mi id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.2" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.2.cmml">τ</mi><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.1" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.1.cmml">∣</mo><mi id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.3" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.5" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.5.cmml">=</mo><mi id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.6" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.6.cmml">τ</mi></mrow><mo id="S2.SS3.3.p2.6.m6.2.2.1.1.1.3" stretchy="false" xref="S2.SS3.3.p2.6.m6.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="S2.SS3.3.p2.6.m6.2.2.2" xref="S2.SS3.3.p2.6.m6.2.2.2.cmml">=</mo><mrow id="S2.SS3.3.p2.6.m6.2.2.3" xref="S2.SS3.3.p2.6.m6.2.2.3.cmml"><mn id="S2.SS3.3.p2.6.m6.2.2.3.2" xref="S2.SS3.3.p2.6.m6.2.2.3.2.cmml">1</mn><mo id="S2.SS3.3.p2.6.m6.2.2.3.1" xref="S2.SS3.3.p2.6.m6.2.2.3.1.cmml">/</mo><mn id="S2.SS3.3.p2.6.m6.2.2.3.3" xref="S2.SS3.3.p2.6.m6.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.6.m6.2b"><apply id="S2.SS3.3.p2.6.m6.2.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2"><eq id="S2.SS3.3.p2.6.m6.2.2.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2.2"></eq><apply id="S2.SS3.3.p2.6.m6.2.2.1.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1"><ci id="S2.SS3.3.p2.6.m6.1.1.cmml" xref="S2.SS3.3.p2.6.m6.1.1">Pr</ci><apply id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1"><and id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1a.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1"></and><apply id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1b.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1"><gt id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.3.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.3"></gt><apply id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.1.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2">superscript</csymbol><ci id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.2">𝑋</ci><ci id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.3.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.2.3">′</ci></apply><apply id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4"><csymbol cd="latexml" id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.1.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.1">conditional</csymbol><ci id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.2.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.2">𝜏</ci><ci id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.3.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1c.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1"><eq id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.5.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS3.3.p2.6.m6.2.2.1.1.1.1.4.cmml" id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1d.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1"></share><ci id="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.6.cmml" xref="S2.SS3.3.p2.6.m6.2.2.1.1.1.1.6">𝜏</ci></apply></apply></apply><apply id="S2.SS3.3.p2.6.m6.2.2.3.cmml" xref="S2.SS3.3.p2.6.m6.2.2.3"><divide id="S2.SS3.3.p2.6.m6.2.2.3.1.cmml" xref="S2.SS3.3.p2.6.m6.2.2.3.1"></divide><cn id="S2.SS3.3.p2.6.m6.2.2.3.2.cmml" type="integer" xref="S2.SS3.3.p2.6.m6.2.2.3.2">1</cn><cn id="S2.SS3.3.p2.6.m6.2.2.3.3.cmml" type="integer" xref="S2.SS3.3.p2.6.m6.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.6.m6.2c">\Pr[X^{\prime}>\tau\mid X=\tau]=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.6.m6.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ ∣ italic_X = italic_τ ] = 1 / 2</annotation></semantics></math>; thus the agent is indifferent between reporting <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.3.p2.7.m7.1"><semantics id="S2.SS3.3.p2.7.m7.1a"><mi id="S2.SS3.3.p2.7.m7.1.1" xref="S2.SS3.3.p2.7.m7.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.7.m7.1b"><ci id="S2.SS3.3.p2.7.m7.1.1.cmml" xref="S2.SS3.3.p2.7.m7.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.7.m7.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.7.m7.1d">italic_L</annotation></semantics></math> or <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.3.p2.8.m8.1"><semantics id="S2.SS3.3.p2.8.m8.1a"><mi id="S2.SS3.3.p2.8.m8.1.1" xref="S2.SS3.3.p2.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.8.m8.1b"><ci id="S2.SS3.3.p2.8.m8.1.1.cmml" xref="S2.SS3.3.p2.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.8.m8.1d">italic_H</annotation></semantics></math>. Moreover, the best response <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.3.p2.9.m9.1"><semantics id="S2.SS3.3.p2.9.m9.1a"><mover accent="true" id="S2.SS3.3.p2.9.m9.1.1" xref="S2.SS3.3.p2.9.m9.1.1.cmml"><mi id="S2.SS3.3.p2.9.m9.1.1.2" xref="S2.SS3.3.p2.9.m9.1.1.2.cmml">τ</mi><mo id="S2.SS3.3.p2.9.m9.1.1.1" xref="S2.SS3.3.p2.9.m9.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.9.m9.1b"><apply id="S2.SS3.3.p2.9.m9.1.1.cmml" xref="S2.SS3.3.p2.9.m9.1.1"><ci id="S2.SS3.3.p2.9.m9.1.1.1.cmml" xref="S2.SS3.3.p2.9.m9.1.1.1">^</ci><ci id="S2.SS3.3.p2.9.m9.1.1.2.cmml" xref="S2.SS3.3.p2.9.m9.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.9.m9.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.9.m9.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> is exactly <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.3.p2.10.m10.1"><semantics id="S2.SS3.3.p2.10.m10.1a"><mi id="S2.SS3.3.p2.10.m10.1.1" xref="S2.SS3.3.p2.10.m10.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.10.m10.1b"><ci id="S2.SS3.3.p2.10.m10.1.1.cmml" xref="S2.SS3.3.p2.10.m10.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.10.m10.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.10.m10.1d">italic_τ</annotation></semantics></math>. It follows the system is at an equilibrium of the dynamics since <math alttext="\dot{\tau}=\hat{\tau}-\tau=0" class="ltx_Math" display="inline" id="S2.SS3.3.p2.11.m11.1"><semantics id="S2.SS3.3.p2.11.m11.1a"><mrow id="S2.SS3.3.p2.11.m11.1.1" xref="S2.SS3.3.p2.11.m11.1.1.cmml"><mover accent="true" id="S2.SS3.3.p2.11.m11.1.1.2" xref="S2.SS3.3.p2.11.m11.1.1.2.cmml"><mi id="S2.SS3.3.p2.11.m11.1.1.2.2" xref="S2.SS3.3.p2.11.m11.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.3.p2.11.m11.1.1.2.1" xref="S2.SS3.3.p2.11.m11.1.1.2.1.cmml">˙</mo></mover><mo id="S2.SS3.3.p2.11.m11.1.1.3" xref="S2.SS3.3.p2.11.m11.1.1.3.cmml">=</mo><mrow id="S2.SS3.3.p2.11.m11.1.1.4" xref="S2.SS3.3.p2.11.m11.1.1.4.cmml"><mover accent="true" id="S2.SS3.3.p2.11.m11.1.1.4.2" xref="S2.SS3.3.p2.11.m11.1.1.4.2.cmml"><mi id="S2.SS3.3.p2.11.m11.1.1.4.2.2" xref="S2.SS3.3.p2.11.m11.1.1.4.2.2.cmml">τ</mi><mo id="S2.SS3.3.p2.11.m11.1.1.4.2.1" xref="S2.SS3.3.p2.11.m11.1.1.4.2.1.cmml">^</mo></mover><mo id="S2.SS3.3.p2.11.m11.1.1.4.1" xref="S2.SS3.3.p2.11.m11.1.1.4.1.cmml">−</mo><mi id="S2.SS3.3.p2.11.m11.1.1.4.3" xref="S2.SS3.3.p2.11.m11.1.1.4.3.cmml">τ</mi></mrow><mo id="S2.SS3.3.p2.11.m11.1.1.5" xref="S2.SS3.3.p2.11.m11.1.1.5.cmml">=</mo><mn id="S2.SS3.3.p2.11.m11.1.1.6" xref="S2.SS3.3.p2.11.m11.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.3.p2.11.m11.1b"><apply id="S2.SS3.3.p2.11.m11.1.1.cmml" xref="S2.SS3.3.p2.11.m11.1.1"><and id="S2.SS3.3.p2.11.m11.1.1a.cmml" xref="S2.SS3.3.p2.11.m11.1.1"></and><apply id="S2.SS3.3.p2.11.m11.1.1b.cmml" xref="S2.SS3.3.p2.11.m11.1.1"><eq id="S2.SS3.3.p2.11.m11.1.1.3.cmml" xref="S2.SS3.3.p2.11.m11.1.1.3"></eq><apply id="S2.SS3.3.p2.11.m11.1.1.2.cmml" xref="S2.SS3.3.p2.11.m11.1.1.2"><ci id="S2.SS3.3.p2.11.m11.1.1.2.1.cmml" xref="S2.SS3.3.p2.11.m11.1.1.2.1">˙</ci><ci id="S2.SS3.3.p2.11.m11.1.1.2.2.cmml" xref="S2.SS3.3.p2.11.m11.1.1.2.2">𝜏</ci></apply><apply id="S2.SS3.3.p2.11.m11.1.1.4.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4"><minus id="S2.SS3.3.p2.11.m11.1.1.4.1.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4.1"></minus><apply id="S2.SS3.3.p2.11.m11.1.1.4.2.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4.2"><ci id="S2.SS3.3.p2.11.m11.1.1.4.2.1.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4.2.1">^</ci><ci id="S2.SS3.3.p2.11.m11.1.1.4.2.2.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4.2.2">𝜏</ci></apply><ci id="S2.SS3.3.p2.11.m11.1.1.4.3.cmml" xref="S2.SS3.3.p2.11.m11.1.1.4.3">𝜏</ci></apply></apply><apply id="S2.SS3.3.p2.11.m11.1.1c.cmml" xref="S2.SS3.3.p2.11.m11.1.1"><eq id="S2.SS3.3.p2.11.m11.1.1.5.cmml" xref="S2.SS3.3.p2.11.m11.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS3.3.p2.11.m11.1.1.4.cmml" id="S2.SS3.3.p2.11.m11.1.1d.cmml" xref="S2.SS3.3.p2.11.m11.1.1"></share><cn id="S2.SS3.3.p2.11.m11.1.1.6.cmml" type="integer" xref="S2.SS3.3.p2.11.m11.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.3.p2.11.m11.1c">\dot{\tau}=\hat{\tau}-\tau=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.3.p2.11.m11.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.4.p3"> <p class="ltx_p" id="S2.SS3.4.p3.12">In the second case, <math alttext="G(\tau)>1/2" class="ltx_Math" display="inline" id="S2.SS3.4.p3.1.m1.1"><semantics id="S2.SS3.4.p3.1.m1.1a"><mrow id="S2.SS3.4.p3.1.m1.1.2" xref="S2.SS3.4.p3.1.m1.1.2.cmml"><mrow id="S2.SS3.4.p3.1.m1.1.2.2" xref="S2.SS3.4.p3.1.m1.1.2.2.cmml"><mi id="S2.SS3.4.p3.1.m1.1.2.2.2" xref="S2.SS3.4.p3.1.m1.1.2.2.2.cmml">G</mi><mo id="S2.SS3.4.p3.1.m1.1.2.2.1" xref="S2.SS3.4.p3.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.4.p3.1.m1.1.2.2.3.2" xref="S2.SS3.4.p3.1.m1.1.2.2.cmml"><mo id="S2.SS3.4.p3.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.4.p3.1.m1.1.2.2.cmml">(</mo><mi id="S2.SS3.4.p3.1.m1.1.1" xref="S2.SS3.4.p3.1.m1.1.1.cmml">τ</mi><mo id="S2.SS3.4.p3.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.4.p3.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.4.p3.1.m1.1.2.1" xref="S2.SS3.4.p3.1.m1.1.2.1.cmml">></mo><mrow id="S2.SS3.4.p3.1.m1.1.2.3" xref="S2.SS3.4.p3.1.m1.1.2.3.cmml"><mn id="S2.SS3.4.p3.1.m1.1.2.3.2" xref="S2.SS3.4.p3.1.m1.1.2.3.2.cmml">1</mn><mo id="S2.SS3.4.p3.1.m1.1.2.3.1" xref="S2.SS3.4.p3.1.m1.1.2.3.1.cmml">/</mo><mn id="S2.SS3.4.p3.1.m1.1.2.3.3" xref="S2.SS3.4.p3.1.m1.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.1.m1.1b"><apply id="S2.SS3.4.p3.1.m1.1.2.cmml" xref="S2.SS3.4.p3.1.m1.1.2"><gt id="S2.SS3.4.p3.1.m1.1.2.1.cmml" xref="S2.SS3.4.p3.1.m1.1.2.1"></gt><apply id="S2.SS3.4.p3.1.m1.1.2.2.cmml" xref="S2.SS3.4.p3.1.m1.1.2.2"><times id="S2.SS3.4.p3.1.m1.1.2.2.1.cmml" xref="S2.SS3.4.p3.1.m1.1.2.2.1"></times><ci id="S2.SS3.4.p3.1.m1.1.2.2.2.cmml" xref="S2.SS3.4.p3.1.m1.1.2.2.2">𝐺</ci><ci id="S2.SS3.4.p3.1.m1.1.1.cmml" xref="S2.SS3.4.p3.1.m1.1.1">𝜏</ci></apply><apply id="S2.SS3.4.p3.1.m1.1.2.3.cmml" xref="S2.SS3.4.p3.1.m1.1.2.3"><divide id="S2.SS3.4.p3.1.m1.1.2.3.1.cmml" xref="S2.SS3.4.p3.1.m1.1.2.3.1"></divide><cn id="S2.SS3.4.p3.1.m1.1.2.3.2.cmml" type="integer" xref="S2.SS3.4.p3.1.m1.1.2.3.2">1</cn><cn id="S2.SS3.4.p3.1.m1.1.2.3.3.cmml" type="integer" xref="S2.SS3.4.p3.1.m1.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.1.m1.1c">G(\tau)>1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.1.m1.1d">italic_G ( italic_τ ) > 1 / 2</annotation></semantics></math>. Then reporting <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.4.p3.2.m2.1"><semantics id="S2.SS3.4.p3.2.m2.1a"><mi id="S2.SS3.4.p3.2.m2.1.1" xref="S2.SS3.4.p3.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.2.m2.1b"><ci id="S2.SS3.4.p3.2.m2.1.1.cmml" xref="S2.SS3.4.p3.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.2.m2.1d">italic_L</annotation></semantics></math> is strictly preferred to reporting <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.4.p3.3.m3.1"><semantics id="S2.SS3.4.p3.3.m3.1a"><mi id="S2.SS3.4.p3.3.m3.1.1" xref="S2.SS3.4.p3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.3.m3.1b"><ci id="S2.SS3.4.p3.3.m3.1.1.cmml" xref="S2.SS3.4.p3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.3.m3.1d">italic_H</annotation></semantics></math>. Since <math alttext="P(\tau;\hat{\tau})=1/2" class="ltx_Math" display="inline" id="S2.SS3.4.p3.4.m4.2"><semantics id="S2.SS3.4.p3.4.m4.2a"><mrow id="S2.SS3.4.p3.4.m4.2.3" xref="S2.SS3.4.p3.4.m4.2.3.cmml"><mrow id="S2.SS3.4.p3.4.m4.2.3.2" xref="S2.SS3.4.p3.4.m4.2.3.2.cmml"><mi id="S2.SS3.4.p3.4.m4.2.3.2.2" xref="S2.SS3.4.p3.4.m4.2.3.2.2.cmml">P</mi><mo id="S2.SS3.4.p3.4.m4.2.3.2.1" xref="S2.SS3.4.p3.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.4.p3.4.m4.2.3.2.3.2" xref="S2.SS3.4.p3.4.m4.2.3.2.3.1.cmml"><mo id="S2.SS3.4.p3.4.m4.2.3.2.3.2.1" stretchy="false" xref="S2.SS3.4.p3.4.m4.2.3.2.3.1.cmml">(</mo><mi id="S2.SS3.4.p3.4.m4.1.1" xref="S2.SS3.4.p3.4.m4.1.1.cmml">τ</mi><mo id="S2.SS3.4.p3.4.m4.2.3.2.3.2.2" xref="S2.SS3.4.p3.4.m4.2.3.2.3.1.cmml">;</mo><mover accent="true" id="S2.SS3.4.p3.4.m4.2.2" xref="S2.SS3.4.p3.4.m4.2.2.cmml"><mi id="S2.SS3.4.p3.4.m4.2.2.2" xref="S2.SS3.4.p3.4.m4.2.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.4.m4.2.2.1" xref="S2.SS3.4.p3.4.m4.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.4.p3.4.m4.2.3.2.3.2.3" stretchy="false" xref="S2.SS3.4.p3.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.4.p3.4.m4.2.3.1" xref="S2.SS3.4.p3.4.m4.2.3.1.cmml">=</mo><mrow id="S2.SS3.4.p3.4.m4.2.3.3" xref="S2.SS3.4.p3.4.m4.2.3.3.cmml"><mn id="S2.SS3.4.p3.4.m4.2.3.3.2" xref="S2.SS3.4.p3.4.m4.2.3.3.2.cmml">1</mn><mo id="S2.SS3.4.p3.4.m4.2.3.3.1" xref="S2.SS3.4.p3.4.m4.2.3.3.1.cmml">/</mo><mn id="S2.SS3.4.p3.4.m4.2.3.3.3" xref="S2.SS3.4.p3.4.m4.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.4.m4.2b"><apply id="S2.SS3.4.p3.4.m4.2.3.cmml" xref="S2.SS3.4.p3.4.m4.2.3"><eq id="S2.SS3.4.p3.4.m4.2.3.1.cmml" xref="S2.SS3.4.p3.4.m4.2.3.1"></eq><apply id="S2.SS3.4.p3.4.m4.2.3.2.cmml" xref="S2.SS3.4.p3.4.m4.2.3.2"><times id="S2.SS3.4.p3.4.m4.2.3.2.1.cmml" xref="S2.SS3.4.p3.4.m4.2.3.2.1"></times><ci id="S2.SS3.4.p3.4.m4.2.3.2.2.cmml" xref="S2.SS3.4.p3.4.m4.2.3.2.2">𝑃</ci><list id="S2.SS3.4.p3.4.m4.2.3.2.3.1.cmml" xref="S2.SS3.4.p3.4.m4.2.3.2.3.2"><ci id="S2.SS3.4.p3.4.m4.1.1.cmml" xref="S2.SS3.4.p3.4.m4.1.1">𝜏</ci><apply id="S2.SS3.4.p3.4.m4.2.2.cmml" xref="S2.SS3.4.p3.4.m4.2.2"><ci id="S2.SS3.4.p3.4.m4.2.2.1.cmml" xref="S2.SS3.4.p3.4.m4.2.2.1">^</ci><ci id="S2.SS3.4.p3.4.m4.2.2.2.cmml" xref="S2.SS3.4.p3.4.m4.2.2.2">𝜏</ci></apply></list></apply><apply id="S2.SS3.4.p3.4.m4.2.3.3.cmml" xref="S2.SS3.4.p3.4.m4.2.3.3"><divide id="S2.SS3.4.p3.4.m4.2.3.3.1.cmml" xref="S2.SS3.4.p3.4.m4.2.3.3.1"></divide><cn id="S2.SS3.4.p3.4.m4.2.3.3.2.cmml" type="integer" xref="S2.SS3.4.p3.4.m4.2.3.3.2">1</cn><cn id="S2.SS3.4.p3.4.m4.2.3.3.3.cmml" type="integer" xref="S2.SS3.4.p3.4.m4.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.4.m4.2c">P(\tau;\hat{\tau})=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.4.m4.2d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = 1 / 2</annotation></semantics></math>, by decreasing monotonicity we have <math alttext="\hat{\tau}>\tau" class="ltx_Math" display="inline" id="S2.SS3.4.p3.5.m5.1"><semantics id="S2.SS3.4.p3.5.m5.1a"><mrow id="S2.SS3.4.p3.5.m5.1.1" xref="S2.SS3.4.p3.5.m5.1.1.cmml"><mover accent="true" id="S2.SS3.4.p3.5.m5.1.1.2" xref="S2.SS3.4.p3.5.m5.1.1.2.cmml"><mi id="S2.SS3.4.p3.5.m5.1.1.2.2" xref="S2.SS3.4.p3.5.m5.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.5.m5.1.1.2.1" xref="S2.SS3.4.p3.5.m5.1.1.2.1.cmml">^</mo></mover><mo id="S2.SS3.4.p3.5.m5.1.1.1" xref="S2.SS3.4.p3.5.m5.1.1.1.cmml">></mo><mi id="S2.SS3.4.p3.5.m5.1.1.3" xref="S2.SS3.4.p3.5.m5.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.5.m5.1b"><apply id="S2.SS3.4.p3.5.m5.1.1.cmml" xref="S2.SS3.4.p3.5.m5.1.1"><gt id="S2.SS3.4.p3.5.m5.1.1.1.cmml" xref="S2.SS3.4.p3.5.m5.1.1.1"></gt><apply id="S2.SS3.4.p3.5.m5.1.1.2.cmml" xref="S2.SS3.4.p3.5.m5.1.1.2"><ci id="S2.SS3.4.p3.5.m5.1.1.2.1.cmml" xref="S2.SS3.4.p3.5.m5.1.1.2.1">^</ci><ci id="S2.SS3.4.p3.5.m5.1.1.2.2.cmml" xref="S2.SS3.4.p3.5.m5.1.1.2.2">𝜏</ci></apply><ci id="S2.SS3.4.p3.5.m5.1.1.3.cmml" xref="S2.SS3.4.p3.5.m5.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.5.m5.1c">\hat{\tau}>\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.5.m5.1d">over^ start_ARG italic_τ end_ARG > italic_τ</annotation></semantics></math>. Thus <math alttext="\dot{\tau}=\hat{\tau}-\tau>0" class="ltx_Math" display="inline" id="S2.SS3.4.p3.6.m6.1"><semantics id="S2.SS3.4.p3.6.m6.1a"><mrow id="S2.SS3.4.p3.6.m6.1.1" xref="S2.SS3.4.p3.6.m6.1.1.cmml"><mover accent="true" id="S2.SS3.4.p3.6.m6.1.1.2" xref="S2.SS3.4.p3.6.m6.1.1.2.cmml"><mi id="S2.SS3.4.p3.6.m6.1.1.2.2" xref="S2.SS3.4.p3.6.m6.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.6.m6.1.1.2.1" xref="S2.SS3.4.p3.6.m6.1.1.2.1.cmml">˙</mo></mover><mo id="S2.SS3.4.p3.6.m6.1.1.3" xref="S2.SS3.4.p3.6.m6.1.1.3.cmml">=</mo><mrow id="S2.SS3.4.p3.6.m6.1.1.4" xref="S2.SS3.4.p3.6.m6.1.1.4.cmml"><mover accent="true" id="S2.SS3.4.p3.6.m6.1.1.4.2" xref="S2.SS3.4.p3.6.m6.1.1.4.2.cmml"><mi id="S2.SS3.4.p3.6.m6.1.1.4.2.2" xref="S2.SS3.4.p3.6.m6.1.1.4.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.6.m6.1.1.4.2.1" xref="S2.SS3.4.p3.6.m6.1.1.4.2.1.cmml">^</mo></mover><mo id="S2.SS3.4.p3.6.m6.1.1.4.1" xref="S2.SS3.4.p3.6.m6.1.1.4.1.cmml">−</mo><mi id="S2.SS3.4.p3.6.m6.1.1.4.3" xref="S2.SS3.4.p3.6.m6.1.1.4.3.cmml">τ</mi></mrow><mo id="S2.SS3.4.p3.6.m6.1.1.5" xref="S2.SS3.4.p3.6.m6.1.1.5.cmml">></mo><mn id="S2.SS3.4.p3.6.m6.1.1.6" xref="S2.SS3.4.p3.6.m6.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.6.m6.1b"><apply id="S2.SS3.4.p3.6.m6.1.1.cmml" xref="S2.SS3.4.p3.6.m6.1.1"><and id="S2.SS3.4.p3.6.m6.1.1a.cmml" xref="S2.SS3.4.p3.6.m6.1.1"></and><apply id="S2.SS3.4.p3.6.m6.1.1b.cmml" xref="S2.SS3.4.p3.6.m6.1.1"><eq id="S2.SS3.4.p3.6.m6.1.1.3.cmml" xref="S2.SS3.4.p3.6.m6.1.1.3"></eq><apply id="S2.SS3.4.p3.6.m6.1.1.2.cmml" xref="S2.SS3.4.p3.6.m6.1.1.2"><ci id="S2.SS3.4.p3.6.m6.1.1.2.1.cmml" xref="S2.SS3.4.p3.6.m6.1.1.2.1">˙</ci><ci id="S2.SS3.4.p3.6.m6.1.1.2.2.cmml" xref="S2.SS3.4.p3.6.m6.1.1.2.2">𝜏</ci></apply><apply id="S2.SS3.4.p3.6.m6.1.1.4.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4"><minus id="S2.SS3.4.p3.6.m6.1.1.4.1.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4.1"></minus><apply id="S2.SS3.4.p3.6.m6.1.1.4.2.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4.2"><ci id="S2.SS3.4.p3.6.m6.1.1.4.2.1.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4.2.1">^</ci><ci id="S2.SS3.4.p3.6.m6.1.1.4.2.2.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4.2.2">𝜏</ci></apply><ci id="S2.SS3.4.p3.6.m6.1.1.4.3.cmml" xref="S2.SS3.4.p3.6.m6.1.1.4.3">𝜏</ci></apply></apply><apply id="S2.SS3.4.p3.6.m6.1.1c.cmml" xref="S2.SS3.4.p3.6.m6.1.1"><gt id="S2.SS3.4.p3.6.m6.1.1.5.cmml" xref="S2.SS3.4.p3.6.m6.1.1.5"></gt><share href="https://arxiv.org/html/2503.16280v1#S2.SS3.4.p3.6.m6.1.1.4.cmml" id="S2.SS3.4.p3.6.m6.1.1d.cmml" xref="S2.SS3.4.p3.6.m6.1.1"></share><cn id="S2.SS3.4.p3.6.m6.1.1.6.cmml" type="integer" xref="S2.SS3.4.p3.6.m6.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.6.m6.1c">\dot{\tau}=\hat{\tau}-\tau>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.6.m6.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ > 0</annotation></semantics></math>. In the third case, <math alttext="G(\tau)<1/2" class="ltx_Math" display="inline" id="S2.SS3.4.p3.7.m7.1"><semantics id="S2.SS3.4.p3.7.m7.1a"><mrow id="S2.SS3.4.p3.7.m7.1.2" xref="S2.SS3.4.p3.7.m7.1.2.cmml"><mrow id="S2.SS3.4.p3.7.m7.1.2.2" xref="S2.SS3.4.p3.7.m7.1.2.2.cmml"><mi id="S2.SS3.4.p3.7.m7.1.2.2.2" xref="S2.SS3.4.p3.7.m7.1.2.2.2.cmml">G</mi><mo id="S2.SS3.4.p3.7.m7.1.2.2.1" xref="S2.SS3.4.p3.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.4.p3.7.m7.1.2.2.3.2" xref="S2.SS3.4.p3.7.m7.1.2.2.cmml"><mo id="S2.SS3.4.p3.7.m7.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.4.p3.7.m7.1.2.2.cmml">(</mo><mi id="S2.SS3.4.p3.7.m7.1.1" xref="S2.SS3.4.p3.7.m7.1.1.cmml">τ</mi><mo id="S2.SS3.4.p3.7.m7.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.4.p3.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.4.p3.7.m7.1.2.1" xref="S2.SS3.4.p3.7.m7.1.2.1.cmml"><</mo><mrow id="S2.SS3.4.p3.7.m7.1.2.3" xref="S2.SS3.4.p3.7.m7.1.2.3.cmml"><mn id="S2.SS3.4.p3.7.m7.1.2.3.2" xref="S2.SS3.4.p3.7.m7.1.2.3.2.cmml">1</mn><mo id="S2.SS3.4.p3.7.m7.1.2.3.1" xref="S2.SS3.4.p3.7.m7.1.2.3.1.cmml">/</mo><mn id="S2.SS3.4.p3.7.m7.1.2.3.3" xref="S2.SS3.4.p3.7.m7.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.7.m7.1b"><apply id="S2.SS3.4.p3.7.m7.1.2.cmml" xref="S2.SS3.4.p3.7.m7.1.2"><lt id="S2.SS3.4.p3.7.m7.1.2.1.cmml" xref="S2.SS3.4.p3.7.m7.1.2.1"></lt><apply id="S2.SS3.4.p3.7.m7.1.2.2.cmml" xref="S2.SS3.4.p3.7.m7.1.2.2"><times id="S2.SS3.4.p3.7.m7.1.2.2.1.cmml" xref="S2.SS3.4.p3.7.m7.1.2.2.1"></times><ci id="S2.SS3.4.p3.7.m7.1.2.2.2.cmml" xref="S2.SS3.4.p3.7.m7.1.2.2.2">𝐺</ci><ci id="S2.SS3.4.p3.7.m7.1.1.cmml" xref="S2.SS3.4.p3.7.m7.1.1">𝜏</ci></apply><apply id="S2.SS3.4.p3.7.m7.1.2.3.cmml" xref="S2.SS3.4.p3.7.m7.1.2.3"><divide id="S2.SS3.4.p3.7.m7.1.2.3.1.cmml" xref="S2.SS3.4.p3.7.m7.1.2.3.1"></divide><cn id="S2.SS3.4.p3.7.m7.1.2.3.2.cmml" type="integer" xref="S2.SS3.4.p3.7.m7.1.2.3.2">1</cn><cn id="S2.SS3.4.p3.7.m7.1.2.3.3.cmml" type="integer" xref="S2.SS3.4.p3.7.m7.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.7.m7.1c">G(\tau)<1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.7.m7.1d">italic_G ( italic_τ ) < 1 / 2</annotation></semantics></math>. Then reporting <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.4.p3.8.m8.1"><semantics id="S2.SS3.4.p3.8.m8.1a"><mi id="S2.SS3.4.p3.8.m8.1.1" xref="S2.SS3.4.p3.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.8.m8.1b"><ci id="S2.SS3.4.p3.8.m8.1.1.cmml" xref="S2.SS3.4.p3.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.8.m8.1d">italic_H</annotation></semantics></math> is strictly preferred to reporting <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.4.p3.9.m9.1"><semantics id="S2.SS3.4.p3.9.m9.1a"><mi id="S2.SS3.4.p3.9.m9.1.1" xref="S2.SS3.4.p3.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.9.m9.1b"><ci id="S2.SS3.4.p3.9.m9.1.1.cmml" xref="S2.SS3.4.p3.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.9.m9.1d">italic_L</annotation></semantics></math>. Since <math alttext="P(\tau;\hat{\tau})=1/2" class="ltx_Math" display="inline" id="S2.SS3.4.p3.10.m10.2"><semantics id="S2.SS3.4.p3.10.m10.2a"><mrow id="S2.SS3.4.p3.10.m10.2.3" xref="S2.SS3.4.p3.10.m10.2.3.cmml"><mrow id="S2.SS3.4.p3.10.m10.2.3.2" xref="S2.SS3.4.p3.10.m10.2.3.2.cmml"><mi id="S2.SS3.4.p3.10.m10.2.3.2.2" xref="S2.SS3.4.p3.10.m10.2.3.2.2.cmml">P</mi><mo id="S2.SS3.4.p3.10.m10.2.3.2.1" xref="S2.SS3.4.p3.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS3.4.p3.10.m10.2.3.2.3.2" xref="S2.SS3.4.p3.10.m10.2.3.2.3.1.cmml"><mo id="S2.SS3.4.p3.10.m10.2.3.2.3.2.1" stretchy="false" xref="S2.SS3.4.p3.10.m10.2.3.2.3.1.cmml">(</mo><mi id="S2.SS3.4.p3.10.m10.1.1" xref="S2.SS3.4.p3.10.m10.1.1.cmml">τ</mi><mo id="S2.SS3.4.p3.10.m10.2.3.2.3.2.2" xref="S2.SS3.4.p3.10.m10.2.3.2.3.1.cmml">;</mo><mover accent="true" id="S2.SS3.4.p3.10.m10.2.2" xref="S2.SS3.4.p3.10.m10.2.2.cmml"><mi id="S2.SS3.4.p3.10.m10.2.2.2" xref="S2.SS3.4.p3.10.m10.2.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.10.m10.2.2.1" xref="S2.SS3.4.p3.10.m10.2.2.1.cmml">^</mo></mover><mo id="S2.SS3.4.p3.10.m10.2.3.2.3.2.3" stretchy="false" xref="S2.SS3.4.p3.10.m10.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS3.4.p3.10.m10.2.3.1" xref="S2.SS3.4.p3.10.m10.2.3.1.cmml">=</mo><mrow id="S2.SS3.4.p3.10.m10.2.3.3" xref="S2.SS3.4.p3.10.m10.2.3.3.cmml"><mn id="S2.SS3.4.p3.10.m10.2.3.3.2" xref="S2.SS3.4.p3.10.m10.2.3.3.2.cmml">1</mn><mo id="S2.SS3.4.p3.10.m10.2.3.3.1" xref="S2.SS3.4.p3.10.m10.2.3.3.1.cmml">/</mo><mn id="S2.SS3.4.p3.10.m10.2.3.3.3" xref="S2.SS3.4.p3.10.m10.2.3.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.10.m10.2b"><apply id="S2.SS3.4.p3.10.m10.2.3.cmml" xref="S2.SS3.4.p3.10.m10.2.3"><eq id="S2.SS3.4.p3.10.m10.2.3.1.cmml" xref="S2.SS3.4.p3.10.m10.2.3.1"></eq><apply id="S2.SS3.4.p3.10.m10.2.3.2.cmml" xref="S2.SS3.4.p3.10.m10.2.3.2"><times id="S2.SS3.4.p3.10.m10.2.3.2.1.cmml" xref="S2.SS3.4.p3.10.m10.2.3.2.1"></times><ci id="S2.SS3.4.p3.10.m10.2.3.2.2.cmml" xref="S2.SS3.4.p3.10.m10.2.3.2.2">𝑃</ci><list id="S2.SS3.4.p3.10.m10.2.3.2.3.1.cmml" xref="S2.SS3.4.p3.10.m10.2.3.2.3.2"><ci id="S2.SS3.4.p3.10.m10.1.1.cmml" xref="S2.SS3.4.p3.10.m10.1.1">𝜏</ci><apply id="S2.SS3.4.p3.10.m10.2.2.cmml" xref="S2.SS3.4.p3.10.m10.2.2"><ci id="S2.SS3.4.p3.10.m10.2.2.1.cmml" xref="S2.SS3.4.p3.10.m10.2.2.1">^</ci><ci id="S2.SS3.4.p3.10.m10.2.2.2.cmml" xref="S2.SS3.4.p3.10.m10.2.2.2">𝜏</ci></apply></list></apply><apply id="S2.SS3.4.p3.10.m10.2.3.3.cmml" xref="S2.SS3.4.p3.10.m10.2.3.3"><divide id="S2.SS3.4.p3.10.m10.2.3.3.1.cmml" xref="S2.SS3.4.p3.10.m10.2.3.3.1"></divide><cn id="S2.SS3.4.p3.10.m10.2.3.3.2.cmml" type="integer" xref="S2.SS3.4.p3.10.m10.2.3.3.2">1</cn><cn id="S2.SS3.4.p3.10.m10.2.3.3.3.cmml" type="integer" xref="S2.SS3.4.p3.10.m10.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.10.m10.2c">P(\tau;\hat{\tau})=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.10.m10.2d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = 1 / 2</annotation></semantics></math>, by decreasing monotonicity we have <math alttext="\tau>\hat{\tau}" class="ltx_Math" display="inline" id="S2.SS3.4.p3.11.m11.1"><semantics id="S2.SS3.4.p3.11.m11.1a"><mrow id="S2.SS3.4.p3.11.m11.1.1" xref="S2.SS3.4.p3.11.m11.1.1.cmml"><mi id="S2.SS3.4.p3.11.m11.1.1.2" xref="S2.SS3.4.p3.11.m11.1.1.2.cmml">τ</mi><mo id="S2.SS3.4.p3.11.m11.1.1.1" xref="S2.SS3.4.p3.11.m11.1.1.1.cmml">></mo><mover accent="true" id="S2.SS3.4.p3.11.m11.1.1.3" xref="S2.SS3.4.p3.11.m11.1.1.3.cmml"><mi id="S2.SS3.4.p3.11.m11.1.1.3.2" xref="S2.SS3.4.p3.11.m11.1.1.3.2.cmml">τ</mi><mo id="S2.SS3.4.p3.11.m11.1.1.3.1" xref="S2.SS3.4.p3.11.m11.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.11.m11.1b"><apply id="S2.SS3.4.p3.11.m11.1.1.cmml" xref="S2.SS3.4.p3.11.m11.1.1"><gt id="S2.SS3.4.p3.11.m11.1.1.1.cmml" xref="S2.SS3.4.p3.11.m11.1.1.1"></gt><ci id="S2.SS3.4.p3.11.m11.1.1.2.cmml" xref="S2.SS3.4.p3.11.m11.1.1.2">𝜏</ci><apply id="S2.SS3.4.p3.11.m11.1.1.3.cmml" xref="S2.SS3.4.p3.11.m11.1.1.3"><ci id="S2.SS3.4.p3.11.m11.1.1.3.1.cmml" xref="S2.SS3.4.p3.11.m11.1.1.3.1">^</ci><ci id="S2.SS3.4.p3.11.m11.1.1.3.2.cmml" xref="S2.SS3.4.p3.11.m11.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.11.m11.1c">\tau>\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.11.m11.1d">italic_τ > over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. Thus <math alttext="\dot{\tau}=\hat{\tau}-\tau<0" class="ltx_Math" display="inline" id="S2.SS3.4.p3.12.m12.1"><semantics id="S2.SS3.4.p3.12.m12.1a"><mrow id="S2.SS3.4.p3.12.m12.1.1" xref="S2.SS3.4.p3.12.m12.1.1.cmml"><mover accent="true" id="S2.SS3.4.p3.12.m12.1.1.2" xref="S2.SS3.4.p3.12.m12.1.1.2.cmml"><mi id="S2.SS3.4.p3.12.m12.1.1.2.2" xref="S2.SS3.4.p3.12.m12.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.12.m12.1.1.2.1" xref="S2.SS3.4.p3.12.m12.1.1.2.1.cmml">˙</mo></mover><mo id="S2.SS3.4.p3.12.m12.1.1.3" xref="S2.SS3.4.p3.12.m12.1.1.3.cmml">=</mo><mrow id="S2.SS3.4.p3.12.m12.1.1.4" xref="S2.SS3.4.p3.12.m12.1.1.4.cmml"><mover accent="true" id="S2.SS3.4.p3.12.m12.1.1.4.2" xref="S2.SS3.4.p3.12.m12.1.1.4.2.cmml"><mi id="S2.SS3.4.p3.12.m12.1.1.4.2.2" xref="S2.SS3.4.p3.12.m12.1.1.4.2.2.cmml">τ</mi><mo id="S2.SS3.4.p3.12.m12.1.1.4.2.1" xref="S2.SS3.4.p3.12.m12.1.1.4.2.1.cmml">^</mo></mover><mo id="S2.SS3.4.p3.12.m12.1.1.4.1" xref="S2.SS3.4.p3.12.m12.1.1.4.1.cmml">−</mo><mi id="S2.SS3.4.p3.12.m12.1.1.4.3" xref="S2.SS3.4.p3.12.m12.1.1.4.3.cmml">τ</mi></mrow><mo id="S2.SS3.4.p3.12.m12.1.1.5" xref="S2.SS3.4.p3.12.m12.1.1.5.cmml"><</mo><mn id="S2.SS3.4.p3.12.m12.1.1.6" xref="S2.SS3.4.p3.12.m12.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.4.p3.12.m12.1b"><apply id="S2.SS3.4.p3.12.m12.1.1.cmml" xref="S2.SS3.4.p3.12.m12.1.1"><and id="S2.SS3.4.p3.12.m12.1.1a.cmml" xref="S2.SS3.4.p3.12.m12.1.1"></and><apply id="S2.SS3.4.p3.12.m12.1.1b.cmml" xref="S2.SS3.4.p3.12.m12.1.1"><eq id="S2.SS3.4.p3.12.m12.1.1.3.cmml" xref="S2.SS3.4.p3.12.m12.1.1.3"></eq><apply id="S2.SS3.4.p3.12.m12.1.1.2.cmml" xref="S2.SS3.4.p3.12.m12.1.1.2"><ci id="S2.SS3.4.p3.12.m12.1.1.2.1.cmml" xref="S2.SS3.4.p3.12.m12.1.1.2.1">˙</ci><ci id="S2.SS3.4.p3.12.m12.1.1.2.2.cmml" xref="S2.SS3.4.p3.12.m12.1.1.2.2">𝜏</ci></apply><apply id="S2.SS3.4.p3.12.m12.1.1.4.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4"><minus id="S2.SS3.4.p3.12.m12.1.1.4.1.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4.1"></minus><apply id="S2.SS3.4.p3.12.m12.1.1.4.2.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4.2"><ci id="S2.SS3.4.p3.12.m12.1.1.4.2.1.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4.2.1">^</ci><ci id="S2.SS3.4.p3.12.m12.1.1.4.2.2.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4.2.2">𝜏</ci></apply><ci id="S2.SS3.4.p3.12.m12.1.1.4.3.cmml" xref="S2.SS3.4.p3.12.m12.1.1.4.3">𝜏</ci></apply></apply><apply id="S2.SS3.4.p3.12.m12.1.1c.cmml" xref="S2.SS3.4.p3.12.m12.1.1"><lt id="S2.SS3.4.p3.12.m12.1.1.5.cmml" xref="S2.SS3.4.p3.12.m12.1.1.5"></lt><share href="https://arxiv.org/html/2503.16280v1#S2.SS3.4.p3.12.m12.1.1.4.cmml" id="S2.SS3.4.p3.12.m12.1.1d.cmml" xref="S2.SS3.4.p3.12.m12.1.1"></share><cn id="S2.SS3.4.p3.12.m12.1.1.6.cmml" type="integer" xref="S2.SS3.4.p3.12.m12.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.4.p3.12.m12.1c">\dot{\tau}=\hat{\tau}-\tau<0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.4.p3.12.m12.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ < 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.5.p4"> <p class="ltx_p" id="S2.SS3.5.p4.11">If <math alttext="G(\tau)" class="ltx_Math" display="inline" id="S2.SS3.5.p4.1.m1.1"><semantics id="S2.SS3.5.p4.1.m1.1a"><mrow id="S2.SS3.5.p4.1.m1.1.2" xref="S2.SS3.5.p4.1.m1.1.2.cmml"><mi id="S2.SS3.5.p4.1.m1.1.2.2" xref="S2.SS3.5.p4.1.m1.1.2.2.cmml">G</mi><mo id="S2.SS3.5.p4.1.m1.1.2.1" xref="S2.SS3.5.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.5.p4.1.m1.1.2.3.2" xref="S2.SS3.5.p4.1.m1.1.2.cmml"><mo id="S2.SS3.5.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.5.p4.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.5.p4.1.m1.1.1" xref="S2.SS3.5.p4.1.m1.1.1.cmml">τ</mi><mo id="S2.SS3.5.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.5.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.1.m1.1b"><apply id="S2.SS3.5.p4.1.m1.1.2.cmml" xref="S2.SS3.5.p4.1.m1.1.2"><times id="S2.SS3.5.p4.1.m1.1.2.1.cmml" xref="S2.SS3.5.p4.1.m1.1.2.1"></times><ci id="S2.SS3.5.p4.1.m1.1.2.2.cmml" xref="S2.SS3.5.p4.1.m1.1.2.2">𝐺</ci><ci id="S2.SS3.5.p4.1.m1.1.1.cmml" xref="S2.SS3.5.p4.1.m1.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.1.m1.1c">G(\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.1.m1.1d">italic_G ( italic_τ )</annotation></semantics></math> is strictly increasing at <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S2.SS3.5.p4.2.m2.1"><semantics id="S2.SS3.5.p4.2.m2.1a"><msup id="S2.SS3.5.p4.2.m2.1.1" xref="S2.SS3.5.p4.2.m2.1.1.cmml"><mi id="S2.SS3.5.p4.2.m2.1.1.2" xref="S2.SS3.5.p4.2.m2.1.1.2.cmml">τ</mi><mo id="S2.SS3.5.p4.2.m2.1.1.3" xref="S2.SS3.5.p4.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.2.m2.1b"><apply id="S2.SS3.5.p4.2.m2.1.1.cmml" xref="S2.SS3.5.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.5.p4.2.m2.1.1.1.cmml" xref="S2.SS3.5.p4.2.m2.1.1">superscript</csymbol><ci id="S2.SS3.5.p4.2.m2.1.1.2.cmml" xref="S2.SS3.5.p4.2.m2.1.1.2">𝜏</ci><times id="S2.SS3.5.p4.2.m2.1.1.3.cmml" xref="S2.SS3.5.p4.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.2.m2.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.2.m2.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, then at any perturbed point <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.5.p4.3.m3.1"><semantics id="S2.SS3.5.p4.3.m3.1a"><mi id="S2.SS3.5.p4.3.m3.1.1" xref="S2.SS3.5.p4.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.3.m3.1b"><ci id="S2.SS3.5.p4.3.m3.1.1.cmml" xref="S2.SS3.5.p4.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.3.m3.1d">italic_τ</annotation></semantics></math> to the left of <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S2.SS3.5.p4.4.m4.1"><semantics id="S2.SS3.5.p4.4.m4.1a"><msup id="S2.SS3.5.p4.4.m4.1.1" xref="S2.SS3.5.p4.4.m4.1.1.cmml"><mi id="S2.SS3.5.p4.4.m4.1.1.2" xref="S2.SS3.5.p4.4.m4.1.1.2.cmml">τ</mi><mo id="S2.SS3.5.p4.4.m4.1.1.3" xref="S2.SS3.5.p4.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.4.m4.1b"><apply id="S2.SS3.5.p4.4.m4.1.1.cmml" xref="S2.SS3.5.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.5.p4.4.m4.1.1.1.cmml" xref="S2.SS3.5.p4.4.m4.1.1">superscript</csymbol><ci id="S2.SS3.5.p4.4.m4.1.1.2.cmml" xref="S2.SS3.5.p4.4.m4.1.1.2">𝜏</ci><times id="S2.SS3.5.p4.4.m4.1.1.3.cmml" xref="S2.SS3.5.p4.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.4.m4.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.4.m4.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we have <math alttext="G(\tau)<1/2" class="ltx_Math" display="inline" id="S2.SS3.5.p4.5.m5.1"><semantics id="S2.SS3.5.p4.5.m5.1a"><mrow id="S2.SS3.5.p4.5.m5.1.2" xref="S2.SS3.5.p4.5.m5.1.2.cmml"><mrow id="S2.SS3.5.p4.5.m5.1.2.2" xref="S2.SS3.5.p4.5.m5.1.2.2.cmml"><mi id="S2.SS3.5.p4.5.m5.1.2.2.2" xref="S2.SS3.5.p4.5.m5.1.2.2.2.cmml">G</mi><mo id="S2.SS3.5.p4.5.m5.1.2.2.1" xref="S2.SS3.5.p4.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.5.p4.5.m5.1.2.2.3.2" xref="S2.SS3.5.p4.5.m5.1.2.2.cmml"><mo id="S2.SS3.5.p4.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.5.p4.5.m5.1.2.2.cmml">(</mo><mi id="S2.SS3.5.p4.5.m5.1.1" xref="S2.SS3.5.p4.5.m5.1.1.cmml">τ</mi><mo id="S2.SS3.5.p4.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.5.p4.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.5.p4.5.m5.1.2.1" xref="S2.SS3.5.p4.5.m5.1.2.1.cmml"><</mo><mrow id="S2.SS3.5.p4.5.m5.1.2.3" xref="S2.SS3.5.p4.5.m5.1.2.3.cmml"><mn id="S2.SS3.5.p4.5.m5.1.2.3.2" xref="S2.SS3.5.p4.5.m5.1.2.3.2.cmml">1</mn><mo id="S2.SS3.5.p4.5.m5.1.2.3.1" xref="S2.SS3.5.p4.5.m5.1.2.3.1.cmml">/</mo><mn id="S2.SS3.5.p4.5.m5.1.2.3.3" xref="S2.SS3.5.p4.5.m5.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.5.m5.1b"><apply id="S2.SS3.5.p4.5.m5.1.2.cmml" xref="S2.SS3.5.p4.5.m5.1.2"><lt id="S2.SS3.5.p4.5.m5.1.2.1.cmml" xref="S2.SS3.5.p4.5.m5.1.2.1"></lt><apply id="S2.SS3.5.p4.5.m5.1.2.2.cmml" xref="S2.SS3.5.p4.5.m5.1.2.2"><times id="S2.SS3.5.p4.5.m5.1.2.2.1.cmml" xref="S2.SS3.5.p4.5.m5.1.2.2.1"></times><ci id="S2.SS3.5.p4.5.m5.1.2.2.2.cmml" xref="S2.SS3.5.p4.5.m5.1.2.2.2">𝐺</ci><ci id="S2.SS3.5.p4.5.m5.1.1.cmml" xref="S2.SS3.5.p4.5.m5.1.1">𝜏</ci></apply><apply id="S2.SS3.5.p4.5.m5.1.2.3.cmml" xref="S2.SS3.5.p4.5.m5.1.2.3"><divide id="S2.SS3.5.p4.5.m5.1.2.3.1.cmml" xref="S2.SS3.5.p4.5.m5.1.2.3.1"></divide><cn id="S2.SS3.5.p4.5.m5.1.2.3.2.cmml" type="integer" xref="S2.SS3.5.p4.5.m5.1.2.3.2">1</cn><cn id="S2.SS3.5.p4.5.m5.1.2.3.3.cmml" type="integer" xref="S2.SS3.5.p4.5.m5.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.5.m5.1c">G(\tau)<1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.5.m5.1d">italic_G ( italic_τ ) < 1 / 2</annotation></semantics></math> and <math alttext="\dot{\tau}<0" class="ltx_Math" display="inline" id="S2.SS3.5.p4.6.m6.1"><semantics id="S2.SS3.5.p4.6.m6.1a"><mrow id="S2.SS3.5.p4.6.m6.1.1" xref="S2.SS3.5.p4.6.m6.1.1.cmml"><mover accent="true" id="S2.SS3.5.p4.6.m6.1.1.2" xref="S2.SS3.5.p4.6.m6.1.1.2.cmml"><mi id="S2.SS3.5.p4.6.m6.1.1.2.2" xref="S2.SS3.5.p4.6.m6.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.5.p4.6.m6.1.1.2.1" xref="S2.SS3.5.p4.6.m6.1.1.2.1.cmml">˙</mo></mover><mo id="S2.SS3.5.p4.6.m6.1.1.1" xref="S2.SS3.5.p4.6.m6.1.1.1.cmml"><</mo><mn id="S2.SS3.5.p4.6.m6.1.1.3" xref="S2.SS3.5.p4.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.6.m6.1b"><apply id="S2.SS3.5.p4.6.m6.1.1.cmml" xref="S2.SS3.5.p4.6.m6.1.1"><lt id="S2.SS3.5.p4.6.m6.1.1.1.cmml" xref="S2.SS3.5.p4.6.m6.1.1.1"></lt><apply id="S2.SS3.5.p4.6.m6.1.1.2.cmml" xref="S2.SS3.5.p4.6.m6.1.1.2"><ci id="S2.SS3.5.p4.6.m6.1.1.2.1.cmml" xref="S2.SS3.5.p4.6.m6.1.1.2.1">˙</ci><ci id="S2.SS3.5.p4.6.m6.1.1.2.2.cmml" xref="S2.SS3.5.p4.6.m6.1.1.2.2">𝜏</ci></apply><cn id="S2.SS3.5.p4.6.m6.1.1.3.cmml" type="integer" xref="S2.SS3.5.p4.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.6.m6.1c">\dot{\tau}<0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.6.m6.1d">over˙ start_ARG italic_τ end_ARG < 0</annotation></semantics></math>; and at any perturbed point <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.5.p4.7.m7.1"><semantics id="S2.SS3.5.p4.7.m7.1a"><mi id="S2.SS3.5.p4.7.m7.1.1" xref="S2.SS3.5.p4.7.m7.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.7.m7.1b"><ci id="S2.SS3.5.p4.7.m7.1.1.cmml" xref="S2.SS3.5.p4.7.m7.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.7.m7.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.7.m7.1d">italic_τ</annotation></semantics></math> to the right of <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S2.SS3.5.p4.8.m8.1"><semantics id="S2.SS3.5.p4.8.m8.1a"><msup id="S2.SS3.5.p4.8.m8.1.1" xref="S2.SS3.5.p4.8.m8.1.1.cmml"><mi id="S2.SS3.5.p4.8.m8.1.1.2" xref="S2.SS3.5.p4.8.m8.1.1.2.cmml">τ</mi><mo id="S2.SS3.5.p4.8.m8.1.1.3" xref="S2.SS3.5.p4.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.8.m8.1b"><apply id="S2.SS3.5.p4.8.m8.1.1.cmml" xref="S2.SS3.5.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.5.p4.8.m8.1.1.1.cmml" xref="S2.SS3.5.p4.8.m8.1.1">superscript</csymbol><ci id="S2.SS3.5.p4.8.m8.1.1.2.cmml" xref="S2.SS3.5.p4.8.m8.1.1.2">𝜏</ci><times id="S2.SS3.5.p4.8.m8.1.1.3.cmml" xref="S2.SS3.5.p4.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.8.m8.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.8.m8.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we have <math alttext="G(\tau)>1/2" class="ltx_Math" display="inline" id="S2.SS3.5.p4.9.m9.1"><semantics id="S2.SS3.5.p4.9.m9.1a"><mrow id="S2.SS3.5.p4.9.m9.1.2" xref="S2.SS3.5.p4.9.m9.1.2.cmml"><mrow id="S2.SS3.5.p4.9.m9.1.2.2" xref="S2.SS3.5.p4.9.m9.1.2.2.cmml"><mi id="S2.SS3.5.p4.9.m9.1.2.2.2" xref="S2.SS3.5.p4.9.m9.1.2.2.2.cmml">G</mi><mo id="S2.SS3.5.p4.9.m9.1.2.2.1" xref="S2.SS3.5.p4.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.5.p4.9.m9.1.2.2.3.2" xref="S2.SS3.5.p4.9.m9.1.2.2.cmml"><mo id="S2.SS3.5.p4.9.m9.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.5.p4.9.m9.1.2.2.cmml">(</mo><mi id="S2.SS3.5.p4.9.m9.1.1" xref="S2.SS3.5.p4.9.m9.1.1.cmml">τ</mi><mo id="S2.SS3.5.p4.9.m9.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.5.p4.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.5.p4.9.m9.1.2.1" xref="S2.SS3.5.p4.9.m9.1.2.1.cmml">></mo><mrow id="S2.SS3.5.p4.9.m9.1.2.3" xref="S2.SS3.5.p4.9.m9.1.2.3.cmml"><mn id="S2.SS3.5.p4.9.m9.1.2.3.2" xref="S2.SS3.5.p4.9.m9.1.2.3.2.cmml">1</mn><mo id="S2.SS3.5.p4.9.m9.1.2.3.1" xref="S2.SS3.5.p4.9.m9.1.2.3.1.cmml">/</mo><mn id="S2.SS3.5.p4.9.m9.1.2.3.3" xref="S2.SS3.5.p4.9.m9.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.9.m9.1b"><apply id="S2.SS3.5.p4.9.m9.1.2.cmml" xref="S2.SS3.5.p4.9.m9.1.2"><gt id="S2.SS3.5.p4.9.m9.1.2.1.cmml" xref="S2.SS3.5.p4.9.m9.1.2.1"></gt><apply id="S2.SS3.5.p4.9.m9.1.2.2.cmml" xref="S2.SS3.5.p4.9.m9.1.2.2"><times id="S2.SS3.5.p4.9.m9.1.2.2.1.cmml" xref="S2.SS3.5.p4.9.m9.1.2.2.1"></times><ci id="S2.SS3.5.p4.9.m9.1.2.2.2.cmml" xref="S2.SS3.5.p4.9.m9.1.2.2.2">𝐺</ci><ci id="S2.SS3.5.p4.9.m9.1.1.cmml" xref="S2.SS3.5.p4.9.m9.1.1">𝜏</ci></apply><apply id="S2.SS3.5.p4.9.m9.1.2.3.cmml" xref="S2.SS3.5.p4.9.m9.1.2.3"><divide id="S2.SS3.5.p4.9.m9.1.2.3.1.cmml" xref="S2.SS3.5.p4.9.m9.1.2.3.1"></divide><cn id="S2.SS3.5.p4.9.m9.1.2.3.2.cmml" type="integer" xref="S2.SS3.5.p4.9.m9.1.2.3.2">1</cn><cn id="S2.SS3.5.p4.9.m9.1.2.3.3.cmml" type="integer" xref="S2.SS3.5.p4.9.m9.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.9.m9.1c">G(\tau)>1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.9.m9.1d">italic_G ( italic_τ ) > 1 / 2</annotation></semantics></math> and <math alttext="\dot{\tau}>0" class="ltx_Math" display="inline" id="S2.SS3.5.p4.10.m10.1"><semantics id="S2.SS3.5.p4.10.m10.1a"><mrow id="S2.SS3.5.p4.10.m10.1.1" xref="S2.SS3.5.p4.10.m10.1.1.cmml"><mover accent="true" id="S2.SS3.5.p4.10.m10.1.1.2" xref="S2.SS3.5.p4.10.m10.1.1.2.cmml"><mi id="S2.SS3.5.p4.10.m10.1.1.2.2" xref="S2.SS3.5.p4.10.m10.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.5.p4.10.m10.1.1.2.1" xref="S2.SS3.5.p4.10.m10.1.1.2.1.cmml">˙</mo></mover><mo id="S2.SS3.5.p4.10.m10.1.1.1" xref="S2.SS3.5.p4.10.m10.1.1.1.cmml">></mo><mn id="S2.SS3.5.p4.10.m10.1.1.3" xref="S2.SS3.5.p4.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.10.m10.1b"><apply id="S2.SS3.5.p4.10.m10.1.1.cmml" xref="S2.SS3.5.p4.10.m10.1.1"><gt id="S2.SS3.5.p4.10.m10.1.1.1.cmml" xref="S2.SS3.5.p4.10.m10.1.1.1"></gt><apply id="S2.SS3.5.p4.10.m10.1.1.2.cmml" xref="S2.SS3.5.p4.10.m10.1.1.2"><ci id="S2.SS3.5.p4.10.m10.1.1.2.1.cmml" xref="S2.SS3.5.p4.10.m10.1.1.2.1">˙</ci><ci id="S2.SS3.5.p4.10.m10.1.1.2.2.cmml" xref="S2.SS3.5.p4.10.m10.1.1.2.2">𝜏</ci></apply><cn id="S2.SS3.5.p4.10.m10.1.1.3.cmml" type="integer" xref="S2.SS3.5.p4.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.10.m10.1c">\dot{\tau}>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.10.m10.1d">over˙ start_ARG italic_τ end_ARG > 0</annotation></semantics></math>. It follows that <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S2.SS3.5.p4.11.m11.1"><semantics id="S2.SS3.5.p4.11.m11.1a"><msup id="S2.SS3.5.p4.11.m11.1.1" xref="S2.SS3.5.p4.11.m11.1.1.cmml"><mi id="S2.SS3.5.p4.11.m11.1.1.2" xref="S2.SS3.5.p4.11.m11.1.1.2.cmml">τ</mi><mo id="S2.SS3.5.p4.11.m11.1.1.3" xref="S2.SS3.5.p4.11.m11.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.5.p4.11.m11.1b"><apply id="S2.SS3.5.p4.11.m11.1.1.cmml" xref="S2.SS3.5.p4.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS3.5.p4.11.m11.1.1.1.cmml" xref="S2.SS3.5.p4.11.m11.1.1">superscript</csymbol><ci id="S2.SS3.5.p4.11.m11.1.1.2.cmml" xref="S2.SS3.5.p4.11.m11.1.1.2">𝜏</ci><times id="S2.SS3.5.p4.11.m11.1.1.3.cmml" xref="S2.SS3.5.p4.11.m11.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.5.p4.11.m11.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.5.p4.11.m11.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is unstable. The same logic implies stability in the strictly decreasing case. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS3.p7"> <p class="ltx_p" id="S2.SS3.p7.9">In most reasonable settings, one would expect <math alttext="G(x)" class="ltx_Math" display="inline" id="S2.SS3.p7.1.m1.1"><semantics id="S2.SS3.p7.1.m1.1a"><mrow id="S2.SS3.p7.1.m1.1.2" xref="S2.SS3.p7.1.m1.1.2.cmml"><mi id="S2.SS3.p7.1.m1.1.2.2" xref="S2.SS3.p7.1.m1.1.2.2.cmml">G</mi><mo id="S2.SS3.p7.1.m1.1.2.1" xref="S2.SS3.p7.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.p7.1.m1.1.2.3.2" xref="S2.SS3.p7.1.m1.1.2.cmml"><mo id="S2.SS3.p7.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.p7.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.p7.1.m1.1.1" xref="S2.SS3.p7.1.m1.1.1.cmml">x</mi><mo id="S2.SS3.p7.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.p7.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.1.m1.1b"><apply id="S2.SS3.p7.1.m1.1.2.cmml" xref="S2.SS3.p7.1.m1.1.2"><times id="S2.SS3.p7.1.m1.1.2.1.cmml" xref="S2.SS3.p7.1.m1.1.2.1"></times><ci id="S2.SS3.p7.1.m1.1.2.2.cmml" xref="S2.SS3.p7.1.m1.1.2.2">𝐺</ci><ci id="S2.SS3.p7.1.m1.1.1.cmml" xref="S2.SS3.p7.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.1.m1.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.1.m1.1d">italic_G ( italic_x )</annotation></semantics></math> to limit toward 1 as <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.2.m2.1"><semantics id="S2.SS3.p7.2.m2.1a"><mi id="S2.SS3.p7.2.m2.1.1" xref="S2.SS3.p7.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.2.m2.1b"><ci id="S2.SS3.p7.2.m2.1.1.cmml" xref="S2.SS3.p7.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.2.m2.1d">italic_x</annotation></semantics></math> approaches <math alttext="\infty" class="ltx_Math" display="inline" id="S2.SS3.p7.3.m3.1"><semantics id="S2.SS3.p7.3.m3.1a"><mi id="S2.SS3.p7.3.m3.1.1" mathvariant="normal" xref="S2.SS3.p7.3.m3.1.1.cmml">∞</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.3.m3.1b"><infinity id="S2.SS3.p7.3.m3.1.1.cmml" xref="S2.SS3.p7.3.m3.1.1"></infinity></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.3.m3.1c">\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.3.m3.1d">∞</annotation></semantics></math>, and toward 0 as <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.4.m4.1"><semantics id="S2.SS3.p7.4.m4.1a"><mi id="S2.SS3.p7.4.m4.1.1" xref="S2.SS3.p7.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.4.m4.1b"><ci id="S2.SS3.p7.4.m4.1.1.cmml" xref="S2.SS3.p7.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.4.m4.1d">italic_x</annotation></semantics></math> approaches <math alttext="-\infty" class="ltx_Math" display="inline" id="S2.SS3.p7.5.m5.1"><semantics id="S2.SS3.p7.5.m5.1a"><mrow id="S2.SS3.p7.5.m5.1.1" xref="S2.SS3.p7.5.m5.1.1.cmml"><mo id="S2.SS3.p7.5.m5.1.1a" xref="S2.SS3.p7.5.m5.1.1.cmml">−</mo><mi id="S2.SS3.p7.5.m5.1.1.2" mathvariant="normal" xref="S2.SS3.p7.5.m5.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.5.m5.1b"><apply id="S2.SS3.p7.5.m5.1.1.cmml" xref="S2.SS3.p7.5.m5.1.1"><minus id="S2.SS3.p7.5.m5.1.1.1.cmml" xref="S2.SS3.p7.5.m5.1.1"></minus><infinity id="S2.SS3.p7.5.m5.1.1.2.cmml" xref="S2.SS3.p7.5.m5.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.5.m5.1c">-\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.5.m5.1d">- ∞</annotation></semantics></math>. Specifically, as a signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.6.m6.1"><semantics id="S2.SS3.p7.6.m6.1a"><mi id="S2.SS3.p7.6.m6.1.1" xref="S2.SS3.p7.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.6.m6.1b"><ci id="S2.SS3.p7.6.m6.1.1.cmml" xref="S2.SS3.p7.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.6.m6.1d">italic_x</annotation></semantics></math> becomes increasingly small, the probability another agent receives a signal smaller than <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.7.m7.1"><semantics id="S2.SS3.p7.7.m7.1a"><mi id="S2.SS3.p7.7.m7.1.1" xref="S2.SS3.p7.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.7.m7.1b"><ci id="S2.SS3.p7.7.m7.1.1.cmml" xref="S2.SS3.p7.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.7.m7.1d">italic_x</annotation></semantics></math> should approach 0. Meanwhile, as a signal <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.8.m8.1"><semantics id="S2.SS3.p7.8.m8.1a"><mi id="S2.SS3.p7.8.m8.1.1" xref="S2.SS3.p7.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.8.m8.1b"><ci id="S2.SS3.p7.8.m8.1.1.cmml" xref="S2.SS3.p7.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.8.m8.1d">italic_x</annotation></semantics></math> grows large, the probability another agent receives a signal smaller than <math alttext="x" class="ltx_Math" display="inline" id="S2.SS3.p7.9.m9.1"><semantics id="S2.SS3.p7.9.m9.1a"><mi id="S2.SS3.p7.9.m9.1.1" xref="S2.SS3.p7.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.9.m9.1b"><ci id="S2.SS3.p7.9.m9.1.1.cmml" xref="S2.SS3.p7.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.9.m9.1d">italic_x</annotation></semantics></math> should approach 1. As the quality of essay increases in a normally distributed class, for example, a grader would believe that the mass of essays below that quality should increase proportionally.</p> </div> <div class="ltx_para" id="S2.SS3.p8"> <p class="ltx_p" id="S2.SS3.p8.3">We show formally in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A1" title="Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">A</span></a> that under such limit behavior of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS3.p8.1.m1.1"><semantics id="S2.SS3.p8.1.m1.1a"><mi id="S2.SS3.p8.1.m1.1.1" xref="S2.SS3.p8.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p8.1.m1.1b"><ci id="S2.SS3.p8.1.m1.1.1.cmml" xref="S2.SS3.p8.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p8.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p8.1.m1.1d">italic_G</annotation></semantics></math>, uninformative equilibria at <math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="S2.SS3.p8.2.m2.1"><semantics id="S2.SS3.p8.2.m2.1a"><mrow id="S2.SS3.p8.2.m2.1.1" xref="S2.SS3.p8.2.m2.1.1.cmml"><msup id="S2.SS3.p8.2.m2.1.1.2" xref="S2.SS3.p8.2.m2.1.1.2.cmml"><mi id="S2.SS3.p8.2.m2.1.1.2.2" xref="S2.SS3.p8.2.m2.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.p8.2.m2.1.1.2.3" xref="S2.SS3.p8.2.m2.1.1.2.3.cmml">∗</mo></msup><mo id="S2.SS3.p8.2.m2.1.1.1" xref="S2.SS3.p8.2.m2.1.1.1.cmml">=</mo><mrow id="S2.SS3.p8.2.m2.1.1.3" xref="S2.SS3.p8.2.m2.1.1.3.cmml"><mo id="S2.SS3.p8.2.m2.1.1.3a" xref="S2.SS3.p8.2.m2.1.1.3.cmml">±</mo><mi id="S2.SS3.p8.2.m2.1.1.3.2" mathvariant="normal" xref="S2.SS3.p8.2.m2.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p8.2.m2.1b"><apply id="S2.SS3.p8.2.m2.1.1.cmml" xref="S2.SS3.p8.2.m2.1.1"><eq id="S2.SS3.p8.2.m2.1.1.1.cmml" xref="S2.SS3.p8.2.m2.1.1.1"></eq><apply id="S2.SS3.p8.2.m2.1.1.2.cmml" xref="S2.SS3.p8.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p8.2.m2.1.1.2.1.cmml" xref="S2.SS3.p8.2.m2.1.1.2">superscript</csymbol><ci id="S2.SS3.p8.2.m2.1.1.2.2.cmml" xref="S2.SS3.p8.2.m2.1.1.2.2">𝜏</ci><times id="S2.SS3.p8.2.m2.1.1.2.3.cmml" xref="S2.SS3.p8.2.m2.1.1.2.3"></times></apply><apply id="S2.SS3.p8.2.m2.1.1.3.cmml" xref="S2.SS3.p8.2.m2.1.1.3"><csymbol cd="latexml" id="S2.SS3.p8.2.m2.1.1.3.1.cmml" xref="S2.SS3.p8.2.m2.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS3.p8.2.m2.1.1.3.2.cmml" xref="S2.SS3.p8.2.m2.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p8.2.m2.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p8.2.m2.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S2.SS3.p8.3.1">stable</em> in OA, while at least one non-infinite equilibrium exists which is unstable. In fact, one would expect in smooth, unimodal settings, as in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a>, that <math alttext="G" class="ltx_Math" display="inline" id="S2.SS3.p8.3.m3.1"><semantics id="S2.SS3.p8.3.m3.1a"><mi id="S2.SS3.p8.3.m3.1.1" xref="S2.SS3.p8.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p8.3.m3.1b"><ci id="S2.SS3.p8.3.m3.1.1.cmml" xref="S2.SS3.p8.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p8.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p8.3.m3.1d">italic_G</annotation></semantics></math> is monotone increasing. In this case, there are three equilibria, and by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem2" title="Theorem 2. ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, the internal equilibrium is unstable while the uninformative equilibria are stable.</p> </div> <div class="ltx_para" id="S2.SS3.p9"> <p class="ltx_p" id="S2.SS3.p9.4">So what does this instability of internal equilibria mean for OA? When signals correspond monotonically to quality of an essay or task, agents will naturally move toward uninformative equilibria. These dynamics only further the fragility of truthfulness in OA: unless the threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS3.p9.1.m1.1"><semantics id="S2.SS3.p9.1.m1.1a"><mi id="S2.SS3.p9.1.m1.1.1" xref="S2.SS3.p9.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.1.m1.1b"><ci id="S2.SS3.p9.1.m1.1.1.cmml" xref="S2.SS3.p9.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.1.m1.1d">italic_τ</annotation></semantics></math> exactly balances out the conditional probabilities that other agents will report <math alttext="H" class="ltx_Math" display="inline" id="S2.SS3.p9.2.m2.1"><semantics id="S2.SS3.p9.2.m2.1a"><mi id="S2.SS3.p9.2.m2.1.1" xref="S2.SS3.p9.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.2.m2.1b"><ci id="S2.SS3.p9.2.m2.1.1.cmml" xref="S2.SS3.p9.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.2.m2.1d">italic_H</annotation></semantics></math> or <math alttext="L" class="ltx_Math" display="inline" id="S2.SS3.p9.3.m3.1"><semantics id="S2.SS3.p9.3.m3.1a"><mi id="S2.SS3.p9.3.m3.1.1" xref="S2.SS3.p9.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.3.m3.1b"><ci id="S2.SS3.p9.3.m3.1.1.cmml" xref="S2.SS3.p9.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.3.m3.1d">italic_L</annotation></semantics></math>, agents will inevitably and increasingly misreport until they reach an uninformative consensus. There are multimodal Bayesian examples where several internal equilibria exist and some are thus topologically required to be locally stable (see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6" title="6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>). However, as long as <math alttext="G" class="ltx_Math" display="inline" id="S2.SS3.p9.4.m4.1"><semantics id="S2.SS3.p9.4.m4.1a"><mi id="S2.SS3.p9.4.m4.1.1" xref="S2.SS3.p9.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p9.4.m4.1b"><ci id="S2.SS3.p9.4.m4.1.1.cmml" xref="S2.SS3.p9.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p9.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p9.4.m4.1d">italic_G</annotation></semantics></math> remains monotone increasing at extreme signals, we expect uninformative equilibria to be stable by our analysis in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A1" title="Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">A</span></a>.</p> </div> <figure class="ltx_figure" id="S2.F2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top" id="S2.F2.9" style="width:195.1pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S2.F2.1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F2.9.9.5.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S2.F2.9.8.4" style="font-size:90%;">Plots of <math alttext="F" class="ltx_Math" display="inline" id="S2.F2.6.5.1.m1.1"><semantics id="S2.F2.6.5.1.m1.1b"><mi id="S2.F2.6.5.1.m1.1.1" xref="S2.F2.6.5.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S2.F2.6.5.1.m1.1c"><ci id="S2.F2.6.5.1.m1.1.1.cmml" xref="S2.F2.6.5.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.6.5.1.m1.1d">F</annotation><annotation encoding="application/x-llamapun" id="S2.F2.6.5.1.m1.1e">italic_F</annotation></semantics></math>, <math alttext="G" class="ltx_Math" display="inline" id="S2.F2.7.6.2.m2.1"><semantics id="S2.F2.7.6.2.m2.1b"><mi id="S2.F2.7.6.2.m2.1.1" xref="S2.F2.7.6.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.F2.7.6.2.m2.1c"><ci id="S2.F2.7.6.2.m2.1.1.cmml" xref="S2.F2.7.6.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.7.6.2.m2.1d">G</annotation><annotation encoding="application/x-llamapun" id="S2.F2.7.6.2.m2.1e">italic_G</annotation></semantics></math>, and <math alttext="Q" class="ltx_Math" display="inline" id="S2.F2.8.7.3.m3.1"><semantics id="S2.F2.8.7.3.m3.1b"><mi id="S2.F2.8.7.3.m3.1.1" xref="S2.F2.8.7.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.F2.8.7.3.m3.1c"><ci id="S2.F2.8.7.3.m3.1.1.cmml" xref="S2.F2.8.7.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.8.7.3.m3.1d">Q</annotation><annotation encoding="application/x-llamapun" id="S2.F2.8.7.3.m3.1e">italic_Q</annotation></semantics></math> in the Gaussian model (§ <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a>) for the parameters <math alttext="a=b=1" class="ltx_Math" display="inline" id="S2.F2.9.8.4.m4.1"><semantics id="S2.F2.9.8.4.m4.1b"><mrow id="S2.F2.9.8.4.m4.1.1" xref="S2.F2.9.8.4.m4.1.1.cmml"><mi id="S2.F2.9.8.4.m4.1.1.2" xref="S2.F2.9.8.4.m4.1.1.2.cmml">a</mi><mo id="S2.F2.9.8.4.m4.1.1.3" xref="S2.F2.9.8.4.m4.1.1.3.cmml">=</mo><mi id="S2.F2.9.8.4.m4.1.1.4" xref="S2.F2.9.8.4.m4.1.1.4.cmml">b</mi><mo id="S2.F2.9.8.4.m4.1.1.5" xref="S2.F2.9.8.4.m4.1.1.5.cmml">=</mo><mn id="S2.F2.9.8.4.m4.1.1.6" xref="S2.F2.9.8.4.m4.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.9.8.4.m4.1c"><apply id="S2.F2.9.8.4.m4.1.1.cmml" xref="S2.F2.9.8.4.m4.1.1"><and id="S2.F2.9.8.4.m4.1.1a.cmml" xref="S2.F2.9.8.4.m4.1.1"></and><apply id="S2.F2.9.8.4.m4.1.1b.cmml" xref="S2.F2.9.8.4.m4.1.1"><eq id="S2.F2.9.8.4.m4.1.1.3.cmml" xref="S2.F2.9.8.4.m4.1.1.3"></eq><ci id="S2.F2.9.8.4.m4.1.1.2.cmml" xref="S2.F2.9.8.4.m4.1.1.2">𝑎</ci><ci id="S2.F2.9.8.4.m4.1.1.4.cmml" xref="S2.F2.9.8.4.m4.1.1.4">𝑏</ci></apply><apply id="S2.F2.9.8.4.m4.1.1c.cmml" xref="S2.F2.9.8.4.m4.1.1"><eq id="S2.F2.9.8.4.m4.1.1.5.cmml" xref="S2.F2.9.8.4.m4.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.F2.9.8.4.m4.1.1.4.cmml" id="S2.F2.9.8.4.m4.1.1d.cmml" xref="S2.F2.9.8.4.m4.1.1"></share><cn id="S2.F2.9.8.4.m4.1.1.6.cmml" type="integer" xref="S2.F2.9.8.4.m4.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.9.8.4.m4.1d">a=b=1</annotation><annotation encoding="application/x-llamapun" id="S2.F2.9.8.4.m4.1e">italic_a = italic_b = 1</annotation></semantics></math>.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top" id="S2.F2.18" style="width:195.1pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S2.F2.10.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F2.18.9.5.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S2.F2.18.8.4" style="font-size:90%;">Plot of <math alttext="G-F" class="ltx_Math" display="inline" id="S2.F2.15.5.1.m1.1"><semantics id="S2.F2.15.5.1.m1.1b"><mrow id="S2.F2.15.5.1.m1.1.1" xref="S2.F2.15.5.1.m1.1.1.cmml"><mi id="S2.F2.15.5.1.m1.1.1.2" xref="S2.F2.15.5.1.m1.1.1.2.cmml">G</mi><mo id="S2.F2.15.5.1.m1.1.1.1" xref="S2.F2.15.5.1.m1.1.1.1.cmml">−</mo><mi id="S2.F2.15.5.1.m1.1.1.3" xref="S2.F2.15.5.1.m1.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.15.5.1.m1.1c"><apply id="S2.F2.15.5.1.m1.1.1.cmml" xref="S2.F2.15.5.1.m1.1.1"><minus id="S2.F2.15.5.1.m1.1.1.1.cmml" xref="S2.F2.15.5.1.m1.1.1.1"></minus><ci id="S2.F2.15.5.1.m1.1.1.2.cmml" xref="S2.F2.15.5.1.m1.1.1.2">𝐺</ci><ci id="S2.F2.15.5.1.m1.1.1.3.cmml" xref="S2.F2.15.5.1.m1.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.15.5.1.m1.1d">G-F</annotation><annotation encoding="application/x-llamapun" id="S2.F2.15.5.1.m1.1e">italic_G - italic_F</annotation></semantics></math> in the Gaussian model (§ <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a>), for parameters <math alttext="a=b=1" class="ltx_Math" display="inline" id="S2.F2.16.6.2.m2.1"><semantics id="S2.F2.16.6.2.m2.1b"><mrow id="S2.F2.16.6.2.m2.1.1" xref="S2.F2.16.6.2.m2.1.1.cmml"><mi id="S2.F2.16.6.2.m2.1.1.2" xref="S2.F2.16.6.2.m2.1.1.2.cmml">a</mi><mo id="S2.F2.16.6.2.m2.1.1.3" xref="S2.F2.16.6.2.m2.1.1.3.cmml">=</mo><mi id="S2.F2.16.6.2.m2.1.1.4" xref="S2.F2.16.6.2.m2.1.1.4.cmml">b</mi><mo id="S2.F2.16.6.2.m2.1.1.5" xref="S2.F2.16.6.2.m2.1.1.5.cmml">=</mo><mn id="S2.F2.16.6.2.m2.1.1.6" xref="S2.F2.16.6.2.m2.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.16.6.2.m2.1c"><apply id="S2.F2.16.6.2.m2.1.1.cmml" xref="S2.F2.16.6.2.m2.1.1"><and id="S2.F2.16.6.2.m2.1.1a.cmml" xref="S2.F2.16.6.2.m2.1.1"></and><apply id="S2.F2.16.6.2.m2.1.1b.cmml" xref="S2.F2.16.6.2.m2.1.1"><eq id="S2.F2.16.6.2.m2.1.1.3.cmml" xref="S2.F2.16.6.2.m2.1.1.3"></eq><ci id="S2.F2.16.6.2.m2.1.1.2.cmml" xref="S2.F2.16.6.2.m2.1.1.2">𝑎</ci><ci id="S2.F2.16.6.2.m2.1.1.4.cmml" xref="S2.F2.16.6.2.m2.1.1.4">𝑏</ci></apply><apply id="S2.F2.16.6.2.m2.1.1c.cmml" xref="S2.F2.16.6.2.m2.1.1"><eq id="S2.F2.16.6.2.m2.1.1.5.cmml" xref="S2.F2.16.6.2.m2.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.F2.16.6.2.m2.1.1.4.cmml" id="S2.F2.16.6.2.m2.1.1d.cmml" xref="S2.F2.16.6.2.m2.1.1"></share><cn id="S2.F2.16.6.2.m2.1.1.6.cmml" type="integer" xref="S2.F2.16.6.2.m2.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.16.6.2.m2.1d">a=b=1</annotation><annotation encoding="application/x-llamapun" id="S2.F2.16.6.2.m2.1e">italic_a = italic_b = 1</annotation></semantics></math>. <math alttext="G(x)-F(x)" class="ltx_Math" display="inline" id="S2.F2.17.7.3.m3.2"><semantics id="S2.F2.17.7.3.m3.2b"><mrow id="S2.F2.17.7.3.m3.2.3" xref="S2.F2.17.7.3.m3.2.3.cmml"><mrow id="S2.F2.17.7.3.m3.2.3.2" xref="S2.F2.17.7.3.m3.2.3.2.cmml"><mi id="S2.F2.17.7.3.m3.2.3.2.2" xref="S2.F2.17.7.3.m3.2.3.2.2.cmml">G</mi><mo id="S2.F2.17.7.3.m3.2.3.2.1" xref="S2.F2.17.7.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S2.F2.17.7.3.m3.2.3.2.3.2" xref="S2.F2.17.7.3.m3.2.3.2.cmml"><mo id="S2.F2.17.7.3.m3.2.3.2.3.2.1" stretchy="false" 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id="S2.F2.17.7.3.m3.2.3.1.cmml" xref="S2.F2.17.7.3.m3.2.3.1"></minus><apply id="S2.F2.17.7.3.m3.2.3.2.cmml" xref="S2.F2.17.7.3.m3.2.3.2"><times id="S2.F2.17.7.3.m3.2.3.2.1.cmml" xref="S2.F2.17.7.3.m3.2.3.2.1"></times><ci id="S2.F2.17.7.3.m3.2.3.2.2.cmml" xref="S2.F2.17.7.3.m3.2.3.2.2">𝐺</ci><ci id="S2.F2.17.7.3.m3.1.1.cmml" xref="S2.F2.17.7.3.m3.1.1">𝑥</ci></apply><apply id="S2.F2.17.7.3.m3.2.3.3.cmml" xref="S2.F2.17.7.3.m3.2.3.3"><times id="S2.F2.17.7.3.m3.2.3.3.1.cmml" xref="S2.F2.17.7.3.m3.2.3.3.1"></times><ci id="S2.F2.17.7.3.m3.2.3.3.2.cmml" xref="S2.F2.17.7.3.m3.2.3.3.2">𝐹</ci><ci id="S2.F2.17.7.3.m3.2.2.cmml" xref="S2.F2.17.7.3.m3.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.17.7.3.m3.2d">G(x)-F(x)</annotation><annotation encoding="application/x-llamapun" id="S2.F2.17.7.3.m3.2e">italic_G ( italic_x ) - italic_F ( italic_x )</annotation></semantics></math> is decreasing at <math alttext="x=0" class="ltx_Math" display="inline" id="S2.F2.18.8.4.m4.1"><semantics id="S2.F2.18.8.4.m4.1b"><mrow id="S2.F2.18.8.4.m4.1.1" xref="S2.F2.18.8.4.m4.1.1.cmml"><mi id="S2.F2.18.8.4.m4.1.1.2" xref="S2.F2.18.8.4.m4.1.1.2.cmml">x</mi><mo id="S2.F2.18.8.4.m4.1.1.1" xref="S2.F2.18.8.4.m4.1.1.1.cmml">=</mo><mn id="S2.F2.18.8.4.m4.1.1.3" xref="S2.F2.18.8.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.18.8.4.m4.1c"><apply id="S2.F2.18.8.4.m4.1.1.cmml" xref="S2.F2.18.8.4.m4.1.1"><eq id="S2.F2.18.8.4.m4.1.1.1.cmml" xref="S2.F2.18.8.4.m4.1.1.1"></eq><ci id="S2.F2.18.8.4.m4.1.1.2.cmml" xref="S2.F2.18.8.4.m4.1.1.2">𝑥</ci><cn id="S2.F2.18.8.4.m4.1.1.3.cmml" type="integer" xref="S2.F2.18.8.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.18.8.4.m4.1d">x=0</annotation><annotation encoding="application/x-llamapun" id="S2.F2.18.8.4.m4.1e">italic_x = 0</annotation></semantics></math>, meaning the corresponding equilibrium in DG is stable under the dynamics.</span></figcaption> </figure> </div> </div> </figure> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4 </span>A Gaussian Model</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.8">Suppose each agent <math alttext="i" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.1"><semantics id="S2.SS4.p1.1.m1.1a"><mi id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.1b"><ci id="S2.SS4.p1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.1d">italic_i</annotation></semantics></math> receives a noisy signal <math alttext="X_{i}" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.1"><semantics id="S2.SS4.p1.2.m2.1a"><msub id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml"><mi id="S2.SS4.p1.2.m2.1.1.2" xref="S2.SS4.p1.2.m2.1.1.2.cmml">X</mi><mi id="S2.SS4.p1.2.m2.1.1.3" xref="S2.SS4.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.1b"><apply id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.2.m2.1.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS4.p1.2.m2.1.1.2.cmml" xref="S2.SS4.p1.2.m2.1.1.2">𝑋</ci><ci id="S2.SS4.p1.2.m2.1.1.3.cmml" xref="S2.SS4.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from a Gaussian distribution, with the noise also normally distributed. That is, <math alttext="X_{i}=aZ+bZ_{i}" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.1"><semantics id="S2.SS4.p1.3.m3.1a"><mrow id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml"><msub id="S2.SS4.p1.3.m3.1.1.2" xref="S2.SS4.p1.3.m3.1.1.2.cmml"><mi id="S2.SS4.p1.3.m3.1.1.2.2" xref="S2.SS4.p1.3.m3.1.1.2.2.cmml">X</mi><mi id="S2.SS4.p1.3.m3.1.1.2.3" xref="S2.SS4.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS4.p1.3.m3.1.1.1" xref="S2.SS4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS4.p1.3.m3.1.1.3" xref="S2.SS4.p1.3.m3.1.1.3.cmml"><mrow id="S2.SS4.p1.3.m3.1.1.3.2" xref="S2.SS4.p1.3.m3.1.1.3.2.cmml"><mi id="S2.SS4.p1.3.m3.1.1.3.2.2" xref="S2.SS4.p1.3.m3.1.1.3.2.2.cmml">a</mi><mo id="S2.SS4.p1.3.m3.1.1.3.2.1" xref="S2.SS4.p1.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS4.p1.3.m3.1.1.3.2.3" xref="S2.SS4.p1.3.m3.1.1.3.2.3.cmml">Z</mi></mrow><mo id="S2.SS4.p1.3.m3.1.1.3.1" xref="S2.SS4.p1.3.m3.1.1.3.1.cmml">+</mo><mrow id="S2.SS4.p1.3.m3.1.1.3.3" xref="S2.SS4.p1.3.m3.1.1.3.3.cmml"><mi id="S2.SS4.p1.3.m3.1.1.3.3.2" xref="S2.SS4.p1.3.m3.1.1.3.3.2.cmml">b</mi><mo id="S2.SS4.p1.3.m3.1.1.3.3.1" xref="S2.SS4.p1.3.m3.1.1.3.3.1.cmml">⁢</mo><msub id="S2.SS4.p1.3.m3.1.1.3.3.3" xref="S2.SS4.p1.3.m3.1.1.3.3.3.cmml"><mi id="S2.SS4.p1.3.m3.1.1.3.3.3.2" xref="S2.SS4.p1.3.m3.1.1.3.3.3.2.cmml">Z</mi><mi id="S2.SS4.p1.3.m3.1.1.3.3.3.3" xref="S2.SS4.p1.3.m3.1.1.3.3.3.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.1b"><apply id="S2.SS4.p1.3.m3.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1"><eq id="S2.SS4.p1.3.m3.1.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1.1"></eq><apply id="S2.SS4.p1.3.m3.1.1.2.cmml" xref="S2.SS4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.p1.3.m3.1.1.2.1.cmml" xref="S2.SS4.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS4.p1.3.m3.1.1.2.2.cmml" xref="S2.SS4.p1.3.m3.1.1.2.2">𝑋</ci><ci id="S2.SS4.p1.3.m3.1.1.2.3.cmml" xref="S2.SS4.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S2.SS4.p1.3.m3.1.1.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3"><plus id="S2.SS4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3.1"></plus><apply id="S2.SS4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2"><times id="S2.SS4.p1.3.m3.1.1.3.2.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2.1"></times><ci id="S2.SS4.p1.3.m3.1.1.3.2.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2.2">𝑎</ci><ci id="S2.SS4.p1.3.m3.1.1.3.2.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2.3">𝑍</ci></apply><apply id="S2.SS4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3"><times id="S2.SS4.p1.3.m3.1.1.3.3.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.1"></times><ci id="S2.SS4.p1.3.m3.1.1.3.3.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.2">𝑏</ci><apply id="S2.SS4.p1.3.m3.1.1.3.3.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.3.m3.1.1.3.3.3.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.3">subscript</csymbol><ci id="S2.SS4.p1.3.m3.1.1.3.3.3.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.3.2">𝑍</ci><ci id="S2.SS4.p1.3.m3.1.1.3.3.3.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.1c">X_{i}=aZ+bZ_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a italic_Z + italic_b italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="Z\sim N(0,1)" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.2"><semantics id="S2.SS4.p1.4.m4.2a"><mrow id="S2.SS4.p1.4.m4.2.3" xref="S2.SS4.p1.4.m4.2.3.cmml"><mi id="S2.SS4.p1.4.m4.2.3.2" xref="S2.SS4.p1.4.m4.2.3.2.cmml">Z</mi><mo id="S2.SS4.p1.4.m4.2.3.1" xref="S2.SS4.p1.4.m4.2.3.1.cmml">∼</mo><mrow id="S2.SS4.p1.4.m4.2.3.3" xref="S2.SS4.p1.4.m4.2.3.3.cmml"><mi id="S2.SS4.p1.4.m4.2.3.3.2" xref="S2.SS4.p1.4.m4.2.3.3.2.cmml">N</mi><mo id="S2.SS4.p1.4.m4.2.3.3.1" xref="S2.SS4.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS4.p1.4.m4.2.3.3.3.2" 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)</annotation></semantics></math>, and <math alttext="a,b\in\mathbb{R}_{>0}" class="ltx_Math" display="inline" id="S2.SS4.p1.6.m6.2"><semantics id="S2.SS4.p1.6.m6.2a"><mrow id="S2.SS4.p1.6.m6.2.3" xref="S2.SS4.p1.6.m6.2.3.cmml"><mrow id="S2.SS4.p1.6.m6.2.3.2.2" xref="S2.SS4.p1.6.m6.2.3.2.1.cmml"><mi id="S2.SS4.p1.6.m6.1.1" xref="S2.SS4.p1.6.m6.1.1.cmml">a</mi><mo id="S2.SS4.p1.6.m6.2.3.2.2.1" xref="S2.SS4.p1.6.m6.2.3.2.1.cmml">,</mo><mi id="S2.SS4.p1.6.m6.2.2" xref="S2.SS4.p1.6.m6.2.2.cmml">b</mi></mrow><mo id="S2.SS4.p1.6.m6.2.3.1" xref="S2.SS4.p1.6.m6.2.3.1.cmml">∈</mo><msub id="S2.SS4.p1.6.m6.2.3.3" xref="S2.SS4.p1.6.m6.2.3.3.cmml"><mi id="S2.SS4.p1.6.m6.2.3.3.2" xref="S2.SS4.p1.6.m6.2.3.3.2.cmml">ℝ</mi><mrow id="S2.SS4.p1.6.m6.2.3.3.3" xref="S2.SS4.p1.6.m6.2.3.3.3.cmml"><mi id="S2.SS4.p1.6.m6.2.3.3.3.2" xref="S2.SS4.p1.6.m6.2.3.3.3.2.cmml"></mi><mo id="S2.SS4.p1.6.m6.2.3.3.3.1" xref="S2.SS4.p1.6.m6.2.3.3.3.1.cmml">></mo><mn id="S2.SS4.p1.6.m6.2.3.3.3.3" xref="S2.SS4.p1.6.m6.2.3.3.3.3.cmml">0</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.6.m6.2b"><apply id="S2.SS4.p1.6.m6.2.3.cmml" xref="S2.SS4.p1.6.m6.2.3"><in id="S2.SS4.p1.6.m6.2.3.1.cmml" xref="S2.SS4.p1.6.m6.2.3.1"></in><list id="S2.SS4.p1.6.m6.2.3.2.1.cmml" xref="S2.SS4.p1.6.m6.2.3.2.2"><ci id="S2.SS4.p1.6.m6.1.1.cmml" xref="S2.SS4.p1.6.m6.1.1">𝑎</ci><ci id="S2.SS4.p1.6.m6.2.2.cmml" xref="S2.SS4.p1.6.m6.2.2">𝑏</ci></list><apply id="S2.SS4.p1.6.m6.2.3.3.cmml" xref="S2.SS4.p1.6.m6.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.p1.6.m6.2.3.3.1.cmml" xref="S2.SS4.p1.6.m6.2.3.3">subscript</csymbol><ci id="S2.SS4.p1.6.m6.2.3.3.2.cmml" xref="S2.SS4.p1.6.m6.2.3.3.2">ℝ</ci><apply id="S2.SS4.p1.6.m6.2.3.3.3.cmml" xref="S2.SS4.p1.6.m6.2.3.3.3"><gt id="S2.SS4.p1.6.m6.2.3.3.3.1.cmml" xref="S2.SS4.p1.6.m6.2.3.3.3.1"></gt><csymbol cd="latexml" id="S2.SS4.p1.6.m6.2.3.3.3.2.cmml" xref="S2.SS4.p1.6.m6.2.3.3.3.2">absent</csymbol><cn id="S2.SS4.p1.6.m6.2.3.3.3.3.cmml" type="integer" xref="S2.SS4.p1.6.m6.2.3.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.6.m6.2c">a,b\in\mathbb{R}_{>0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.m6.2d">italic_a , italic_b ∈ blackboard_R start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT</annotation></semantics></math>. This model e.g. realistically captures peer grading settings where there is a more fine-grained evaluation of essays anchored in a standard bell-curve distribution. It follows that the marginal distribution for each agent is <math alttext="F(x)=N(x\mid 0,a^{2}+b^{2})" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m7.3"><semantics id="S2.SS4.p1.7.m7.3a"><mrow id="S2.SS4.p1.7.m7.3.3" xref="S2.SS4.p1.7.m7.3.3.cmml"><mrow id="S2.SS4.p1.7.m7.3.3.3" xref="S2.SS4.p1.7.m7.3.3.3.cmml"><mi id="S2.SS4.p1.7.m7.3.3.3.2" xref="S2.SS4.p1.7.m7.3.3.3.2.cmml">F</mi><mo id="S2.SS4.p1.7.m7.3.3.3.1" xref="S2.SS4.p1.7.m7.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS4.p1.7.m7.3.3.3.3.2" xref="S2.SS4.p1.7.m7.3.3.3.cmml"><mo id="S2.SS4.p1.7.m7.3.3.3.3.2.1" stretchy="false" xref="S2.SS4.p1.7.m7.3.3.3.cmml">(</mo><mi id="S2.SS4.p1.7.m7.1.1" xref="S2.SS4.p1.7.m7.1.1.cmml">x</mi><mo id="S2.SS4.p1.7.m7.3.3.3.3.2.2" stretchy="false" xref="S2.SS4.p1.7.m7.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p1.7.m7.3.3.2" xref="S2.SS4.p1.7.m7.3.3.2.cmml">=</mo><mrow id="S2.SS4.p1.7.m7.3.3.1" xref="S2.SS4.p1.7.m7.3.3.1.cmml"><mi id="S2.SS4.p1.7.m7.3.3.1.3" xref="S2.SS4.p1.7.m7.3.3.1.3.cmml">N</mi><mo id="S2.SS4.p1.7.m7.3.3.1.2" xref="S2.SS4.p1.7.m7.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS4.p1.7.m7.3.3.1.1.1" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.cmml"><mo id="S2.SS4.p1.7.m7.3.3.1.1.1.2" stretchy="false" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.p1.7.m7.3.3.1.1.1.1" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.cmml"><mi id="S2.SS4.p1.7.m7.3.3.1.1.1.1.3" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.3.cmml">x</mi><mo id="S2.SS4.p1.7.m7.3.3.1.1.1.1.2" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.2.cmml">∣</mo><mrow id="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.2.cmml"><mn id="S2.SS4.p1.7.m7.2.2" xref="S2.SS4.p1.7.m7.2.2.cmml">0</mn><mo id="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.2" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.2.cmml">,</mo><mrow id="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.1" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.1.cmml"><msup id="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.1.2" xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.1.2.cmml"><mi 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xref="S2.SS4.p1.7.m7.3.3.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m7.3c">F(x)=N(x\mid 0,a^{2}+b^{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m7.3d">italic_F ( italic_x ) = italic_N ( italic_x ∣ 0 , italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>. This model allows for a continuum of agents with the same <math alttext="F" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m8.1"><semantics id="S2.SS4.p1.8.m8.1a"><mi id="S2.SS4.p1.8.m8.1.1" xref="S2.SS4.p1.8.m8.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m8.1b"><ci id="S2.SS4.p1.8.m8.1.1.cmml" xref="S2.SS4.p1.8.m8.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m8.1c">F</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m8.1d">italic_F</annotation></semantics></math>, so the setting is consistent with our dynamics.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.8">The correlation coefficient for two signals <math alttext="X,X^{\prime}" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.2"><semantics id="S2.SS4.p2.1.m1.2a"><mrow id="S2.SS4.p2.1.m1.2.2.1" xref="S2.SS4.p2.1.m1.2.2.2.cmml"><mi id="S2.SS4.p2.1.m1.1.1" xref="S2.SS4.p2.1.m1.1.1.cmml">X</mi><mo id="S2.SS4.p2.1.m1.2.2.1.2" xref="S2.SS4.p2.1.m1.2.2.2.cmml">,</mo><msup id="S2.SS4.p2.1.m1.2.2.1.1" xref="S2.SS4.p2.1.m1.2.2.1.1.cmml"><mi id="S2.SS4.p2.1.m1.2.2.1.1.2" xref="S2.SS4.p2.1.m1.2.2.1.1.2.cmml">X</mi><mo id="S2.SS4.p2.1.m1.2.2.1.1.3" xref="S2.SS4.p2.1.m1.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.2b"><list id="S2.SS4.p2.1.m1.2.2.2.cmml" xref="S2.SS4.p2.1.m1.2.2.1"><ci id="S2.SS4.p2.1.m1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1">𝑋</ci><apply id="S2.SS4.p2.1.m1.2.2.1.1.cmml" xref="S2.SS4.p2.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.1.m1.2.2.1.1.1.cmml" xref="S2.SS4.p2.1.m1.2.2.1.1">superscript</csymbol><ci id="S2.SS4.p2.1.m1.2.2.1.1.2.cmml" xref="S2.SS4.p2.1.m1.2.2.1.1.2">𝑋</ci><ci id="S2.SS4.p2.1.m1.2.2.1.1.3.cmml" xref="S2.SS4.p2.1.m1.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.2c">X,X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.2d">italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\rho=\frac{a^{2}}{a^{2}+b^{2}}" class="ltx_Math" display="inline" id="S2.SS4.p2.2.m2.1"><semantics id="S2.SS4.p2.2.m2.1a"><mrow id="S2.SS4.p2.2.m2.1.1" xref="S2.SS4.p2.2.m2.1.1.cmml"><mi id="S2.SS4.p2.2.m2.1.1.2" xref="S2.SS4.p2.2.m2.1.1.2.cmml">ρ</mi><mo id="S2.SS4.p2.2.m2.1.1.1" xref="S2.SS4.p2.2.m2.1.1.1.cmml">=</mo><mfrac id="S2.SS4.p2.2.m2.1.1.3" xref="S2.SS4.p2.2.m2.1.1.3.cmml"><msup id="S2.SS4.p2.2.m2.1.1.3.2" xref="S2.SS4.p2.2.m2.1.1.3.2.cmml"><mi id="S2.SS4.p2.2.m2.1.1.3.2.2" xref="S2.SS4.p2.2.m2.1.1.3.2.2.cmml">a</mi><mn id="S2.SS4.p2.2.m2.1.1.3.2.3" xref="S2.SS4.p2.2.m2.1.1.3.2.3.cmml">2</mn></msup><mrow id="S2.SS4.p2.2.m2.1.1.3.3" xref="S2.SS4.p2.2.m2.1.1.3.3.cmml"><msup id="S2.SS4.p2.2.m2.1.1.3.3.2" xref="S2.SS4.p2.2.m2.1.1.3.3.2.cmml"><mi id="S2.SS4.p2.2.m2.1.1.3.3.2.2" xref="S2.SS4.p2.2.m2.1.1.3.3.2.2.cmml">a</mi><mn id="S2.SS4.p2.2.m2.1.1.3.3.2.3" xref="S2.SS4.p2.2.m2.1.1.3.3.2.3.cmml">2</mn></msup><mo id="S2.SS4.p2.2.m2.1.1.3.3.1" xref="S2.SS4.p2.2.m2.1.1.3.3.1.cmml">+</mo><msup id="S2.SS4.p2.2.m2.1.1.3.3.3" xref="S2.SS4.p2.2.m2.1.1.3.3.3.cmml"><mi id="S2.SS4.p2.2.m2.1.1.3.3.3.2" xref="S2.SS4.p2.2.m2.1.1.3.3.3.2.cmml">b</mi><mn id="S2.SS4.p2.2.m2.1.1.3.3.3.3" xref="S2.SS4.p2.2.m2.1.1.3.3.3.3.cmml">2</mn></msup></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.1b"><apply id="S2.SS4.p2.2.m2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1"><eq id="S2.SS4.p2.2.m2.1.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1.1"></eq><ci id="S2.SS4.p2.2.m2.1.1.2.cmml" xref="S2.SS4.p2.2.m2.1.1.2">𝜌</ci><apply id="S2.SS4.p2.2.m2.1.1.3.cmml" xref="S2.SS4.p2.2.m2.1.1.3"><divide id="S2.SS4.p2.2.m2.1.1.3.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3"></divide><apply id="S2.SS4.p2.2.m2.1.1.3.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.3.2.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.SS4.p2.2.m2.1.1.3.2.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.2.2">𝑎</ci><cn id="S2.SS4.p2.2.m2.1.1.3.2.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3.2.3">2</cn></apply><apply id="S2.SS4.p2.2.m2.1.1.3.3.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3"><plus id="S2.SS4.p2.2.m2.1.1.3.3.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.1"></plus><apply id="S2.SS4.p2.2.m2.1.1.3.3.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.3.3.2.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.2">superscript</csymbol><ci id="S2.SS4.p2.2.m2.1.1.3.3.2.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.2.2">𝑎</ci><cn id="S2.SS4.p2.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3.3.2.3">2</cn></apply><apply id="S2.SS4.p2.2.m2.1.1.3.3.3.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS4.p2.2.m2.1.1.3.3.3.1.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.3">superscript</csymbol><ci id="S2.SS4.p2.2.m2.1.1.3.3.3.2.cmml" xref="S2.SS4.p2.2.m2.1.1.3.3.3.2">𝑏</ci><cn id="S2.SS4.p2.2.m2.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.1c">\rho=\frac{a^{2}}{a^{2}+b^{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.1d">italic_ρ = divide start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>. 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∣ over^ start_ARG italic_μ end_ARG ( italic_x ) , over^ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, where <math alttext="\hat{\mu}(x)=\rho x" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m5.1"><semantics id="S2.SS4.p2.5.m5.1a"><mrow id="S2.SS4.p2.5.m5.1.2" xref="S2.SS4.p2.5.m5.1.2.cmml"><mrow id="S2.SS4.p2.5.m5.1.2.2" xref="S2.SS4.p2.5.m5.1.2.2.cmml"><mover accent="true" id="S2.SS4.p2.5.m5.1.2.2.2" xref="S2.SS4.p2.5.m5.1.2.2.2.cmml"><mi id="S2.SS4.p2.5.m5.1.2.2.2.2" xref="S2.SS4.p2.5.m5.1.2.2.2.2.cmml">μ</mi><mo id="S2.SS4.p2.5.m5.1.2.2.2.1" xref="S2.SS4.p2.5.m5.1.2.2.2.1.cmml">^</mo></mover><mo id="S2.SS4.p2.5.m5.1.2.2.1" xref="S2.SS4.p2.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.p2.5.m5.1.2.2.3.2" xref="S2.SS4.p2.5.m5.1.2.2.cmml"><mo id="S2.SS4.p2.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.p2.5.m5.1.2.2.cmml">(</mo><mi id="S2.SS4.p2.5.m5.1.1" 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id="S2.SS4.p2.5.m5.1.2.2.2.2.cmml" xref="S2.SS4.p2.5.m5.1.2.2.2.2">𝜇</ci></apply><ci id="S2.SS4.p2.5.m5.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1">𝑥</ci></apply><apply id="S2.SS4.p2.5.m5.1.2.3.cmml" xref="S2.SS4.p2.5.m5.1.2.3"><times id="S2.SS4.p2.5.m5.1.2.3.1.cmml" xref="S2.SS4.p2.5.m5.1.2.3.1"></times><ci id="S2.SS4.p2.5.m5.1.2.3.2.cmml" xref="S2.SS4.p2.5.m5.1.2.3.2">𝜌</ci><ci id="S2.SS4.p2.5.m5.1.2.3.3.cmml" xref="S2.SS4.p2.5.m5.1.2.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m5.1c">\hat{\mu}(x)=\rho x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m5.1d">over^ start_ARG italic_μ end_ARG ( italic_x ) = italic_ρ italic_x</annotation></semantics></math> and <math alttext="\hat{\sigma}^{2}=b^{2}(1+\rho)." class="ltx_Math" display="inline" id="S2.SS4.p2.6.m6.1"><semantics id="S2.SS4.p2.6.m6.1a"><mrow id="S2.SS4.p2.6.m6.1.1.1" xref="S2.SS4.p2.6.m6.1.1.1.1.cmml"><mrow id="S2.SS4.p2.6.m6.1.1.1.1" 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xref="S2.SS4.p2.6.m6.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.6.m6.1.1.1.1.3.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.3">superscript</csymbol><apply id="S2.SS4.p2.6.m6.1.1.1.1.3.2.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.3.2"><ci id="S2.SS4.p2.6.m6.1.1.1.1.3.2.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.3.2.1">^</ci><ci id="S2.SS4.p2.6.m6.1.1.1.1.3.2.2.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.3.2.2">𝜎</ci></apply><cn id="S2.SS4.p2.6.m6.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.p2.6.m6.1.1.1.1.3.3">2</cn></apply><apply id="S2.SS4.p2.6.m6.1.1.1.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1"><times id="S2.SS4.p2.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.2"></times><apply id="S2.SS4.p2.6.m6.1.1.1.1.1.3.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.6.m6.1.1.1.1.1.3.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS4.p2.6.m6.1.1.1.1.1.3.2.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.3.2">𝑏</ci><cn id="S2.SS4.p2.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.p2.6.m6.1.1.1.1.1.3.3">2</cn></apply><apply id="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.1.1"><plus id="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.1"></plus><cn id="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.2">1</cn><ci id="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS4.p2.6.m6.1.1.1.1.1.1.1.1.3">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.6.m6.1c">\hat{\sigma}^{2}=b^{2}(1+\rho).</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.6.m6.1d">over^ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ρ ) .</annotation></semantics></math> Denoting the CDF of a standard Normal distribution by <math alttext="\Phi" class="ltx_Math" display="inline" id="S2.SS4.p2.7.m7.1"><semantics id="S2.SS4.p2.7.m7.1a"><mi id="S2.SS4.p2.7.m7.1.1" mathvariant="normal" xref="S2.SS4.p2.7.m7.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.7.m7.1b"><ci id="S2.SS4.p2.7.m7.1.1.cmml" xref="S2.SS4.p2.7.m7.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.7.m7.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.7.m7.1d">roman_Φ</annotation></semantics></math>, it follows that for a fixed <math alttext="\tau" class="ltx_Math" display="inline" id="S2.SS4.p2.8.m8.1"><semantics id="S2.SS4.p2.8.m8.1a"><mi id="S2.SS4.p2.8.m8.1.1" xref="S2.SS4.p2.8.m8.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.8.m8.1b"><ci id="S2.SS4.p2.8.m8.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.8.m8.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.8.m8.1d">italic_τ</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <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(\tau;x)=\Phi\left(\frac{\tau-\rho x}{b\sqrt{1+\rho}}\right)." class="ltx_Math" display="block" id="S2.E7.m1.4"><semantics id="S2.E7.m1.4a"><mrow id="S2.E7.m1.4.4.1" xref="S2.E7.m1.4.4.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1" xref="S2.E7.m1.4.4.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1.2" xref="S2.E7.m1.4.4.1.1.2.cmml"><mi id="S2.E7.m1.4.4.1.1.2.2" xref="S2.E7.m1.4.4.1.1.2.2.cmml">P</mi><mo id="S2.E7.m1.4.4.1.1.2.1" xref="S2.E7.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S2.E7.m1.4.4.1.1.2.3.2" xref="S2.E7.m1.4.4.1.1.2.3.1.cmml"><mo id="S2.E7.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S2.E7.m1.4.4.1.1.2.3.1.cmml">(</mo><mi id="S2.E7.m1.1.1" xref="S2.E7.m1.1.1.cmml">τ</mi><mo id="S2.E7.m1.4.4.1.1.2.3.2.2" xref="S2.E7.m1.4.4.1.1.2.3.1.cmml">;</mo><mi id="S2.E7.m1.2.2" xref="S2.E7.m1.2.2.cmml">x</mi><mo id="S2.E7.m1.4.4.1.1.2.3.2.3" stretchy="false" xref="S2.E7.m1.4.4.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.4.4.1.1.1" xref="S2.E7.m1.4.4.1.1.1.cmml">=</mo><mrow id="S2.E7.m1.4.4.1.1.3" xref="S2.E7.m1.4.4.1.1.3.cmml"><mi id="S2.E7.m1.4.4.1.1.3.2" mathvariant="normal" xref="S2.E7.m1.4.4.1.1.3.2.cmml">Φ</mi><mo id="S2.E7.m1.4.4.1.1.3.1" xref="S2.E7.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E7.m1.4.4.1.1.3.3.2" xref="S2.E7.m1.3.3.cmml"><mo id="S2.E7.m1.4.4.1.1.3.3.2.1" xref="S2.E7.m1.3.3.cmml">(</mo><mfrac id="S2.E7.m1.3.3" xref="S2.E7.m1.3.3.cmml"><mrow id="S2.E7.m1.3.3.2" xref="S2.E7.m1.3.3.2.cmml"><mi id="S2.E7.m1.3.3.2.2" xref="S2.E7.m1.3.3.2.2.cmml">τ</mi><mo id="S2.E7.m1.3.3.2.1" xref="S2.E7.m1.3.3.2.1.cmml">−</mo><mrow id="S2.E7.m1.3.3.2.3" xref="S2.E7.m1.3.3.2.3.cmml"><mi id="S2.E7.m1.3.3.2.3.2" xref="S2.E7.m1.3.3.2.3.2.cmml">ρ</mi><mo id="S2.E7.m1.3.3.2.3.1" xref="S2.E7.m1.3.3.2.3.1.cmml">⁢</mo><mi id="S2.E7.m1.3.3.2.3.3" xref="S2.E7.m1.3.3.2.3.3.cmml">x</mi></mrow></mrow><mrow id="S2.E7.m1.3.3.3" xref="S2.E7.m1.3.3.3.cmml"><mi id="S2.E7.m1.3.3.3.2" xref="S2.E7.m1.3.3.3.2.cmml">b</mi><mo id="S2.E7.m1.3.3.3.1" xref="S2.E7.m1.3.3.3.1.cmml">⁢</mo><msqrt id="S2.E7.m1.3.3.3.3" xref="S2.E7.m1.3.3.3.3.cmml"><mrow id="S2.E7.m1.3.3.3.3.2" xref="S2.E7.m1.3.3.3.3.2.cmml"><mn id="S2.E7.m1.3.3.3.3.2.2" xref="S2.E7.m1.3.3.3.3.2.2.cmml">1</mn><mo id="S2.E7.m1.3.3.3.3.2.1" xref="S2.E7.m1.3.3.3.3.2.1.cmml">+</mo><mi id="S2.E7.m1.3.3.3.3.2.3" xref="S2.E7.m1.3.3.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac><mo id="S2.E7.m1.4.4.1.1.3.3.2.2" xref="S2.E7.m1.3.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E7.m1.4.4.1.2" lspace="0em" xref="S2.E7.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.m1.4b"><apply id="S2.E7.m1.4.4.1.1.cmml" xref="S2.E7.m1.4.4.1"><eq id="S2.E7.m1.4.4.1.1.1.cmml" xref="S2.E7.m1.4.4.1.1.1"></eq><apply id="S2.E7.m1.4.4.1.1.2.cmml" xref="S2.E7.m1.4.4.1.1.2"><times id="S2.E7.m1.4.4.1.1.2.1.cmml" xref="S2.E7.m1.4.4.1.1.2.1"></times><ci id="S2.E7.m1.4.4.1.1.2.2.cmml" xref="S2.E7.m1.4.4.1.1.2.2">𝑃</ci><list id="S2.E7.m1.4.4.1.1.2.3.1.cmml" xref="S2.E7.m1.4.4.1.1.2.3.2"><ci id="S2.E7.m1.1.1.cmml" xref="S2.E7.m1.1.1">𝜏</ci><ci id="S2.E7.m1.2.2.cmml" xref="S2.E7.m1.2.2">𝑥</ci></list></apply><apply id="S2.E7.m1.4.4.1.1.3.cmml" xref="S2.E7.m1.4.4.1.1.3"><times id="S2.E7.m1.4.4.1.1.3.1.cmml" xref="S2.E7.m1.4.4.1.1.3.1"></times><ci id="S2.E7.m1.4.4.1.1.3.2.cmml" xref="S2.E7.m1.4.4.1.1.3.2">Φ</ci><apply id="S2.E7.m1.3.3.cmml" xref="S2.E7.m1.4.4.1.1.3.3.2"><divide id="S2.E7.m1.3.3.1.cmml" xref="S2.E7.m1.4.4.1.1.3.3.2"></divide><apply id="S2.E7.m1.3.3.2.cmml" xref="S2.E7.m1.3.3.2"><minus id="S2.E7.m1.3.3.2.1.cmml" xref="S2.E7.m1.3.3.2.1"></minus><ci id="S2.E7.m1.3.3.2.2.cmml" xref="S2.E7.m1.3.3.2.2">𝜏</ci><apply id="S2.E7.m1.3.3.2.3.cmml" xref="S2.E7.m1.3.3.2.3"><times id="S2.E7.m1.3.3.2.3.1.cmml" xref="S2.E7.m1.3.3.2.3.1"></times><ci id="S2.E7.m1.3.3.2.3.2.cmml" xref="S2.E7.m1.3.3.2.3.2">𝜌</ci><ci id="S2.E7.m1.3.3.2.3.3.cmml" xref="S2.E7.m1.3.3.2.3.3">𝑥</ci></apply></apply><apply id="S2.E7.m1.3.3.3.cmml" xref="S2.E7.m1.3.3.3"><times id="S2.E7.m1.3.3.3.1.cmml" xref="S2.E7.m1.3.3.3.1"></times><ci id="S2.E7.m1.3.3.3.2.cmml" xref="S2.E7.m1.3.3.3.2">𝑏</ci><apply id="S2.E7.m1.3.3.3.3.cmml" xref="S2.E7.m1.3.3.3.3"><root id="S2.E7.m1.3.3.3.3a.cmml" xref="S2.E7.m1.3.3.3.3"></root><apply id="S2.E7.m1.3.3.3.3.2.cmml" xref="S2.E7.m1.3.3.3.3.2"><plus id="S2.E7.m1.3.3.3.3.2.1.cmml" xref="S2.E7.m1.3.3.3.3.2.1"></plus><cn id="S2.E7.m1.3.3.3.3.2.2.cmml" type="integer" xref="S2.E7.m1.3.3.3.3.2.2">1</cn><ci id="S2.E7.m1.3.3.3.3.2.3.cmml" xref="S2.E7.m1.3.3.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.4c">P(\tau;x)=\Phi\left(\frac{\tau-\rho x}{b\sqrt{1+\rho}}\right).</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.4d">italic_P ( italic_τ ; italic_x ) = roman_Φ ( divide start_ARG italic_τ - italic_ρ italic_x end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p2.9">We can now also write</p> <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F(x)=\Phi\left(\frac{x}{\sqrt{a^{2}+b^{2}}}\right)=\Phi\left(\frac{\sqrt{\rho}% }{a}\;x\right)\mbox{ and }G(x)=\Phi\left(\frac{x-\rho x}{b\sqrt{1+\rho}}\right% )=\Phi\left(\frac{(1-\rho)}{b\sqrt{1+\rho}}\;x\right)~{}." class="ltx_Math" display="block" 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id="S2.E8.m1.1.1.3.3.2.3.cmml" xref="S2.E8.m1.1.1.3.3.2.3">𝜌</ci></apply></apply></apply></apply><ci id="S2.E8.m1.6.6.1.1.2.1.1.1.2.cmml" xref="S2.E8.m1.6.6.1.1.2.1.1.1.2">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.6c">F(x)=\Phi\left(\frac{x}{\sqrt{a^{2}+b^{2}}}\right)=\Phi\left(\frac{\sqrt{\rho}% }{a}\;x\right)\mbox{ and }G(x)=\Phi\left(\frac{x-\rho x}{b\sqrt{1+\rho}}\right% )=\Phi\left(\frac{(1-\rho)}{b\sqrt{1+\rho}}\;x\right)~{}.</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.6d">italic_F ( italic_x ) = roman_Φ ( divide start_ARG italic_x end_ARG start_ARG square-root start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ) = roman_Φ ( divide start_ARG square-root start_ARG italic_ρ end_ARG end_ARG start_ARG italic_a end_ARG italic_x ) and italic_G ( italic_x ) = roman_Φ ( divide start_ARG italic_x - italic_ρ italic_x end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG ) = roman_Φ ( divide start_ARG ( 1 - italic_ρ ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS4.p3"> <p class="ltx_p" id="S2.SS4.p3.1">We then immediately observe existence of equilibria according to our general results:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary1.1.1.1">Corollary 1</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary1.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary1.p1"> <p class="ltx_p" id="Thmcorollary1.p1.2">In the Gaussian model under OA, we have three equilibria at <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmcorollary1.p1.1.m1.1"><semantics id="Thmcorollary1.p1.1.m1.1a"><mrow id="Thmcorollary1.p1.1.m1.1.1" xref="Thmcorollary1.p1.1.m1.1.1.cmml"><mi id="Thmcorollary1.p1.1.m1.1.1.2" xref="Thmcorollary1.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmcorollary1.p1.1.m1.1.1.1" xref="Thmcorollary1.p1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary1.p1.1.m1.1.1.3" xref="Thmcorollary1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary1.p1.1.m1.1b"><apply id="Thmcorollary1.p1.1.m1.1.1.cmml" xref="Thmcorollary1.p1.1.m1.1.1"><eq id="Thmcorollary1.p1.1.m1.1.1.1.cmml" xref="Thmcorollary1.p1.1.m1.1.1.1"></eq><ci id="Thmcorollary1.p1.1.m1.1.1.2.cmml" xref="Thmcorollary1.p1.1.m1.1.1.2">𝜏</ci><cn id="Thmcorollary1.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary1.p1.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary1.p1.1.m1.1d">italic_τ = 0</annotation></semantics></math> and <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmcorollary1.p1.2.m2.1"><semantics id="Thmcorollary1.p1.2.m2.1a"><mrow id="Thmcorollary1.p1.2.m2.1.1" xref="Thmcorollary1.p1.2.m2.1.1.cmml"><mo id="Thmcorollary1.p1.2.m2.1.1a" xref="Thmcorollary1.p1.2.m2.1.1.cmml">±</mo><mi id="Thmcorollary1.p1.2.m2.1.1.2" mathvariant="normal" xref="Thmcorollary1.p1.2.m2.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary1.p1.2.m2.1b"><apply id="Thmcorollary1.p1.2.m2.1.1.cmml" xref="Thmcorollary1.p1.2.m2.1.1"><csymbol cd="latexml" id="Thmcorollary1.p1.2.m2.1.1.1.cmml" xref="Thmcorollary1.p1.2.m2.1.1">plus-or-minus</csymbol><infinity id="Thmcorollary1.p1.2.m2.1.1.2.cmml" xref="Thmcorollary1.p1.2.m2.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary1.p1.2.m2.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary1.p1.2.m2.1d">± ∞</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S2.SS4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS4.1.p1"> <p class="ltx_p" id="S2.SS4.1.p1.8">Existence of equilibria at <math alttext="\tau=\pm\infty" class="ltx_Math" display="inline" id="S2.SS4.1.p1.1.m1.1"><semantics id="S2.SS4.1.p1.1.m1.1a"><mrow id="S2.SS4.1.p1.1.m1.1.1" xref="S2.SS4.1.p1.1.m1.1.1.cmml"><mi id="S2.SS4.1.p1.1.m1.1.1.2" xref="S2.SS4.1.p1.1.m1.1.1.2.cmml">τ</mi><mo id="S2.SS4.1.p1.1.m1.1.1.1" xref="S2.SS4.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS4.1.p1.1.m1.1.1.3" xref="S2.SS4.1.p1.1.m1.1.1.3.cmml"><mo id="S2.SS4.1.p1.1.m1.1.1.3a" xref="S2.SS4.1.p1.1.m1.1.1.3.cmml">±</mo><mi id="S2.SS4.1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.SS4.1.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.1.m1.1b"><apply id="S2.SS4.1.p1.1.m1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1"><eq id="S2.SS4.1.p1.1.m1.1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.1"></eq><ci id="S2.SS4.1.p1.1.m1.1.1.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.2">𝜏</ci><apply id="S2.SS4.1.p1.1.m1.1.1.3.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S2.SS4.1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS4.1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.1.m1.1c">\tau=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.1.m1.1d">italic_τ = ± ∞</annotation></semantics></math> follows from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition1" title="Proposition 1. ‣ Results. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>. Note that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S2.SS4.1.p1.2.m2.2"><semantics id="S2.SS4.1.p1.2.m2.2a"><mrow id="S2.SS4.1.p1.2.m2.2.3" xref="S2.SS4.1.p1.2.m2.2.3.cmml"><mi id="S2.SS4.1.p1.2.m2.2.3.2" xref="S2.SS4.1.p1.2.m2.2.3.2.cmml">P</mi><mo id="S2.SS4.1.p1.2.m2.2.3.1" xref="S2.SS4.1.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.1.p1.2.m2.2.3.3.2" xref="S2.SS4.1.p1.2.m2.2.3.3.1.cmml"><mo id="S2.SS4.1.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS4.1.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS4.1.p1.2.m2.1.1" xref="S2.SS4.1.p1.2.m2.1.1.cmml">τ</mi><mo id="S2.SS4.1.p1.2.m2.2.3.3.2.2" xref="S2.SS4.1.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="S2.SS4.1.p1.2.m2.2.2" xref="S2.SS4.1.p1.2.m2.2.2.cmml">x</mi><mo id="S2.SS4.1.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS4.1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.2.m2.2b"><apply id="S2.SS4.1.p1.2.m2.2.3.cmml" xref="S2.SS4.1.p1.2.m2.2.3"><times id="S2.SS4.1.p1.2.m2.2.3.1.cmml" xref="S2.SS4.1.p1.2.m2.2.3.1"></times><ci id="S2.SS4.1.p1.2.m2.2.3.2.cmml" xref="S2.SS4.1.p1.2.m2.2.3.2">𝑃</ci><list id="S2.SS4.1.p1.2.m2.2.3.3.1.cmml" xref="S2.SS4.1.p1.2.m2.2.3.3.2"><ci id="S2.SS4.1.p1.2.m2.1.1.cmml" xref="S2.SS4.1.p1.2.m2.1.1">𝜏</ci><ci id="S2.SS4.1.p1.2.m2.2.2.cmml" xref="S2.SS4.1.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly monotone decreasing and continuous in <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.1.p1.3.m3.1"><semantics id="S2.SS4.1.p1.3.m3.1a"><mi id="S2.SS4.1.p1.3.m3.1.1" xref="S2.SS4.1.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.3.m3.1b"><ci id="S2.SS4.1.p1.3.m3.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.3.m3.1d">italic_x</annotation></semantics></math> so that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem1" title="Theorem 1. ‣ Results. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> applies. As <math alttext="G(0)=\Phi(0)=\tfrac{1}{2}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.4.m4.2"><semantics id="S2.SS4.1.p1.4.m4.2a"><mrow id="S2.SS4.1.p1.4.m4.2.3" xref="S2.SS4.1.p1.4.m4.2.3.cmml"><mrow id="S2.SS4.1.p1.4.m4.2.3.2" xref="S2.SS4.1.p1.4.m4.2.3.2.cmml"><mi id="S2.SS4.1.p1.4.m4.2.3.2.2" xref="S2.SS4.1.p1.4.m4.2.3.2.2.cmml">G</mi><mo id="S2.SS4.1.p1.4.m4.2.3.2.1" xref="S2.SS4.1.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS4.1.p1.4.m4.2.3.2.3.2" xref="S2.SS4.1.p1.4.m4.2.3.2.cmml"><mo id="S2.SS4.1.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S2.SS4.1.p1.4.m4.2.3.2.cmml">(</mo><mn id="S2.SS4.1.p1.4.m4.1.1" xref="S2.SS4.1.p1.4.m4.1.1.cmml">0</mn><mo id="S2.SS4.1.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="S2.SS4.1.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.4.m4.2.3.3" xref="S2.SS4.1.p1.4.m4.2.3.3.cmml">=</mo><mrow id="S2.SS4.1.p1.4.m4.2.3.4" xref="S2.SS4.1.p1.4.m4.2.3.4.cmml"><mi id="S2.SS4.1.p1.4.m4.2.3.4.2" mathvariant="normal" xref="S2.SS4.1.p1.4.m4.2.3.4.2.cmml">Φ</mi><mo id="S2.SS4.1.p1.4.m4.2.3.4.1" xref="S2.SS4.1.p1.4.m4.2.3.4.1.cmml">⁢</mo><mrow id="S2.SS4.1.p1.4.m4.2.3.4.3.2" xref="S2.SS4.1.p1.4.m4.2.3.4.cmml"><mo id="S2.SS4.1.p1.4.m4.2.3.4.3.2.1" stretchy="false" xref="S2.SS4.1.p1.4.m4.2.3.4.cmml">(</mo><mn id="S2.SS4.1.p1.4.m4.2.2" xref="S2.SS4.1.p1.4.m4.2.2.cmml">0</mn><mo id="S2.SS4.1.p1.4.m4.2.3.4.3.2.2" stretchy="false" xref="S2.SS4.1.p1.4.m4.2.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.4.m4.2.3.5" xref="S2.SS4.1.p1.4.m4.2.3.5.cmml">=</mo><mfrac id="S2.SS4.1.p1.4.m4.2.3.6" xref="S2.SS4.1.p1.4.m4.2.3.6.cmml"><mn id="S2.SS4.1.p1.4.m4.2.3.6.2" xref="S2.SS4.1.p1.4.m4.2.3.6.2.cmml">1</mn><mn id="S2.SS4.1.p1.4.m4.2.3.6.3" xref="S2.SS4.1.p1.4.m4.2.3.6.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.4.m4.2b"><apply id="S2.SS4.1.p1.4.m4.2.3.cmml" xref="S2.SS4.1.p1.4.m4.2.3"><and id="S2.SS4.1.p1.4.m4.2.3a.cmml" xref="S2.SS4.1.p1.4.m4.2.3"></and><apply id="S2.SS4.1.p1.4.m4.2.3b.cmml" xref="S2.SS4.1.p1.4.m4.2.3"><eq id="S2.SS4.1.p1.4.m4.2.3.3.cmml" xref="S2.SS4.1.p1.4.m4.2.3.3"></eq><apply id="S2.SS4.1.p1.4.m4.2.3.2.cmml" xref="S2.SS4.1.p1.4.m4.2.3.2"><times id="S2.SS4.1.p1.4.m4.2.3.2.1.cmml" xref="S2.SS4.1.p1.4.m4.2.3.2.1"></times><ci id="S2.SS4.1.p1.4.m4.2.3.2.2.cmml" xref="S2.SS4.1.p1.4.m4.2.3.2.2">𝐺</ci><cn id="S2.SS4.1.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.1.1">0</cn></apply><apply id="S2.SS4.1.p1.4.m4.2.3.4.cmml" xref="S2.SS4.1.p1.4.m4.2.3.4"><times id="S2.SS4.1.p1.4.m4.2.3.4.1.cmml" xref="S2.SS4.1.p1.4.m4.2.3.4.1"></times><ci id="S2.SS4.1.p1.4.m4.2.3.4.2.cmml" xref="S2.SS4.1.p1.4.m4.2.3.4.2">Φ</ci><cn id="S2.SS4.1.p1.4.m4.2.2.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.2.2">0</cn></apply></apply><apply id="S2.SS4.1.p1.4.m4.2.3c.cmml" xref="S2.SS4.1.p1.4.m4.2.3"><eq id="S2.SS4.1.p1.4.m4.2.3.5.cmml" xref="S2.SS4.1.p1.4.m4.2.3.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S2.SS4.1.p1.4.m4.2.3.4.cmml" id="S2.SS4.1.p1.4.m4.2.3d.cmml" xref="S2.SS4.1.p1.4.m4.2.3"></share><apply id="S2.SS4.1.p1.4.m4.2.3.6.cmml" xref="S2.SS4.1.p1.4.m4.2.3.6"><divide id="S2.SS4.1.p1.4.m4.2.3.6.1.cmml" xref="S2.SS4.1.p1.4.m4.2.3.6"></divide><cn id="S2.SS4.1.p1.4.m4.2.3.6.2.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.2.3.6.2">1</cn><cn id="S2.SS4.1.p1.4.m4.2.3.6.3.cmml" type="integer" xref="S2.SS4.1.p1.4.m4.2.3.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.4.m4.2c">G(0)=\Phi(0)=\tfrac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.4.m4.2d">italic_G ( 0 ) = roman_Φ ( 0 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>, <math alttext="\tau=0" class="ltx_Math" display="inline" id="S2.SS4.1.p1.5.m5.1"><semantics id="S2.SS4.1.p1.5.m5.1a"><mrow id="S2.SS4.1.p1.5.m5.1.1" xref="S2.SS4.1.p1.5.m5.1.1.cmml"><mi id="S2.SS4.1.p1.5.m5.1.1.2" xref="S2.SS4.1.p1.5.m5.1.1.2.cmml">τ</mi><mo id="S2.SS4.1.p1.5.m5.1.1.1" xref="S2.SS4.1.p1.5.m5.1.1.1.cmml">=</mo><mn id="S2.SS4.1.p1.5.m5.1.1.3" xref="S2.SS4.1.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.5.m5.1b"><apply id="S2.SS4.1.p1.5.m5.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1"><eq id="S2.SS4.1.p1.5.m5.1.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1.1"></eq><ci id="S2.SS4.1.p1.5.m5.1.1.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2">𝜏</ci><cn id="S2.SS4.1.p1.5.m5.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.5.m5.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.5.m5.1d">italic_τ = 0</annotation></semantics></math> is an equilibrium for OA. Note also that <math alttext="\Phi" class="ltx_Math" display="inline" id="S2.SS4.1.p1.6.m6.1"><semantics id="S2.SS4.1.p1.6.m6.1a"><mi id="S2.SS4.1.p1.6.m6.1.1" mathvariant="normal" xref="S2.SS4.1.p1.6.m6.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.6.m6.1b"><ci id="S2.SS4.1.p1.6.m6.1.1.cmml" xref="S2.SS4.1.p1.6.m6.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.6.m6.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.6.m6.1d">roman_Φ</annotation></semantics></math> is strictly monotone, so <math alttext="G(x)" class="ltx_Math" display="inline" id="S2.SS4.1.p1.7.m7.1"><semantics id="S2.SS4.1.p1.7.m7.1a"><mrow id="S2.SS4.1.p1.7.m7.1.2" xref="S2.SS4.1.p1.7.m7.1.2.cmml"><mi id="S2.SS4.1.p1.7.m7.1.2.2" xref="S2.SS4.1.p1.7.m7.1.2.2.cmml">G</mi><mo id="S2.SS4.1.p1.7.m7.1.2.1" xref="S2.SS4.1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.1.p1.7.m7.1.2.3.2" xref="S2.SS4.1.p1.7.m7.1.2.cmml"><mo id="S2.SS4.1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS4.1.p1.7.m7.1.2.cmml">(</mo><mi id="S2.SS4.1.p1.7.m7.1.1" xref="S2.SS4.1.p1.7.m7.1.1.cmml">x</mi><mo id="S2.SS4.1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS4.1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.7.m7.1b"><apply id="S2.SS4.1.p1.7.m7.1.2.cmml" xref="S2.SS4.1.p1.7.m7.1.2"><times id="S2.SS4.1.p1.7.m7.1.2.1.cmml" xref="S2.SS4.1.p1.7.m7.1.2.1"></times><ci id="S2.SS4.1.p1.7.m7.1.2.2.cmml" xref="S2.SS4.1.p1.7.m7.1.2.2">𝐺</ci><ci id="S2.SS4.1.p1.7.m7.1.1.cmml" xref="S2.SS4.1.p1.7.m7.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.7.m7.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.7.m7.1d">italic_G ( italic_x )</annotation></semantics></math> only crosses 1/2 once and <math alttext="\tau=0" class="ltx_Math" display="inline" id="S2.SS4.1.p1.8.m8.1"><semantics id="S2.SS4.1.p1.8.m8.1a"><mrow id="S2.SS4.1.p1.8.m8.1.1" xref="S2.SS4.1.p1.8.m8.1.1.cmml"><mi id="S2.SS4.1.p1.8.m8.1.1.2" xref="S2.SS4.1.p1.8.m8.1.1.2.cmml">τ</mi><mo id="S2.SS4.1.p1.8.m8.1.1.1" xref="S2.SS4.1.p1.8.m8.1.1.1.cmml">=</mo><mn id="S2.SS4.1.p1.8.m8.1.1.3" xref="S2.SS4.1.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.8.m8.1b"><apply id="S2.SS4.1.p1.8.m8.1.1.cmml" xref="S2.SS4.1.p1.8.m8.1.1"><eq id="S2.SS4.1.p1.8.m8.1.1.1.cmml" xref="S2.SS4.1.p1.8.m8.1.1.1"></eq><ci id="S2.SS4.1.p1.8.m8.1.1.2.cmml" xref="S2.SS4.1.p1.8.m8.1.1.2">𝜏</ci><cn id="S2.SS4.1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.8.m8.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.8.m8.1d">italic_τ = 0</annotation></semantics></math> is therefore the <em class="ltx_emph ltx_font_italic" id="S2.SS4.1.p1.8.1">only</em> non-infinite equilibrium (see a visualization in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.F2" title="Figure 2 ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>). ∎</p> </div> </div> <div class="ltx_para" id="S2.SS4.p4"> <p class="ltx_p" id="S2.SS4.p4.1">Now, consider the stability of these equilibria.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary2.1.1.1">Corollary 2</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary2.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary2.p1"> <p class="ltx_p" id="Thmcorollary2.p1.2">In the Gaussian model under OA, the threshold equilibrium <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmcorollary2.p1.1.m1.1"><semantics id="Thmcorollary2.p1.1.m1.1a"><mrow id="Thmcorollary2.p1.1.m1.1.1" xref="Thmcorollary2.p1.1.m1.1.1.cmml"><mi id="Thmcorollary2.p1.1.m1.1.1.2" xref="Thmcorollary2.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmcorollary2.p1.1.m1.1.1.1" xref="Thmcorollary2.p1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary2.p1.1.m1.1.1.3" xref="Thmcorollary2.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary2.p1.1.m1.1b"><apply id="Thmcorollary2.p1.1.m1.1.1.cmml" xref="Thmcorollary2.p1.1.m1.1.1"><eq id="Thmcorollary2.p1.1.m1.1.1.1.cmml" xref="Thmcorollary2.p1.1.m1.1.1.1"></eq><ci id="Thmcorollary2.p1.1.m1.1.1.2.cmml" xref="Thmcorollary2.p1.1.m1.1.1.2">𝜏</ci><cn id="Thmcorollary2.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary2.p1.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary2.p1.1.m1.1d">italic_τ = 0</annotation></semantics></math> is unstable, while the uninformative equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmcorollary2.p1.2.m2.1"><semantics id="Thmcorollary2.p1.2.m2.1a"><mrow id="Thmcorollary2.p1.2.m2.1.1" xref="Thmcorollary2.p1.2.m2.1.1.cmml"><mo id="Thmcorollary2.p1.2.m2.1.1a" xref="Thmcorollary2.p1.2.m2.1.1.cmml">±</mo><mi id="Thmcorollary2.p1.2.m2.1.1.2" mathvariant="normal" xref="Thmcorollary2.p1.2.m2.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary2.p1.2.m2.1b"><apply id="Thmcorollary2.p1.2.m2.1.1.cmml" xref="Thmcorollary2.p1.2.m2.1.1"><csymbol cd="latexml" id="Thmcorollary2.p1.2.m2.1.1.1.cmml" xref="Thmcorollary2.p1.2.m2.1.1">plus-or-minus</csymbol><infinity id="Thmcorollary2.p1.2.m2.1.1.2.cmml" xref="Thmcorollary2.p1.2.m2.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary2.p1.2.m2.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary2.p1.2.m2.1d">± ∞</annotation></semantics></math> are stable.</p> </div> </div> <div class="ltx_proof" id="S2.SS4.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS4.2.p1"> <p class="ltx_p" id="S2.SS4.2.p1.9">Let <math alttext="c_{G}(\rho)=\frac{(1-\rho)}{b\sqrt{1+\rho}}" class="ltx_Math" display="inline" id="S2.SS4.2.p1.1.m1.2"><semantics id="S2.SS4.2.p1.1.m1.2a"><mrow id="S2.SS4.2.p1.1.m1.2.3" xref="S2.SS4.2.p1.1.m1.2.3.cmml"><mrow id="S2.SS4.2.p1.1.m1.2.3.2" xref="S2.SS4.2.p1.1.m1.2.3.2.cmml"><msub id="S2.SS4.2.p1.1.m1.2.3.2.2" xref="S2.SS4.2.p1.1.m1.2.3.2.2.cmml"><mi id="S2.SS4.2.p1.1.m1.2.3.2.2.2" xref="S2.SS4.2.p1.1.m1.2.3.2.2.2.cmml">c</mi><mi id="S2.SS4.2.p1.1.m1.2.3.2.2.3" xref="S2.SS4.2.p1.1.m1.2.3.2.2.3.cmml">G</mi></msub><mo id="S2.SS4.2.p1.1.m1.2.3.2.1" xref="S2.SS4.2.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.1.m1.2.3.2.3.2" xref="S2.SS4.2.p1.1.m1.2.3.2.cmml"><mo id="S2.SS4.2.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S2.SS4.2.p1.1.m1.2.3.2.cmml">(</mo><mi id="S2.SS4.2.p1.1.m1.2.2" xref="S2.SS4.2.p1.1.m1.2.2.cmml">ρ</mi><mo id="S2.SS4.2.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S2.SS4.2.p1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.1.m1.2.3.1" xref="S2.SS4.2.p1.1.m1.2.3.1.cmml">=</mo><mfrac id="S2.SS4.2.p1.1.m1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.cmml"><mrow id="S2.SS4.2.p1.1.m1.1.1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.2.p1.1.m1.1.1.1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.cmml"><mn id="S2.SS4.2.p1.1.m1.1.1.1.1.1.2" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS4.2.p1.1.m1.1.1.1.1.1.1" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS4.2.p1.1.m1.1.1.1.1.1.3" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="S2.SS4.2.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS4.2.p1.1.m1.1.1.3" xref="S2.SS4.2.p1.1.m1.1.1.3.cmml"><mi id="S2.SS4.2.p1.1.m1.1.1.3.2" xref="S2.SS4.2.p1.1.m1.1.1.3.2.cmml">b</mi><mo id="S2.SS4.2.p1.1.m1.1.1.3.1" xref="S2.SS4.2.p1.1.m1.1.1.3.1.cmml">⁢</mo><msqrt id="S2.SS4.2.p1.1.m1.1.1.3.3" xref="S2.SS4.2.p1.1.m1.1.1.3.3.cmml"><mrow id="S2.SS4.2.p1.1.m1.1.1.3.3.2" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.cmml"><mn id="S2.SS4.2.p1.1.m1.1.1.3.3.2.2" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.2.cmml">1</mn><mo id="S2.SS4.2.p1.1.m1.1.1.3.3.2.1" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.1.cmml">+</mo><mi id="S2.SS4.2.p1.1.m1.1.1.3.3.2.3" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.1.m1.2b"><apply id="S2.SS4.2.p1.1.m1.2.3.cmml" xref="S2.SS4.2.p1.1.m1.2.3"><eq id="S2.SS4.2.p1.1.m1.2.3.1.cmml" xref="S2.SS4.2.p1.1.m1.2.3.1"></eq><apply id="S2.SS4.2.p1.1.m1.2.3.2.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2"><times id="S2.SS4.2.p1.1.m1.2.3.2.1.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2.1"></times><apply id="S2.SS4.2.p1.1.m1.2.3.2.2.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.1.m1.2.3.2.2.1.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2.2">subscript</csymbol><ci id="S2.SS4.2.p1.1.m1.2.3.2.2.2.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2.2.2">𝑐</ci><ci id="S2.SS4.2.p1.1.m1.2.3.2.2.3.cmml" xref="S2.SS4.2.p1.1.m1.2.3.2.2.3">𝐺</ci></apply><ci id="S2.SS4.2.p1.1.m1.2.2.cmml" xref="S2.SS4.2.p1.1.m1.2.2">𝜌</ci></apply><apply id="S2.SS4.2.p1.1.m1.1.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1"><divide id="S2.SS4.2.p1.1.m1.1.1.2.cmml" xref="S2.SS4.2.p1.1.m1.1.1"></divide><apply id="S2.SS4.2.p1.1.m1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.1.1"><minus id="S2.SS4.2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.1"></minus><cn id="S2.SS4.2.p1.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.2">1</cn><ci id="S2.SS4.2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.1.1.1.3">𝜌</ci></apply><apply id="S2.SS4.2.p1.1.m1.1.1.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3"><times id="S2.SS4.2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.1"></times><ci id="S2.SS4.2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.2">𝑏</ci><apply id="S2.SS4.2.p1.1.m1.1.1.3.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3"><root id="S2.SS4.2.p1.1.m1.1.1.3.3a.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3"></root><apply id="S2.SS4.2.p1.1.m1.1.1.3.3.2.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2"><plus id="S2.SS4.2.p1.1.m1.1.1.3.3.2.1.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.1"></plus><cn id="S2.SS4.2.p1.1.m1.1.1.3.3.2.2.cmml" type="integer" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.2">1</cn><ci id="S2.SS4.2.p1.1.m1.1.1.3.3.2.3.cmml" xref="S2.SS4.2.p1.1.m1.1.1.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.1.m1.2c">c_{G}(\rho)=\frac{(1-\rho)}{b\sqrt{1+\rho}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.1.m1.2d">italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = divide start_ARG ( 1 - italic_ρ ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG</annotation></semantics></math> be the coefficient of <math alttext="x" class="ltx_Math" display="inline" id="S2.SS4.2.p1.2.m2.1"><semantics id="S2.SS4.2.p1.2.m2.1a"><mi id="S2.SS4.2.p1.2.m2.1.1" xref="S2.SS4.2.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.2.m2.1b"><ci id="S2.SS4.2.p1.2.m2.1.1.cmml" xref="S2.SS4.2.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.2.m2.1d">italic_x</annotation></semantics></math> in <math alttext="G(x)" class="ltx_Math" display="inline" id="S2.SS4.2.p1.3.m3.1"><semantics id="S2.SS4.2.p1.3.m3.1a"><mrow id="S2.SS4.2.p1.3.m3.1.2" xref="S2.SS4.2.p1.3.m3.1.2.cmml"><mi id="S2.SS4.2.p1.3.m3.1.2.2" xref="S2.SS4.2.p1.3.m3.1.2.2.cmml">G</mi><mo id="S2.SS4.2.p1.3.m3.1.2.1" xref="S2.SS4.2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.3.m3.1.2.3.2" xref="S2.SS4.2.p1.3.m3.1.2.cmml"><mo id="S2.SS4.2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS4.2.p1.3.m3.1.2.cmml">(</mo><mi id="S2.SS4.2.p1.3.m3.1.1" xref="S2.SS4.2.p1.3.m3.1.1.cmml">x</mi><mo id="S2.SS4.2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS4.2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.3.m3.1b"><apply id="S2.SS4.2.p1.3.m3.1.2.cmml" xref="S2.SS4.2.p1.3.m3.1.2"><times id="S2.SS4.2.p1.3.m3.1.2.1.cmml" xref="S2.SS4.2.p1.3.m3.1.2.1"></times><ci id="S2.SS4.2.p1.3.m3.1.2.2.cmml" xref="S2.SS4.2.p1.3.m3.1.2.2">𝐺</ci><ci id="S2.SS4.2.p1.3.m3.1.1.cmml" xref="S2.SS4.2.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.3.m3.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.3.m3.1d">italic_G ( italic_x )</annotation></semantics></math>. Then <math alttext="G^{\prime}(x)=c_{G}(\rho)\phi(c_{G}x)" class="ltx_Math" display="inline" id="S2.SS4.2.p1.4.m4.3"><semantics id="S2.SS4.2.p1.4.m4.3a"><mrow id="S2.SS4.2.p1.4.m4.3.3" xref="S2.SS4.2.p1.4.m4.3.3.cmml"><mrow id="S2.SS4.2.p1.4.m4.3.3.3" xref="S2.SS4.2.p1.4.m4.3.3.3.cmml"><msup id="S2.SS4.2.p1.4.m4.3.3.3.2" xref="S2.SS4.2.p1.4.m4.3.3.3.2.cmml"><mi id="S2.SS4.2.p1.4.m4.3.3.3.2.2" xref="S2.SS4.2.p1.4.m4.3.3.3.2.2.cmml">G</mi><mo id="S2.SS4.2.p1.4.m4.3.3.3.2.3" xref="S2.SS4.2.p1.4.m4.3.3.3.2.3.cmml">′</mo></msup><mo id="S2.SS4.2.p1.4.m4.3.3.3.1" xref="S2.SS4.2.p1.4.m4.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.4.m4.3.3.3.3.2" xref="S2.SS4.2.p1.4.m4.3.3.3.cmml"><mo id="S2.SS4.2.p1.4.m4.3.3.3.3.2.1" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.3.cmml">(</mo><mi id="S2.SS4.2.p1.4.m4.1.1" xref="S2.SS4.2.p1.4.m4.1.1.cmml">x</mi><mo id="S2.SS4.2.p1.4.m4.3.3.3.3.2.2" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.4.m4.3.3.2" xref="S2.SS4.2.p1.4.m4.3.3.2.cmml">=</mo><mrow id="S2.SS4.2.p1.4.m4.3.3.1" xref="S2.SS4.2.p1.4.m4.3.3.1.cmml"><msub id="S2.SS4.2.p1.4.m4.3.3.1.3" xref="S2.SS4.2.p1.4.m4.3.3.1.3.cmml"><mi id="S2.SS4.2.p1.4.m4.3.3.1.3.2" xref="S2.SS4.2.p1.4.m4.3.3.1.3.2.cmml">c</mi><mi id="S2.SS4.2.p1.4.m4.3.3.1.3.3" xref="S2.SS4.2.p1.4.m4.3.3.1.3.3.cmml">G</mi></msub><mo id="S2.SS4.2.p1.4.m4.3.3.1.2" xref="S2.SS4.2.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.4.m4.3.3.1.4.2" xref="S2.SS4.2.p1.4.m4.3.3.1.cmml"><mo id="S2.SS4.2.p1.4.m4.3.3.1.4.2.1" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.1.cmml">(</mo><mi id="S2.SS4.2.p1.4.m4.2.2" xref="S2.SS4.2.p1.4.m4.2.2.cmml">ρ</mi><mo id="S2.SS4.2.p1.4.m4.3.3.1.4.2.2" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.1.cmml">)</mo></mrow><mo id="S2.SS4.2.p1.4.m4.3.3.1.2a" xref="S2.SS4.2.p1.4.m4.3.3.1.2.cmml">⁢</mo><mi id="S2.SS4.2.p1.4.m4.3.3.1.5" xref="S2.SS4.2.p1.4.m4.3.3.1.5.cmml">ϕ</mi><mo id="S2.SS4.2.p1.4.m4.3.3.1.2b" xref="S2.SS4.2.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS4.2.p1.4.m4.3.3.1.1.1" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="S2.SS4.2.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml"><msub id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.cmml"><mi id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.2" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.2.cmml">c</mi><mi id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.3" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.3.cmml">G</mi></msub><mo id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.1" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.3" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.SS4.2.p1.4.m4.3.3.1.1.1.3" stretchy="false" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.4.m4.3b"><apply id="S2.SS4.2.p1.4.m4.3.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3"><eq id="S2.SS4.2.p1.4.m4.3.3.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.2"></eq><apply id="S2.SS4.2.p1.4.m4.3.3.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3"><times id="S2.SS4.2.p1.4.m4.3.3.3.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3.1"></times><apply id="S2.SS4.2.p1.4.m4.3.3.3.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.3.3.3.2.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3.2">superscript</csymbol><ci id="S2.SS4.2.p1.4.m4.3.3.3.2.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3.2.2">𝐺</ci><ci id="S2.SS4.2.p1.4.m4.3.3.3.2.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.3.2.3">′</ci></apply><ci id="S2.SS4.2.p1.4.m4.1.1.cmml" xref="S2.SS4.2.p1.4.m4.1.1">𝑥</ci></apply><apply id="S2.SS4.2.p1.4.m4.3.3.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1"><times id="S2.SS4.2.p1.4.m4.3.3.1.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.2"></times><apply id="S2.SS4.2.p1.4.m4.3.3.1.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.3.3.1.3.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.3">subscript</csymbol><ci id="S2.SS4.2.p1.4.m4.3.3.1.3.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.3.2">𝑐</ci><ci id="S2.SS4.2.p1.4.m4.3.3.1.3.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.3.3">𝐺</ci></apply><ci id="S2.SS4.2.p1.4.m4.2.2.cmml" xref="S2.SS4.2.p1.4.m4.2.2">𝜌</ci><ci id="S2.SS4.2.p1.4.m4.3.3.1.5.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.5">italic-ϕ</ci><apply id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1"><times id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.1"></times><apply id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.1.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2">subscript</csymbol><ci id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.2.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.2">𝑐</ci><ci id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.2.3">𝐺</ci></apply><ci id="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.3.cmml" xref="S2.SS4.2.p1.4.m4.3.3.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.4.m4.3c">G^{\prime}(x)=c_{G}(\rho)\phi(c_{G}x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.4.m4.3d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) = italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) italic_ϕ ( italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT italic_x )</annotation></semantics></math> for <math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS4.2.p1.5.m5.1"><semantics id="S2.SS4.2.p1.5.m5.1a"><mi id="S2.SS4.2.p1.5.m5.1.1" xref="S2.SS4.2.p1.5.m5.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.5.m5.1b"><ci id="S2.SS4.2.p1.5.m5.1.1.cmml" xref="S2.SS4.2.p1.5.m5.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.5.m5.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.5.m5.1d">italic_ϕ</annotation></semantics></math> the PDF of the standard Normal, and so <math alttext="G^{\prime}(0)=c_{G}(\rho)\phi(0)>0" class="ltx_Math" display="inline" id="S2.SS4.2.p1.6.m6.3"><semantics id="S2.SS4.2.p1.6.m6.3a"><mrow id="S2.SS4.2.p1.6.m6.3.4" xref="S2.SS4.2.p1.6.m6.3.4.cmml"><mrow id="S2.SS4.2.p1.6.m6.3.4.2" xref="S2.SS4.2.p1.6.m6.3.4.2.cmml"><msup id="S2.SS4.2.p1.6.m6.3.4.2.2" xref="S2.SS4.2.p1.6.m6.3.4.2.2.cmml"><mi id="S2.SS4.2.p1.6.m6.3.4.2.2.2" xref="S2.SS4.2.p1.6.m6.3.4.2.2.2.cmml">G</mi><mo id="S2.SS4.2.p1.6.m6.3.4.2.2.3" xref="S2.SS4.2.p1.6.m6.3.4.2.2.3.cmml">′</mo></msup><mo id="S2.SS4.2.p1.6.m6.3.4.2.1" xref="S2.SS4.2.p1.6.m6.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.6.m6.3.4.2.3.2" xref="S2.SS4.2.p1.6.m6.3.4.2.cmml"><mo id="S2.SS4.2.p1.6.m6.3.4.2.3.2.1" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.2.cmml">(</mo><mn id="S2.SS4.2.p1.6.m6.1.1" xref="S2.SS4.2.p1.6.m6.1.1.cmml">0</mn><mo id="S2.SS4.2.p1.6.m6.3.4.2.3.2.2" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.6.m6.3.4.3" xref="S2.SS4.2.p1.6.m6.3.4.3.cmml">=</mo><mrow id="S2.SS4.2.p1.6.m6.3.4.4" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml"><msub id="S2.SS4.2.p1.6.m6.3.4.4.2" xref="S2.SS4.2.p1.6.m6.3.4.4.2.cmml"><mi id="S2.SS4.2.p1.6.m6.3.4.4.2.2" xref="S2.SS4.2.p1.6.m6.3.4.4.2.2.cmml">c</mi><mi id="S2.SS4.2.p1.6.m6.3.4.4.2.3" xref="S2.SS4.2.p1.6.m6.3.4.4.2.3.cmml">G</mi></msub><mo id="S2.SS4.2.p1.6.m6.3.4.4.1" xref="S2.SS4.2.p1.6.m6.3.4.4.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.6.m6.3.4.4.3.2" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml"><mo id="S2.SS4.2.p1.6.m6.3.4.4.3.2.1" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml">(</mo><mi id="S2.SS4.2.p1.6.m6.2.2" xref="S2.SS4.2.p1.6.m6.2.2.cmml">ρ</mi><mo id="S2.SS4.2.p1.6.m6.3.4.4.3.2.2" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml">)</mo></mrow><mo id="S2.SS4.2.p1.6.m6.3.4.4.1a" xref="S2.SS4.2.p1.6.m6.3.4.4.1.cmml">⁢</mo><mi id="S2.SS4.2.p1.6.m6.3.4.4.4" xref="S2.SS4.2.p1.6.m6.3.4.4.4.cmml">ϕ</mi><mo id="S2.SS4.2.p1.6.m6.3.4.4.1b" xref="S2.SS4.2.p1.6.m6.3.4.4.1.cmml">⁢</mo><mrow id="S2.SS4.2.p1.6.m6.3.4.4.5.2" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml"><mo id="S2.SS4.2.p1.6.m6.3.4.4.5.2.1" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml">(</mo><mn id="S2.SS4.2.p1.6.m6.3.3" xref="S2.SS4.2.p1.6.m6.3.3.cmml">0</mn><mo id="S2.SS4.2.p1.6.m6.3.4.4.5.2.2" stretchy="false" xref="S2.SS4.2.p1.6.m6.3.4.4.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p1.6.m6.3.4.5" xref="S2.SS4.2.p1.6.m6.3.4.5.cmml">></mo><mn id="S2.SS4.2.p1.6.m6.3.4.6" xref="S2.SS4.2.p1.6.m6.3.4.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.6.m6.3b"><apply id="S2.SS4.2.p1.6.m6.3.4.cmml" xref="S2.SS4.2.p1.6.m6.3.4"><and id="S2.SS4.2.p1.6.m6.3.4a.cmml" xref="S2.SS4.2.p1.6.m6.3.4"></and><apply id="S2.SS4.2.p1.6.m6.3.4b.cmml" xref="S2.SS4.2.p1.6.m6.3.4"><eq id="S2.SS4.2.p1.6.m6.3.4.3.cmml" xref="S2.SS4.2.p1.6.m6.3.4.3"></eq><apply id="S2.SS4.2.p1.6.m6.3.4.2.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2"><times id="S2.SS4.2.p1.6.m6.3.4.2.1.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2.1"></times><apply id="S2.SS4.2.p1.6.m6.3.4.2.2.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.6.m6.3.4.2.2.1.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2.2">superscript</csymbol><ci id="S2.SS4.2.p1.6.m6.3.4.2.2.2.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2.2.2">𝐺</ci><ci id="S2.SS4.2.p1.6.m6.3.4.2.2.3.cmml" xref="S2.SS4.2.p1.6.m6.3.4.2.2.3">′</ci></apply><cn id="S2.SS4.2.p1.6.m6.1.1.cmml" type="integer" xref="S2.SS4.2.p1.6.m6.1.1">0</cn></apply><apply id="S2.SS4.2.p1.6.m6.3.4.4.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4"><times id="S2.SS4.2.p1.6.m6.3.4.4.1.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.1"></times><apply id="S2.SS4.2.p1.6.m6.3.4.4.2.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.2"><csymbol cd="ambiguous" id="S2.SS4.2.p1.6.m6.3.4.4.2.1.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.2">subscript</csymbol><ci id="S2.SS4.2.p1.6.m6.3.4.4.2.2.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.2.2">𝑐</ci><ci id="S2.SS4.2.p1.6.m6.3.4.4.2.3.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.2.3">𝐺</ci></apply><ci id="S2.SS4.2.p1.6.m6.2.2.cmml" xref="S2.SS4.2.p1.6.m6.2.2">𝜌</ci><ci id="S2.SS4.2.p1.6.m6.3.4.4.4.cmml" xref="S2.SS4.2.p1.6.m6.3.4.4.4">italic-ϕ</ci><cn id="S2.SS4.2.p1.6.m6.3.3.cmml" type="integer" xref="S2.SS4.2.p1.6.m6.3.3">0</cn></apply></apply><apply id="S2.SS4.2.p1.6.m6.3.4c.cmml" xref="S2.SS4.2.p1.6.m6.3.4"><gt id="S2.SS4.2.p1.6.m6.3.4.5.cmml" xref="S2.SS4.2.p1.6.m6.3.4.5"></gt><share href="https://arxiv.org/html/2503.16280v1#S2.SS4.2.p1.6.m6.3.4.4.cmml" id="S2.SS4.2.p1.6.m6.3.4d.cmml" xref="S2.SS4.2.p1.6.m6.3.4"></share><cn id="S2.SS4.2.p1.6.m6.3.4.6.cmml" type="integer" xref="S2.SS4.2.p1.6.m6.3.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.6.m6.3c">G^{\prime}(0)=c_{G}(\rho)\phi(0)>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.6.m6.3d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) italic_ϕ ( 0 ) > 0</annotation></semantics></math>. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem2" title="Theorem 2. ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, then, <math alttext="\tau=0" class="ltx_Math" display="inline" id="S2.SS4.2.p1.7.m7.1"><semantics id="S2.SS4.2.p1.7.m7.1a"><mrow id="S2.SS4.2.p1.7.m7.1.1" xref="S2.SS4.2.p1.7.m7.1.1.cmml"><mi id="S2.SS4.2.p1.7.m7.1.1.2" xref="S2.SS4.2.p1.7.m7.1.1.2.cmml">τ</mi><mo id="S2.SS4.2.p1.7.m7.1.1.1" xref="S2.SS4.2.p1.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS4.2.p1.7.m7.1.1.3" xref="S2.SS4.2.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.7.m7.1b"><apply id="S2.SS4.2.p1.7.m7.1.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1"><eq id="S2.SS4.2.p1.7.m7.1.1.1.cmml" xref="S2.SS4.2.p1.7.m7.1.1.1"></eq><ci id="S2.SS4.2.p1.7.m7.1.1.2.cmml" xref="S2.SS4.2.p1.7.m7.1.1.2">𝜏</ci><cn id="S2.SS4.2.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS4.2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.7.m7.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.7.m7.1d">italic_τ = 0</annotation></semantics></math> is unstable. Since <math alttext="0" class="ltx_Math" display="inline" id="S2.SS4.2.p1.8.m8.1"><semantics id="S2.SS4.2.p1.8.m8.1a"><mn id="S2.SS4.2.p1.8.m8.1.1" xref="S2.SS4.2.p1.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.8.m8.1b"><cn id="S2.SS4.2.p1.8.m8.1.1.cmml" type="integer" xref="S2.SS4.2.p1.8.m8.1.1">0</cn></annotation-xml></semantics></math> is the only non-infinite equilibrium, the equilibria <math alttext="\tau=\pm\infty" class="ltx_Math" display="inline" id="S2.SS4.2.p1.9.m9.1"><semantics id="S2.SS4.2.p1.9.m9.1a"><mrow id="S2.SS4.2.p1.9.m9.1.1" xref="S2.SS4.2.p1.9.m9.1.1.cmml"><mi id="S2.SS4.2.p1.9.m9.1.1.2" xref="S2.SS4.2.p1.9.m9.1.1.2.cmml">τ</mi><mo id="S2.SS4.2.p1.9.m9.1.1.1" xref="S2.SS4.2.p1.9.m9.1.1.1.cmml">=</mo><mrow id="S2.SS4.2.p1.9.m9.1.1.3" xref="S2.SS4.2.p1.9.m9.1.1.3.cmml"><mo id="S2.SS4.2.p1.9.m9.1.1.3a" xref="S2.SS4.2.p1.9.m9.1.1.3.cmml">±</mo><mi id="S2.SS4.2.p1.9.m9.1.1.3.2" mathvariant="normal" xref="S2.SS4.2.p1.9.m9.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p1.9.m9.1b"><apply id="S2.SS4.2.p1.9.m9.1.1.cmml" xref="S2.SS4.2.p1.9.m9.1.1"><eq id="S2.SS4.2.p1.9.m9.1.1.1.cmml" xref="S2.SS4.2.p1.9.m9.1.1.1"></eq><ci id="S2.SS4.2.p1.9.m9.1.1.2.cmml" xref="S2.SS4.2.p1.9.m9.1.1.2">𝜏</ci><apply id="S2.SS4.2.p1.9.m9.1.1.3.cmml" xref="S2.SS4.2.p1.9.m9.1.1.3"><csymbol cd="latexml" id="S2.SS4.2.p1.9.m9.1.1.3.1.cmml" xref="S2.SS4.2.p1.9.m9.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS4.2.p1.9.m9.1.1.3.2.cmml" xref="S2.SS4.2.p1.9.m9.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p1.9.m9.1c">\tau=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p1.9.m9.1d">italic_τ = ± ∞</annotation></semantics></math> are stable by topological necessity. ∎</p> </div> </div> <div class="ltx_para" id="S2.SS4.p5"> <p class="ltx_p" id="S2.SS4.p5.2">For any finite starting point <math alttext="\tau(0)\neq 0" class="ltx_Math" display="inline" id="S2.SS4.p5.1.m1.1"><semantics id="S2.SS4.p5.1.m1.1a"><mrow id="S2.SS4.p5.1.m1.1.2" xref="S2.SS4.p5.1.m1.1.2.cmml"><mrow id="S2.SS4.p5.1.m1.1.2.2" xref="S2.SS4.p5.1.m1.1.2.2.cmml"><mi id="S2.SS4.p5.1.m1.1.2.2.2" xref="S2.SS4.p5.1.m1.1.2.2.2.cmml">τ</mi><mo id="S2.SS4.p5.1.m1.1.2.2.1" xref="S2.SS4.p5.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.1.m1.1.2.2.3.2" xref="S2.SS4.p5.1.m1.1.2.2.cmml"><mo id="S2.SS4.p5.1.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.p5.1.m1.1.2.2.cmml">(</mo><mn id="S2.SS4.p5.1.m1.1.1" xref="S2.SS4.p5.1.m1.1.1.cmml">0</mn><mo id="S2.SS4.p5.1.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS4.p5.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p5.1.m1.1.2.1" xref="S2.SS4.p5.1.m1.1.2.1.cmml">≠</mo><mn id="S2.SS4.p5.1.m1.1.2.3" xref="S2.SS4.p5.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.1.m1.1b"><apply id="S2.SS4.p5.1.m1.1.2.cmml" xref="S2.SS4.p5.1.m1.1.2"><neq id="S2.SS4.p5.1.m1.1.2.1.cmml" xref="S2.SS4.p5.1.m1.1.2.1"></neq><apply id="S2.SS4.p5.1.m1.1.2.2.cmml" xref="S2.SS4.p5.1.m1.1.2.2"><times id="S2.SS4.p5.1.m1.1.2.2.1.cmml" xref="S2.SS4.p5.1.m1.1.2.2.1"></times><ci id="S2.SS4.p5.1.m1.1.2.2.2.cmml" xref="S2.SS4.p5.1.m1.1.2.2.2">𝜏</ci><cn id="S2.SS4.p5.1.m1.1.1.cmml" type="integer" xref="S2.SS4.p5.1.m1.1.1">0</cn></apply><cn id="S2.SS4.p5.1.m1.1.2.3.cmml" type="integer" xref="S2.SS4.p5.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.1.m1.1c">\tau(0)\neq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.1.m1.1d">italic_τ ( 0 ) ≠ 0</annotation></semantics></math>, then, we have <math alttext="\lim_{t\to\infty}\tau(t)=\pm\infty" class="ltx_Math" display="inline" id="S2.SS4.p5.2.m2.1"><semantics id="S2.SS4.p5.2.m2.1a"><mrow id="S2.SS4.p5.2.m2.1.2" xref="S2.SS4.p5.2.m2.1.2.cmml"><mrow id="S2.SS4.p5.2.m2.1.2.2" xref="S2.SS4.p5.2.m2.1.2.2.cmml"><msub id="S2.SS4.p5.2.m2.1.2.2.1" xref="S2.SS4.p5.2.m2.1.2.2.1.cmml"><mo id="S2.SS4.p5.2.m2.1.2.2.1.2" xref="S2.SS4.p5.2.m2.1.2.2.1.2.cmml">lim</mo><mrow id="S2.SS4.p5.2.m2.1.2.2.1.3" xref="S2.SS4.p5.2.m2.1.2.2.1.3.cmml"><mi id="S2.SS4.p5.2.m2.1.2.2.1.3.2" xref="S2.SS4.p5.2.m2.1.2.2.1.3.2.cmml">t</mi><mo id="S2.SS4.p5.2.m2.1.2.2.1.3.1" stretchy="false" xref="S2.SS4.p5.2.m2.1.2.2.1.3.1.cmml">→</mo><mi id="S2.SS4.p5.2.m2.1.2.2.1.3.3" mathvariant="normal" xref="S2.SS4.p5.2.m2.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S2.SS4.p5.2.m2.1.2.2.2" xref="S2.SS4.p5.2.m2.1.2.2.2.cmml"><mi id="S2.SS4.p5.2.m2.1.2.2.2.2" xref="S2.SS4.p5.2.m2.1.2.2.2.2.cmml">τ</mi><mo id="S2.SS4.p5.2.m2.1.2.2.2.1" xref="S2.SS4.p5.2.m2.1.2.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.p5.2.m2.1.2.2.2.3.2" xref="S2.SS4.p5.2.m2.1.2.2.2.cmml"><mo id="S2.SS4.p5.2.m2.1.2.2.2.3.2.1" stretchy="false" xref="S2.SS4.p5.2.m2.1.2.2.2.cmml">(</mo><mi id="S2.SS4.p5.2.m2.1.1" xref="S2.SS4.p5.2.m2.1.1.cmml">t</mi><mo id="S2.SS4.p5.2.m2.1.2.2.2.3.2.2" stretchy="false" xref="S2.SS4.p5.2.m2.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS4.p5.2.m2.1.2.1" xref="S2.SS4.p5.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS4.p5.2.m2.1.2.3" xref="S2.SS4.p5.2.m2.1.2.3.cmml"><mo id="S2.SS4.p5.2.m2.1.2.3a" xref="S2.SS4.p5.2.m2.1.2.3.cmml">±</mo><mi id="S2.SS4.p5.2.m2.1.2.3.2" mathvariant="normal" xref="S2.SS4.p5.2.m2.1.2.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p5.2.m2.1b"><apply id="S2.SS4.p5.2.m2.1.2.cmml" xref="S2.SS4.p5.2.m2.1.2"><eq id="S2.SS4.p5.2.m2.1.2.1.cmml" xref="S2.SS4.p5.2.m2.1.2.1"></eq><apply id="S2.SS4.p5.2.m2.1.2.2.cmml" xref="S2.SS4.p5.2.m2.1.2.2"><apply id="S2.SS4.p5.2.m2.1.2.2.1.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1"><csymbol cd="ambiguous" id="S2.SS4.p5.2.m2.1.2.2.1.1.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1">subscript</csymbol><limit id="S2.SS4.p5.2.m2.1.2.2.1.2.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1.2"></limit><apply id="S2.SS4.p5.2.m2.1.2.2.1.3.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1.3"><ci id="S2.SS4.p5.2.m2.1.2.2.1.3.1.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1.3.1">→</ci><ci id="S2.SS4.p5.2.m2.1.2.2.1.3.2.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1.3.2">𝑡</ci><infinity id="S2.SS4.p5.2.m2.1.2.2.1.3.3.cmml" xref="S2.SS4.p5.2.m2.1.2.2.1.3.3"></infinity></apply></apply><apply id="S2.SS4.p5.2.m2.1.2.2.2.cmml" xref="S2.SS4.p5.2.m2.1.2.2.2"><times id="S2.SS4.p5.2.m2.1.2.2.2.1.cmml" xref="S2.SS4.p5.2.m2.1.2.2.2.1"></times><ci id="S2.SS4.p5.2.m2.1.2.2.2.2.cmml" xref="S2.SS4.p5.2.m2.1.2.2.2.2">𝜏</ci><ci id="S2.SS4.p5.2.m2.1.1.cmml" xref="S2.SS4.p5.2.m2.1.1">𝑡</ci></apply></apply><apply id="S2.SS4.p5.2.m2.1.2.3.cmml" xref="S2.SS4.p5.2.m2.1.2.3"><csymbol cd="latexml" id="S2.SS4.p5.2.m2.1.2.3.1.cmml" xref="S2.SS4.p5.2.m2.1.2.3">plus-or-minus</csymbol><infinity id="S2.SS4.p5.2.m2.1.2.3.2.cmml" xref="S2.SS4.p5.2.m2.1.2.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p5.2.m2.1c">\lim_{t\to\infty}\tau(t)=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p5.2.m2.1d">roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_τ ( italic_t ) = ± ∞</annotation></semantics></math>. Therefore, we find in settings where graders receive a noisy, normally distributed version of information about an essay, Output Agreement incentivizes agents to stabilize at an equilibrium where they all submit the same report. This uninformative consensus is in contrast to the truthful equilibrium guarantees of the binary signal model.<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>In the Gaussian model, strong diagonalization holds within a reasonable range of thresholds around 0, depending on <math alttext="\rho" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">\rho</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_ρ</annotation></semantics></math>.</span></span></span></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>Dasgupta-Ghosh</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.4">We next consider the mechanism proposed by <cite class="ltx_cite ltx_citemacro_citet">Dasgupta and Ghosh [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib4" title="">2013</a>]</cite>, which we will refer to as the Dasgupta-Ghosh (DG) mechanism. DG is a multi-task mechanism: agents participate in a set of peer prediction tasks, and their payment is a function of their reports across these tasks. For a given task, agent <math alttext="i" 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">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_i</annotation></semantics></math> receives a payment of one if their report <math alttext="r_{i}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">r</mi><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝑟</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">r_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> matches that of another agent <math alttext="j" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_j</annotation></semantics></math> on that task, as in Output Agreement. However, they are also <em class="ltx_emph ltx_font_italic" id="S3.p1.4.1">penalized</em> if their report matches that of the other agent on some unrelated task agent <math alttext="i" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_i</annotation></semantics></math> did not complete, with the intution that agents should be paid only for their performance over and above what would be expected by chance.<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>There have been a variety of ways considered to calculate this penalty. This version corresponds to their <math alttext="d=1" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mrow id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml"><mi id="footnote3.m1.1.1.2" xref="footnote3.m1.1.1.2.cmml">d</mi><mo id="footnote3.m1.1.1.1" xref="footnote3.m1.1.1.1.cmml">=</mo><mn id="footnote3.m1.1.1.3" xref="footnote3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><apply id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1"><eq id="footnote3.m1.1.1.1.cmml" xref="footnote3.m1.1.1.1"></eq><ci id="footnote3.m1.1.1.2.cmml" xref="footnote3.m1.1.1.2">𝑑</ci><cn id="footnote3.m1.1.1.3.cmml" type="integer" xref="footnote3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">d=1</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_d = 1</annotation></semantics></math> option.</span></span></span> In the binary signal setting, this penalty ensures that uninformative equilibria in OA where all agents agree to report H or L now have zero value, thereby ensuring strategic agents are not incentivized toward them.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.4">Formally, if <math alttext="r_{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">r</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">𝑟</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">r_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> agent <math alttext="i" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mi id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><ci id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_i</annotation></semantics></math>’s report on the task, <math alttext="r_{j}" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><msub id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml"><mi id="S3.p2.3.m3.1.1.2" xref="S3.p2.3.m3.1.1.2.cmml">r</mi><mi id="S3.p2.3.m3.1.1.3" xref="S3.p2.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><apply id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.3.m3.1.1.1.cmml" xref="S3.p2.3.m3.1.1">subscript</csymbol><ci id="S3.p2.3.m3.1.1.2.cmml" xref="S3.p2.3.m3.1.1.2">𝑟</ci><ci id="S3.p2.3.m3.1.1.3.cmml" xref="S3.p2.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">r_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is the peer report on the task, and <math alttext="r_{k}" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><msub id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><mi id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml">r</mi><mi id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" 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start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.8">Assume the task <math alttext="r_{k}" class="ltx_Math" display="inline" id="S3.p2.5.m1.1"><semantics id="S3.p2.5.m1.1a"><msub id="S3.p2.5.m1.1.1" xref="S3.p2.5.m1.1.1.cmml"><mi id="S3.p2.5.m1.1.1.2" xref="S3.p2.5.m1.1.1.2.cmml">r</mi><mi id="S3.p2.5.m1.1.1.3" xref="S3.p2.5.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.5.m1.1b"><apply id="S3.p2.5.m1.1.1.cmml" xref="S3.p2.5.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.5.m1.1.1.1.cmml" xref="S3.p2.5.m1.1.1">subscript</csymbol><ci id="S3.p2.5.m1.1.1.2.cmml" xref="S3.p2.5.m1.1.1.2">𝑟</ci><ci id="S3.p2.5.m1.1.1.3.cmml" xref="S3.p2.5.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m1.1c">r_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m1.1d">italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> comes from is i.i.d. to the task being scored and, as with our analysis of OA, all agents are using symmetric strategies. Then agent <math alttext="i" class="ltx_Math" display="inline" id="S3.p2.6.m2.1"><semantics id="S3.p2.6.m2.1a"><mi id="S3.p2.6.m2.1.1" xref="S3.p2.6.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p2.6.m2.1b"><ci id="S3.p2.6.m2.1.1.cmml" xref="S3.p2.6.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m2.1d">italic_i</annotation></semantics></math>’s expected penalty is <math alttext="\mathop{\mathbb{E}}\mathbf{1}[r_{i}=\sigma(X)]" class="ltx_Math" display="inline" id="S3.p2.7.m3.2"><semantics id="S3.p2.7.m3.2a"><mrow id="S3.p2.7.m3.2.2" xref="S3.p2.7.m3.2.2.cmml"><mo id="S3.p2.7.m3.2.2.2" rspace="0.167em" xref="S3.p2.7.m3.2.2.2.cmml">𝔼</mo><mrow id="S3.p2.7.m3.2.2.1" xref="S3.p2.7.m3.2.2.1.cmml"><mn id="S3.p2.7.m3.2.2.1.3" xref="S3.p2.7.m3.2.2.1.3.cmml">𝟏</mn><mo id="S3.p2.7.m3.2.2.1.2" xref="S3.p2.7.m3.2.2.1.2.cmml">⁢</mo><mrow id="S3.p2.7.m3.2.2.1.1.1" xref="S3.p2.7.m3.2.2.1.1.2.cmml"><mo id="S3.p2.7.m3.2.2.1.1.1.2" stretchy="false" xref="S3.p2.7.m3.2.2.1.1.2.1.cmml">[</mo><mrow id="S3.p2.7.m3.2.2.1.1.1.1" xref="S3.p2.7.m3.2.2.1.1.1.1.cmml"><msub id="S3.p2.7.m3.2.2.1.1.1.1.2" xref="S3.p2.7.m3.2.2.1.1.1.1.2.cmml"><mi id="S3.p2.7.m3.2.2.1.1.1.1.2.2" xref="S3.p2.7.m3.2.2.1.1.1.1.2.2.cmml">r</mi><mi id="S3.p2.7.m3.2.2.1.1.1.1.2.3" xref="S3.p2.7.m3.2.2.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.p2.7.m3.2.2.1.1.1.1.1" xref="S3.p2.7.m3.2.2.1.1.1.1.1.cmml">=</mo><mrow id="S3.p2.7.m3.2.2.1.1.1.1.3" xref="S3.p2.7.m3.2.2.1.1.1.1.3.cmml"><mi id="S3.p2.7.m3.2.2.1.1.1.1.3.2" xref="S3.p2.7.m3.2.2.1.1.1.1.3.2.cmml">σ</mi><mo id="S3.p2.7.m3.2.2.1.1.1.1.3.1" xref="S3.p2.7.m3.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.p2.7.m3.2.2.1.1.1.1.3.3.2" xref="S3.p2.7.m3.2.2.1.1.1.1.3.cmml"><mo id="S3.p2.7.m3.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S3.p2.7.m3.2.2.1.1.1.1.3.cmml">(</mo><mi id="S3.p2.7.m3.1.1" xref="S3.p2.7.m3.1.1.cmml">X</mi><mo id="S3.p2.7.m3.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S3.p2.7.m3.2.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.p2.7.m3.2.2.1.1.1.3" stretchy="false" xref="S3.p2.7.m3.2.2.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.7.m3.2b"><apply id="S3.p2.7.m3.2.2.cmml" xref="S3.p2.7.m3.2.2"><ci id="S3.p2.7.m3.2.2.2.cmml" xref="S3.p2.7.m3.2.2.2">𝔼</ci><apply id="S3.p2.7.m3.2.2.1.cmml" xref="S3.p2.7.m3.2.2.1"><times id="S3.p2.7.m3.2.2.1.2.cmml" xref="S3.p2.7.m3.2.2.1.2"></times><cn id="S3.p2.7.m3.2.2.1.3.cmml" type="integer" xref="S3.p2.7.m3.2.2.1.3">1</cn><apply id="S3.p2.7.m3.2.2.1.1.2.cmml" xref="S3.p2.7.m3.2.2.1.1.1"><csymbol cd="latexml" id="S3.p2.7.m3.2.2.1.1.2.1.cmml" xref="S3.p2.7.m3.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S3.p2.7.m3.2.2.1.1.1.1.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1"><eq id="S3.p2.7.m3.2.2.1.1.1.1.1.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.1"></eq><apply id="S3.p2.7.m3.2.2.1.1.1.1.2.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.p2.7.m3.2.2.1.1.1.1.2.1.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.2">subscript</csymbol><ci id="S3.p2.7.m3.2.2.1.1.1.1.2.2.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.2.2">𝑟</ci><ci id="S3.p2.7.m3.2.2.1.1.1.1.2.3.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.2.3">𝑖</ci></apply><apply id="S3.p2.7.m3.2.2.1.1.1.1.3.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.3"><times id="S3.p2.7.m3.2.2.1.1.1.1.3.1.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.3.1"></times><ci id="S3.p2.7.m3.2.2.1.1.1.1.3.2.cmml" xref="S3.p2.7.m3.2.2.1.1.1.1.3.2">𝜎</ci><ci id="S3.p2.7.m3.1.1.cmml" xref="S3.p2.7.m3.1.1">𝑋</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m3.2c">\mathop{\mathbb{E}}\mathbf{1}[r_{i}=\sigma(X)]</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m3.2d">blackboard_E bold_1 [ italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ ( italic_X ) ]</annotation></semantics></math>. That is, the expected penalty is exactly the prior probability of a signal that leads to a report of <math alttext="r_{i}" class="ltx_Math" display="inline" id="S3.p2.8.m4.1"><semantics id="S3.p2.8.m4.1a"><msub id="S3.p2.8.m4.1.1" xref="S3.p2.8.m4.1.1.cmml"><mi id="S3.p2.8.m4.1.1.2" xref="S3.p2.8.m4.1.1.2.cmml">r</mi><mi id="S3.p2.8.m4.1.1.3" xref="S3.p2.8.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.8.m4.1b"><apply id="S3.p2.8.m4.1.1.cmml" xref="S3.p2.8.m4.1.1"><csymbol cd="ambiguous" id="S3.p2.8.m4.1.1.1.cmml" xref="S3.p2.8.m4.1.1">subscript</csymbol><ci id="S3.p2.8.m4.1.1.2.cmml" xref="S3.p2.8.m4.1.1.2">𝑟</ci><ci id="S3.p2.8.m4.1.1.3.cmml" xref="S3.p2.8.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m4.1c">r_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m4.1d">italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. We denote this as</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{L}=\Pr[\sigma(X)=L]=\Pr[X\leq\tau]=F(\tau)," class="ltx_Math" display="block" id="S3.Ex8.m1.5"><semantics id="S3.Ex8.m1.5a"><mrow id="S3.Ex8.m1.5.5.1" xref="S3.Ex8.m1.5.5.1.1.cmml"><mrow id="S3.Ex8.m1.5.5.1.1" xref="S3.Ex8.m1.5.5.1.1.cmml"><msub id="S3.Ex8.m1.5.5.1.1.4" xref="S3.Ex8.m1.5.5.1.1.4.cmml"><mi id="S3.Ex8.m1.5.5.1.1.4.2" xref="S3.Ex8.m1.5.5.1.1.4.2.cmml">π</mi><mi id="S3.Ex8.m1.5.5.1.1.4.3" xref="S3.Ex8.m1.5.5.1.1.4.3.cmml">L</mi></msub><mo id="S3.Ex8.m1.5.5.1.1.5" xref="S3.Ex8.m1.5.5.1.1.5.cmml">=</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml"><mi id="S3.Ex8.m1.2.2" xref="S3.Ex8.m1.2.2.cmml">Pr</mi><mo id="S3.Ex8.m1.5.5.1.1.1.1a" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml">⁡</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml"><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml">[</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.2" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.2.cmml">σ</mi><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.1" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.3.2" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml">(</mo><mi id="S3.Ex8.m1.1.1" xref="S3.Ex8.m1.1.1.cmml">X</mi><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml">=</mo><mi id="S3.Ex8.m1.5.5.1.1.1.1.1.1.3" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="S3.Ex8.m1.5.5.1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.1.2.cmml">]</mo></mrow></mrow><mo id="S3.Ex8.m1.5.5.1.1.6" xref="S3.Ex8.m1.5.5.1.1.6.cmml">=</mo><mrow id="S3.Ex8.m1.5.5.1.1.2.1" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml"><mi id="S3.Ex8.m1.3.3" xref="S3.Ex8.m1.3.3.cmml">Pr</mi><mo id="S3.Ex8.m1.5.5.1.1.2.1a" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml">⁡</mo><mrow id="S3.Ex8.m1.5.5.1.1.2.1.1" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml"><mo id="S3.Ex8.m1.5.5.1.1.2.1.1.2" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml">[</mo><mrow id="S3.Ex8.m1.5.5.1.1.2.1.1.1" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.cmml"><mi id="S3.Ex8.m1.5.5.1.1.2.1.1.1.2" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.2.cmml">X</mi><mo id="S3.Ex8.m1.5.5.1.1.2.1.1.1.1" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.1.cmml">≤</mo><mi id="S3.Ex8.m1.5.5.1.1.2.1.1.1.3" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.3.cmml">τ</mi></mrow><mo id="S3.Ex8.m1.5.5.1.1.2.1.1.3" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.2.2.cmml">]</mo></mrow></mrow><mo id="S3.Ex8.m1.5.5.1.1.7" xref="S3.Ex8.m1.5.5.1.1.7.cmml">=</mo><mrow id="S3.Ex8.m1.5.5.1.1.8" xref="S3.Ex8.m1.5.5.1.1.8.cmml"><mi id="S3.Ex8.m1.5.5.1.1.8.2" xref="S3.Ex8.m1.5.5.1.1.8.2.cmml">F</mi><mo id="S3.Ex8.m1.5.5.1.1.8.1" xref="S3.Ex8.m1.5.5.1.1.8.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.5.5.1.1.8.3.2" xref="S3.Ex8.m1.5.5.1.1.8.cmml"><mo id="S3.Ex8.m1.5.5.1.1.8.3.2.1" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.8.cmml">(</mo><mi id="S3.Ex8.m1.4.4" xref="S3.Ex8.m1.4.4.cmml">τ</mi><mo id="S3.Ex8.m1.5.5.1.1.8.3.2.2" stretchy="false" xref="S3.Ex8.m1.5.5.1.1.8.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.5.5.1.2" xref="S3.Ex8.m1.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex8.m1.5b"><apply id="S3.Ex8.m1.5.5.1.1.cmml" xref="S3.Ex8.m1.5.5.1"><and id="S3.Ex8.m1.5.5.1.1a.cmml" xref="S3.Ex8.m1.5.5.1"></and><apply id="S3.Ex8.m1.5.5.1.1b.cmml" xref="S3.Ex8.m1.5.5.1"><eq id="S3.Ex8.m1.5.5.1.1.5.cmml" xref="S3.Ex8.m1.5.5.1.1.5"></eq><apply id="S3.Ex8.m1.5.5.1.1.4.cmml" xref="S3.Ex8.m1.5.5.1.1.4"><csymbol cd="ambiguous" id="S3.Ex8.m1.5.5.1.1.4.1.cmml" xref="S3.Ex8.m1.5.5.1.1.4">subscript</csymbol><ci id="S3.Ex8.m1.5.5.1.1.4.2.cmml" xref="S3.Ex8.m1.5.5.1.1.4.2">𝜋</ci><ci id="S3.Ex8.m1.5.5.1.1.4.3.cmml" xref="S3.Ex8.m1.5.5.1.1.4.3">𝐿</ci></apply><apply id="S3.Ex8.m1.5.5.1.1.1.2.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1"><ci id="S3.Ex8.m1.2.2.cmml" xref="S3.Ex8.m1.2.2">Pr</ci><apply id="S3.Ex8.m1.5.5.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1"><eq id="S3.Ex8.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.1"></eq><apply id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2"><times id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.1"></times><ci id="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.2.2">𝜎</ci><ci id="S3.Ex8.m1.1.1.cmml" xref="S3.Ex8.m1.1.1">𝑋</ci></apply><ci id="S3.Ex8.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S3.Ex8.m1.5.5.1.1.1.1.1.1.3">𝐿</ci></apply></apply></apply><apply id="S3.Ex8.m1.5.5.1.1c.cmml" xref="S3.Ex8.m1.5.5.1"><eq id="S3.Ex8.m1.5.5.1.1.6.cmml" xref="S3.Ex8.m1.5.5.1.1.6"></eq><share href="https://arxiv.org/html/2503.16280v1#S3.Ex8.m1.5.5.1.1.1.cmml" id="S3.Ex8.m1.5.5.1.1d.cmml" xref="S3.Ex8.m1.5.5.1"></share><apply id="S3.Ex8.m1.5.5.1.1.2.2.cmml" xref="S3.Ex8.m1.5.5.1.1.2.1"><ci id="S3.Ex8.m1.3.3.cmml" xref="S3.Ex8.m1.3.3">Pr</ci><apply id="S3.Ex8.m1.5.5.1.1.2.1.1.1.cmml" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1"><leq id="S3.Ex8.m1.5.5.1.1.2.1.1.1.1.cmml" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.1"></leq><ci id="S3.Ex8.m1.5.5.1.1.2.1.1.1.2.cmml" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.2">𝑋</ci><ci id="S3.Ex8.m1.5.5.1.1.2.1.1.1.3.cmml" xref="S3.Ex8.m1.5.5.1.1.2.1.1.1.3">𝜏</ci></apply></apply></apply><apply id="S3.Ex8.m1.5.5.1.1e.cmml" xref="S3.Ex8.m1.5.5.1"><eq id="S3.Ex8.m1.5.5.1.1.7.cmml" xref="S3.Ex8.m1.5.5.1.1.7"></eq><share href="https://arxiv.org/html/2503.16280v1#S3.Ex8.m1.5.5.1.1.2.cmml" id="S3.Ex8.m1.5.5.1.1f.cmml" xref="S3.Ex8.m1.5.5.1"></share><apply id="S3.Ex8.m1.5.5.1.1.8.cmml" xref="S3.Ex8.m1.5.5.1.1.8"><times id="S3.Ex8.m1.5.5.1.1.8.1.cmml" xref="S3.Ex8.m1.5.5.1.1.8.1"></times><ci id="S3.Ex8.m1.5.5.1.1.8.2.cmml" xref="S3.Ex8.m1.5.5.1.1.8.2">𝐹</ci><ci id="S3.Ex8.m1.4.4.cmml" xref="S3.Ex8.m1.4.4">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.5c">\pi_{L}=\Pr[\sigma(X)=L]=\Pr[X\leq\tau]=F(\tau),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.5d">italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = roman_Pr [ italic_σ ( italic_X ) = italic_L ] = roman_Pr [ italic_X ≤ italic_τ ] = italic_F ( italic_τ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.9">with <math alttext="\pi_{H}=1-\pi_{L}" class="ltx_Math" display="inline" id="S3.p2.9.m1.1"><semantics id="S3.p2.9.m1.1a"><mrow id="S3.p2.9.m1.1.1" xref="S3.p2.9.m1.1.1.cmml"><msub id="S3.p2.9.m1.1.1.2" xref="S3.p2.9.m1.1.1.2.cmml"><mi id="S3.p2.9.m1.1.1.2.2" xref="S3.p2.9.m1.1.1.2.2.cmml">π</mi><mi id="S3.p2.9.m1.1.1.2.3" xref="S3.p2.9.m1.1.1.2.3.cmml">H</mi></msub><mo id="S3.p2.9.m1.1.1.1" xref="S3.p2.9.m1.1.1.1.cmml">=</mo><mrow id="S3.p2.9.m1.1.1.3" xref="S3.p2.9.m1.1.1.3.cmml"><mn id="S3.p2.9.m1.1.1.3.2" xref="S3.p2.9.m1.1.1.3.2.cmml">1</mn><mo id="S3.p2.9.m1.1.1.3.1" xref="S3.p2.9.m1.1.1.3.1.cmml">−</mo><msub id="S3.p2.9.m1.1.1.3.3" xref="S3.p2.9.m1.1.1.3.3.cmml"><mi id="S3.p2.9.m1.1.1.3.3.2" xref="S3.p2.9.m1.1.1.3.3.2.cmml">π</mi><mi id="S3.p2.9.m1.1.1.3.3.3" xref="S3.p2.9.m1.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.9.m1.1b"><apply id="S3.p2.9.m1.1.1.cmml" xref="S3.p2.9.m1.1.1"><eq id="S3.p2.9.m1.1.1.1.cmml" xref="S3.p2.9.m1.1.1.1"></eq><apply id="S3.p2.9.m1.1.1.2.cmml" xref="S3.p2.9.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p2.9.m1.1.1.2.1.cmml" xref="S3.p2.9.m1.1.1.2">subscript</csymbol><ci id="S3.p2.9.m1.1.1.2.2.cmml" xref="S3.p2.9.m1.1.1.2.2">𝜋</ci><ci id="S3.p2.9.m1.1.1.2.3.cmml" xref="S3.p2.9.m1.1.1.2.3">𝐻</ci></apply><apply id="S3.p2.9.m1.1.1.3.cmml" xref="S3.p2.9.m1.1.1.3"><minus id="S3.p2.9.m1.1.1.3.1.cmml" xref="S3.p2.9.m1.1.1.3.1"></minus><cn id="S3.p2.9.m1.1.1.3.2.cmml" type="integer" xref="S3.p2.9.m1.1.1.3.2">1</cn><apply id="S3.p2.9.m1.1.1.3.3.cmml" xref="S3.p2.9.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.p2.9.m1.1.1.3.3.1.cmml" xref="S3.p2.9.m1.1.1.3.3">subscript</csymbol><ci id="S3.p2.9.m1.1.1.3.3.2.cmml" xref="S3.p2.9.m1.1.1.3.3.2">𝜋</ci><ci id="S3.p2.9.m1.1.1.3.3.3.cmml" xref="S3.p2.9.m1.1.1.3.3.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m1.1c">\pi_{H}=1-\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m1.1d">italic_π start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = 1 - italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, we instead work with the following formulation of DG which has the same expected payment:</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M_{\textrm{DG}}(r_{i},r_{j},\pi_{L})=\mathbf{1}[r_{i}=r_{j}]-\pi_{r_{i}}." class="ltx_Math" display="block" id="S3.Ex9.m1.1"><semantics id="S3.Ex9.m1.1a"><mrow id="S3.Ex9.m1.1.1.1" xref="S3.Ex9.m1.1.1.1.1.cmml"><mrow id="S3.Ex9.m1.1.1.1.1" xref="S3.Ex9.m1.1.1.1.1.cmml"><mrow id="S3.Ex9.m1.1.1.1.1.3" xref="S3.Ex9.m1.1.1.1.1.3.cmml"><msub id="S3.Ex9.m1.1.1.1.1.3.5" xref="S3.Ex9.m1.1.1.1.1.3.5.cmml"><mi id="S3.Ex9.m1.1.1.1.1.3.5.2" xref="S3.Ex9.m1.1.1.1.1.3.5.2.cmml">M</mi><mtext id="S3.Ex9.m1.1.1.1.1.3.5.3" xref="S3.Ex9.m1.1.1.1.1.3.5.3a.cmml">DG</mtext></msub><mo id="S3.Ex9.m1.1.1.1.1.3.4" xref="S3.Ex9.m1.1.1.1.1.3.4.cmml">⁢</mo><mrow 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.10">Despite the nature of DG as a multi-task mechanism, this means that (under our assumption of symmetry and if other agents use the same strategy across tasks) we can essentially analyze each task in isolation. More formally, this means we can adopt the same model from § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS1" title="2.1 Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.1</span></a> we used to analyze OA, changing only the payment rule and updating the quantities that derive from it accordingly.</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>Equilibrium Characterization</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.3">For DG, the (ex-interim) expected utility for playing strategy <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><msub 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" xref="S3.SS1.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">i</mi></msub><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">subscript</csymbol><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝜎</ci><ci id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> depends on <math alttext="\sigma_{j}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><msub id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">𝜎</ci><ci id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\sigma_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> both directly in the bonus for agreement and indirectly through <math alttext="\pi_{L}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><msub id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">π</mi><mi id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝜋</ci><ci id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E9"> <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 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id="S3.E9.m1.2.2.1.2.cmml" xref="S3.E9.m1.2.2.1.2"></times><apply id="S3.E9.m1.2.2.1.3.cmml" xref="S3.E9.m1.2.2.1.3"><csymbol cd="ambiguous" id="S3.E9.m1.2.2.1.3.1.cmml" xref="S3.E9.m1.2.2.1.3">subscript</csymbol><ci id="S3.E9.m1.2.2.1.3.2.cmml" xref="S3.E9.m1.2.2.1.3.2">𝜎</ci><ci id="S3.E9.m1.2.2.1.3.3.cmml" xref="S3.E9.m1.2.2.1.3.3">𝑖</ci></apply><ci id="S3.E9.m1.2.2.1.1.cmml" xref="S3.E9.m1.2.2.1.1">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9.m1.6c">U_{i}(\sigma_{i},\sigma,x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf% {1}[\sigma_{i}(x)=\sigma(x^{\prime})]-\pi_{\sigma_{i}(x)}.</annotation><annotation encoding="application/x-llamapun" id="S3.E9.m1.6d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_σ , italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] - italic_π start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) 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">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.5">The condition for a symmetric Bayes-Nash equilibrium can then be written as</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall x\in\mathbb{R},\;\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1% }[\sigma(x)=\sigma(x^{\prime})]-\pi_{\sigma(x)}\geq\mathop{\mathbb{E}}_{x^{% \prime}\sim\beta(x)}\mathbf{1}[\overline{\sigma(x)}=\sigma(x^{\prime})]-\pi_{% \overline{\sigma(x)}}." class="ltx_Math" display="block" id="S3.Ex10.m1.7"><semantics id="S3.Ex10.m1.7a"><mrow id="S3.Ex10.m1.7.7.1"><mrow id="S3.Ex10.m1.7.7.1.1.2" xref="S3.Ex10.m1.7.7.1.1.3.cmml"><mrow id="S3.Ex10.m1.7.7.1.1.1.1" xref="S3.Ex10.m1.7.7.1.1.1.1.cmml"><mrow id="S3.Ex10.m1.7.7.1.1.1.1.2" xref="S3.Ex10.m1.7.7.1.1.1.1.2.cmml"><mo id="S3.Ex10.m1.7.7.1.1.1.1.2.1" rspace="0.167em" xref="S3.Ex10.m1.7.7.1.1.1.1.2.1.cmml">∀</mo><mi id="S3.Ex10.m1.7.7.1.1.1.1.2.2" xref="S3.Ex10.m1.7.7.1.1.1.1.2.2.cmml">x</mi></mrow><mo id="S3.Ex10.m1.7.7.1.1.1.1.1" xref="S3.Ex10.m1.7.7.1.1.1.1.1.cmml">∈</mo><mi id="S3.Ex10.m1.7.7.1.1.1.1.3" xref="S3.Ex10.m1.7.7.1.1.1.1.3.cmml">ℝ</mi></mrow><mo id="S3.Ex10.m1.7.7.1.1.2.3" xref="S3.Ex10.m1.7.7.1.1.3a.cmml">,</mo><mrow id="S3.Ex10.m1.7.7.1.1.2.2" xref="S3.Ex10.m1.7.7.1.1.2.2.cmml"><mrow 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id="S3.Ex10.m1.7.7.1.1.2.2.1.3.2" xref="S3.Ex10.m1.7.7.1.1.2.2.1.3.2.cmml">π</mi><mrow id="S3.Ex10.m1.2.2.1" xref="S3.Ex10.m1.2.2.1.cmml"><mi id="S3.Ex10.m1.2.2.1.3" xref="S3.Ex10.m1.2.2.1.3.cmml">σ</mi><mo id="S3.Ex10.m1.2.2.1.2" xref="S3.Ex10.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.Ex10.m1.2.2.1.4.2" xref="S3.Ex10.m1.2.2.1.cmml"><mo id="S3.Ex10.m1.2.2.1.4.2.1" stretchy="false" xref="S3.Ex10.m1.2.2.1.cmml">(</mo><mi id="S3.Ex10.m1.2.2.1.1" xref="S3.Ex10.m1.2.2.1.1.cmml">x</mi><mo id="S3.Ex10.m1.2.2.1.4.2.2" stretchy="false" xref="S3.Ex10.m1.2.2.1.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.Ex10.m1.7.7.1.1.2.2.3" rspace="0.1389em" xref="S3.Ex10.m1.7.7.1.1.2.2.3.cmml">≥</mo><mrow id="S3.Ex10.m1.7.7.1.1.2.2.2" xref="S3.Ex10.m1.7.7.1.1.2.2.2.cmml"><mrow id="S3.Ex10.m1.7.7.1.1.2.2.2.1" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.cmml"><munder id="S3.Ex10.m1.7.7.1.1.2.2.2.1.2" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.2.cmml"><mo id="S3.Ex10.m1.7.7.1.1.2.2.2.1.2.2" lspace="0.1389em" movablelimits="false" 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xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.3">𝜎</ci><apply id="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></apply></apply><apply id="S3.Ex10.m1.7.7.1.1.2.2.2.3.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.Ex10.m1.7.7.1.1.2.2.2.3.1.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.3">subscript</csymbol><ci id="S3.Ex10.m1.7.7.1.1.2.2.2.3.2.cmml" xref="S3.Ex10.m1.7.7.1.1.2.2.2.3.2">𝜋</ci><apply id="S3.Ex10.m1.5.5.1.cmml" xref="S3.Ex10.m1.5.5.1"><ci id="S3.Ex10.m1.5.5.1.2.cmml" xref="S3.Ex10.m1.5.5.1.2">¯</ci><apply id="S3.Ex10.m1.5.5.1.1.1.cmml" xref="S3.Ex10.m1.5.5.1.1.1"><times id="S3.Ex10.m1.5.5.1.1.1.2.cmml" xref="S3.Ex10.m1.5.5.1.1.1.2"></times><ci id="S3.Ex10.m1.5.5.1.1.1.3.cmml" xref="S3.Ex10.m1.5.5.1.1.1.3">𝜎</ci><ci id="S3.Ex10.m1.5.5.1.1.1.1.cmml" xref="S3.Ex10.m1.5.5.1.1.1.1">𝑥</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.7c">\forall x\in\mathbb{R},\;\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1% }[\sigma(x)=\sigma(x^{\prime})]-\pi_{\sigma(x)}\geq\mathop{\mathbb{E}}_{x^{% \prime}\sim\beta(x)}\mathbf{1}[\overline{\sigma(x)}=\sigma(x^{\prime})]-\pi_{% \overline{\sigma(x)}}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.7d">∀ italic_x ∈ blackboard_R , blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ ( italic_x ) = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] - italic_π start_POSTSUBSCRIPT italic_σ ( italic_x ) end_POSTSUBSCRIPT ≥ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ over¯ start_ARG italic_σ ( italic_x ) end_ARG = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] - italic_π start_POSTSUBSCRIPT over¯ start_ARG italic_σ ( italic_x ) end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.4">Since <math alttext="\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[\sigma(x)=\sigma(x^{% \prime})]=1-\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[\overline{% \sigma(x)}=\sigma(x^{\prime})]" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m1.6"><semantics id="S3.SS1.p1.4.m1.6a"><mrow id="S3.SS1.p1.4.m1.6.6" xref="S3.SS1.p1.4.m1.6.6.cmml"><mrow id="S3.SS1.p1.4.m1.5.5.1" xref="S3.SS1.p1.4.m1.5.5.1.cmml"><msub id="S3.SS1.p1.4.m1.5.5.1.2" xref="S3.SS1.p1.4.m1.5.5.1.2.cmml"><mo id="S3.SS1.p1.4.m1.5.5.1.2.2" xref="S3.SS1.p1.4.m1.5.5.1.2.2.cmml">𝔼</mo><mrow id="S3.SS1.p1.4.m1.1.1.1" xref="S3.SS1.p1.4.m1.1.1.1.cmml"><msup id="S3.SS1.p1.4.m1.1.1.1.3" xref="S3.SS1.p1.4.m1.1.1.1.3.cmml"><mi id="S3.SS1.p1.4.m1.1.1.1.3.2" xref="S3.SS1.p1.4.m1.1.1.1.3.2.cmml">x</mi><mo id="S3.SS1.p1.4.m1.1.1.1.3.3" xref="S3.SS1.p1.4.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S3.SS1.p1.4.m1.1.1.1.2" xref="S3.SS1.p1.4.m1.1.1.1.2.cmml">∼</mo><mrow id="S3.SS1.p1.4.m1.1.1.1.4" xref="S3.SS1.p1.4.m1.1.1.1.4.cmml"><mi id="S3.SS1.p1.4.m1.1.1.1.4.2" xref="S3.SS1.p1.4.m1.1.1.1.4.2.cmml">β</mi><mo id="S3.SS1.p1.4.m1.1.1.1.4.1" xref="S3.SS1.p1.4.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.SS1.p1.4.m1.1.1.1.4.3.2" xref="S3.SS1.p1.4.m1.1.1.1.4.cmml"><mo id="S3.SS1.p1.4.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.SS1.p1.4.m1.1.1.1.4.cmml">(</mo><mi id="S3.SS1.p1.4.m1.1.1.1.1" 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xref="S3.SS1.p1.4.m1.6.6.2.1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m1.6c">\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[\sigma(x)=\sigma(x^{% \prime})]=1-\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1}[\overline{% \sigma(x)}=\sigma(x^{\prime})]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m1.6d">blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ ( italic_x ) = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] = 1 - blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ over¯ start_ARG italic_σ ( italic_x ) end_ARG = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ]</annotation></semantics></math>, this condition simplifies to</p> <table class="ltx_equation ltx_eqn_table" id="S3.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall x\in\mathbb{R},\;\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1% }[\sigma(x)=\sigma(x^{\prime})]\geq\pi_{\sigma(x)}." class="ltx_Math" display="block" id="S3.E10.m1.4"><semantics id="S3.E10.m1.4a"><mrow id="S3.E10.m1.4.4.1"><mrow id="S3.E10.m1.4.4.1.1.2" xref="S3.E10.m1.4.4.1.1.3.cmml"><mrow id="S3.E10.m1.4.4.1.1.1.1" xref="S3.E10.m1.4.4.1.1.1.1.cmml"><mrow id="S3.E10.m1.4.4.1.1.1.1.2" xref="S3.E10.m1.4.4.1.1.1.1.2.cmml"><mo id="S3.E10.m1.4.4.1.1.1.1.2.1" rspace="0.167em" xref="S3.E10.m1.4.4.1.1.1.1.2.1.cmml">∀</mo><mi id="S3.E10.m1.4.4.1.1.1.1.2.2" xref="S3.E10.m1.4.4.1.1.1.1.2.2.cmml">x</mi></mrow><mo id="S3.E10.m1.4.4.1.1.1.1.1" xref="S3.E10.m1.4.4.1.1.1.1.1.cmml">∈</mo><mi id="S3.E10.m1.4.4.1.1.1.1.3" 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xref="S3.E10.m1.2.2.1.1">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.4c">\forall x\in\mathbb{R},\;\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}\mathbf{1% }[\sigma(x)=\sigma(x^{\prime})]\geq\pi_{\sigma(x)}.</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.4d">∀ italic_x ∈ blackboard_R , blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ ( italic_x ) = italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] ≥ italic_π start_POSTSUBSCRIPT italic_σ ( italic_x ) 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">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.6">For a threshold equilibrium, Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E10" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">10</span></a> is equivalent to:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx3"> <tbody id="S3.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall x\leq\tau,P(\tau;x)\geq\pi_{L}" class="ltx_Math" display="inline" id="S3.E11.m1.4"><semantics id="S3.E11.m1.4a"><mrow id="S3.E11.m1.4.4.2" xref="S3.E11.m1.4.4.3.cmml"><mrow id="S3.E11.m1.3.3.1.1" xref="S3.E11.m1.3.3.1.1.cmml"><mrow id="S3.E11.m1.3.3.1.1.2" xref="S3.E11.m1.3.3.1.1.2.cmml"><mo id="S3.E11.m1.3.3.1.1.2.1" rspace="0.167em" xref="S3.E11.m1.3.3.1.1.2.1.cmml">∀</mo><mi id="S3.E11.m1.3.3.1.1.2.2" xref="S3.E11.m1.3.3.1.1.2.2.cmml">x</mi></mrow><mo id="S3.E11.m1.3.3.1.1.1" xref="S3.E11.m1.3.3.1.1.1.cmml">≤</mo><mi id="S3.E11.m1.3.3.1.1.3" xref="S3.E11.m1.3.3.1.1.3.cmml">τ</mi></mrow><mo id="S3.E11.m1.4.4.2.3" xref="S3.E11.m1.4.4.3a.cmml">,</mo><mrow id="S3.E11.m1.4.4.2.2" xref="S3.E11.m1.4.4.2.2.cmml"><mrow id="S3.E11.m1.4.4.2.2.2" xref="S3.E11.m1.4.4.2.2.2.cmml"><mi id="S3.E11.m1.4.4.2.2.2.2" xref="S3.E11.m1.4.4.2.2.2.2.cmml">P</mi><mo id="S3.E11.m1.4.4.2.2.2.1" xref="S3.E11.m1.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S3.E11.m1.4.4.2.2.2.3.2" xref="S3.E11.m1.4.4.2.2.2.3.1.cmml"><mo id="S3.E11.m1.4.4.2.2.2.3.2.1" stretchy="false" xref="S3.E11.m1.4.4.2.2.2.3.1.cmml">(</mo><mi id="S3.E11.m1.1.1" xref="S3.E11.m1.1.1.cmml">τ</mi><mo id="S3.E11.m1.4.4.2.2.2.3.2.2" xref="S3.E11.m1.4.4.2.2.2.3.1.cmml">;</mo><mi id="S3.E11.m1.2.2" xref="S3.E11.m1.2.2.cmml">x</mi><mo id="S3.E11.m1.4.4.2.2.2.3.2.3" stretchy="false" xref="S3.E11.m1.4.4.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo 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encoding="application/x-llamapun" id="S3.E11.m1.4d">∀ italic_x ≤ italic_τ , italic_P ( italic_τ ; italic_x ) ≥ italic_π start_POSTSUBSCRIPT italic_L 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">(11)</span></td> </tr></tbody> <tbody id="S3.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\forall x>\tau,P(\tau;x)\leq\pi_{L}." class="ltx_Math" display="inline" id="S3.E12.m1.3"><semantics id="S3.E12.m1.3a"><mrow id="S3.E12.m1.3.3.1"><mrow id="S3.E12.m1.3.3.1.1.2" xref="S3.E12.m1.3.3.1.1.3.cmml"><mrow id="S3.E12.m1.3.3.1.1.1.1" xref="S3.E12.m1.3.3.1.1.1.1.cmml"><mrow id="S3.E12.m1.3.3.1.1.1.1.2" xref="S3.E12.m1.3.3.1.1.1.1.2.cmml"><mo 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ltx_title_paragraph">Equilibrium results.</h4> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p1.1">Now we have the tools to analyze threhsold equilibria in DG; we leave proofs to § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2" title="Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">B</span></a>, since they largely follow the same logic as our proofs for OA. We first observe that uninformative strategies <math alttext="\tau^{*}=\pm\infty" 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"><msup 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">τ</mi><mo 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">∗</mo></msup><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><mrow id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml"><mo id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3a" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml">±</mo><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" 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id="S3.SS1.SSS0.Px1.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> remain equilibria in the DG mechanism.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition3.1.1.1">Proposition 3</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition3.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition3.p1"> <p class="ltx_p" id="Thmproposition3.p1.1"><math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="Thmproposition3.p1.1.m1.1"><semantics id="Thmproposition3.p1.1.m1.1a"><mrow id="Thmproposition3.p1.1.m1.1.1" xref="Thmproposition3.p1.1.m1.1.1.cmml"><msup id="Thmproposition3.p1.1.m1.1.1.2" 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id="Thmproposition3.p1.1.m1.1.1.2.1.cmml" xref="Thmproposition3.p1.1.m1.1.1.2">superscript</csymbol><ci id="Thmproposition3.p1.1.m1.1.1.2.2.cmml" xref="Thmproposition3.p1.1.m1.1.1.2.2">𝜏</ci><times id="Thmproposition3.p1.1.m1.1.1.2.3.cmml" xref="Thmproposition3.p1.1.m1.1.1.2.3"></times></apply><apply id="Thmproposition3.p1.1.m1.1.1.3.cmml" xref="Thmproposition3.p1.1.m1.1.1.3"><csymbol cd="latexml" id="Thmproposition3.p1.1.m1.1.1.3.1.cmml" xref="Thmproposition3.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="Thmproposition3.p1.1.m1.1.1.3.2.cmml" xref="Thmproposition3.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition3.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition3.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are both always threshold equilibria under DG.</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p2.1">We can also provide necessary and sufficient conditions for a finite threshold equilibrium for DG. Both their form and proof follow the same logic we saw for OA, adapted to account for the additional <math alttext="\pi_{L}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p2.1.m1.1"><semantics id="S3.SS1.SSS0.Px1.p2.1.m1.1a"><msub id="S3.SS1.SSS0.Px1.p2.1.m1.1.1" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2.cmml">π</mi><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p2.1.m1.1b"><apply id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.2">𝜋</ci><ci id="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p2.1.m1.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.1.m1.1c">\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.1.m1.1d">italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> term also depending on the choice of threshold.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem3.1.1.1">Theorem 3</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem3.p1"> <p class="ltx_p" id="Thmtheorem3.p1.10">Let finite threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem3.p1.1.m1.1"><semantics id="Thmtheorem3.p1.1.m1.1a"><mi id="Thmtheorem3.p1.1.m1.1.1" xref="Thmtheorem3.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.1.m1.1b"><ci id="Thmtheorem3.p1.1.m1.1.1.cmml" xref="Thmtheorem3.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.1.m1.1d">italic_τ</annotation></semantics></math> be given and <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.2.m2.2"><semantics id="Thmtheorem3.p1.2.m2.2a"><mrow id="Thmtheorem3.p1.2.m2.2.3" xref="Thmtheorem3.p1.2.m2.2.3.cmml"><mi id="Thmtheorem3.p1.2.m2.2.3.2" xref="Thmtheorem3.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmtheorem3.p1.2.m2.2.3.1" xref="Thmtheorem3.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.2.m2.2.3.3.2" xref="Thmtheorem3.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem3.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem3.p1.2.m2.1.1" xref="Thmtheorem3.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmtheorem3.p1.2.m2.2.3.3.2.2" xref="Thmtheorem3.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmtheorem3.p1.2.m2.2.2" xref="Thmtheorem3.p1.2.m2.2.2.cmml">x</mi><mo id="Thmtheorem3.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem3.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.2.m2.2b"><apply id="Thmtheorem3.p1.2.m2.2.3.cmml" xref="Thmtheorem3.p1.2.m2.2.3"><times id="Thmtheorem3.p1.2.m2.2.3.1.cmml" xref="Thmtheorem3.p1.2.m2.2.3.1"></times><ci id="Thmtheorem3.p1.2.m2.2.3.2.cmml" xref="Thmtheorem3.p1.2.m2.2.3.2">𝑃</ci><list id="Thmtheorem3.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem3.p1.2.m2.2.3.3.2"><ci id="Thmtheorem3.p1.2.m2.1.1.cmml" xref="Thmtheorem3.p1.2.m2.1.1">𝜏</ci><ci id="Thmtheorem3.p1.2.m2.2.2.cmml" xref="Thmtheorem3.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> be continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem3.p1.3.m3.1"><semantics id="Thmtheorem3.p1.3.m3.1a"><mi id="Thmtheorem3.p1.3.m3.1.1" xref="Thmtheorem3.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.3.m3.1b"><ci id="Thmtheorem3.p1.3.m3.1.1.cmml" xref="Thmtheorem3.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.3.m3.1d">italic_x</annotation></semantics></math>. If <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem3.p1.4.m4.1"><semantics id="Thmtheorem3.p1.4.m4.1a"><mi id="Thmtheorem3.p1.4.m4.1.1" xref="Thmtheorem3.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.4.m4.1b"><ci id="Thmtheorem3.p1.4.m4.1.1.cmml" xref="Thmtheorem3.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.4.m4.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium under the DG mechanism then <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.5.m5.2"><semantics id="Thmtheorem3.p1.5.m5.2a"><mrow id="Thmtheorem3.p1.5.m5.2.3" xref="Thmtheorem3.p1.5.m5.2.3.cmml"><mrow id="Thmtheorem3.p1.5.m5.2.3.2" xref="Thmtheorem3.p1.5.m5.2.3.2.cmml"><mi id="Thmtheorem3.p1.5.m5.2.3.2.2" xref="Thmtheorem3.p1.5.m5.2.3.2.2.cmml">G</mi><mo id="Thmtheorem3.p1.5.m5.2.3.2.1" xref="Thmtheorem3.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.5.m5.2.3.2.3.2" xref="Thmtheorem3.p1.5.m5.2.3.2.cmml"><mo id="Thmtheorem3.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem3.p1.5.m5.2.3.2.cmml">(</mo><mi id="Thmtheorem3.p1.5.m5.1.1" xref="Thmtheorem3.p1.5.m5.1.1.cmml">τ</mi><mo id="Thmtheorem3.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem3.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem3.p1.5.m5.2.3.1" xref="Thmtheorem3.p1.5.m5.2.3.1.cmml">=</mo><mrow id="Thmtheorem3.p1.5.m5.2.3.3" xref="Thmtheorem3.p1.5.m5.2.3.3.cmml"><mi id="Thmtheorem3.p1.5.m5.2.3.3.2" xref="Thmtheorem3.p1.5.m5.2.3.3.2.cmml">F</mi><mo id="Thmtheorem3.p1.5.m5.2.3.3.1" xref="Thmtheorem3.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.5.m5.2.3.3.3.2" xref="Thmtheorem3.p1.5.m5.2.3.3.cmml"><mo id="Thmtheorem3.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.5.m5.2.3.3.cmml">(</mo><mi id="Thmtheorem3.p1.5.m5.2.2" xref="Thmtheorem3.p1.5.m5.2.2.cmml">τ</mi><mo id="Thmtheorem3.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem3.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.5.m5.2b"><apply id="Thmtheorem3.p1.5.m5.2.3.cmml" xref="Thmtheorem3.p1.5.m5.2.3"><eq id="Thmtheorem3.p1.5.m5.2.3.1.cmml" xref="Thmtheorem3.p1.5.m5.2.3.1"></eq><apply id="Thmtheorem3.p1.5.m5.2.3.2.cmml" xref="Thmtheorem3.p1.5.m5.2.3.2"><times id="Thmtheorem3.p1.5.m5.2.3.2.1.cmml" xref="Thmtheorem3.p1.5.m5.2.3.2.1"></times><ci id="Thmtheorem3.p1.5.m5.2.3.2.2.cmml" xref="Thmtheorem3.p1.5.m5.2.3.2.2">𝐺</ci><ci id="Thmtheorem3.p1.5.m5.1.1.cmml" xref="Thmtheorem3.p1.5.m5.1.1">𝜏</ci></apply><apply id="Thmtheorem3.p1.5.m5.2.3.3.cmml" xref="Thmtheorem3.p1.5.m5.2.3.3"><times id="Thmtheorem3.p1.5.m5.2.3.3.1.cmml" xref="Thmtheorem3.p1.5.m5.2.3.3.1"></times><ci id="Thmtheorem3.p1.5.m5.2.3.3.2.cmml" xref="Thmtheorem3.p1.5.m5.2.3.3.2">𝐹</ci><ci id="Thmtheorem3.p1.5.m5.2.2.cmml" xref="Thmtheorem3.p1.5.m5.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.5.m5.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.5.m5.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>. Conversely, if <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.6.m6.2"><semantics id="Thmtheorem3.p1.6.m6.2a"><mrow id="Thmtheorem3.p1.6.m6.2.3" xref="Thmtheorem3.p1.6.m6.2.3.cmml"><mrow id="Thmtheorem3.p1.6.m6.2.3.2" xref="Thmtheorem3.p1.6.m6.2.3.2.cmml"><mi id="Thmtheorem3.p1.6.m6.2.3.2.2" xref="Thmtheorem3.p1.6.m6.2.3.2.2.cmml">G</mi><mo id="Thmtheorem3.p1.6.m6.2.3.2.1" xref="Thmtheorem3.p1.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.6.m6.2.3.2.3.2" xref="Thmtheorem3.p1.6.m6.2.3.2.cmml"><mo id="Thmtheorem3.p1.6.m6.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem3.p1.6.m6.2.3.2.cmml">(</mo><mi id="Thmtheorem3.p1.6.m6.1.1" xref="Thmtheorem3.p1.6.m6.1.1.cmml">τ</mi><mo id="Thmtheorem3.p1.6.m6.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem3.p1.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem3.p1.6.m6.2.3.1" xref="Thmtheorem3.p1.6.m6.2.3.1.cmml">=</mo><mrow id="Thmtheorem3.p1.6.m6.2.3.3" xref="Thmtheorem3.p1.6.m6.2.3.3.cmml"><mi id="Thmtheorem3.p1.6.m6.2.3.3.2" xref="Thmtheorem3.p1.6.m6.2.3.3.2.cmml">F</mi><mo id="Thmtheorem3.p1.6.m6.2.3.3.1" xref="Thmtheorem3.p1.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.6.m6.2.3.3.3.2" xref="Thmtheorem3.p1.6.m6.2.3.3.cmml"><mo id="Thmtheorem3.p1.6.m6.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.6.m6.2.3.3.cmml">(</mo><mi id="Thmtheorem3.p1.6.m6.2.2" xref="Thmtheorem3.p1.6.m6.2.2.cmml">τ</mi><mo id="Thmtheorem3.p1.6.m6.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem3.p1.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.6.m6.2b"><apply id="Thmtheorem3.p1.6.m6.2.3.cmml" xref="Thmtheorem3.p1.6.m6.2.3"><eq id="Thmtheorem3.p1.6.m6.2.3.1.cmml" xref="Thmtheorem3.p1.6.m6.2.3.1"></eq><apply id="Thmtheorem3.p1.6.m6.2.3.2.cmml" xref="Thmtheorem3.p1.6.m6.2.3.2"><times id="Thmtheorem3.p1.6.m6.2.3.2.1.cmml" xref="Thmtheorem3.p1.6.m6.2.3.2.1"></times><ci id="Thmtheorem3.p1.6.m6.2.3.2.2.cmml" xref="Thmtheorem3.p1.6.m6.2.3.2.2">𝐺</ci><ci id="Thmtheorem3.p1.6.m6.1.1.cmml" xref="Thmtheorem3.p1.6.m6.1.1">𝜏</ci></apply><apply id="Thmtheorem3.p1.6.m6.2.3.3.cmml" xref="Thmtheorem3.p1.6.m6.2.3.3"><times id="Thmtheorem3.p1.6.m6.2.3.3.1.cmml" xref="Thmtheorem3.p1.6.m6.2.3.3.1"></times><ci id="Thmtheorem3.p1.6.m6.2.3.3.2.cmml" xref="Thmtheorem3.p1.6.m6.2.3.3.2">𝐹</ci><ci id="Thmtheorem3.p1.6.m6.2.2.cmml" xref="Thmtheorem3.p1.6.m6.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.6.m6.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.6.m6.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math> and either (a) <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem3.p1.7.m7.2"><semantics id="Thmtheorem3.p1.7.m7.2a"><mrow id="Thmtheorem3.p1.7.m7.2.3" xref="Thmtheorem3.p1.7.m7.2.3.cmml"><mi id="Thmtheorem3.p1.7.m7.2.3.2" xref="Thmtheorem3.p1.7.m7.2.3.2.cmml">P</mi><mo id="Thmtheorem3.p1.7.m7.2.3.1" xref="Thmtheorem3.p1.7.m7.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.7.m7.2.3.3.2" xref="Thmtheorem3.p1.7.m7.2.3.3.1.cmml"><mo id="Thmtheorem3.p1.7.m7.2.3.3.2.1" stretchy="false" xref="Thmtheorem3.p1.7.m7.2.3.3.1.cmml">(</mo><mi id="Thmtheorem3.p1.7.m7.1.1" xref="Thmtheorem3.p1.7.m7.1.1.cmml">τ</mi><mo id="Thmtheorem3.p1.7.m7.2.3.3.2.2" xref="Thmtheorem3.p1.7.m7.2.3.3.1.cmml">;</mo><mi id="Thmtheorem3.p1.7.m7.2.2" xref="Thmtheorem3.p1.7.m7.2.2.cmml">x</mi><mo id="Thmtheorem3.p1.7.m7.2.3.3.2.3" stretchy="false" xref="Thmtheorem3.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.7.m7.2b"><apply id="Thmtheorem3.p1.7.m7.2.3.cmml" xref="Thmtheorem3.p1.7.m7.2.3"><times id="Thmtheorem3.p1.7.m7.2.3.1.cmml" xref="Thmtheorem3.p1.7.m7.2.3.1"></times><ci id="Thmtheorem3.p1.7.m7.2.3.2.cmml" xref="Thmtheorem3.p1.7.m7.2.3.2">𝑃</ci><list id="Thmtheorem3.p1.7.m7.2.3.3.1.cmml" xref="Thmtheorem3.p1.7.m7.2.3.3.2"><ci id="Thmtheorem3.p1.7.m7.1.1.cmml" xref="Thmtheorem3.p1.7.m7.1.1">𝜏</ci><ci id="Thmtheorem3.p1.7.m7.2.2.cmml" xref="Thmtheorem3.p1.7.m7.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.7.m7.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.7.m7.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is monotone decreasing in x or (b) <math alttext="P(\tau;x)-\pi_{L}" class="ltx_Math" display="inline" id="Thmtheorem3.p1.8.m8.2"><semantics id="Thmtheorem3.p1.8.m8.2a"><mrow id="Thmtheorem3.p1.8.m8.2.3" xref="Thmtheorem3.p1.8.m8.2.3.cmml"><mrow id="Thmtheorem3.p1.8.m8.2.3.2" xref="Thmtheorem3.p1.8.m8.2.3.2.cmml"><mi id="Thmtheorem3.p1.8.m8.2.3.2.2" xref="Thmtheorem3.p1.8.m8.2.3.2.2.cmml">P</mi><mo id="Thmtheorem3.p1.8.m8.2.3.2.1" xref="Thmtheorem3.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem3.p1.8.m8.2.3.2.3.2" xref="Thmtheorem3.p1.8.m8.2.3.2.3.1.cmml"><mo id="Thmtheorem3.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem3.p1.8.m8.2.3.2.3.1.cmml">(</mo><mi id="Thmtheorem3.p1.8.m8.1.1" xref="Thmtheorem3.p1.8.m8.1.1.cmml">τ</mi><mo id="Thmtheorem3.p1.8.m8.2.3.2.3.2.2" xref="Thmtheorem3.p1.8.m8.2.3.2.3.1.cmml">;</mo><mi id="Thmtheorem3.p1.8.m8.2.2" xref="Thmtheorem3.p1.8.m8.2.2.cmml">x</mi><mo id="Thmtheorem3.p1.8.m8.2.3.2.3.2.3" stretchy="false" xref="Thmtheorem3.p1.8.m8.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem3.p1.8.m8.2.3.1" xref="Thmtheorem3.p1.8.m8.2.3.1.cmml">−</mo><msub id="Thmtheorem3.p1.8.m8.2.3.3" xref="Thmtheorem3.p1.8.m8.2.3.3.cmml"><mi id="Thmtheorem3.p1.8.m8.2.3.3.2" xref="Thmtheorem3.p1.8.m8.2.3.3.2.cmml">π</mi><mi id="Thmtheorem3.p1.8.m8.2.3.3.3" xref="Thmtheorem3.p1.8.m8.2.3.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.8.m8.2b"><apply id="Thmtheorem3.p1.8.m8.2.3.cmml" xref="Thmtheorem3.p1.8.m8.2.3"><minus id="Thmtheorem3.p1.8.m8.2.3.1.cmml" xref="Thmtheorem3.p1.8.m8.2.3.1"></minus><apply id="Thmtheorem3.p1.8.m8.2.3.2.cmml" xref="Thmtheorem3.p1.8.m8.2.3.2"><times id="Thmtheorem3.p1.8.m8.2.3.2.1.cmml" xref="Thmtheorem3.p1.8.m8.2.3.2.1"></times><ci id="Thmtheorem3.p1.8.m8.2.3.2.2.cmml" xref="Thmtheorem3.p1.8.m8.2.3.2.2">𝑃</ci><list id="Thmtheorem3.p1.8.m8.2.3.2.3.1.cmml" xref="Thmtheorem3.p1.8.m8.2.3.2.3.2"><ci id="Thmtheorem3.p1.8.m8.1.1.cmml" xref="Thmtheorem3.p1.8.m8.1.1">𝜏</ci><ci id="Thmtheorem3.p1.8.m8.2.2.cmml" xref="Thmtheorem3.p1.8.m8.2.2">𝑥</ci></list></apply><apply id="Thmtheorem3.p1.8.m8.2.3.3.cmml" xref="Thmtheorem3.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="Thmtheorem3.p1.8.m8.2.3.3.1.cmml" xref="Thmtheorem3.p1.8.m8.2.3.3">subscript</csymbol><ci id="Thmtheorem3.p1.8.m8.2.3.3.2.cmml" xref="Thmtheorem3.p1.8.m8.2.3.3.2">𝜋</ci><ci id="Thmtheorem3.p1.8.m8.2.3.3.3.cmml" xref="Thmtheorem3.p1.8.m8.2.3.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.8.m8.2c">P(\tau;x)-\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.8.m8.2d">italic_P ( italic_τ ; italic_x ) - italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> has a single crossing of <math alttext="0" class="ltx_Math" display="inline" id="Thmtheorem3.p1.9.m9.1"><semantics id="Thmtheorem3.p1.9.m9.1a"><mn id="Thmtheorem3.p1.9.m9.1.1" xref="Thmtheorem3.p1.9.m9.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.9.m9.1b"><cn id="Thmtheorem3.p1.9.m9.1.1.cmml" type="integer" xref="Thmtheorem3.p1.9.m9.1.1">0</cn></annotation-xml></semantics></math> from positive to negative then <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem3.p1.10.m10.1"><semantics id="Thmtheorem3.p1.10.m10.1a"><mi id="Thmtheorem3.p1.10.m10.1.1" xref="Thmtheorem3.p1.10.m10.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem3.p1.10.m10.1b"><ci id="Thmtheorem3.p1.10.m10.1.1.cmml" xref="Thmtheorem3.p1.10.m10.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem3.p1.10.m10.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem3.p1.10.m10.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium under the DG mechanism.</p> </div> </div> </section> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Dynamics</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">We consider stability of equilibria for DG under the dynamics described by Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E6" title="In 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>). Again, we leave proofs in this section to § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2" title="Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">B</span></a>. Note that as in OA, when everyone else is playing according to <math alttext="\tau" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_τ</annotation></semantics></math>, there exists a unique best response which is also a threshold strategy.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition4.1.1.1">Proposition 4</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition4.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition4.p1"> <p class="ltx_p" id="Thmproposition4.p1.7">Assume that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmproposition4.p1.1.m1.2"><semantics id="Thmproposition4.p1.1.m1.2a"><mrow id="Thmproposition4.p1.1.m1.2.3" xref="Thmproposition4.p1.1.m1.2.3.cmml"><mi id="Thmproposition4.p1.1.m1.2.3.2" xref="Thmproposition4.p1.1.m1.2.3.2.cmml">P</mi><mo id="Thmproposition4.p1.1.m1.2.3.1" xref="Thmproposition4.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="Thmproposition4.p1.1.m1.2.3.3.2" xref="Thmproposition4.p1.1.m1.2.3.3.1.cmml"><mo id="Thmproposition4.p1.1.m1.2.3.3.2.1" stretchy="false" xref="Thmproposition4.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="Thmproposition4.p1.1.m1.1.1" xref="Thmproposition4.p1.1.m1.1.1.cmml">τ</mi><mo id="Thmproposition4.p1.1.m1.2.3.3.2.2" xref="Thmproposition4.p1.1.m1.2.3.3.1.cmml">;</mo><mi id="Thmproposition4.p1.1.m1.2.2" xref="Thmproposition4.p1.1.m1.2.2.cmml">x</mi><mo id="Thmproposition4.p1.1.m1.2.3.3.2.3" stretchy="false" xref="Thmproposition4.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.1.m1.2b"><apply id="Thmproposition4.p1.1.m1.2.3.cmml" xref="Thmproposition4.p1.1.m1.2.3"><times id="Thmproposition4.p1.1.m1.2.3.1.cmml" xref="Thmproposition4.p1.1.m1.2.3.1"></times><ci id="Thmproposition4.p1.1.m1.2.3.2.cmml" xref="Thmproposition4.p1.1.m1.2.3.2">𝑃</ci><list id="Thmproposition4.p1.1.m1.2.3.3.1.cmml" xref="Thmproposition4.p1.1.m1.2.3.3.2"><ci id="Thmproposition4.p1.1.m1.1.1.cmml" xref="Thmproposition4.p1.1.m1.1.1">𝜏</ci><ci id="Thmproposition4.p1.1.m1.2.2.cmml" xref="Thmproposition4.p1.1.m1.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.1.m1.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.1.m1.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmproposition4.p1.2.m2.1"><semantics id="Thmproposition4.p1.2.m2.1a"><mi id="Thmproposition4.p1.2.m2.1.1" xref="Thmproposition4.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.2.m2.1b"><ci id="Thmproposition4.p1.2.m2.1.1.cmml" xref="Thmproposition4.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.2.m2.1d">italic_x</annotation></semantics></math> for a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmproposition4.p1.3.m3.1"><semantics id="Thmproposition4.p1.3.m3.1a"><mi id="Thmproposition4.p1.3.m3.1.1" xref="Thmproposition4.p1.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.3.m3.1b"><ci id="Thmproposition4.p1.3.m3.1.1.cmml" xref="Thmproposition4.p1.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.3.m3.1d">italic_τ</annotation></semantics></math>. If all agents are playing according threshold strategy <math alttext="\tau" class="ltx_Math" display="inline" id="Thmproposition4.p1.4.m4.1"><semantics id="Thmproposition4.p1.4.m4.1a"><mi id="Thmproposition4.p1.4.m4.1.1" xref="Thmproposition4.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.4.m4.1b"><ci id="Thmproposition4.p1.4.m4.1.1.cmml" xref="Thmproposition4.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.4.m4.1d">italic_τ</annotation></semantics></math>, the unique best response of an agent across all strategies <math alttext="\sigma:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="Thmproposition4.p1.5.m5.1"><semantics id="Thmproposition4.p1.5.m5.1a"><mrow id="Thmproposition4.p1.5.m5.1.1" xref="Thmproposition4.p1.5.m5.1.1.cmml"><mi id="Thmproposition4.p1.5.m5.1.1.2" xref="Thmproposition4.p1.5.m5.1.1.2.cmml">σ</mi><mo id="Thmproposition4.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmproposition4.p1.5.m5.1.1.1.cmml">:</mo><mrow id="Thmproposition4.p1.5.m5.1.1.3" xref="Thmproposition4.p1.5.m5.1.1.3.cmml"><mi id="Thmproposition4.p1.5.m5.1.1.3.2" xref="Thmproposition4.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="Thmproposition4.p1.5.m5.1.1.3.1" stretchy="false" xref="Thmproposition4.p1.5.m5.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="Thmproposition4.p1.5.m5.1.1.3.3" xref="Thmproposition4.p1.5.m5.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.5.m5.1b"><apply id="Thmproposition4.p1.5.m5.1.1.cmml" xref="Thmproposition4.p1.5.m5.1.1"><ci id="Thmproposition4.p1.5.m5.1.1.1.cmml" xref="Thmproposition4.p1.5.m5.1.1.1">:</ci><ci id="Thmproposition4.p1.5.m5.1.1.2.cmml" xref="Thmproposition4.p1.5.m5.1.1.2">𝜎</ci><apply id="Thmproposition4.p1.5.m5.1.1.3.cmml" xref="Thmproposition4.p1.5.m5.1.1.3"><ci id="Thmproposition4.p1.5.m5.1.1.3.1.cmml" xref="Thmproposition4.p1.5.m5.1.1.3.1">→</ci><ci id="Thmproposition4.p1.5.m5.1.1.3.2.cmml" xref="Thmproposition4.p1.5.m5.1.1.3.2">ℝ</ci><ci id="Thmproposition4.p1.5.m5.1.1.3.3.cmml" xref="Thmproposition4.p1.5.m5.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.5.m5.1c">\sigma:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.5.m5.1d">italic_σ : blackboard_R → caligraphic_R</annotation></semantics></math> is to play according to threshold strategy <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="Thmproposition4.p1.6.m6.1"><semantics id="Thmproposition4.p1.6.m6.1a"><mover accent="true" id="Thmproposition4.p1.6.m6.1.1" xref="Thmproposition4.p1.6.m6.1.1.cmml"><mi id="Thmproposition4.p1.6.m6.1.1.2" xref="Thmproposition4.p1.6.m6.1.1.2.cmml">τ</mi><mo id="Thmproposition4.p1.6.m6.1.1.1" xref="Thmproposition4.p1.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.6.m6.1b"><apply id="Thmproposition4.p1.6.m6.1.1.cmml" xref="Thmproposition4.p1.6.m6.1.1"><ci id="Thmproposition4.p1.6.m6.1.1.1.cmml" xref="Thmproposition4.p1.6.m6.1.1.1">^</ci><ci id="Thmproposition4.p1.6.m6.1.1.2.cmml" xref="Thmproposition4.p1.6.m6.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.6.m6.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.6.m6.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> satisfying <math alttext="P(\tau;\hat{\tau})=F(\tau)" class="ltx_Math" display="inline" id="Thmproposition4.p1.7.m7.3"><semantics id="Thmproposition4.p1.7.m7.3a"><mrow id="Thmproposition4.p1.7.m7.3.4" xref="Thmproposition4.p1.7.m7.3.4.cmml"><mrow id="Thmproposition4.p1.7.m7.3.4.2" xref="Thmproposition4.p1.7.m7.3.4.2.cmml"><mi id="Thmproposition4.p1.7.m7.3.4.2.2" xref="Thmproposition4.p1.7.m7.3.4.2.2.cmml">P</mi><mo id="Thmproposition4.p1.7.m7.3.4.2.1" xref="Thmproposition4.p1.7.m7.3.4.2.1.cmml">⁢</mo><mrow id="Thmproposition4.p1.7.m7.3.4.2.3.2" xref="Thmproposition4.p1.7.m7.3.4.2.3.1.cmml"><mo id="Thmproposition4.p1.7.m7.3.4.2.3.2.1" stretchy="false" xref="Thmproposition4.p1.7.m7.3.4.2.3.1.cmml">(</mo><mi id="Thmproposition4.p1.7.m7.1.1" xref="Thmproposition4.p1.7.m7.1.1.cmml">τ</mi><mo id="Thmproposition4.p1.7.m7.3.4.2.3.2.2" xref="Thmproposition4.p1.7.m7.3.4.2.3.1.cmml">;</mo><mover accent="true" id="Thmproposition4.p1.7.m7.2.2" xref="Thmproposition4.p1.7.m7.2.2.cmml"><mi id="Thmproposition4.p1.7.m7.2.2.2" xref="Thmproposition4.p1.7.m7.2.2.2.cmml">τ</mi><mo id="Thmproposition4.p1.7.m7.2.2.1" xref="Thmproposition4.p1.7.m7.2.2.1.cmml">^</mo></mover><mo id="Thmproposition4.p1.7.m7.3.4.2.3.2.3" stretchy="false" xref="Thmproposition4.p1.7.m7.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmproposition4.p1.7.m7.3.4.1" xref="Thmproposition4.p1.7.m7.3.4.1.cmml">=</mo><mrow id="Thmproposition4.p1.7.m7.3.4.3" xref="Thmproposition4.p1.7.m7.3.4.3.cmml"><mi id="Thmproposition4.p1.7.m7.3.4.3.2" xref="Thmproposition4.p1.7.m7.3.4.3.2.cmml">F</mi><mo id="Thmproposition4.p1.7.m7.3.4.3.1" xref="Thmproposition4.p1.7.m7.3.4.3.1.cmml">⁢</mo><mrow id="Thmproposition4.p1.7.m7.3.4.3.3.2" xref="Thmproposition4.p1.7.m7.3.4.3.cmml"><mo id="Thmproposition4.p1.7.m7.3.4.3.3.2.1" stretchy="false" xref="Thmproposition4.p1.7.m7.3.4.3.cmml">(</mo><mi id="Thmproposition4.p1.7.m7.3.3" xref="Thmproposition4.p1.7.m7.3.3.cmml">τ</mi><mo id="Thmproposition4.p1.7.m7.3.4.3.3.2.2" stretchy="false" xref="Thmproposition4.p1.7.m7.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition4.p1.7.m7.3b"><apply id="Thmproposition4.p1.7.m7.3.4.cmml" xref="Thmproposition4.p1.7.m7.3.4"><eq id="Thmproposition4.p1.7.m7.3.4.1.cmml" xref="Thmproposition4.p1.7.m7.3.4.1"></eq><apply id="Thmproposition4.p1.7.m7.3.4.2.cmml" xref="Thmproposition4.p1.7.m7.3.4.2"><times id="Thmproposition4.p1.7.m7.3.4.2.1.cmml" xref="Thmproposition4.p1.7.m7.3.4.2.1"></times><ci id="Thmproposition4.p1.7.m7.3.4.2.2.cmml" xref="Thmproposition4.p1.7.m7.3.4.2.2">𝑃</ci><list id="Thmproposition4.p1.7.m7.3.4.2.3.1.cmml" xref="Thmproposition4.p1.7.m7.3.4.2.3.2"><ci id="Thmproposition4.p1.7.m7.1.1.cmml" xref="Thmproposition4.p1.7.m7.1.1">𝜏</ci><apply id="Thmproposition4.p1.7.m7.2.2.cmml" xref="Thmproposition4.p1.7.m7.2.2"><ci id="Thmproposition4.p1.7.m7.2.2.1.cmml" xref="Thmproposition4.p1.7.m7.2.2.1">^</ci><ci id="Thmproposition4.p1.7.m7.2.2.2.cmml" xref="Thmproposition4.p1.7.m7.2.2.2">𝜏</ci></apply></list></apply><apply id="Thmproposition4.p1.7.m7.3.4.3.cmml" xref="Thmproposition4.p1.7.m7.3.4.3"><times id="Thmproposition4.p1.7.m7.3.4.3.1.cmml" xref="Thmproposition4.p1.7.m7.3.4.3.1"></times><ci id="Thmproposition4.p1.7.m7.3.4.3.2.cmml" xref="Thmproposition4.p1.7.m7.3.4.3.2">𝐹</ci><ci id="Thmproposition4.p1.7.m7.3.3.cmml" xref="Thmproposition4.p1.7.m7.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition4.p1.7.m7.3c">P(\tau;\hat{\tau})=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmproposition4.p1.7.m7.3d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = italic_F ( italic_τ )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">Now, as in OA, we can characterize the stability of equilibria.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem4.1.1.1">Theorem 4</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem4.p1"> <p class="ltx_p" id="Thmtheorem4.p1.7">Assume for all <math alttext="\tau\in\mathbb{R}" class="ltx_Math" display="inline" id="Thmtheorem4.p1.1.m1.1"><semantics id="Thmtheorem4.p1.1.m1.1a"><mrow id="Thmtheorem4.p1.1.m1.1.1" xref="Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="Thmtheorem4.p1.1.m1.1.1.2" xref="Thmtheorem4.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmtheorem4.p1.1.m1.1.1.1" xref="Thmtheorem4.p1.1.m1.1.1.1.cmml">∈</mo><mi id="Thmtheorem4.p1.1.m1.1.1.3" xref="Thmtheorem4.p1.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.1.m1.1b"><apply id="Thmtheorem4.p1.1.m1.1.1.cmml" xref="Thmtheorem4.p1.1.m1.1.1"><in id="Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="Thmtheorem4.p1.1.m1.1.1.1"></in><ci id="Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="Thmtheorem4.p1.1.m1.1.1.2">𝜏</ci><ci id="Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="Thmtheorem4.p1.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.1.m1.1c">\tau\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.1.m1.1d">italic_τ ∈ blackboard_R</annotation></semantics></math> that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem4.p1.2.m2.2"><semantics id="Thmtheorem4.p1.2.m2.2a"><mrow id="Thmtheorem4.p1.2.m2.2.3" xref="Thmtheorem4.p1.2.m2.2.3.cmml"><mi id="Thmtheorem4.p1.2.m2.2.3.2" xref="Thmtheorem4.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmtheorem4.p1.2.m2.2.3.1" xref="Thmtheorem4.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem4.p1.2.m2.2.3.3.2" xref="Thmtheorem4.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem4.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem4.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem4.p1.2.m2.1.1" xref="Thmtheorem4.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmtheorem4.p1.2.m2.2.3.3.2.2" xref="Thmtheorem4.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmtheorem4.p1.2.m2.2.2" xref="Thmtheorem4.p1.2.m2.2.2.cmml">x</mi><mo id="Thmtheorem4.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem4.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.2.m2.2b"><apply id="Thmtheorem4.p1.2.m2.2.3.cmml" xref="Thmtheorem4.p1.2.m2.2.3"><times id="Thmtheorem4.p1.2.m2.2.3.1.cmml" xref="Thmtheorem4.p1.2.m2.2.3.1"></times><ci id="Thmtheorem4.p1.2.m2.2.3.2.cmml" xref="Thmtheorem4.p1.2.m2.2.3.2">𝑃</ci><list id="Thmtheorem4.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem4.p1.2.m2.2.3.3.2"><ci id="Thmtheorem4.p1.2.m2.1.1.cmml" xref="Thmtheorem4.p1.2.m2.1.1">𝜏</ci><ci id="Thmtheorem4.p1.2.m2.2.2.cmml" xref="Thmtheorem4.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem4.p1.3.m3.1"><semantics id="Thmtheorem4.p1.3.m3.1a"><mi id="Thmtheorem4.p1.3.m3.1.1" xref="Thmtheorem4.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.3.m3.1b"><ci id="Thmtheorem4.p1.3.m3.1.1.cmml" xref="Thmtheorem4.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.3.m3.1d">italic_x</annotation></semantics></math>. Then if <math alttext="G(\tau)-F(\tau)" class="ltx_Math" display="inline" id="Thmtheorem4.p1.4.m4.2"><semantics id="Thmtheorem4.p1.4.m4.2a"><mrow id="Thmtheorem4.p1.4.m4.2.3" xref="Thmtheorem4.p1.4.m4.2.3.cmml"><mrow id="Thmtheorem4.p1.4.m4.2.3.2" xref="Thmtheorem4.p1.4.m4.2.3.2.cmml"><mi id="Thmtheorem4.p1.4.m4.2.3.2.2" xref="Thmtheorem4.p1.4.m4.2.3.2.2.cmml">G</mi><mo id="Thmtheorem4.p1.4.m4.2.3.2.1" xref="Thmtheorem4.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem4.p1.4.m4.2.3.2.3.2" xref="Thmtheorem4.p1.4.m4.2.3.2.cmml"><mo id="Thmtheorem4.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem4.p1.4.m4.2.3.2.cmml">(</mo><mi id="Thmtheorem4.p1.4.m4.1.1" xref="Thmtheorem4.p1.4.m4.1.1.cmml">τ</mi><mo id="Thmtheorem4.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem4.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem4.p1.4.m4.2.3.1" xref="Thmtheorem4.p1.4.m4.2.3.1.cmml">−</mo><mrow id="Thmtheorem4.p1.4.m4.2.3.3" xref="Thmtheorem4.p1.4.m4.2.3.3.cmml"><mi id="Thmtheorem4.p1.4.m4.2.3.3.2" xref="Thmtheorem4.p1.4.m4.2.3.3.2.cmml">F</mi><mo id="Thmtheorem4.p1.4.m4.2.3.3.1" xref="Thmtheorem4.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem4.p1.4.m4.2.3.3.3.2" xref="Thmtheorem4.p1.4.m4.2.3.3.cmml"><mo id="Thmtheorem4.p1.4.m4.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem4.p1.4.m4.2.3.3.cmml">(</mo><mi id="Thmtheorem4.p1.4.m4.2.2" xref="Thmtheorem4.p1.4.m4.2.2.cmml">τ</mi><mo id="Thmtheorem4.p1.4.m4.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem4.p1.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.4.m4.2b"><apply id="Thmtheorem4.p1.4.m4.2.3.cmml" xref="Thmtheorem4.p1.4.m4.2.3"><minus id="Thmtheorem4.p1.4.m4.2.3.1.cmml" xref="Thmtheorem4.p1.4.m4.2.3.1"></minus><apply id="Thmtheorem4.p1.4.m4.2.3.2.cmml" xref="Thmtheorem4.p1.4.m4.2.3.2"><times id="Thmtheorem4.p1.4.m4.2.3.2.1.cmml" xref="Thmtheorem4.p1.4.m4.2.3.2.1"></times><ci id="Thmtheorem4.p1.4.m4.2.3.2.2.cmml" xref="Thmtheorem4.p1.4.m4.2.3.2.2">𝐺</ci><ci id="Thmtheorem4.p1.4.m4.1.1.cmml" xref="Thmtheorem4.p1.4.m4.1.1">𝜏</ci></apply><apply id="Thmtheorem4.p1.4.m4.2.3.3.cmml" xref="Thmtheorem4.p1.4.m4.2.3.3"><times id="Thmtheorem4.p1.4.m4.2.3.3.1.cmml" xref="Thmtheorem4.p1.4.m4.2.3.3.1"></times><ci id="Thmtheorem4.p1.4.m4.2.3.3.2.cmml" xref="Thmtheorem4.p1.4.m4.2.3.3.2">𝐹</ci><ci id="Thmtheorem4.p1.4.m4.2.2.cmml" xref="Thmtheorem4.p1.4.m4.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.4.m4.2c">G(\tau)-F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.4.m4.2d">italic_G ( italic_τ ) - italic_F ( italic_τ )</annotation></semantics></math> is strictly decreasing at equilibrium point <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem4.p1.5.m5.1"><semantics id="Thmtheorem4.p1.5.m5.1a"><msup id="Thmtheorem4.p1.5.m5.1.1" xref="Thmtheorem4.p1.5.m5.1.1.cmml"><mi id="Thmtheorem4.p1.5.m5.1.1.2" xref="Thmtheorem4.p1.5.m5.1.1.2.cmml">τ</mi><mo id="Thmtheorem4.p1.5.m5.1.1.3" xref="Thmtheorem4.p1.5.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.5.m5.1b"><apply id="Thmtheorem4.p1.5.m5.1.1.cmml" xref="Thmtheorem4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="Thmtheorem4.p1.5.m5.1.1.1.cmml" xref="Thmtheorem4.p1.5.m5.1.1">superscript</csymbol><ci id="Thmtheorem4.p1.5.m5.1.1.2.cmml" xref="Thmtheorem4.p1.5.m5.1.1.2">𝜏</ci><times id="Thmtheorem4.p1.5.m5.1.1.3.cmml" xref="Thmtheorem4.p1.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.5.m5.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.5.m5.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem4.p1.6.m6.1"><semantics id="Thmtheorem4.p1.6.m6.1a"><msup id="Thmtheorem4.p1.6.m6.1.1" xref="Thmtheorem4.p1.6.m6.1.1.cmml"><mi id="Thmtheorem4.p1.6.m6.1.1.2" xref="Thmtheorem4.p1.6.m6.1.1.2.cmml">τ</mi><mo id="Thmtheorem4.p1.6.m6.1.1.3" xref="Thmtheorem4.p1.6.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.6.m6.1b"><apply id="Thmtheorem4.p1.6.m6.1.1.cmml" xref="Thmtheorem4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="Thmtheorem4.p1.6.m6.1.1.1.cmml" xref="Thmtheorem4.p1.6.m6.1.1">superscript</csymbol><ci id="Thmtheorem4.p1.6.m6.1.1.2.cmml" xref="Thmtheorem4.p1.6.m6.1.1.2">𝜏</ci><times id="Thmtheorem4.p1.6.m6.1.1.3.cmml" xref="Thmtheorem4.p1.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.6.m6.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.6.m6.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is stable. Similarly, if it is strictly increasing <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmtheorem4.p1.7.m7.1"><semantics id="Thmtheorem4.p1.7.m7.1a"><msup id="Thmtheorem4.p1.7.m7.1.1" xref="Thmtheorem4.p1.7.m7.1.1.cmml"><mi id="Thmtheorem4.p1.7.m7.1.1.2" xref="Thmtheorem4.p1.7.m7.1.1.2.cmml">τ</mi><mo id="Thmtheorem4.p1.7.m7.1.1.3" xref="Thmtheorem4.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmtheorem4.p1.7.m7.1b"><apply id="Thmtheorem4.p1.7.m7.1.1.cmml" xref="Thmtheorem4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmtheorem4.p1.7.m7.1.1.1.cmml" xref="Thmtheorem4.p1.7.m7.1.1">superscript</csymbol><ci id="Thmtheorem4.p1.7.m7.1.1.2.cmml" xref="Thmtheorem4.p1.7.m7.1.1.2">𝜏</ci><times id="Thmtheorem4.p1.7.m7.1.1.3.cmml" xref="Thmtheorem4.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem4.p1.7.m7.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem4.p1.7.m7.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is stable.</p> </div> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.4">The difference between <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_G</annotation></semantics></math> in the OA case and <math alttext="G-F" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml">G</mi><mo id="S3.SS2.p3.2.m2.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.cmml">−</mo><mi id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><minus id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1"></minus><ci id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2">𝐺</ci><ci id="S3.SS2.p3.2.m2.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">G-F</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_G - italic_F</annotation></semantics></math> here is important. We saw that <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mi id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><ci id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">italic_G</annotation></semantics></math> was strictly increasing in natural settings like the Gaussian model. In contrast, in similar settings <math alttext="G-F" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><mrow id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml"><mi id="S3.SS2.p3.4.m4.1.1.2" xref="S3.SS2.p3.4.m4.1.1.2.cmml">G</mi><mo id="S3.SS2.p3.4.m4.1.1.1" xref="S3.SS2.p3.4.m4.1.1.1.cmml">−</mo><mi id="S3.SS2.p3.4.m4.1.1.3" xref="S3.SS2.p3.4.m4.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><apply id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1"><minus id="S3.SS2.p3.4.m4.1.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1.1"></minus><ci id="S3.SS2.p3.4.m4.1.1.2.cmml" xref="S3.SS2.p3.4.m4.1.1.2">𝐺</ci><ci id="S3.SS2.p3.4.m4.1.1.3.cmml" xref="S3.SS2.p3.4.m4.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">G-F</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">italic_G - italic_F</annotation></semantics></math> is strictly decreasing at a unique finite equilibrium point. While not all settings will behave as nicely as our Gaussian example, under reasonable behavior in the tails we expect existence of a stable nontrivial equilibria, while the uninformative equilibria remain unstable. We discuss such generalizations in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A2" title="Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">B</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Gaussian Model</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">We revisit the Gaussian setting introduced in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a> under the DG mechanism. First note that the same equilibria occur under DG as in OA.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary3.1.1.1">Corollary 3</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary3.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary3.p1"> <p class="ltx_p" id="Thmcorollary3.p1.2">In the Gaussian model under DG, we have three equilibria at <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmcorollary3.p1.1.m1.1"><semantics id="Thmcorollary3.p1.1.m1.1a"><mrow id="Thmcorollary3.p1.1.m1.1.1" xref="Thmcorollary3.p1.1.m1.1.1.cmml"><mi id="Thmcorollary3.p1.1.m1.1.1.2" xref="Thmcorollary3.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmcorollary3.p1.1.m1.1.1.1" xref="Thmcorollary3.p1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary3.p1.1.m1.1.1.3" xref="Thmcorollary3.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary3.p1.1.m1.1b"><apply id="Thmcorollary3.p1.1.m1.1.1.cmml" xref="Thmcorollary3.p1.1.m1.1.1"><eq id="Thmcorollary3.p1.1.m1.1.1.1.cmml" xref="Thmcorollary3.p1.1.m1.1.1.1"></eq><ci id="Thmcorollary3.p1.1.m1.1.1.2.cmml" xref="Thmcorollary3.p1.1.m1.1.1.2">𝜏</ci><cn id="Thmcorollary3.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary3.p1.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary3.p1.1.m1.1d">italic_τ = 0</annotation></semantics></math> and <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmcorollary3.p1.2.m2.1"><semantics id="Thmcorollary3.p1.2.m2.1a"><mrow id="Thmcorollary3.p1.2.m2.1.1" xref="Thmcorollary3.p1.2.m2.1.1.cmml"><mo id="Thmcorollary3.p1.2.m2.1.1a" xref="Thmcorollary3.p1.2.m2.1.1.cmml">±</mo><mi id="Thmcorollary3.p1.2.m2.1.1.2" mathvariant="normal" xref="Thmcorollary3.p1.2.m2.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary3.p1.2.m2.1b"><apply id="Thmcorollary3.p1.2.m2.1.1.cmml" xref="Thmcorollary3.p1.2.m2.1.1"><csymbol cd="latexml" id="Thmcorollary3.p1.2.m2.1.1.1.cmml" xref="Thmcorollary3.p1.2.m2.1.1">plus-or-minus</csymbol><infinity id="Thmcorollary3.p1.2.m2.1.1.2.cmml" xref="Thmcorollary3.p1.2.m2.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary3.p1.2.m2.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary3.p1.2.m2.1d">± ∞</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S3.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.1.p1"> <p class="ltx_p" id="S3.SS3.1.p1.1">Existence of equilibria at <math alttext="\tau=\pm\infty" class="ltx_Math" display="inline" id="S3.SS3.1.p1.1.m1.1"><semantics id="S3.SS3.1.p1.1.m1.1a"><mrow id="S3.SS3.1.p1.1.m1.1.1" xref="S3.SS3.1.p1.1.m1.1.1.cmml"><mi id="S3.SS3.1.p1.1.m1.1.1.2" xref="S3.SS3.1.p1.1.m1.1.1.2.cmml">τ</mi><mo id="S3.SS3.1.p1.1.m1.1.1.1" xref="S3.SS3.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.SS3.1.p1.1.m1.1.1.3" xref="S3.SS3.1.p1.1.m1.1.1.3.cmml"><mo id="S3.SS3.1.p1.1.m1.1.1.3a" xref="S3.SS3.1.p1.1.m1.1.1.3.cmml">±</mo><mi id="S3.SS3.1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.SS3.1.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.1.p1.1.m1.1b"><apply id="S3.SS3.1.p1.1.m1.1.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1"><eq id="S3.SS3.1.p1.1.m1.1.1.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1.1"></eq><ci id="S3.SS3.1.p1.1.m1.1.1.2.cmml" xref="S3.SS3.1.p1.1.m1.1.1.2">𝜏</ci><apply id="S3.SS3.1.p1.1.m1.1.1.3.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S3.SS3.1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="S3.SS3.1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.1.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.1.p1.1.m1.1c">\tau=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.1.p1.1.m1.1d">italic_τ = ± ∞</annotation></semantics></math> immediately follows from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition3" title="Proposition 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <div class="ltx_para" id="S3.SS3.2.p2"> <p class="ltx_p" id="S3.SS3.2.p2.7">Now, note that <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S3.SS3.2.p2.1.m1.2"><semantics id="S3.SS3.2.p2.1.m1.2a"><mrow id="S3.SS3.2.p2.1.m1.2.3" xref="S3.SS3.2.p2.1.m1.2.3.cmml"><mi id="S3.SS3.2.p2.1.m1.2.3.2" xref="S3.SS3.2.p2.1.m1.2.3.2.cmml">P</mi><mo id="S3.SS3.2.p2.1.m1.2.3.1" xref="S3.SS3.2.p2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S3.SS3.2.p2.1.m1.2.3.3.2" xref="S3.SS3.2.p2.1.m1.2.3.3.1.cmml"><mo id="S3.SS3.2.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S3.SS3.2.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.SS3.2.p2.1.m1.1.1" xref="S3.SS3.2.p2.1.m1.1.1.cmml">τ</mi><mo id="S3.SS3.2.p2.1.m1.2.3.3.2.2" xref="S3.SS3.2.p2.1.m1.2.3.3.1.cmml">;</mo><mi id="S3.SS3.2.p2.1.m1.2.2" xref="S3.SS3.2.p2.1.m1.2.2.cmml">x</mi><mo id="S3.SS3.2.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S3.SS3.2.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.1.m1.2b"><apply id="S3.SS3.2.p2.1.m1.2.3.cmml" xref="S3.SS3.2.p2.1.m1.2.3"><times id="S3.SS3.2.p2.1.m1.2.3.1.cmml" xref="S3.SS3.2.p2.1.m1.2.3.1"></times><ci id="S3.SS3.2.p2.1.m1.2.3.2.cmml" xref="S3.SS3.2.p2.1.m1.2.3.2">𝑃</ci><list id="S3.SS3.2.p2.1.m1.2.3.3.1.cmml" xref="S3.SS3.2.p2.1.m1.2.3.3.2"><ci id="S3.SS3.2.p2.1.m1.1.1.cmml" xref="S3.SS3.2.p2.1.m1.1.1">𝜏</ci><ci id="S3.SS3.2.p2.1.m1.2.2.cmml" xref="S3.SS3.2.p2.1.m1.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.1.m1.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.1.m1.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> (Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E7" title="In 2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a>)) is strictly decreasing and continuous so that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem3" title="Theorem 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a> applies. By Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E8" title="In 2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a>), <math alttext="G(0)=F(0)=\Phi(0)" class="ltx_Math" display="inline" id="S3.SS3.2.p2.2.m2.3"><semantics id="S3.SS3.2.p2.2.m2.3a"><mrow id="S3.SS3.2.p2.2.m2.3.4" xref="S3.SS3.2.p2.2.m2.3.4.cmml"><mrow id="S3.SS3.2.p2.2.m2.3.4.2" xref="S3.SS3.2.p2.2.m2.3.4.2.cmml"><mi id="S3.SS3.2.p2.2.m2.3.4.2.2" xref="S3.SS3.2.p2.2.m2.3.4.2.2.cmml">G</mi><mo id="S3.SS3.2.p2.2.m2.3.4.2.1" xref="S3.SS3.2.p2.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS3.2.p2.2.m2.3.4.2.3.2" xref="S3.SS3.2.p2.2.m2.3.4.2.cmml"><mo id="S3.SS3.2.p2.2.m2.3.4.2.3.2.1" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.2.cmml">(</mo><mn id="S3.SS3.2.p2.2.m2.1.1" xref="S3.SS3.2.p2.2.m2.1.1.cmml">0</mn><mo id="S3.SS3.2.p2.2.m2.3.4.2.3.2.2" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.2.p2.2.m2.3.4.3" xref="S3.SS3.2.p2.2.m2.3.4.3.cmml">=</mo><mrow id="S3.SS3.2.p2.2.m2.3.4.4" xref="S3.SS3.2.p2.2.m2.3.4.4.cmml"><mi id="S3.SS3.2.p2.2.m2.3.4.4.2" xref="S3.SS3.2.p2.2.m2.3.4.4.2.cmml">F</mi><mo id="S3.SS3.2.p2.2.m2.3.4.4.1" xref="S3.SS3.2.p2.2.m2.3.4.4.1.cmml">⁢</mo><mrow id="S3.SS3.2.p2.2.m2.3.4.4.3.2" xref="S3.SS3.2.p2.2.m2.3.4.4.cmml"><mo id="S3.SS3.2.p2.2.m2.3.4.4.3.2.1" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.4.cmml">(</mo><mn id="S3.SS3.2.p2.2.m2.2.2" xref="S3.SS3.2.p2.2.m2.2.2.cmml">0</mn><mo id="S3.SS3.2.p2.2.m2.3.4.4.3.2.2" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.4.cmml">)</mo></mrow></mrow><mo id="S3.SS3.2.p2.2.m2.3.4.5" xref="S3.SS3.2.p2.2.m2.3.4.5.cmml">=</mo><mrow id="S3.SS3.2.p2.2.m2.3.4.6" xref="S3.SS3.2.p2.2.m2.3.4.6.cmml"><mi id="S3.SS3.2.p2.2.m2.3.4.6.2" mathvariant="normal" xref="S3.SS3.2.p2.2.m2.3.4.6.2.cmml">Φ</mi><mo id="S3.SS3.2.p2.2.m2.3.4.6.1" xref="S3.SS3.2.p2.2.m2.3.4.6.1.cmml">⁢</mo><mrow id="S3.SS3.2.p2.2.m2.3.4.6.3.2" xref="S3.SS3.2.p2.2.m2.3.4.6.cmml"><mo id="S3.SS3.2.p2.2.m2.3.4.6.3.2.1" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.6.cmml">(</mo><mn id="S3.SS3.2.p2.2.m2.3.3" xref="S3.SS3.2.p2.2.m2.3.3.cmml">0</mn><mo id="S3.SS3.2.p2.2.m2.3.4.6.3.2.2" stretchy="false" xref="S3.SS3.2.p2.2.m2.3.4.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.2.m2.3b"><apply id="S3.SS3.2.p2.2.m2.3.4.cmml" xref="S3.SS3.2.p2.2.m2.3.4"><and id="S3.SS3.2.p2.2.m2.3.4a.cmml" xref="S3.SS3.2.p2.2.m2.3.4"></and><apply id="S3.SS3.2.p2.2.m2.3.4b.cmml" xref="S3.SS3.2.p2.2.m2.3.4"><eq id="S3.SS3.2.p2.2.m2.3.4.3.cmml" xref="S3.SS3.2.p2.2.m2.3.4.3"></eq><apply id="S3.SS3.2.p2.2.m2.3.4.2.cmml" xref="S3.SS3.2.p2.2.m2.3.4.2"><times id="S3.SS3.2.p2.2.m2.3.4.2.1.cmml" xref="S3.SS3.2.p2.2.m2.3.4.2.1"></times><ci id="S3.SS3.2.p2.2.m2.3.4.2.2.cmml" xref="S3.SS3.2.p2.2.m2.3.4.2.2">𝐺</ci><cn id="S3.SS3.2.p2.2.m2.1.1.cmml" type="integer" xref="S3.SS3.2.p2.2.m2.1.1">0</cn></apply><apply id="S3.SS3.2.p2.2.m2.3.4.4.cmml" xref="S3.SS3.2.p2.2.m2.3.4.4"><times id="S3.SS3.2.p2.2.m2.3.4.4.1.cmml" xref="S3.SS3.2.p2.2.m2.3.4.4.1"></times><ci id="S3.SS3.2.p2.2.m2.3.4.4.2.cmml" xref="S3.SS3.2.p2.2.m2.3.4.4.2">𝐹</ci><cn id="S3.SS3.2.p2.2.m2.2.2.cmml" type="integer" xref="S3.SS3.2.p2.2.m2.2.2">0</cn></apply></apply><apply id="S3.SS3.2.p2.2.m2.3.4c.cmml" xref="S3.SS3.2.p2.2.m2.3.4"><eq id="S3.SS3.2.p2.2.m2.3.4.5.cmml" xref="S3.SS3.2.p2.2.m2.3.4.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S3.SS3.2.p2.2.m2.3.4.4.cmml" id="S3.SS3.2.p2.2.m2.3.4d.cmml" xref="S3.SS3.2.p2.2.m2.3.4"></share><apply id="S3.SS3.2.p2.2.m2.3.4.6.cmml" xref="S3.SS3.2.p2.2.m2.3.4.6"><times id="S3.SS3.2.p2.2.m2.3.4.6.1.cmml" xref="S3.SS3.2.p2.2.m2.3.4.6.1"></times><ci id="S3.SS3.2.p2.2.m2.3.4.6.2.cmml" xref="S3.SS3.2.p2.2.m2.3.4.6.2">Φ</ci><cn id="S3.SS3.2.p2.2.m2.3.3.cmml" type="integer" xref="S3.SS3.2.p2.2.m2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.2.m2.3c">G(0)=F(0)=\Phi(0)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.2.m2.3d">italic_G ( 0 ) = italic_F ( 0 ) = roman_Φ ( 0 )</annotation></semantics></math>, so <math alttext="\tau=0" class="ltx_Math" display="inline" id="S3.SS3.2.p2.3.m3.1"><semantics id="S3.SS3.2.p2.3.m3.1a"><mrow id="S3.SS3.2.p2.3.m3.1.1" xref="S3.SS3.2.p2.3.m3.1.1.cmml"><mi id="S3.SS3.2.p2.3.m3.1.1.2" xref="S3.SS3.2.p2.3.m3.1.1.2.cmml">τ</mi><mo id="S3.SS3.2.p2.3.m3.1.1.1" xref="S3.SS3.2.p2.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS3.2.p2.3.m3.1.1.3" xref="S3.SS3.2.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.3.m3.1b"><apply id="S3.SS3.2.p2.3.m3.1.1.cmml" xref="S3.SS3.2.p2.3.m3.1.1"><eq id="S3.SS3.2.p2.3.m3.1.1.1.cmml" xref="S3.SS3.2.p2.3.m3.1.1.1"></eq><ci id="S3.SS3.2.p2.3.m3.1.1.2.cmml" xref="S3.SS3.2.p2.3.m3.1.1.2">𝜏</ci><cn id="S3.SS3.2.p2.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.2.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.3.m3.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.3.m3.1d">italic_τ = 0</annotation></semantics></math> is an equilibrium under DG. Moreover, there could not be another finite equilibrium unless the coefficients of <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.2.p2.4.m4.1"><semantics id="S3.SS3.2.p2.4.m4.1a"><mi id="S3.SS3.2.p2.4.m4.1.1" xref="S3.SS3.2.p2.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.4.m4.1b"><ci id="S3.SS3.2.p2.4.m4.1.1.cmml" xref="S3.SS3.2.p2.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.4.m4.1d">italic_x</annotation></semantics></math> in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.E8" title="In 2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a>) were equal. Letting <math alttext="c_{F}=\frac{\sqrt{\rho}}{a}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.5.m5.1"><semantics id="S3.SS3.2.p2.5.m5.1a"><mrow id="S3.SS3.2.p2.5.m5.1.1" xref="S3.SS3.2.p2.5.m5.1.1.cmml"><msub id="S3.SS3.2.p2.5.m5.1.1.2" xref="S3.SS3.2.p2.5.m5.1.1.2.cmml"><mi id="S3.SS3.2.p2.5.m5.1.1.2.2" xref="S3.SS3.2.p2.5.m5.1.1.2.2.cmml">c</mi><mi id="S3.SS3.2.p2.5.m5.1.1.2.3" xref="S3.SS3.2.p2.5.m5.1.1.2.3.cmml">F</mi></msub><mo id="S3.SS3.2.p2.5.m5.1.1.1" xref="S3.SS3.2.p2.5.m5.1.1.1.cmml">=</mo><mfrac id="S3.SS3.2.p2.5.m5.1.1.3" xref="S3.SS3.2.p2.5.m5.1.1.3.cmml"><msqrt id="S3.SS3.2.p2.5.m5.1.1.3.2" xref="S3.SS3.2.p2.5.m5.1.1.3.2.cmml"><mi id="S3.SS3.2.p2.5.m5.1.1.3.2.2" xref="S3.SS3.2.p2.5.m5.1.1.3.2.2.cmml">ρ</mi></msqrt><mi id="S3.SS3.2.p2.5.m5.1.1.3.3" xref="S3.SS3.2.p2.5.m5.1.1.3.3.cmml">a</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.5.m5.1b"><apply id="S3.SS3.2.p2.5.m5.1.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1"><eq id="S3.SS3.2.p2.5.m5.1.1.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1.1"></eq><apply id="S3.SS3.2.p2.5.m5.1.1.2.cmml" xref="S3.SS3.2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.2.p2.5.m5.1.1.2.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS3.2.p2.5.m5.1.1.2.2.cmml" xref="S3.SS3.2.p2.5.m5.1.1.2.2">𝑐</ci><ci id="S3.SS3.2.p2.5.m5.1.1.2.3.cmml" xref="S3.SS3.2.p2.5.m5.1.1.2.3">𝐹</ci></apply><apply id="S3.SS3.2.p2.5.m5.1.1.3.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3"><divide id="S3.SS3.2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3"></divide><apply id="S3.SS3.2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3.2"><root id="S3.SS3.2.p2.5.m5.1.1.3.2a.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3.2"></root><ci id="S3.SS3.2.p2.5.m5.1.1.3.2.2.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3.2.2">𝜌</ci></apply><ci id="S3.SS3.2.p2.5.m5.1.1.3.3.cmml" xref="S3.SS3.2.p2.5.m5.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.5.m5.1c">c_{F}=\frac{\sqrt{\rho}}{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.5.m5.1d">italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = divide start_ARG square-root start_ARG italic_ρ end_ARG end_ARG start_ARG italic_a end_ARG</annotation></semantics></math> and <math alttext="c_{G}=\frac{(1-\rho)}{b\sqrt{1+\rho}}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.6.m6.1"><semantics id="S3.SS3.2.p2.6.m6.1a"><mrow id="S3.SS3.2.p2.6.m6.1.2" xref="S3.SS3.2.p2.6.m6.1.2.cmml"><msub id="S3.SS3.2.p2.6.m6.1.2.2" xref="S3.SS3.2.p2.6.m6.1.2.2.cmml"><mi id="S3.SS3.2.p2.6.m6.1.2.2.2" xref="S3.SS3.2.p2.6.m6.1.2.2.2.cmml">c</mi><mi id="S3.SS3.2.p2.6.m6.1.2.2.3" xref="S3.SS3.2.p2.6.m6.1.2.2.3.cmml">G</mi></msub><mo id="S3.SS3.2.p2.6.m6.1.2.1" xref="S3.SS3.2.p2.6.m6.1.2.1.cmml">=</mo><mfrac id="S3.SS3.2.p2.6.m6.1.1" xref="S3.SS3.2.p2.6.m6.1.1.cmml"><mrow id="S3.SS3.2.p2.6.m6.1.1.1.1" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.cmml"><mo 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id="S3.SS3.2.p2.6.m6.1.1.3.3.2.1" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2.1.cmml">+</mo><mi id="S3.SS3.2.p2.6.m6.1.1.3.3.2.3" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.6.m6.1b"><apply id="S3.SS3.2.p2.6.m6.1.2.cmml" xref="S3.SS3.2.p2.6.m6.1.2"><eq id="S3.SS3.2.p2.6.m6.1.2.1.cmml" xref="S3.SS3.2.p2.6.m6.1.2.1"></eq><apply id="S3.SS3.2.p2.6.m6.1.2.2.cmml" xref="S3.SS3.2.p2.6.m6.1.2.2"><csymbol cd="ambiguous" id="S3.SS3.2.p2.6.m6.1.2.2.1.cmml" xref="S3.SS3.2.p2.6.m6.1.2.2">subscript</csymbol><ci id="S3.SS3.2.p2.6.m6.1.2.2.2.cmml" xref="S3.SS3.2.p2.6.m6.1.2.2.2">𝑐</ci><ci id="S3.SS3.2.p2.6.m6.1.2.2.3.cmml" xref="S3.SS3.2.p2.6.m6.1.2.2.3">𝐺</ci></apply><apply id="S3.SS3.2.p2.6.m6.1.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1"><divide id="S3.SS3.2.p2.6.m6.1.1.2.cmml" xref="S3.SS3.2.p2.6.m6.1.1"></divide><apply id="S3.SS3.2.p2.6.m6.1.1.1.1.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1"><minus id="S3.SS3.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.1"></minus><cn id="S3.SS3.2.p2.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.2">1</cn><ci id="S3.SS3.2.p2.6.m6.1.1.1.1.1.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.1.1.1.3">𝜌</ci></apply><apply id="S3.SS3.2.p2.6.m6.1.1.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3"><times id="S3.SS3.2.p2.6.m6.1.1.3.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.1"></times><ci id="S3.SS3.2.p2.6.m6.1.1.3.2.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.2">𝑏</ci><apply id="S3.SS3.2.p2.6.m6.1.1.3.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.3"><root id="S3.SS3.2.p2.6.m6.1.1.3.3a.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.3"></root><apply id="S3.SS3.2.p2.6.m6.1.1.3.3.2.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2"><plus id="S3.SS3.2.p2.6.m6.1.1.3.3.2.1.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2.1"></plus><cn id="S3.SS3.2.p2.6.m6.1.1.3.3.2.2.cmml" type="integer" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2.2">1</cn><ci id="S3.SS3.2.p2.6.m6.1.1.3.3.2.3.cmml" xref="S3.SS3.2.p2.6.m6.1.1.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.6.m6.1c">c_{G}=\frac{(1-\rho)}{b\sqrt{1+\rho}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.6.m6.1d">italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT = divide start_ARG ( 1 - italic_ρ ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG</annotation></semantics></math> be these coefficients, and substituting <math alttext="a^{2}=\frac{\rho}{1-\rho}b^{2}" class="ltx_Math" display="inline" id="S3.SS3.2.p2.7.m7.1"><semantics id="S3.SS3.2.p2.7.m7.1a"><mrow id="S3.SS3.2.p2.7.m7.1.1" xref="S3.SS3.2.p2.7.m7.1.1.cmml"><msup id="S3.SS3.2.p2.7.m7.1.1.2" xref="S3.SS3.2.p2.7.m7.1.1.2.cmml"><mi id="S3.SS3.2.p2.7.m7.1.1.2.2" xref="S3.SS3.2.p2.7.m7.1.1.2.2.cmml">a</mi><mn id="S3.SS3.2.p2.7.m7.1.1.2.3" xref="S3.SS3.2.p2.7.m7.1.1.2.3.cmml">2</mn></msup><mo id="S3.SS3.2.p2.7.m7.1.1.1" xref="S3.SS3.2.p2.7.m7.1.1.1.cmml">=</mo><mrow id="S3.SS3.2.p2.7.m7.1.1.3" xref="S3.SS3.2.p2.7.m7.1.1.3.cmml"><mfrac id="S3.SS3.2.p2.7.m7.1.1.3.2" xref="S3.SS3.2.p2.7.m7.1.1.3.2.cmml"><mi id="S3.SS3.2.p2.7.m7.1.1.3.2.2" xref="S3.SS3.2.p2.7.m7.1.1.3.2.2.cmml">ρ</mi><mrow id="S3.SS3.2.p2.7.m7.1.1.3.2.3" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.cmml"><mn id="S3.SS3.2.p2.7.m7.1.1.3.2.3.2" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.2.cmml">1</mn><mo id="S3.SS3.2.p2.7.m7.1.1.3.2.3.1" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.1.cmml">−</mo><mi id="S3.SS3.2.p2.7.m7.1.1.3.2.3.3" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.3.cmml">ρ</mi></mrow></mfrac><mo id="S3.SS3.2.p2.7.m7.1.1.3.1" xref="S3.SS3.2.p2.7.m7.1.1.3.1.cmml">⁢</mo><msup id="S3.SS3.2.p2.7.m7.1.1.3.3" xref="S3.SS3.2.p2.7.m7.1.1.3.3.cmml"><mi id="S3.SS3.2.p2.7.m7.1.1.3.3.2" xref="S3.SS3.2.p2.7.m7.1.1.3.3.2.cmml">b</mi><mn id="S3.SS3.2.p2.7.m7.1.1.3.3.3" xref="S3.SS3.2.p2.7.m7.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.7.m7.1b"><apply id="S3.SS3.2.p2.7.m7.1.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1"><eq id="S3.SS3.2.p2.7.m7.1.1.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.1"></eq><apply id="S3.SS3.2.p2.7.m7.1.1.2.cmml" xref="S3.SS3.2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.2.p2.7.m7.1.1.2.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S3.SS3.2.p2.7.m7.1.1.2.2.cmml" xref="S3.SS3.2.p2.7.m7.1.1.2.2">𝑎</ci><cn id="S3.SS3.2.p2.7.m7.1.1.2.3.cmml" type="integer" xref="S3.SS3.2.p2.7.m7.1.1.2.3">2</cn></apply><apply id="S3.SS3.2.p2.7.m7.1.1.3.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3"><times id="S3.SS3.2.p2.7.m7.1.1.3.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.1"></times><apply id="S3.SS3.2.p2.7.m7.1.1.3.2.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2"><divide id="S3.SS3.2.p2.7.m7.1.1.3.2.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2"></divide><ci id="S3.SS3.2.p2.7.m7.1.1.3.2.2.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2.2">𝜌</ci><apply id="S3.SS3.2.p2.7.m7.1.1.3.2.3.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3"><minus id="S3.SS3.2.p2.7.m7.1.1.3.2.3.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.1"></minus><cn id="S3.SS3.2.p2.7.m7.1.1.3.2.3.2.cmml" type="integer" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.2">1</cn><ci id="S3.SS3.2.p2.7.m7.1.1.3.2.3.3.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.2.3.3">𝜌</ci></apply></apply><apply id="S3.SS3.2.p2.7.m7.1.1.3.3.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.2.p2.7.m7.1.1.3.3.1.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.3">superscript</csymbol><ci id="S3.SS3.2.p2.7.m7.1.1.3.3.2.cmml" xref="S3.SS3.2.p2.7.m7.1.1.3.3.2">𝑏</ci><cn id="S3.SS3.2.p2.7.m7.1.1.3.3.3.cmml" type="integer" xref="S3.SS3.2.p2.7.m7.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.7.m7.1c">a^{2}=\frac{\rho}{1-\rho}b^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.7.m7.1d">italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_ρ end_ARG start_ARG 1 - italic_ρ end_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx4"> <tbody id="S3.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\left(\frac{c_{F}}{c_{G}}\right)^{2}=\frac{\rho b^{2}(1+\rho)}{a^% {2}(1-\rho)^{2}}=\frac{\rho b^{2}(1+\rho)}{\frac{\rho}{1-\rho}b^{2}(1-\rho)^{2% }}=\frac{1+\rho}{1-\rho}>1~{}." class="ltx_Math" display="inline" id="S3.E13.m1.6"><semantics id="S3.E13.m1.6a"><mrow id="S3.E13.m1.6.6.1" xref="S3.E13.m1.6.6.1.1.cmml"><mrow id="S3.E13.m1.6.6.1.1" xref="S3.E13.m1.6.6.1.1.cmml"><msup id="S3.E13.m1.6.6.1.1.2" xref="S3.E13.m1.6.6.1.1.2.cmml"><mrow id="S3.E13.m1.6.6.1.1.2.2.2" xref="S3.E13.m1.5.5.cmml"><mo id="S3.E13.m1.6.6.1.1.2.2.2.1" xref="S3.E13.m1.5.5.cmml">(</mo><mstyle displaystyle="true" id="S3.E13.m1.5.5" xref="S3.E13.m1.5.5.cmml"><mfrac id="S3.E13.m1.5.5a" xref="S3.E13.m1.5.5.cmml"><msub id="S3.E13.m1.5.5.2" xref="S3.E13.m1.5.5.2.cmml"><mi id="S3.E13.m1.5.5.2.2" xref="S3.E13.m1.5.5.2.2.cmml">c</mi><mi id="S3.E13.m1.5.5.2.3" xref="S3.E13.m1.5.5.2.3.cmml">F</mi></msub><msub id="S3.E13.m1.5.5.3" xref="S3.E13.m1.5.5.3.cmml"><mi id="S3.E13.m1.5.5.3.2" xref="S3.E13.m1.5.5.3.2.cmml">c</mi><mi id="S3.E13.m1.5.5.3.3" xref="S3.E13.m1.5.5.3.3.cmml">G</mi></msub></mfrac></mstyle><mo id="S3.E13.m1.6.6.1.1.2.2.2.2" xref="S3.E13.m1.5.5.cmml">)</mo></mrow><mn id="S3.E13.m1.6.6.1.1.2.3" xref="S3.E13.m1.6.6.1.1.2.3.cmml">2</mn></msup><mo id="S3.E13.m1.6.6.1.1.3" xref="S3.E13.m1.6.6.1.1.3.cmml">=</mo><mstyle displaystyle="true" id="S3.E13.m1.2.2" xref="S3.E13.m1.2.2.cmml"><mfrac id="S3.E13.m1.2.2a" xref="S3.E13.m1.2.2.cmml"><mrow id="S3.E13.m1.1.1.1" xref="S3.E13.m1.1.1.1.cmml"><mi id="S3.E13.m1.1.1.1.3" xref="S3.E13.m1.1.1.1.3.cmml">ρ</mi><mo id="S3.E13.m1.1.1.1.2" xref="S3.E13.m1.1.1.1.2.cmml">⁢</mo><msup id="S3.E13.m1.1.1.1.4" xref="S3.E13.m1.1.1.1.4.cmml"><mi id="S3.E13.m1.1.1.1.4.2" xref="S3.E13.m1.1.1.1.4.2.cmml">b</mi><mn id="S3.E13.m1.1.1.1.4.3" xref="S3.E13.m1.1.1.1.4.3.cmml">2</mn></msup><mo id="S3.E13.m1.1.1.1.2a" xref="S3.E13.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E13.m1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.cmml"><mo id="S3.E13.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E13.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.cmml"><mn id="S3.E13.m1.1.1.1.1.1.1.2" xref="S3.E13.m1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.E13.m1.1.1.1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.E13.m1.1.1.1.1.1.1.3" xref="S3.E13.m1.1.1.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="S3.E13.m1.1.1.1.1.1.3" stretchy="false" xref="S3.E13.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S3.E13.m1.2.2.2" xref="S3.E13.m1.2.2.2.cmml"><msup id="S3.E13.m1.2.2.2.3" xref="S3.E13.m1.2.2.2.3.cmml"><mi id="S3.E13.m1.2.2.2.3.2" xref="S3.E13.m1.2.2.2.3.2.cmml">a</mi><mn id="S3.E13.m1.2.2.2.3.3" xref="S3.E13.m1.2.2.2.3.3.cmml">2</mn></msup><mo id="S3.E13.m1.2.2.2.2" xref="S3.E13.m1.2.2.2.2.cmml">⁢</mo><msup id="S3.E13.m1.2.2.2.1" xref="S3.E13.m1.2.2.2.1.cmml"><mrow id="S3.E13.m1.2.2.2.1.1.1" xref="S3.E13.m1.2.2.2.1.1.1.1.cmml"><mo id="S3.E13.m1.2.2.2.1.1.1.2" stretchy="false" xref="S3.E13.m1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.2.2.2.1.1.1.1" xref="S3.E13.m1.2.2.2.1.1.1.1.cmml"><mn id="S3.E13.m1.2.2.2.1.1.1.1.2" xref="S3.E13.m1.2.2.2.1.1.1.1.2.cmml">1</mn><mo id="S3.E13.m1.2.2.2.1.1.1.1.1" xref="S3.E13.m1.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S3.E13.m1.2.2.2.1.1.1.1.3" xref="S3.E13.m1.2.2.2.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="S3.E13.m1.2.2.2.1.1.1.3" stretchy="false" xref="S3.E13.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow><mn id="S3.E13.m1.2.2.2.1.3" xref="S3.E13.m1.2.2.2.1.3.cmml">2</mn></msup></mrow></mfrac></mstyle><mo id="S3.E13.m1.6.6.1.1.4" xref="S3.E13.m1.6.6.1.1.4.cmml">=</mo><mstyle displaystyle="true" id="S3.E13.m1.4.4" xref="S3.E13.m1.4.4.cmml"><mfrac id="S3.E13.m1.4.4a" xref="S3.E13.m1.4.4.cmml"><mrow id="S3.E13.m1.3.3.1" xref="S3.E13.m1.3.3.1.cmml"><mi id="S3.E13.m1.3.3.1.3" xref="S3.E13.m1.3.3.1.3.cmml">ρ</mi><mo id="S3.E13.m1.3.3.1.2" xref="S3.E13.m1.3.3.1.2.cmml">⁢</mo><msup id="S3.E13.m1.3.3.1.4" xref="S3.E13.m1.3.3.1.4.cmml"><mi id="S3.E13.m1.3.3.1.4.2" xref="S3.E13.m1.3.3.1.4.2.cmml">b</mi><mn id="S3.E13.m1.3.3.1.4.3" xref="S3.E13.m1.3.3.1.4.3.cmml">2</mn></msup><mo id="S3.E13.m1.3.3.1.2a" xref="S3.E13.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.E13.m1.3.3.1.1.1" xref="S3.E13.m1.3.3.1.1.1.1.cmml"><mo id="S3.E13.m1.3.3.1.1.1.2" stretchy="false" xref="S3.E13.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.3.3.1.1.1.1" xref="S3.E13.m1.3.3.1.1.1.1.cmml"><mn id="S3.E13.m1.3.3.1.1.1.1.2" xref="S3.E13.m1.3.3.1.1.1.1.2.cmml">1</mn><mo id="S3.E13.m1.3.3.1.1.1.1.1" xref="S3.E13.m1.3.3.1.1.1.1.1.cmml">+</mo><mi id="S3.E13.m1.3.3.1.1.1.1.3" xref="S3.E13.m1.3.3.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="S3.E13.m1.3.3.1.1.1.3" stretchy="false" xref="S3.E13.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S3.E13.m1.4.4.2" xref="S3.E13.m1.4.4.2.cmml"><mfrac id="S3.E13.m1.4.4.2.3" xref="S3.E13.m1.4.4.2.3.cmml"><mi id="S3.E13.m1.4.4.2.3.2" xref="S3.E13.m1.4.4.2.3.2.cmml">ρ</mi><mrow id="S3.E13.m1.4.4.2.3.3" xref="S3.E13.m1.4.4.2.3.3.cmml"><mn id="S3.E13.m1.4.4.2.3.3.2" xref="S3.E13.m1.4.4.2.3.3.2.cmml">1</mn><mo id="S3.E13.m1.4.4.2.3.3.1" xref="S3.E13.m1.4.4.2.3.3.1.cmml">−</mo><mi id="S3.E13.m1.4.4.2.3.3.3" xref="S3.E13.m1.4.4.2.3.3.3.cmml">ρ</mi></mrow></mfrac><mo id="S3.E13.m1.4.4.2.2" xref="S3.E13.m1.4.4.2.2.cmml">⁢</mo><msup id="S3.E13.m1.4.4.2.4" xref="S3.E13.m1.4.4.2.4.cmml"><mi id="S3.E13.m1.4.4.2.4.2" xref="S3.E13.m1.4.4.2.4.2.cmml">b</mi><mn id="S3.E13.m1.4.4.2.4.3" xref="S3.E13.m1.4.4.2.4.3.cmml">2</mn></msup><mo id="S3.E13.m1.4.4.2.2a" xref="S3.E13.m1.4.4.2.2.cmml">⁢</mo><msup id="S3.E13.m1.4.4.2.1" xref="S3.E13.m1.4.4.2.1.cmml"><mrow id="S3.E13.m1.4.4.2.1.1.1" xref="S3.E13.m1.4.4.2.1.1.1.1.cmml"><mo id="S3.E13.m1.4.4.2.1.1.1.2" stretchy="false" xref="S3.E13.m1.4.4.2.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.4.4.2.1.1.1.1" xref="S3.E13.m1.4.4.2.1.1.1.1.cmml"><mn id="S3.E13.m1.4.4.2.1.1.1.1.2" xref="S3.E13.m1.4.4.2.1.1.1.1.2.cmml">1</mn><mo id="S3.E13.m1.4.4.2.1.1.1.1.1" xref="S3.E13.m1.4.4.2.1.1.1.1.1.cmml">−</mo><mi id="S3.E13.m1.4.4.2.1.1.1.1.3" xref="S3.E13.m1.4.4.2.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="S3.E13.m1.4.4.2.1.1.1.3" stretchy="false" xref="S3.E13.m1.4.4.2.1.1.1.1.cmml">)</mo></mrow><mn id="S3.E13.m1.4.4.2.1.3" xref="S3.E13.m1.4.4.2.1.3.cmml">2</mn></msup></mrow></mfrac></mstyle><mo id="S3.E13.m1.6.6.1.1.5" xref="S3.E13.m1.6.6.1.1.5.cmml">=</mo><mstyle displaystyle="true" id="S3.E13.m1.6.6.1.1.6" xref="S3.E13.m1.6.6.1.1.6.cmml"><mfrac id="S3.E13.m1.6.6.1.1.6a" xref="S3.E13.m1.6.6.1.1.6.cmml"><mrow id="S3.E13.m1.6.6.1.1.6.2" xref="S3.E13.m1.6.6.1.1.6.2.cmml"><mn id="S3.E13.m1.6.6.1.1.6.2.2" xref="S3.E13.m1.6.6.1.1.6.2.2.cmml">1</mn><mo id="S3.E13.m1.6.6.1.1.6.2.1" xref="S3.E13.m1.6.6.1.1.6.2.1.cmml">+</mo><mi id="S3.E13.m1.6.6.1.1.6.2.3" xref="S3.E13.m1.6.6.1.1.6.2.3.cmml">ρ</mi></mrow><mrow id="S3.E13.m1.6.6.1.1.6.3" xref="S3.E13.m1.6.6.1.1.6.3.cmml"><mn id="S3.E13.m1.6.6.1.1.6.3.2" xref="S3.E13.m1.6.6.1.1.6.3.2.cmml">1</mn><mo id="S3.E13.m1.6.6.1.1.6.3.1" xref="S3.E13.m1.6.6.1.1.6.3.1.cmml">−</mo><mi id="S3.E13.m1.6.6.1.1.6.3.3" xref="S3.E13.m1.6.6.1.1.6.3.3.cmml">ρ</mi></mrow></mfrac></mstyle><mo id="S3.E13.m1.6.6.1.1.7" xref="S3.E13.m1.6.6.1.1.7.cmml">></mo><mn id="S3.E13.m1.6.6.1.1.8" xref="S3.E13.m1.6.6.1.1.8.cmml">1</mn></mrow><mo id="S3.E13.m1.6.6.1.2" lspace="0.330em" xref="S3.E13.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E13.m1.6b"><apply id="S3.E13.m1.6.6.1.1.cmml" xref="S3.E13.m1.6.6.1"><and id="S3.E13.m1.6.6.1.1a.cmml" xref="S3.E13.m1.6.6.1"></and><apply id="S3.E13.m1.6.6.1.1b.cmml" 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xref="S3.E13.m1.6.6.1"><gt id="S3.E13.m1.6.6.1.1.7.cmml" xref="S3.E13.m1.6.6.1.1.7"></gt><share href="https://arxiv.org/html/2503.16280v1#S3.E13.m1.6.6.1.1.6.cmml" id="S3.E13.m1.6.6.1.1h.cmml" xref="S3.E13.m1.6.6.1"></share><cn id="S3.E13.m1.6.6.1.1.8.cmml" type="integer" xref="S3.E13.m1.6.6.1.1.8">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.6c">\displaystyle\left(\frac{c_{F}}{c_{G}}\right)^{2}=\frac{\rho b^{2}(1+\rho)}{a^% {2}(1-\rho)^{2}}=\frac{\rho b^{2}(1+\rho)}{\frac{\rho}{1-\rho}b^{2}(1-\rho)^{2% }}=\frac{1+\rho}{1-\rho}>1~{}.</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.6d">( divide start_ARG italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_ρ italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ρ ) end_ARG start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG italic_ρ italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ρ ) end_ARG start_ARG divide start_ARG italic_ρ end_ARG start_ARG 1 - italic_ρ end_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG 1 + italic_ρ end_ARG start_ARG 1 - italic_ρ end_ARG > 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.2.p2.8">Uniqueness of <math alttext="\tau=0" class="ltx_Math" display="inline" id="S3.SS3.2.p2.8.m1.1"><semantics id="S3.SS3.2.p2.8.m1.1a"><mrow id="S3.SS3.2.p2.8.m1.1.1" xref="S3.SS3.2.p2.8.m1.1.1.cmml"><mi id="S3.SS3.2.p2.8.m1.1.1.2" xref="S3.SS3.2.p2.8.m1.1.1.2.cmml">τ</mi><mo id="S3.SS3.2.p2.8.m1.1.1.1" xref="S3.SS3.2.p2.8.m1.1.1.1.cmml">=</mo><mn id="S3.SS3.2.p2.8.m1.1.1.3" xref="S3.SS3.2.p2.8.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.2.p2.8.m1.1b"><apply id="S3.SS3.2.p2.8.m1.1.1.cmml" xref="S3.SS3.2.p2.8.m1.1.1"><eq id="S3.SS3.2.p2.8.m1.1.1.1.cmml" xref="S3.SS3.2.p2.8.m1.1.1.1"></eq><ci id="S3.SS3.2.p2.8.m1.1.1.2.cmml" xref="S3.SS3.2.p2.8.m1.1.1.2">𝜏</ci><cn id="S3.SS3.2.p2.8.m1.1.1.3.cmml" type="integer" xref="S3.SS3.2.p2.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.2.p2.8.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.2.p2.8.m1.1d">italic_τ = 0</annotation></semantics></math> as a finite threshold equilibrium follows. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">Now, consider the stability of these equilibria.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary4.1.1.1">Corollary 4</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary4.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary4.p1"> <p class="ltx_p" id="Thmcorollary4.p1.2">In the Gaussian model under DG, the threshold equilibrium <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmcorollary4.p1.1.m1.1"><semantics id="Thmcorollary4.p1.1.m1.1a"><mrow id="Thmcorollary4.p1.1.m1.1.1" xref="Thmcorollary4.p1.1.m1.1.1.cmml"><mi id="Thmcorollary4.p1.1.m1.1.1.2" xref="Thmcorollary4.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmcorollary4.p1.1.m1.1.1.1" xref="Thmcorollary4.p1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary4.p1.1.m1.1.1.3" xref="Thmcorollary4.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.1.m1.1b"><apply id="Thmcorollary4.p1.1.m1.1.1.cmml" xref="Thmcorollary4.p1.1.m1.1.1"><eq id="Thmcorollary4.p1.1.m1.1.1.1.cmml" xref="Thmcorollary4.p1.1.m1.1.1.1"></eq><ci id="Thmcorollary4.p1.1.m1.1.1.2.cmml" xref="Thmcorollary4.p1.1.m1.1.1.2">𝜏</ci><cn id="Thmcorollary4.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary4.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.1.m1.1d">italic_τ = 0</annotation></semantics></math> is stable, while the uninformative equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmcorollary4.p1.2.m2.1"><semantics id="Thmcorollary4.p1.2.m2.1a"><mrow id="Thmcorollary4.p1.2.m2.1.1" xref="Thmcorollary4.p1.2.m2.1.1.cmml"><mo id="Thmcorollary4.p1.2.m2.1.1a" xref="Thmcorollary4.p1.2.m2.1.1.cmml">±</mo><mi id="Thmcorollary4.p1.2.m2.1.1.2" mathvariant="normal" xref="Thmcorollary4.p1.2.m2.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary4.p1.2.m2.1b"><apply id="Thmcorollary4.p1.2.m2.1.1.cmml" xref="Thmcorollary4.p1.2.m2.1.1"><csymbol cd="latexml" id="Thmcorollary4.p1.2.m2.1.1.1.cmml" xref="Thmcorollary4.p1.2.m2.1.1">plus-or-minus</csymbol><infinity id="Thmcorollary4.p1.2.m2.1.1.2.cmml" xref="Thmcorollary4.p1.2.m2.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary4.p1.2.m2.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary4.p1.2.m2.1d">± ∞</annotation></semantics></math> are unstable.</p> </div> </div> <div class="ltx_proof" id="S3.SS3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS3.3.p1"> <p class="ltx_p" id="S3.SS3.3.p1.8">As depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.F2" title="Figure 2 ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, <math alttext="G(x)-F(x)" class="ltx_Math" display="inline" id="S3.SS3.3.p1.1.m1.2"><semantics id="S3.SS3.3.p1.1.m1.2a"><mrow id="S3.SS3.3.p1.1.m1.2.3" xref="S3.SS3.3.p1.1.m1.2.3.cmml"><mrow id="S3.SS3.3.p1.1.m1.2.3.2" xref="S3.SS3.3.p1.1.m1.2.3.2.cmml"><mi id="S3.SS3.3.p1.1.m1.2.3.2.2" xref="S3.SS3.3.p1.1.m1.2.3.2.2.cmml">G</mi><mo id="S3.SS3.3.p1.1.m1.2.3.2.1" xref="S3.SS3.3.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS3.3.p1.1.m1.2.3.2.3.2" xref="S3.SS3.3.p1.1.m1.2.3.2.cmml"><mo id="S3.SS3.3.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S3.SS3.3.p1.1.m1.2.3.2.cmml">(</mo><mi id="S3.SS3.3.p1.1.m1.1.1" xref="S3.SS3.3.p1.1.m1.1.1.cmml">x</mi><mo id="S3.SS3.3.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S3.SS3.3.p1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.3.p1.1.m1.2.3.1" xref="S3.SS3.3.p1.1.m1.2.3.1.cmml">−</mo><mrow id="S3.SS3.3.p1.1.m1.2.3.3" xref="S3.SS3.3.p1.1.m1.2.3.3.cmml"><mi id="S3.SS3.3.p1.1.m1.2.3.3.2" xref="S3.SS3.3.p1.1.m1.2.3.3.2.cmml">F</mi><mo id="S3.SS3.3.p1.1.m1.2.3.3.1" xref="S3.SS3.3.p1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS3.3.p1.1.m1.2.3.3.3.2" xref="S3.SS3.3.p1.1.m1.2.3.3.cmml"><mo id="S3.SS3.3.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S3.SS3.3.p1.1.m1.2.3.3.cmml">(</mo><mi id="S3.SS3.3.p1.1.m1.2.2" xref="S3.SS3.3.p1.1.m1.2.2.cmml">x</mi><mo id="S3.SS3.3.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S3.SS3.3.p1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.1.m1.2b"><apply id="S3.SS3.3.p1.1.m1.2.3.cmml" xref="S3.SS3.3.p1.1.m1.2.3"><minus id="S3.SS3.3.p1.1.m1.2.3.1.cmml" xref="S3.SS3.3.p1.1.m1.2.3.1"></minus><apply id="S3.SS3.3.p1.1.m1.2.3.2.cmml" xref="S3.SS3.3.p1.1.m1.2.3.2"><times id="S3.SS3.3.p1.1.m1.2.3.2.1.cmml" xref="S3.SS3.3.p1.1.m1.2.3.2.1"></times><ci id="S3.SS3.3.p1.1.m1.2.3.2.2.cmml" xref="S3.SS3.3.p1.1.m1.2.3.2.2">𝐺</ci><ci id="S3.SS3.3.p1.1.m1.1.1.cmml" xref="S3.SS3.3.p1.1.m1.1.1">𝑥</ci></apply><apply id="S3.SS3.3.p1.1.m1.2.3.3.cmml" xref="S3.SS3.3.p1.1.m1.2.3.3"><times id="S3.SS3.3.p1.1.m1.2.3.3.1.cmml" xref="S3.SS3.3.p1.1.m1.2.3.3.1"></times><ci id="S3.SS3.3.p1.1.m1.2.3.3.2.cmml" xref="S3.SS3.3.p1.1.m1.2.3.3.2">𝐹</ci><ci id="S3.SS3.3.p1.1.m1.2.2.cmml" xref="S3.SS3.3.p1.1.m1.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.1.m1.2c">G(x)-F(x)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.1.m1.2d">italic_G ( italic_x ) - italic_F ( italic_x )</annotation></semantics></math> is strictly decreasing at <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.2.m2.1"><semantics id="S3.SS3.3.p1.2.m2.1a"><mn id="S3.SS3.3.p1.2.m2.1.1" xref="S3.SS3.3.p1.2.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.2.m2.1b"><cn id="S3.SS3.3.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS3.3.p1.2.m2.1.1">0</cn></annotation-xml></semantics></math>. Formally, <math alttext="G^{\prime}(x)-F^{\prime}(x)=c_{G}\phi(c_{G}x)-c_{F}\phi(c_{F}x)" class="ltx_Math" display="inline" id="S3.SS3.3.p1.3.m3.4"><semantics id="S3.SS3.3.p1.3.m3.4a"><mrow id="S3.SS3.3.p1.3.m3.4.4" xref="S3.SS3.3.p1.3.m3.4.4.cmml"><mrow id="S3.SS3.3.p1.3.m3.4.4.4" xref="S3.SS3.3.p1.3.m3.4.4.4.cmml"><mrow id="S3.SS3.3.p1.3.m3.4.4.4.2" xref="S3.SS3.3.p1.3.m3.4.4.4.2.cmml"><msup id="S3.SS3.3.p1.3.m3.4.4.4.2.2" xref="S3.SS3.3.p1.3.m3.4.4.4.2.2.cmml"><mi id="S3.SS3.3.p1.3.m3.4.4.4.2.2.2" xref="S3.SS3.3.p1.3.m3.4.4.4.2.2.2.cmml">G</mi><mo id="S3.SS3.3.p1.3.m3.4.4.4.2.2.3" xref="S3.SS3.3.p1.3.m3.4.4.4.2.2.3.cmml">′</mo></msup><mo id="S3.SS3.3.p1.3.m3.4.4.4.2.1" xref="S3.SS3.3.p1.3.m3.4.4.4.2.1.cmml">⁢</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.4.2.3.2" xref="S3.SS3.3.p1.3.m3.4.4.4.2.cmml"><mo id="S3.SS3.3.p1.3.m3.4.4.4.2.3.2.1" stretchy="false" xref="S3.SS3.3.p1.3.m3.4.4.4.2.cmml">(</mo><mi id="S3.SS3.3.p1.3.m3.1.1" xref="S3.SS3.3.p1.3.m3.1.1.cmml">x</mi><mo id="S3.SS3.3.p1.3.m3.4.4.4.2.3.2.2" stretchy="false" xref="S3.SS3.3.p1.3.m3.4.4.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.3.p1.3.m3.4.4.4.1" xref="S3.SS3.3.p1.3.m3.4.4.4.1.cmml">−</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.4.3" xref="S3.SS3.3.p1.3.m3.4.4.4.3.cmml"><msup id="S3.SS3.3.p1.3.m3.4.4.4.3.2" xref="S3.SS3.3.p1.3.m3.4.4.4.3.2.cmml"><mi id="S3.SS3.3.p1.3.m3.4.4.4.3.2.2" xref="S3.SS3.3.p1.3.m3.4.4.4.3.2.2.cmml">F</mi><mo id="S3.SS3.3.p1.3.m3.4.4.4.3.2.3" xref="S3.SS3.3.p1.3.m3.4.4.4.3.2.3.cmml">′</mo></msup><mo id="S3.SS3.3.p1.3.m3.4.4.4.3.1" xref="S3.SS3.3.p1.3.m3.4.4.4.3.1.cmml">⁢</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.4.3.3.2" xref="S3.SS3.3.p1.3.m3.4.4.4.3.cmml"><mo id="S3.SS3.3.p1.3.m3.4.4.4.3.3.2.1" stretchy="false" xref="S3.SS3.3.p1.3.m3.4.4.4.3.cmml">(</mo><mi id="S3.SS3.3.p1.3.m3.2.2" xref="S3.SS3.3.p1.3.m3.2.2.cmml">x</mi><mo id="S3.SS3.3.p1.3.m3.4.4.4.3.3.2.2" stretchy="false" xref="S3.SS3.3.p1.3.m3.4.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS3.3.p1.3.m3.4.4.3" xref="S3.SS3.3.p1.3.m3.4.4.3.cmml">=</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.2" xref="S3.SS3.3.p1.3.m3.4.4.2.cmml"><mrow id="S3.SS3.3.p1.3.m3.3.3.1.1" xref="S3.SS3.3.p1.3.m3.3.3.1.1.cmml"><msub id="S3.SS3.3.p1.3.m3.3.3.1.1.3" xref="S3.SS3.3.p1.3.m3.3.3.1.1.3.cmml"><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.3.2" xref="S3.SS3.3.p1.3.m3.3.3.1.1.3.2.cmml">c</mi><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.3.3" xref="S3.SS3.3.p1.3.m3.3.3.1.1.3.3.cmml">G</mi></msub><mo id="S3.SS3.3.p1.3.m3.3.3.1.1.2" xref="S3.SS3.3.p1.3.m3.3.3.1.1.2.cmml">⁢</mo><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.4" xref="S3.SS3.3.p1.3.m3.3.3.1.1.4.cmml">ϕ</mi><mo id="S3.SS3.3.p1.3.m3.3.3.1.1.2a" xref="S3.SS3.3.p1.3.m3.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.cmml"><msub id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2.cmml"><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2.2" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2.2.cmml">c</mi><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2.3" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.2.3.cmml">G</mi></msub><mo id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.1" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.3" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S3.SS3.3.p1.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.3.p1.3.m3.4.4.2.3" xref="S3.SS3.3.p1.3.m3.4.4.2.3.cmml">−</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.2.2" xref="S3.SS3.3.p1.3.m3.4.4.2.2.cmml"><msub id="S3.SS3.3.p1.3.m3.4.4.2.2.3" xref="S3.SS3.3.p1.3.m3.4.4.2.2.3.cmml"><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.3.2" xref="S3.SS3.3.p1.3.m3.4.4.2.2.3.2.cmml">c</mi><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.3.3" xref="S3.SS3.3.p1.3.m3.4.4.2.2.3.3.cmml">F</mi></msub><mo id="S3.SS3.3.p1.3.m3.4.4.2.2.2" xref="S3.SS3.3.p1.3.m3.4.4.2.2.2.cmml">⁢</mo><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.4" xref="S3.SS3.3.p1.3.m3.4.4.2.2.4.cmml">ϕ</mi><mo id="S3.SS3.3.p1.3.m3.4.4.2.2.2a" xref="S3.SS3.3.p1.3.m3.4.4.2.2.2.cmml">⁢</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.cmml"><mo id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.2" stretchy="false" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.cmml"><msub id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2.cmml"><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2.2" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2.2.cmml">c</mi><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2.3" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.2.3.cmml">F</mi></msub><mo id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.1" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.3" xref="S3.SS3.3.p1.3.m3.4.4.2.2.1.1.1.3.cmml">x</mi></mrow><mo 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= italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT italic_ϕ ( italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT italic_x ) - italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT italic_ϕ ( italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT italic_x )</annotation></semantics></math>, so <math alttext="G^{\prime}(0)-F^{\prime}(0)=(c_{G}-c_{F})\phi(0)<0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.4.m4.4"><semantics id="S3.SS3.3.p1.4.m4.4a"><mrow id="S3.SS3.3.p1.4.m4.4.4" xref="S3.SS3.3.p1.4.m4.4.4.cmml"><mrow id="S3.SS3.3.p1.4.m4.4.4.3" xref="S3.SS3.3.p1.4.m4.4.4.3.cmml"><mrow id="S3.SS3.3.p1.4.m4.4.4.3.2" xref="S3.SS3.3.p1.4.m4.4.4.3.2.cmml"><msup id="S3.SS3.3.p1.4.m4.4.4.3.2.2" xref="S3.SS3.3.p1.4.m4.4.4.3.2.2.cmml"><mi id="S3.SS3.3.p1.4.m4.4.4.3.2.2.2" xref="S3.SS3.3.p1.4.m4.4.4.3.2.2.2.cmml">G</mi><mo id="S3.SS3.3.p1.4.m4.4.4.3.2.2.3" xref="S3.SS3.3.p1.4.m4.4.4.3.2.2.3.cmml">′</mo></msup><mo id="S3.SS3.3.p1.4.m4.4.4.3.2.1" 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id="S3.SS3.3.p1.4.m4.4.4.1.1.1.1.3.2.cmml" xref="S3.SS3.3.p1.4.m4.4.4.1.1.1.1.3.2">𝑐</ci><ci id="S3.SS3.3.p1.4.m4.4.4.1.1.1.1.3.3.cmml" xref="S3.SS3.3.p1.4.m4.4.4.1.1.1.1.3.3">𝐹</ci></apply></apply><ci id="S3.SS3.3.p1.4.m4.4.4.1.3.cmml" xref="S3.SS3.3.p1.4.m4.4.4.1.3">italic-ϕ</ci><cn id="S3.SS3.3.p1.4.m4.3.3.cmml" type="integer" xref="S3.SS3.3.p1.4.m4.3.3">0</cn></apply></apply><apply id="S3.SS3.3.p1.4.m4.4.4c.cmml" xref="S3.SS3.3.p1.4.m4.4.4"><lt id="S3.SS3.3.p1.4.m4.4.4.5.cmml" xref="S3.SS3.3.p1.4.m4.4.4.5"></lt><share href="https://arxiv.org/html/2503.16280v1#S3.SS3.3.p1.4.m4.4.4.1.cmml" id="S3.SS3.3.p1.4.m4.4.4d.cmml" xref="S3.SS3.3.p1.4.m4.4.4"></share><cn id="S3.SS3.3.p1.4.m4.4.4.6.cmml" type="integer" xref="S3.SS3.3.p1.4.m4.4.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.4.m4.4c">G^{\prime}(0)-F^{\prime}(0)=(c_{G}-c_{F})\phi(0)<0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.4.m4.4d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) - italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = ( italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ) italic_ϕ ( 0 ) < 0</annotation></semantics></math> (since <math alttext="c_{G}<c_{F}" class="ltx_Math" display="inline" id="S3.SS3.3.p1.5.m5.1"><semantics id="S3.SS3.3.p1.5.m5.1a"><mrow id="S3.SS3.3.p1.5.m5.1.1" xref="S3.SS3.3.p1.5.m5.1.1.cmml"><msub id="S3.SS3.3.p1.5.m5.1.1.2" xref="S3.SS3.3.p1.5.m5.1.1.2.cmml"><mi id="S3.SS3.3.p1.5.m5.1.1.2.2" xref="S3.SS3.3.p1.5.m5.1.1.2.2.cmml">c</mi><mi id="S3.SS3.3.p1.5.m5.1.1.2.3" xref="S3.SS3.3.p1.5.m5.1.1.2.3.cmml">G</mi></msub><mo id="S3.SS3.3.p1.5.m5.1.1.1" xref="S3.SS3.3.p1.5.m5.1.1.1.cmml"><</mo><msub id="S3.SS3.3.p1.5.m5.1.1.3" xref="S3.SS3.3.p1.5.m5.1.1.3.cmml"><mi id="S3.SS3.3.p1.5.m5.1.1.3.2" xref="S3.SS3.3.p1.5.m5.1.1.3.2.cmml">c</mi><mi id="S3.SS3.3.p1.5.m5.1.1.3.3" xref="S3.SS3.3.p1.5.m5.1.1.3.3.cmml">F</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.5.m5.1b"><apply id="S3.SS3.3.p1.5.m5.1.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1"><lt id="S3.SS3.3.p1.5.m5.1.1.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1.1"></lt><apply id="S3.SS3.3.p1.5.m5.1.1.2.cmml" xref="S3.SS3.3.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.3.p1.5.m5.1.1.2.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS3.3.p1.5.m5.1.1.2.2.cmml" xref="S3.SS3.3.p1.5.m5.1.1.2.2">𝑐</ci><ci id="S3.SS3.3.p1.5.m5.1.1.2.3.cmml" xref="S3.SS3.3.p1.5.m5.1.1.2.3">𝐺</ci></apply><apply id="S3.SS3.3.p1.5.m5.1.1.3.cmml" xref="S3.SS3.3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.3.p1.5.m5.1.1.3.1.cmml" xref="S3.SS3.3.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS3.3.p1.5.m5.1.1.3.2.cmml" xref="S3.SS3.3.p1.5.m5.1.1.3.2">𝑐</ci><ci id="S3.SS3.3.p1.5.m5.1.1.3.3.cmml" xref="S3.SS3.3.p1.5.m5.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.5.m5.1c">c_{G}<c_{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.5.m5.1d">italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT < italic_c start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> by (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E13" title="In Proof. ‣ 3.3 Gaussian Model ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">13</span></a>)). Thus, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem4" title="Theorem 4. ‣ 3.2 Dynamics ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>, the equilibrium at <math alttext="0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.6.m6.1"><semantics id="S3.SS3.3.p1.6.m6.1a"><mn id="S3.SS3.3.p1.6.m6.1.1" xref="S3.SS3.3.p1.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.6.m6.1b"><cn id="S3.SS3.3.p1.6.m6.1.1.cmml" type="integer" xref="S3.SS3.3.p1.6.m6.1.1">0</cn></annotation-xml></semantics></math> is stable. Since <math alttext="\tau=0" class="ltx_Math" display="inline" id="S3.SS3.3.p1.7.m7.1"><semantics id="S3.SS3.3.p1.7.m7.1a"><mrow id="S3.SS3.3.p1.7.m7.1.1" xref="S3.SS3.3.p1.7.m7.1.1.cmml"><mi id="S3.SS3.3.p1.7.m7.1.1.2" xref="S3.SS3.3.p1.7.m7.1.1.2.cmml">τ</mi><mo id="S3.SS3.3.p1.7.m7.1.1.1" xref="S3.SS3.3.p1.7.m7.1.1.1.cmml">=</mo><mn id="S3.SS3.3.p1.7.m7.1.1.3" xref="S3.SS3.3.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.7.m7.1b"><apply id="S3.SS3.3.p1.7.m7.1.1.cmml" xref="S3.SS3.3.p1.7.m7.1.1"><eq id="S3.SS3.3.p1.7.m7.1.1.1.cmml" xref="S3.SS3.3.p1.7.m7.1.1.1"></eq><ci id="S3.SS3.3.p1.7.m7.1.1.2.cmml" xref="S3.SS3.3.p1.7.m7.1.1.2">𝜏</ci><cn id="S3.SS3.3.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.SS3.3.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.7.m7.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.7.m7.1d">italic_τ = 0</annotation></semantics></math> is stable, the equilibria <math alttext="\tau=\pm\infty" class="ltx_Math" display="inline" id="S3.SS3.3.p1.8.m8.1"><semantics id="S3.SS3.3.p1.8.m8.1a"><mrow id="S3.SS3.3.p1.8.m8.1.1" xref="S3.SS3.3.p1.8.m8.1.1.cmml"><mi id="S3.SS3.3.p1.8.m8.1.1.2" xref="S3.SS3.3.p1.8.m8.1.1.2.cmml">τ</mi><mo id="S3.SS3.3.p1.8.m8.1.1.1" xref="S3.SS3.3.p1.8.m8.1.1.1.cmml">=</mo><mrow id="S3.SS3.3.p1.8.m8.1.1.3" xref="S3.SS3.3.p1.8.m8.1.1.3.cmml"><mo id="S3.SS3.3.p1.8.m8.1.1.3a" xref="S3.SS3.3.p1.8.m8.1.1.3.cmml">±</mo><mi id="S3.SS3.3.p1.8.m8.1.1.3.2" mathvariant="normal" xref="S3.SS3.3.p1.8.m8.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.3.p1.8.m8.1b"><apply id="S3.SS3.3.p1.8.m8.1.1.cmml" xref="S3.SS3.3.p1.8.m8.1.1"><eq id="S3.SS3.3.p1.8.m8.1.1.1.cmml" xref="S3.SS3.3.p1.8.m8.1.1.1"></eq><ci id="S3.SS3.3.p1.8.m8.1.1.2.cmml" xref="S3.SS3.3.p1.8.m8.1.1.2">𝜏</ci><apply id="S3.SS3.3.p1.8.m8.1.1.3.cmml" xref="S3.SS3.3.p1.8.m8.1.1.3"><csymbol cd="latexml" id="S3.SS3.3.p1.8.m8.1.1.3.1.cmml" xref="S3.SS3.3.p1.8.m8.1.1.3">plus-or-minus</csymbol><infinity id="S3.SS3.3.p1.8.m8.1.1.3.2.cmml" xref="S3.SS3.3.p1.8.m8.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.3.p1.8.m8.1c">\tau=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.3.p1.8.m8.1d">italic_τ = ± ∞</annotation></semantics></math> are unstable by topological necessity. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.5">Since <math alttext="\tau=0" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mrow id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">τ</mi><mo id="S3.SS3.p3.1.m1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><eq id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1"></eq><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">𝜏</ci><cn id="S3.SS3.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_τ = 0</annotation></semantics></math> is the only stable equilibrium, it is in fact globally attracting: for any finite starting threshold <math alttext="\tau(0)" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mrow id="S3.SS3.p3.2.m2.1.2" xref="S3.SS3.p3.2.m2.1.2.cmml"><mi id="S3.SS3.p3.2.m2.1.2.2" xref="S3.SS3.p3.2.m2.1.2.2.cmml">τ</mi><mo id="S3.SS3.p3.2.m2.1.2.1" xref="S3.SS3.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS3.p3.2.m2.1.2.3.2" xref="S3.SS3.p3.2.m2.1.2.cmml"><mo id="S3.SS3.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS3.p3.2.m2.1.2.cmml">(</mo><mn id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml">0</mn><mo id="S3.SS3.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS3.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><apply id="S3.SS3.p3.2.m2.1.2.cmml" xref="S3.SS3.p3.2.m2.1.2"><times id="S3.SS3.p3.2.m2.1.2.1.cmml" xref="S3.SS3.p3.2.m2.1.2.1"></times><ci id="S3.SS3.p3.2.m2.1.2.2.cmml" xref="S3.SS3.p3.2.m2.1.2.2">𝜏</ci><cn id="S3.SS3.p3.2.m2.1.1.cmml" type="integer" xref="S3.SS3.p3.2.m2.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">\tau(0)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">italic_τ ( 0 )</annotation></semantics></math>, we have <math alttext="\lim_{t\to\infty}\tau(t)=0" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mrow id="S3.SS3.p3.3.m3.1.2" xref="S3.SS3.p3.3.m3.1.2.cmml"><mrow id="S3.SS3.p3.3.m3.1.2.2" xref="S3.SS3.p3.3.m3.1.2.2.cmml"><msub id="S3.SS3.p3.3.m3.1.2.2.1" xref="S3.SS3.p3.3.m3.1.2.2.1.cmml"><mo id="S3.SS3.p3.3.m3.1.2.2.1.2" xref="S3.SS3.p3.3.m3.1.2.2.1.2.cmml">lim</mo><mrow id="S3.SS3.p3.3.m3.1.2.2.1.3" xref="S3.SS3.p3.3.m3.1.2.2.1.3.cmml"><mi id="S3.SS3.p3.3.m3.1.2.2.1.3.2" xref="S3.SS3.p3.3.m3.1.2.2.1.3.2.cmml">t</mi><mo id="S3.SS3.p3.3.m3.1.2.2.1.3.1" stretchy="false" xref="S3.SS3.p3.3.m3.1.2.2.1.3.1.cmml">→</mo><mi id="S3.SS3.p3.3.m3.1.2.2.1.3.3" mathvariant="normal" xref="S3.SS3.p3.3.m3.1.2.2.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S3.SS3.p3.3.m3.1.2.2.2" xref="S3.SS3.p3.3.m3.1.2.2.2.cmml"><mi id="S3.SS3.p3.3.m3.1.2.2.2.2" xref="S3.SS3.p3.3.m3.1.2.2.2.2.cmml">τ</mi><mo id="S3.SS3.p3.3.m3.1.2.2.2.1" xref="S3.SS3.p3.3.m3.1.2.2.2.1.cmml">⁢</mo><mrow id="S3.SS3.p3.3.m3.1.2.2.2.3.2" xref="S3.SS3.p3.3.m3.1.2.2.2.cmml"><mo id="S3.SS3.p3.3.m3.1.2.2.2.3.2.1" stretchy="false" xref="S3.SS3.p3.3.m3.1.2.2.2.cmml">(</mo><mi id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml">t</mi><mo id="S3.SS3.p3.3.m3.1.2.2.2.3.2.2" stretchy="false" xref="S3.SS3.p3.3.m3.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS3.p3.3.m3.1.2.1" xref="S3.SS3.p3.3.m3.1.2.1.cmml">=</mo><mn id="S3.SS3.p3.3.m3.1.2.3" xref="S3.SS3.p3.3.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><apply id="S3.SS3.p3.3.m3.1.2.cmml" xref="S3.SS3.p3.3.m3.1.2"><eq id="S3.SS3.p3.3.m3.1.2.1.cmml" xref="S3.SS3.p3.3.m3.1.2.1"></eq><apply id="S3.SS3.p3.3.m3.1.2.2.cmml" xref="S3.SS3.p3.3.m3.1.2.2"><apply id="S3.SS3.p3.3.m3.1.2.2.1.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.1.2.2.1.1.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1">subscript</csymbol><limit id="S3.SS3.p3.3.m3.1.2.2.1.2.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1.2"></limit><apply id="S3.SS3.p3.3.m3.1.2.2.1.3.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1.3"><ci id="S3.SS3.p3.3.m3.1.2.2.1.3.1.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1.3.1">→</ci><ci id="S3.SS3.p3.3.m3.1.2.2.1.3.2.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1.3.2">𝑡</ci><infinity id="S3.SS3.p3.3.m3.1.2.2.1.3.3.cmml" xref="S3.SS3.p3.3.m3.1.2.2.1.3.3"></infinity></apply></apply><apply id="S3.SS3.p3.3.m3.1.2.2.2.cmml" xref="S3.SS3.p3.3.m3.1.2.2.2"><times id="S3.SS3.p3.3.m3.1.2.2.2.1.cmml" xref="S3.SS3.p3.3.m3.1.2.2.2.1"></times><ci id="S3.SS3.p3.3.m3.1.2.2.2.2.cmml" xref="S3.SS3.p3.3.m3.1.2.2.2.2">𝜏</ci><ci id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">𝑡</ci></apply></apply><cn id="S3.SS3.p3.3.m3.1.2.3.cmml" type="integer" xref="S3.SS3.p3.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">\lim_{t\to\infty}\tau(t)=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">roman_lim start_POSTSUBSCRIPT italic_t → ∞ end_POSTSUBSCRIPT italic_τ ( italic_t ) = 0</annotation></semantics></math>. In contrast to OA, this means that the behavior of DG in the Gaussian case is quite robust: unless the initial threshold starts exactly at an uninformative equilibrium it inevitably converges toward an informative one where <math alttext="H" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S3.SS3.p3.5.m5.1"><semantics id="S3.SS3.p3.5.m5.1a"><mi id="S3.SS3.p3.5.m5.1.1" xref="S3.SS3.p3.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.5.m5.1b"><ci id="S3.SS3.p3.5.m5.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.5.m5.1d">italic_L</annotation></semantics></math> each get reported half the time. In this sense, DG behaves in a way one might hope based on its traditional analysis: it discourages uninformative equilibria.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.3">While robust, this equilibrium is also inflexible: the mechanism designer cannot specify a desired threshold, and must settle for <math alttext="\tau=0" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mrow id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">τ</mi><mo id="S3.SS3.p4.1.m1.1.1.1" xref="S3.SS3.p4.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS3.p4.1.m1.1.1.3" xref="S3.SS3.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><eq id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1.1"></eq><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">𝜏</ci><cn id="S3.SS3.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.SS3.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_τ = 0</annotation></semantics></math>. Thus the semantics of <math alttext="H" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mi id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><ci id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><mi id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><ci id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">italic_L</annotation></semantics></math> relative to the underlying signal are predetermined, as half of the tasks must be classified each way. In peer grading, for example, half the assignments would be labeled unsatisfactory, which might not be the desired outcome.<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>See § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S7" title="7 Discussion ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a> for futher discussion.</span></span></span> We expect this general phenomenon, that there is a unique equilibrium threshold for DG which is determined by the underlying signal distribution, extends beyond the Gaussian case; see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6" title="6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Determinant-based Mutual Information (DMI) Mechanism</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">The Determinant-based Mutual Information (DMI) mechanism was introduced by <cite class="ltx_cite ltx_citemacro_citet">Kong [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib10" title="">2024</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib9" title="">2020</a>]</cite>. It is a minimal multi-task mechanism with several strong guarantees, such as being dominantly truthful under consistent strategies (see below). In the case of binary reports, the simplest version of the DMI mechanism collects 4 reports from a pair of agents, and computes a score based on a measure of their mutual information. To construct an unbiased estimator for this mutual information measure, the pairs of reports are split into two groups, and the determinants of the count matrices associated to each group are multiplied together.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.4">Formally, the mechanism is given as follows. Define <math alttext="\mathbf{1}_{H}=(1,0)" class="ltx_Math" display="inline" id="S4.p2.1.m1.2"><semantics id="S4.p2.1.m1.2a"><mrow id="S4.p2.1.m1.2.3" xref="S4.p2.1.m1.2.3.cmml"><msub id="S4.p2.1.m1.2.3.2" xref="S4.p2.1.m1.2.3.2.cmml"><mn id="S4.p2.1.m1.2.3.2.2" xref="S4.p2.1.m1.2.3.2.2.cmml">𝟏</mn><mi id="S4.p2.1.m1.2.3.2.3" xref="S4.p2.1.m1.2.3.2.3.cmml">H</mi></msub><mo id="S4.p2.1.m1.2.3.1" xref="S4.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S4.p2.1.m1.2.3.3.2" xref="S4.p2.1.m1.2.3.3.1.cmml"><mo id="S4.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.p2.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">1</mn><mo id="S4.p2.1.m1.2.3.3.2.2" xref="S4.p2.1.m1.2.3.3.1.cmml">,</mo><mn id="S4.p2.1.m1.2.2" xref="S4.p2.1.m1.2.2.cmml">0</mn><mo id="S4.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.2b"><apply id="S4.p2.1.m1.2.3.cmml" xref="S4.p2.1.m1.2.3"><eq id="S4.p2.1.m1.2.3.1.cmml" xref="S4.p2.1.m1.2.3.1"></eq><apply id="S4.p2.1.m1.2.3.2.cmml" xref="S4.p2.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.3.2.1.cmml" xref="S4.p2.1.m1.2.3.2">subscript</csymbol><cn id="S4.p2.1.m1.2.3.2.2.cmml" type="integer" xref="S4.p2.1.m1.2.3.2.2">1</cn><ci id="S4.p2.1.m1.2.3.2.3.cmml" xref="S4.p2.1.m1.2.3.2.3">𝐻</ci></apply><interval closure="open" id="S4.p2.1.m1.2.3.3.1.cmml" xref="S4.p2.1.m1.2.3.3.2"><cn id="S4.p2.1.m1.1.1.cmml" type="integer" xref="S4.p2.1.m1.1.1">1</cn><cn id="S4.p2.1.m1.2.2.cmml" type="integer" xref="S4.p2.1.m1.2.2">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.2c">\mathbf{1}_{H}=(1,0)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.2d">bold_1 start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = ( 1 , 0 )</annotation></semantics></math> and <math alttext="\mathbf{1}_{L}=(0,1)" class="ltx_Math" display="inline" id="S4.p2.2.m2.2"><semantics id="S4.p2.2.m2.2a"><mrow id="S4.p2.2.m2.2.3" xref="S4.p2.2.m2.2.3.cmml"><msub id="S4.p2.2.m2.2.3.2" xref="S4.p2.2.m2.2.3.2.cmml"><mn id="S4.p2.2.m2.2.3.2.2" xref="S4.p2.2.m2.2.3.2.2.cmml">𝟏</mn><mi id="S4.p2.2.m2.2.3.2.3" xref="S4.p2.2.m2.2.3.2.3.cmml">L</mi></msub><mo id="S4.p2.2.m2.2.3.1" xref="S4.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.p2.2.m2.2.3.3.2" xref="S4.p2.2.m2.2.3.3.1.cmml"><mo id="S4.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S4.p2.2.m2.2.3.3.1.cmml">(</mo><mn id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">0</mn><mo id="S4.p2.2.m2.2.3.3.2.2" xref="S4.p2.2.m2.2.3.3.1.cmml">,</mo><mn id="S4.p2.2.m2.2.2" xref="S4.p2.2.m2.2.2.cmml">1</mn><mo id="S4.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S4.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.2b"><apply id="S4.p2.2.m2.2.3.cmml" xref="S4.p2.2.m2.2.3"><eq id="S4.p2.2.m2.2.3.1.cmml" xref="S4.p2.2.m2.2.3.1"></eq><apply id="S4.p2.2.m2.2.3.2.cmml" xref="S4.p2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.p2.2.m2.2.3.2.1.cmml" xref="S4.p2.2.m2.2.3.2">subscript</csymbol><cn id="S4.p2.2.m2.2.3.2.2.cmml" type="integer" xref="S4.p2.2.m2.2.3.2.2">1</cn><ci id="S4.p2.2.m2.2.3.2.3.cmml" xref="S4.p2.2.m2.2.3.2.3">𝐿</ci></apply><interval closure="open" id="S4.p2.2.m2.2.3.3.1.cmml" xref="S4.p2.2.m2.2.3.3.2"><cn id="S4.p2.2.m2.1.1.cmml" type="integer" xref="S4.p2.2.m2.1.1">0</cn><cn id="S4.p2.2.m2.2.2.cmml" type="integer" xref="S4.p2.2.m2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.2c">\mathbf{1}_{L}=(0,1)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.2d">bold_1 start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = ( 0 , 1 )</annotation></semantics></math> to be indicator vectors in <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><msup id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml"><mi id="S4.p2.3.m3.1.1.2" xref="S4.p2.3.m3.1.1.2.cmml">ℝ</mi><mn id="S4.p2.3.m3.1.1.3" xref="S4.p2.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><apply id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p2.3.m3.1.1.1.cmml" xref="S4.p2.3.m3.1.1">superscript</csymbol><ci id="S4.p2.3.m3.1.1.2.cmml" xref="S4.p2.3.m3.1.1.2">ℝ</ci><cn id="S4.p2.3.m3.1.1.3.cmml" type="integer" xref="S4.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="M_{\textrm{DMI}}:\{L,H\}^{4}\times\{L,H\}^{4}\to\mathbb{R}" class="ltx_Math" display="inline" id="S4.p2.4.m4.4"><semantics id="S4.p2.4.m4.4a"><mrow id="S4.p2.4.m4.4.5" xref="S4.p2.4.m4.4.5.cmml"><msub id="S4.p2.4.m4.4.5.2" xref="S4.p2.4.m4.4.5.2.cmml"><mi id="S4.p2.4.m4.4.5.2.2" xref="S4.p2.4.m4.4.5.2.2.cmml">M</mi><mtext id="S4.p2.4.m4.4.5.2.3" xref="S4.p2.4.m4.4.5.2.3a.cmml">DMI</mtext></msub><mo id="S4.p2.4.m4.4.5.1" lspace="0.278em" rspace="0.278em" xref="S4.p2.4.m4.4.5.1.cmml">:</mo><mrow id="S4.p2.4.m4.4.5.3" xref="S4.p2.4.m4.4.5.3.cmml"><mrow id="S4.p2.4.m4.4.5.3.2" xref="S4.p2.4.m4.4.5.3.2.cmml"><msup id="S4.p2.4.m4.4.5.3.2.2" xref="S4.p2.4.m4.4.5.3.2.2.cmml"><mrow id="S4.p2.4.m4.4.5.3.2.2.2.2" xref="S4.p2.4.m4.4.5.3.2.2.2.1.cmml"><mo id="S4.p2.4.m4.4.5.3.2.2.2.2.1" stretchy="false" xref="S4.p2.4.m4.4.5.3.2.2.2.1.cmml">{</mo><mi id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml">L</mi><mo id="S4.p2.4.m4.4.5.3.2.2.2.2.2" xref="S4.p2.4.m4.4.5.3.2.2.2.1.cmml">,</mo><mi id="S4.p2.4.m4.2.2" xref="S4.p2.4.m4.2.2.cmml">H</mi><mo id="S4.p2.4.m4.4.5.3.2.2.2.2.3" rspace="0.055em" stretchy="false" xref="S4.p2.4.m4.4.5.3.2.2.2.1.cmml">}</mo></mrow><mn id="S4.p2.4.m4.4.5.3.2.2.3" xref="S4.p2.4.m4.4.5.3.2.2.3.cmml">4</mn></msup><mo id="S4.p2.4.m4.4.5.3.2.1" rspace="0.222em" xref="S4.p2.4.m4.4.5.3.2.1.cmml">×</mo><msup id="S4.p2.4.m4.4.5.3.2.3" xref="S4.p2.4.m4.4.5.3.2.3.cmml"><mrow id="S4.p2.4.m4.4.5.3.2.3.2.2" xref="S4.p2.4.m4.4.5.3.2.3.2.1.cmml"><mo id="S4.p2.4.m4.4.5.3.2.3.2.2.1" stretchy="false" xref="S4.p2.4.m4.4.5.3.2.3.2.1.cmml">{</mo><mi id="S4.p2.4.m4.3.3" xref="S4.p2.4.m4.3.3.cmml">L</mi><mo id="S4.p2.4.m4.4.5.3.2.3.2.2.2" xref="S4.p2.4.m4.4.5.3.2.3.2.1.cmml">,</mo><mi id="S4.p2.4.m4.4.4" xref="S4.p2.4.m4.4.4.cmml">H</mi><mo id="S4.p2.4.m4.4.5.3.2.3.2.2.3" stretchy="false" xref="S4.p2.4.m4.4.5.3.2.3.2.1.cmml">}</mo></mrow><mn id="S4.p2.4.m4.4.5.3.2.3.3" xref="S4.p2.4.m4.4.5.3.2.3.3.cmml">4</mn></msup></mrow><mo id="S4.p2.4.m4.4.5.3.1" stretchy="false" xref="S4.p2.4.m4.4.5.3.1.cmml">→</mo><mi id="S4.p2.4.m4.4.5.3.3" xref="S4.p2.4.m4.4.5.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.4b"><apply id="S4.p2.4.m4.4.5.cmml" xref="S4.p2.4.m4.4.5"><ci id="S4.p2.4.m4.4.5.1.cmml" xref="S4.p2.4.m4.4.5.1">:</ci><apply id="S4.p2.4.m4.4.5.2.cmml" xref="S4.p2.4.m4.4.5.2"><csymbol cd="ambiguous" id="S4.p2.4.m4.4.5.2.1.cmml" xref="S4.p2.4.m4.4.5.2">subscript</csymbol><ci id="S4.p2.4.m4.4.5.2.2.cmml" xref="S4.p2.4.m4.4.5.2.2">𝑀</ci><ci id="S4.p2.4.m4.4.5.2.3a.cmml" xref="S4.p2.4.m4.4.5.2.3"><mtext id="S4.p2.4.m4.4.5.2.3.cmml" mathsize="70%" xref="S4.p2.4.m4.4.5.2.3">DMI</mtext></ci></apply><apply id="S4.p2.4.m4.4.5.3.cmml" xref="S4.p2.4.m4.4.5.3"><ci id="S4.p2.4.m4.4.5.3.1.cmml" xref="S4.p2.4.m4.4.5.3.1">→</ci><apply id="S4.p2.4.m4.4.5.3.2.cmml" xref="S4.p2.4.m4.4.5.3.2"><times id="S4.p2.4.m4.4.5.3.2.1.cmml" xref="S4.p2.4.m4.4.5.3.2.1"></times><apply id="S4.p2.4.m4.4.5.3.2.2.cmml" xref="S4.p2.4.m4.4.5.3.2.2"><csymbol cd="ambiguous" id="S4.p2.4.m4.4.5.3.2.2.1.cmml" xref="S4.p2.4.m4.4.5.3.2.2">superscript</csymbol><set id="S4.p2.4.m4.4.5.3.2.2.2.1.cmml" xref="S4.p2.4.m4.4.5.3.2.2.2.2"><ci id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1">𝐿</ci><ci id="S4.p2.4.m4.2.2.cmml" xref="S4.p2.4.m4.2.2">𝐻</ci></set><cn id="S4.p2.4.m4.4.5.3.2.2.3.cmml" type="integer" xref="S4.p2.4.m4.4.5.3.2.2.3">4</cn></apply><apply id="S4.p2.4.m4.4.5.3.2.3.cmml" xref="S4.p2.4.m4.4.5.3.2.3"><csymbol cd="ambiguous" id="S4.p2.4.m4.4.5.3.2.3.1.cmml" xref="S4.p2.4.m4.4.5.3.2.3">superscript</csymbol><set id="S4.p2.4.m4.4.5.3.2.3.2.1.cmml" xref="S4.p2.4.m4.4.5.3.2.3.2.2"><ci id="S4.p2.4.m4.3.3.cmml" xref="S4.p2.4.m4.3.3">𝐿</ci><ci id="S4.p2.4.m4.4.4.cmml" xref="S4.p2.4.m4.4.4">𝐻</ci></set><cn id="S4.p2.4.m4.4.5.3.2.3.3.cmml" type="integer" xref="S4.p2.4.m4.4.5.3.2.3.3">4</cn></apply></apply><ci id="S4.p2.4.m4.4.5.3.3.cmml" xref="S4.p2.4.m4.4.5.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.4c">M_{\textrm{DMI}}:\{L,H\}^{4}\times\{L,H\}^{4}\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.4d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT : { italic_L , italic_H } start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT × { italic_L , italic_H } start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT → blackboard_R</annotation></semantics></math> is the function</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx5"> <tbody id="S4.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle M_{\textrm{DMI}}(r_{1},\ldots,r_{4},r_{1}^{\prime},\ldots,r_{4}^% {\prime})=\det M_{12}\det M_{34}~{}," class="ltx_Math" display="inline" id="S4.Ex11.m1.3"><semantics id="S4.Ex11.m1.3a"><mrow id="S4.Ex11.m1.3.3.1" xref="S4.Ex11.m1.3.3.1.1.cmml"><mrow id="S4.Ex11.m1.3.3.1.1" xref="S4.Ex11.m1.3.3.1.1.cmml"><mrow id="S4.Ex11.m1.3.3.1.1.4" xref="S4.Ex11.m1.3.3.1.1.4.cmml"><msub id="S4.Ex11.m1.3.3.1.1.4.6" xref="S4.Ex11.m1.3.3.1.1.4.6.cmml"><mi id="S4.Ex11.m1.3.3.1.1.4.6.2" xref="S4.Ex11.m1.3.3.1.1.4.6.2.cmml">M</mi><mtext id="S4.Ex11.m1.3.3.1.1.4.6.3" xref="S4.Ex11.m1.3.3.1.1.4.6.3a.cmml">DMI</mtext></msub><mo id="S4.Ex11.m1.3.3.1.1.4.5" xref="S4.Ex11.m1.3.3.1.1.4.5.cmml">⁢</mo><mrow id="S4.Ex11.m1.3.3.1.1.4.4.4" xref="S4.Ex11.m1.3.3.1.1.4.4.5.cmml"><mo id="S4.Ex11.m1.3.3.1.1.4.4.4.5" stretchy="false" xref="S4.Ex11.m1.3.3.1.1.4.4.5.cmml">(</mo><msub id="S4.Ex11.m1.3.3.1.1.1.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.Ex11.m1.3.3.1.1.1.1.1.1.2" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.2.cmml">r</mi><mn id="S4.Ex11.m1.3.3.1.1.1.1.1.1.3" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex11.m1.3.3.1.1.4.4.4.6" xref="S4.Ex11.m1.3.3.1.1.4.4.5.cmml">,</mo><mi 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xref="S4.Ex11.m1.3.3.1.1.4.4.4.4.3">′</ci></apply></vector></apply><apply id="S4.Ex11.m1.3.3.1.1.6.cmml" xref="S4.Ex11.m1.3.3.1.1.6"><determinant id="S4.Ex11.m1.3.3.1.1.6.1.cmml" xref="S4.Ex11.m1.3.3.1.1.6.1"></determinant><apply id="S4.Ex11.m1.3.3.1.1.6.2.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2"><times id="S4.Ex11.m1.3.3.1.1.6.2.1.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.1"></times><apply id="S4.Ex11.m1.3.3.1.1.6.2.2.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.3.3.1.1.6.2.2.1.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.2">subscript</csymbol><ci id="S4.Ex11.m1.3.3.1.1.6.2.2.2.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.2.2">𝑀</ci><cn id="S4.Ex11.m1.3.3.1.1.6.2.2.3.cmml" type="integer" xref="S4.Ex11.m1.3.3.1.1.6.2.2.3">12</cn></apply><apply id="S4.Ex11.m1.3.3.1.1.6.2.3.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.3"><determinant id="S4.Ex11.m1.3.3.1.1.6.2.3.1.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.3.1"></determinant><apply id="S4.Ex11.m1.3.3.1.1.6.2.3.2.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.3.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.3.3.1.1.6.2.3.2.1.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.3.2">subscript</csymbol><ci id="S4.Ex11.m1.3.3.1.1.6.2.3.2.2.cmml" xref="S4.Ex11.m1.3.3.1.1.6.2.3.2.2">𝑀</ci><cn id="S4.Ex11.m1.3.3.1.1.6.2.3.2.3.cmml" type="integer" xref="S4.Ex11.m1.3.3.1.1.6.2.3.2.3">34</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.3c">\displaystyle M_{\textrm{DMI}}(r_{1},\ldots,r_{4},r_{1}^{\prime},\ldots,r_{4}^% {\prime})=\det M_{12}\det M_{34}~{},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.3d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT roman_det italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.7">where <math alttext="M_{ij}=\mathbf{1}_{r_{i}}\mathbf{1}_{r_{i}^{\prime}}^{T}+\mathbf{1}_{r_{j}}% \mathbf{1}_{r_{j}^{\prime}}^{T}" class="ltx_Math" display="inline" id="S4.p2.5.m1.1"><semantics id="S4.p2.5.m1.1a"><mrow id="S4.p2.5.m1.1.1" xref="S4.p2.5.m1.1.1.cmml"><msub id="S4.p2.5.m1.1.1.2" xref="S4.p2.5.m1.1.1.2.cmml"><mi id="S4.p2.5.m1.1.1.2.2" xref="S4.p2.5.m1.1.1.2.2.cmml">M</mi><mrow id="S4.p2.5.m1.1.1.2.3" xref="S4.p2.5.m1.1.1.2.3.cmml"><mi id="S4.p2.5.m1.1.1.2.3.2" xref="S4.p2.5.m1.1.1.2.3.2.cmml">i</mi><mo id="S4.p2.5.m1.1.1.2.3.1" xref="S4.p2.5.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m1.1.1.2.3.3" xref="S4.p2.5.m1.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S4.p2.5.m1.1.1.1" xref="S4.p2.5.m1.1.1.1.cmml">=</mo><mrow id="S4.p2.5.m1.1.1.3" xref="S4.p2.5.m1.1.1.3.cmml"><mrow id="S4.p2.5.m1.1.1.3.2" xref="S4.p2.5.m1.1.1.3.2.cmml"><msub id="S4.p2.5.m1.1.1.3.2.2" xref="S4.p2.5.m1.1.1.3.2.2.cmml"><mn id="S4.p2.5.m1.1.1.3.2.2.2" xref="S4.p2.5.m1.1.1.3.2.2.2.cmml">𝟏</mn><msub id="S4.p2.5.m1.1.1.3.2.2.3" xref="S4.p2.5.m1.1.1.3.2.2.3.cmml"><mi id="S4.p2.5.m1.1.1.3.2.2.3.2" xref="S4.p2.5.m1.1.1.3.2.2.3.2.cmml">r</mi><mi id="S4.p2.5.m1.1.1.3.2.2.3.3" xref="S4.p2.5.m1.1.1.3.2.2.3.3.cmml">i</mi></msub></msub><mo id="S4.p2.5.m1.1.1.3.2.1" xref="S4.p2.5.m1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S4.p2.5.m1.1.1.3.2.3" xref="S4.p2.5.m1.1.1.3.2.3.cmml"><mn id="S4.p2.5.m1.1.1.3.2.3.2.2" xref="S4.p2.5.m1.1.1.3.2.3.2.2.cmml">𝟏</mn><msubsup id="S4.p2.5.m1.1.1.3.2.3.2.3" xref="S4.p2.5.m1.1.1.3.2.3.2.3.cmml"><mi id="S4.p2.5.m1.1.1.3.2.3.2.3.2.2" xref="S4.p2.5.m1.1.1.3.2.3.2.3.2.2.cmml">r</mi><mi id="S4.p2.5.m1.1.1.3.2.3.2.3.2.3" xref="S4.p2.5.m1.1.1.3.2.3.2.3.2.3.cmml">i</mi><mo 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xref="S4.p2.5.m1.1.1.3.3.3.2.3.3">′</ci></apply></apply><ci id="S4.p2.5.m1.1.1.3.3.3.3.cmml" xref="S4.p2.5.m1.1.1.3.3.3.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m1.1c">M_{ij}=\mathbf{1}_{r_{i}}\mathbf{1}_{r_{i}^{\prime}}^{T}+\mathbf{1}_{r_{j}}% \mathbf{1}_{r_{j}^{\prime}}^{T}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m1.1d">italic_M start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = bold_1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT + bold_1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> is the count matrix for tasks <math alttext="i" class="ltx_Math" display="inline" id="S4.p2.6.m2.1"><semantics id="S4.p2.6.m2.1a"><mi id="S4.p2.6.m2.1.1" xref="S4.p2.6.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.p2.6.m2.1b"><ci id="S4.p2.6.m2.1.1.cmml" xref="S4.p2.6.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m2.1d">italic_i</annotation></semantics></math> and <math alttext="j" class="ltx_Math" display="inline" id="S4.p2.7.m3.1"><semantics id="S4.p2.7.m3.1a"><mi id="S4.p2.7.m3.1.1" xref="S4.p2.7.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.p2.7.m3.1b"><ci id="S4.p2.7.m3.1.1.cmml" xref="S4.p2.7.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m3.1d">italic_j</annotation></semantics></math>.</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>Equilibrium Characterization</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.2">Unlike DG, where we could compute the payment for a single task in isolation (albeit requiring multiple tasks from a different agent), DMI is inherently a multi-task mechanism. As in the original analysis of DMI, here we restrict to <em class="ltx_emph ltx_font_italic" id="S4.SS1.p1.2.1">consistent</em> strategies, where the same <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_σ</annotation></semantics></math> is applied to each <math alttext="X_{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><msub 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">X</mi><mi id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml">i</mi></msub><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">subscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝑋</ci><ci id="S4.SS1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. (See below for behavior beyond this case.) Even under this restriction, threshold equilibria are much more complex in DMI given the interactions between four different signals, making it unclear how to reason directly from the definition of a threshold equilibrium. Instead, we restrict all agents to playing threshold strategies. 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start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.4">be the (ex-ante) expected utility for playing threshold strategy <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m1.1"><semantics id="S4.SS1.p1.3.m1.1a"><msub id="S4.SS1.p1.3.m1.1.1" xref="S4.SS1.p1.3.m1.1.1.cmml"><mi id="S4.SS1.p1.3.m1.1.1.2" xref="S4.SS1.p1.3.m1.1.1.2.cmml">σ</mi><mi id="S4.SS1.p1.3.m1.1.1.3" xref="S4.SS1.p1.3.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m1.1b"><apply id="S4.SS1.p1.3.m1.1.1.cmml" xref="S4.SS1.p1.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m1.1.1.1.cmml" xref="S4.SS1.p1.3.m1.1.1">subscript</csymbol><ci id="S4.SS1.p1.3.m1.1.1.2.cmml" xref="S4.SS1.p1.3.m1.1.1.2">𝜎</ci><ci id="S4.SS1.p1.3.m1.1.1.3.cmml" xref="S4.SS1.p1.3.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m1.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m1.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> given the other agent using <math alttext="\sigma_{j}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m2.1"><semantics id="S4.SS1.p1.4.m2.1a"><msub id="S4.SS1.p1.4.m2.1.1" xref="S4.SS1.p1.4.m2.1.1.cmml"><mi id="S4.SS1.p1.4.m2.1.1.2" xref="S4.SS1.p1.4.m2.1.1.2.cmml">σ</mi><mi id="S4.SS1.p1.4.m2.1.1.3" xref="S4.SS1.p1.4.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m2.1b"><apply id="S4.SS1.p1.4.m2.1.1.cmml" xref="S4.SS1.p1.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m2.1.1.1.cmml" xref="S4.SS1.p1.4.m2.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m2.1.1.2.cmml" xref="S4.SS1.p1.4.m2.1.1.2">𝜎</ci><ci id="S4.SS1.p1.4.m2.1.1.3.cmml" xref="S4.SS1.p1.4.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m2.1c">\sigma_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m2.1d">italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmdefinition2.1.1.1">Definition 2</span></span><span class="ltx_text ltx_font_bold" id="Thmdefinition2.2.2">.</span> </h6> <div class="ltx_para" id="Thmdefinition2.p1"> <p class="ltx_p" id="Thmdefinition2.p1.3">A threshold strategy <math alttext="\sigma:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="Thmdefinition2.p1.1.m1.1"><semantics id="Thmdefinition2.p1.1.m1.1a"><mrow id="Thmdefinition2.p1.1.m1.1.1" xref="Thmdefinition2.p1.1.m1.1.1.cmml"><mi id="Thmdefinition2.p1.1.m1.1.1.2" xref="Thmdefinition2.p1.1.m1.1.1.2.cmml">σ</mi><mo id="Thmdefinition2.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmdefinition2.p1.1.m1.1.1.1.cmml">:</mo><mrow id="Thmdefinition2.p1.1.m1.1.1.3" xref="Thmdefinition2.p1.1.m1.1.1.3.cmml"><mi id="Thmdefinition2.p1.1.m1.1.1.3.2" xref="Thmdefinition2.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="Thmdefinition2.p1.1.m1.1.1.3.1" stretchy="false" xref="Thmdefinition2.p1.1.m1.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition2.p1.1.m1.1.1.3.3" xref="Thmdefinition2.p1.1.m1.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.1.m1.1b"><apply id="Thmdefinition2.p1.1.m1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1"><ci id="Thmdefinition2.p1.1.m1.1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1.1">:</ci><ci id="Thmdefinition2.p1.1.m1.1.1.2.cmml" xref="Thmdefinition2.p1.1.m1.1.1.2">𝜎</ci><apply id="Thmdefinition2.p1.1.m1.1.1.3.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3"><ci id="Thmdefinition2.p1.1.m1.1.1.3.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3.1">→</ci><ci id="Thmdefinition2.p1.1.m1.1.1.3.2.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3.2">ℝ</ci><ci id="Thmdefinition2.p1.1.m1.1.1.3.3.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.1.m1.1c">\sigma:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.1.m1.1d">italic_σ : blackboard_R → caligraphic_R</annotation></semantics></math> is a <span class="ltx_text ltx_font_italic" id="Thmdefinition2.p1.3.1">equilibrium restricted to threshold strategies</span> under DMI if for all threshold strategies <math alttext="\hat{\sigma}:\mathbb{R}\to\mathcal{R}" class="ltx_Math" display="inline" id="Thmdefinition2.p1.2.m2.1"><semantics id="Thmdefinition2.p1.2.m2.1a"><mrow id="Thmdefinition2.p1.2.m2.1.1" xref="Thmdefinition2.p1.2.m2.1.1.cmml"><mover accent="true" id="Thmdefinition2.p1.2.m2.1.1.2" xref="Thmdefinition2.p1.2.m2.1.1.2.cmml"><mi id="Thmdefinition2.p1.2.m2.1.1.2.2" xref="Thmdefinition2.p1.2.m2.1.1.2.2.cmml">σ</mi><mo id="Thmdefinition2.p1.2.m2.1.1.2.1" xref="Thmdefinition2.p1.2.m2.1.1.2.1.cmml">^</mo></mover><mo id="Thmdefinition2.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="Thmdefinition2.p1.2.m2.1.1.1.cmml">:</mo><mrow id="Thmdefinition2.p1.2.m2.1.1.3" xref="Thmdefinition2.p1.2.m2.1.1.3.cmml"><mi id="Thmdefinition2.p1.2.m2.1.1.3.2" xref="Thmdefinition2.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="Thmdefinition2.p1.2.m2.1.1.3.1" stretchy="false" xref="Thmdefinition2.p1.2.m2.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition2.p1.2.m2.1.1.3.3" xref="Thmdefinition2.p1.2.m2.1.1.3.3.cmml">ℛ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.2.m2.1b"><apply id="Thmdefinition2.p1.2.m2.1.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1"><ci id="Thmdefinition2.p1.2.m2.1.1.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1.1">:</ci><apply id="Thmdefinition2.p1.2.m2.1.1.2.cmml" xref="Thmdefinition2.p1.2.m2.1.1.2"><ci id="Thmdefinition2.p1.2.m2.1.1.2.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1.2.1">^</ci><ci id="Thmdefinition2.p1.2.m2.1.1.2.2.cmml" xref="Thmdefinition2.p1.2.m2.1.1.2.2">𝜎</ci></apply><apply id="Thmdefinition2.p1.2.m2.1.1.3.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3"><ci id="Thmdefinition2.p1.2.m2.1.1.3.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3.1">→</ci><ci id="Thmdefinition2.p1.2.m2.1.1.3.2.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3.2">ℝ</ci><ci id="Thmdefinition2.p1.2.m2.1.1.3.3.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3.3">ℛ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.2.m2.1c">\hat{\sigma}:\mathbb{R}\to\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.2.m2.1d">over^ start_ARG italic_σ end_ARG : blackboard_R → caligraphic_R</annotation></semantics></math>, <math alttext="U_{i}(\sigma,\sigma)\geq U_{i}(\hat{\sigma},\sigma)" class="ltx_Math" display="inline" id="Thmdefinition2.p1.3.m3.4"><semantics id="Thmdefinition2.p1.3.m3.4a"><mrow id="Thmdefinition2.p1.3.m3.4.5" xref="Thmdefinition2.p1.3.m3.4.5.cmml"><mrow id="Thmdefinition2.p1.3.m3.4.5.2" xref="Thmdefinition2.p1.3.m3.4.5.2.cmml"><msub id="Thmdefinition2.p1.3.m3.4.5.2.2" xref="Thmdefinition2.p1.3.m3.4.5.2.2.cmml"><mi id="Thmdefinition2.p1.3.m3.4.5.2.2.2" xref="Thmdefinition2.p1.3.m3.4.5.2.2.2.cmml">U</mi><mi id="Thmdefinition2.p1.3.m3.4.5.2.2.3" xref="Thmdefinition2.p1.3.m3.4.5.2.2.3.cmml">i</mi></msub><mo id="Thmdefinition2.p1.3.m3.4.5.2.1" xref="Thmdefinition2.p1.3.m3.4.5.2.1.cmml">⁢</mo><mrow id="Thmdefinition2.p1.3.m3.4.5.2.3.2" xref="Thmdefinition2.p1.3.m3.4.5.2.3.1.cmml"><mo id="Thmdefinition2.p1.3.m3.4.5.2.3.2.1" stretchy="false" xref="Thmdefinition2.p1.3.m3.4.5.2.3.1.cmml">(</mo><mi id="Thmdefinition2.p1.3.m3.1.1" xref="Thmdefinition2.p1.3.m3.1.1.cmml">σ</mi><mo id="Thmdefinition2.p1.3.m3.4.5.2.3.2.2" xref="Thmdefinition2.p1.3.m3.4.5.2.3.1.cmml">,</mo><mi id="Thmdefinition2.p1.3.m3.2.2" xref="Thmdefinition2.p1.3.m3.2.2.cmml">σ</mi><mo id="Thmdefinition2.p1.3.m3.4.5.2.3.2.3" stretchy="false" xref="Thmdefinition2.p1.3.m3.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmdefinition2.p1.3.m3.4.5.1" xref="Thmdefinition2.p1.3.m3.4.5.1.cmml">≥</mo><mrow id="Thmdefinition2.p1.3.m3.4.5.3" xref="Thmdefinition2.p1.3.m3.4.5.3.cmml"><msub id="Thmdefinition2.p1.3.m3.4.5.3.2" xref="Thmdefinition2.p1.3.m3.4.5.3.2.cmml"><mi id="Thmdefinition2.p1.3.m3.4.5.3.2.2" xref="Thmdefinition2.p1.3.m3.4.5.3.2.2.cmml">U</mi><mi id="Thmdefinition2.p1.3.m3.4.5.3.2.3" xref="Thmdefinition2.p1.3.m3.4.5.3.2.3.cmml">i</mi></msub><mo id="Thmdefinition2.p1.3.m3.4.5.3.1" xref="Thmdefinition2.p1.3.m3.4.5.3.1.cmml">⁢</mo><mrow id="Thmdefinition2.p1.3.m3.4.5.3.3.2" xref="Thmdefinition2.p1.3.m3.4.5.3.3.1.cmml"><mo id="Thmdefinition2.p1.3.m3.4.5.3.3.2.1" stretchy="false" xref="Thmdefinition2.p1.3.m3.4.5.3.3.1.cmml">(</mo><mover accent="true" id="Thmdefinition2.p1.3.m3.3.3" xref="Thmdefinition2.p1.3.m3.3.3.cmml"><mi id="Thmdefinition2.p1.3.m3.3.3.2" xref="Thmdefinition2.p1.3.m3.3.3.2.cmml">σ</mi><mo id="Thmdefinition2.p1.3.m3.3.3.1" xref="Thmdefinition2.p1.3.m3.3.3.1.cmml">^</mo></mover><mo id="Thmdefinition2.p1.3.m3.4.5.3.3.2.2" xref="Thmdefinition2.p1.3.m3.4.5.3.3.1.cmml">,</mo><mi id="Thmdefinition2.p1.3.m3.4.4" xref="Thmdefinition2.p1.3.m3.4.4.cmml">σ</mi><mo id="Thmdefinition2.p1.3.m3.4.5.3.3.2.3" stretchy="false" xref="Thmdefinition2.p1.3.m3.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.3.m3.4b"><apply id="Thmdefinition2.p1.3.m3.4.5.cmml" xref="Thmdefinition2.p1.3.m3.4.5"><geq id="Thmdefinition2.p1.3.m3.4.5.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.1"></geq><apply id="Thmdefinition2.p1.3.m3.4.5.2.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2"><times id="Thmdefinition2.p1.3.m3.4.5.2.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.1"></times><apply id="Thmdefinition2.p1.3.m3.4.5.2.2.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.2"><csymbol cd="ambiguous" id="Thmdefinition2.p1.3.m3.4.5.2.2.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.2">subscript</csymbol><ci id="Thmdefinition2.p1.3.m3.4.5.2.2.2.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.2.2">𝑈</ci><ci id="Thmdefinition2.p1.3.m3.4.5.2.2.3.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.2.3">𝑖</ci></apply><interval closure="open" id="Thmdefinition2.p1.3.m3.4.5.2.3.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.2.3.2"><ci id="Thmdefinition2.p1.3.m3.1.1.cmml" xref="Thmdefinition2.p1.3.m3.1.1">𝜎</ci><ci id="Thmdefinition2.p1.3.m3.2.2.cmml" xref="Thmdefinition2.p1.3.m3.2.2">𝜎</ci></interval></apply><apply id="Thmdefinition2.p1.3.m3.4.5.3.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3"><times id="Thmdefinition2.p1.3.m3.4.5.3.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.1"></times><apply id="Thmdefinition2.p1.3.m3.4.5.3.2.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.2"><csymbol cd="ambiguous" id="Thmdefinition2.p1.3.m3.4.5.3.2.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.2">subscript</csymbol><ci id="Thmdefinition2.p1.3.m3.4.5.3.2.2.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.2.2">𝑈</ci><ci id="Thmdefinition2.p1.3.m3.4.5.3.2.3.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.2.3">𝑖</ci></apply><interval closure="open" id="Thmdefinition2.p1.3.m3.4.5.3.3.1.cmml" xref="Thmdefinition2.p1.3.m3.4.5.3.3.2"><apply id="Thmdefinition2.p1.3.m3.3.3.cmml" xref="Thmdefinition2.p1.3.m3.3.3"><ci id="Thmdefinition2.p1.3.m3.3.3.1.cmml" xref="Thmdefinition2.p1.3.m3.3.3.1">^</ci><ci id="Thmdefinition2.p1.3.m3.3.3.2.cmml" xref="Thmdefinition2.p1.3.m3.3.3.2">𝜎</ci></apply><ci id="Thmdefinition2.p1.3.m3.4.4.cmml" xref="Thmdefinition2.p1.3.m3.4.4">𝜎</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.3.m3.4c">U_{i}(\sigma,\sigma)\geq U_{i}(\hat{\sigma},\sigma)</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.3.m3.4d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ , italic_σ ) ≥ italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over^ start_ARG italic_σ end_ARG , italic_σ )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">This is a weaker condition than being a threshold equilibrium becacuse it rules out deviations to non-threshold strategies, an issue we will return to. However, by restricting to this space we can provide an equilibrium characterization in the same spirit as we did for OA and DG.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.4">As described above, the DMI mechanism is designed as an unbiased estimator of the (squared) determinant-based mutual information <math alttext="\textrm{DMI}(Y;Y^{\prime})^{2}=(\det U_{Y;Y^{\prime}})^{2}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.5"><semantics id="S4.SS1.p3.1.m1.5a"><mrow id="S4.SS1.p3.1.m1.5.5" xref="S4.SS1.p3.1.m1.5.5.cmml"><mrow id="S4.SS1.p3.1.m1.4.4.1" xref="S4.SS1.p3.1.m1.4.4.1.cmml"><mtext id="S4.SS1.p3.1.m1.4.4.1.3" xref="S4.SS1.p3.1.m1.4.4.1.3a.cmml">DMI</mtext><mo id="S4.SS1.p3.1.m1.4.4.1.2" xref="S4.SS1.p3.1.m1.4.4.1.2.cmml">⁢</mo><msup id="S4.SS1.p3.1.m1.4.4.1.1" xref="S4.SS1.p3.1.m1.4.4.1.1.cmml"><mrow id="S4.SS1.p3.1.m1.4.4.1.1.1.1" xref="S4.SS1.p3.1.m1.4.4.1.1.1.2.cmml"><mo id="S4.SS1.p3.1.m1.4.4.1.1.1.1.2" stretchy="false" 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xref="S4.SS1.p3.1.m1.5.5.2.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.5b"><apply id="S4.SS1.p3.1.m1.5.5.cmml" xref="S4.SS1.p3.1.m1.5.5"><eq id="S4.SS1.p3.1.m1.5.5.3.cmml" xref="S4.SS1.p3.1.m1.5.5.3"></eq><apply id="S4.SS1.p3.1.m1.4.4.1.cmml" xref="S4.SS1.p3.1.m1.4.4.1"><times id="S4.SS1.p3.1.m1.4.4.1.2.cmml" xref="S4.SS1.p3.1.m1.4.4.1.2"></times><ci id="S4.SS1.p3.1.m1.4.4.1.3a.cmml" xref="S4.SS1.p3.1.m1.4.4.1.3"><mtext id="S4.SS1.p3.1.m1.4.4.1.3.cmml" xref="S4.SS1.p3.1.m1.4.4.1.3">DMI</mtext></ci><apply id="S4.SS1.p3.1.m1.4.4.1.1.cmml" xref="S4.SS1.p3.1.m1.4.4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.4.4.1.1.2.cmml" xref="S4.SS1.p3.1.m1.4.4.1.1">superscript</csymbol><list id="S4.SS1.p3.1.m1.4.4.1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.4.4.1.1.1.1"><ci id="S4.SS1.p3.1.m1.3.3.cmml" xref="S4.SS1.p3.1.m1.3.3">𝑌</ci><apply id="S4.SS1.p3.1.m1.4.4.1.1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" 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xref="S4.SS1.p3.1.m1.5.5.2.1.1.1.2.2">𝑈</ci><list id="S4.SS1.p3.1.m1.2.2.2.3.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2"><ci id="S4.SS1.p3.1.m1.1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1.1.1">𝑌</ci><apply id="S4.SS1.p3.1.m1.2.2.2.2.1.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.2.2.2.2.1.1.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2.1">superscript</csymbol><ci id="S4.SS1.p3.1.m1.2.2.2.2.1.2.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2.1.2">𝑌</ci><ci id="S4.SS1.p3.1.m1.2.2.2.2.1.3.cmml" xref="S4.SS1.p3.1.m1.2.2.2.2.1.3">′</ci></apply></list></apply></apply><cn id="S4.SS1.p3.1.m1.5.5.2.3.cmml" type="integer" xref="S4.SS1.p3.1.m1.5.5.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.5c">\textrm{DMI}(Y;Y^{\prime})^{2}=(\det U_{Y;Y^{\prime}})^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.5d">DMI ( italic_Y ; italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( roman_det italic_U start_POSTSUBSCRIPT italic_Y ; italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="U_{Y,Y^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.2"><semantics id="S4.SS1.p3.2.m2.2a"><msub id="S4.SS1.p3.2.m2.2.3" xref="S4.SS1.p3.2.m2.2.3.cmml"><mi id="S4.SS1.p3.2.m2.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml">U</mi><mrow id="S4.SS1.p3.2.m2.2.2.2.2" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.p3.2.m2.1.1.1.1" xref="S4.SS1.p3.2.m2.1.1.1.1.cmml">Y</mi><mo id="S4.SS1.p3.2.m2.2.2.2.2.2" xref="S4.SS1.p3.2.m2.2.2.2.3.cmml">,</mo><msup id="S4.SS1.p3.2.m2.2.2.2.2.1" xref="S4.SS1.p3.2.m2.2.2.2.2.1.cmml"><mi id="S4.SS1.p3.2.m2.2.2.2.2.1.2" xref="S4.SS1.p3.2.m2.2.2.2.2.1.2.cmml">Y</mi><mo id="S4.SS1.p3.2.m2.2.2.2.2.1.3" xref="S4.SS1.p3.2.m2.2.2.2.2.1.3.cmml">′</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.2b"><apply id="S4.SS1.p3.2.m2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.2.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3">subscript</csymbol><ci id="S4.SS1.p3.2.m2.2.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2">𝑈</ci><list id="S4.SS1.p3.2.m2.2.2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2"><ci id="S4.SS1.p3.2.m2.1.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1.1.1">𝑌</ci><apply id="S4.SS1.p3.2.m2.2.2.2.2.1.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.2.2.2.2.1.1.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1">superscript</csymbol><ci id="S4.SS1.p3.2.m2.2.2.2.2.1.2.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1.2">𝑌</ci><ci id="S4.SS1.p3.2.m2.2.2.2.2.1.3.cmml" xref="S4.SS1.p3.2.m2.2.2.2.2.1.3">′</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.2c">U_{Y,Y^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.2d">italic_U start_POSTSUBSCRIPT italic_Y , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the joint distribution matrix of the <math alttext="\{H,L\}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.2"><semantics id="S4.SS1.p3.3.m3.2a"><mrow id="S4.SS1.p3.3.m3.2.3.2" xref="S4.SS1.p3.3.m3.2.3.1.cmml"><mo id="S4.SS1.p3.3.m3.2.3.2.1" stretchy="false" xref="S4.SS1.p3.3.m3.2.3.1.cmml">{</mo><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">H</mi><mo id="S4.SS1.p3.3.m3.2.3.2.2" xref="S4.SS1.p3.3.m3.2.3.1.cmml">,</mo><mi id="S4.SS1.p3.3.m3.2.2" xref="S4.SS1.p3.3.m3.2.2.cmml">L</mi><mo id="S4.SS1.p3.3.m3.2.3.2.3" stretchy="false" xref="S4.SS1.p3.3.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.2b"><set id="S4.SS1.p3.3.m3.2.3.1.cmml" xref="S4.SS1.p3.3.m3.2.3.2"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝐻</ci><ci id="S4.SS1.p3.3.m3.2.2.cmml" xref="S4.SS1.p3.3.m3.2.2">𝐿</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.2c">\{H,L\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.2d">{ italic_H , italic_L }</annotation></semantics></math>-valued random variables <math alttext="Y,Y^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.2"><semantics id="S4.SS1.p3.4.m4.2a"><mrow id="S4.SS1.p3.4.m4.2.2.1" xref="S4.SS1.p3.4.m4.2.2.2.cmml"><mi id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml">Y</mi><mo id="S4.SS1.p3.4.m4.2.2.1.2" xref="S4.SS1.p3.4.m4.2.2.2.cmml">,</mo><msup id="S4.SS1.p3.4.m4.2.2.1.1" xref="S4.SS1.p3.4.m4.2.2.1.1.cmml"><mi id="S4.SS1.p3.4.m4.2.2.1.1.2" xref="S4.SS1.p3.4.m4.2.2.1.1.2.cmml">Y</mi><mo id="S4.SS1.p3.4.m4.2.2.1.1.3" xref="S4.SS1.p3.4.m4.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.2b"><list id="S4.SS1.p3.4.m4.2.2.2.cmml" xref="S4.SS1.p3.4.m4.2.2.1"><ci id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">𝑌</ci><apply id="S4.SS1.p3.4.m4.2.2.1.1.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.2.2.1.1.1.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1">superscript</csymbol><ci id="S4.SS1.p3.4.m4.2.2.1.1.2.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1.2">𝑌</ci><ci id="S4.SS1.p3.4.m4.2.2.1.1.3.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.2c">Y,Y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.2d">italic_Y , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, given by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx6"> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle U_{Y,Y^{\prime}}=\begin{bmatrix}\Pr[Y=H,Y^{\prime}=H]&\Pr[Y=H,Y^% {\prime}=L]\\ \Pr[Y=L,Y^{\prime}=H]&\Pr[Y=L,Y^{\prime}=L]\end{bmatrix}~{}." class="ltx_Math" display="inline" id="S4.Ex12.m1.4"><semantics id="S4.Ex12.m1.4a"><mrow id="S4.Ex12.m1.4.4.1" xref="S4.Ex12.m1.4.4.1.1.cmml"><mrow id="S4.Ex12.m1.4.4.1.1" xref="S4.Ex12.m1.4.4.1.1.cmml"><msub id="S4.Ex12.m1.4.4.1.1.2" xref="S4.Ex12.m1.4.4.1.1.2.cmml"><mi id="S4.Ex12.m1.4.4.1.1.2.2" xref="S4.Ex12.m1.4.4.1.1.2.2.cmml">U</mi><mrow id="S4.Ex12.m1.3.3.2.2" xref="S4.Ex12.m1.3.3.2.3.cmml"><mi id="S4.Ex12.m1.2.2.1.1" xref="S4.Ex12.m1.2.2.1.1.cmml">Y</mi><mo id="S4.Ex12.m1.3.3.2.2.2" xref="S4.Ex12.m1.3.3.2.3.cmml">,</mo><msup id="S4.Ex12.m1.3.3.2.2.1" xref="S4.Ex12.m1.3.3.2.2.1.cmml"><mi id="S4.Ex12.m1.3.3.2.2.1.2" xref="S4.Ex12.m1.3.3.2.2.1.2.cmml">Y</mi><mo id="S4.Ex12.m1.3.3.2.2.1.3" xref="S4.Ex12.m1.3.3.2.2.1.3.cmml">′</mo></msup></mrow></msub><mo id="S4.Ex12.m1.4.4.1.1.1" xref="S4.Ex12.m1.4.4.1.1.1.cmml">=</mo><mrow id="S4.Ex12.m1.1.1a.3" xref="S4.Ex12.m1.1.1a.2.cmml"><mo id="S4.Ex12.m1.1.1a.3.1" 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xref="S4.Ex12.m1.1.1.1.1.12.12.6.3.3.2.2.2.3">′</ci></apply><ci id="S4.Ex12.m1.1.1.1.1.12.12.6.3.3.2.2.3.cmml" xref="S4.Ex12.m1.1.1.1.1.12.12.6.3.3.2.2.3">𝐿</ci></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.4c">\displaystyle U_{Y,Y^{\prime}}=\begin{bmatrix}\Pr[Y=H,Y^{\prime}=H]&\Pr[Y=H,Y^% {\prime}=L]\\ \Pr[Y=L,Y^{\prime}=H]&\Pr[Y=L,Y^{\prime}=L]\end{bmatrix}~{}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.4d">italic_U start_POSTSUBSCRIPT italic_Y , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL roman_Pr [ italic_Y = italic_H , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_H ] end_CELL start_CELL roman_Pr [ italic_Y = italic_H , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_L ] end_CELL end_ROW start_ROW start_CELL roman_Pr [ italic_Y = italic_L , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_H ] end_CELL start_CELL roman_Pr [ italic_Y = italic_L , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_L ] end_CELL end_ROW 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.SS1.p3.7">This unbiasedness is implied by <cite class="ltx_cite ltx_citemacro_citet">Kong [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib10" title="">2024</a>, Claim 4.4]</cite>, which we paraphrase for this binary-report context: for any <math alttext="Y,Y^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m1.2"><semantics id="S4.SS1.p3.5.m1.2a"><mrow id="S4.SS1.p3.5.m1.2.2.1" xref="S4.SS1.p3.5.m1.2.2.2.cmml"><mi id="S4.SS1.p3.5.m1.1.1" xref="S4.SS1.p3.5.m1.1.1.cmml">Y</mi><mo id="S4.SS1.p3.5.m1.2.2.1.2" xref="S4.SS1.p3.5.m1.2.2.2.cmml">,</mo><msup id="S4.SS1.p3.5.m1.2.2.1.1" xref="S4.SS1.p3.5.m1.2.2.1.1.cmml"><mi id="S4.SS1.p3.5.m1.2.2.1.1.2" xref="S4.SS1.p3.5.m1.2.2.1.1.2.cmml">Y</mi><mo id="S4.SS1.p3.5.m1.2.2.1.1.3" xref="S4.SS1.p3.5.m1.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m1.2b"><list id="S4.SS1.p3.5.m1.2.2.2.cmml" xref="S4.SS1.p3.5.m1.2.2.1"><ci id="S4.SS1.p3.5.m1.1.1.cmml" xref="S4.SS1.p3.5.m1.1.1">𝑌</ci><apply id="S4.SS1.p3.5.m1.2.2.1.1.cmml" xref="S4.SS1.p3.5.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m1.2.2.1.1.1.cmml" xref="S4.SS1.p3.5.m1.2.2.1.1">superscript</csymbol><ci id="S4.SS1.p3.5.m1.2.2.1.1.2.cmml" xref="S4.SS1.p3.5.m1.2.2.1.1.2">𝑌</ci><ci id="S4.SS1.p3.5.m1.2.2.1.1.3.cmml" xref="S4.SS1.p3.5.m1.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m1.2c">Y,Y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m1.2d">italic_Y , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists a constant <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m2.1"><semantics id="S4.SS1.p3.6.m2.1a"><mrow id="S4.SS1.p3.6.m2.1.1" xref="S4.SS1.p3.6.m2.1.1.cmml"><mi id="S4.SS1.p3.6.m2.1.1.2" xref="S4.SS1.p3.6.m2.1.1.2.cmml">α</mi><mo id="S4.SS1.p3.6.m2.1.1.1" xref="S4.SS1.p3.6.m2.1.1.1.cmml">></mo><mn id="S4.SS1.p3.6.m2.1.1.3" xref="S4.SS1.p3.6.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m2.1b"><apply id="S4.SS1.p3.6.m2.1.1.cmml" xref="S4.SS1.p3.6.m2.1.1"><gt id="S4.SS1.p3.6.m2.1.1.1.cmml" xref="S4.SS1.p3.6.m2.1.1.1"></gt><ci id="S4.SS1.p3.6.m2.1.1.2.cmml" xref="S4.SS1.p3.6.m2.1.1.2">𝛼</ci><cn id="S4.SS1.p3.6.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p3.6.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m2.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m2.1d">italic_α > 0</annotation></semantics></math> such that <math alttext="\mathop{\mathbb{E}}\det\sum_{i\in S}\mathbf{1}_{Y}\mathbf{1}_{Y^{\prime}}^{T}=% \alpha\det U_{Y,Y^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m3.2"><semantics id="S4.SS1.p3.7.m3.2a"><mrow id="S4.SS1.p3.7.m3.2.3" xref="S4.SS1.p3.7.m3.2.3.cmml"><mrow id="S4.SS1.p3.7.m3.2.3.2" xref="S4.SS1.p3.7.m3.2.3.2.cmml"><mo id="S4.SS1.p3.7.m3.2.3.2.1" rspace="0.0835em" xref="S4.SS1.p3.7.m3.2.3.2.1.cmml">𝔼</mo><mrow id="S4.SS1.p3.7.m3.2.3.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.cmml"><mo id="S4.SS1.p3.7.m3.2.3.2.2.1" lspace="0.0835em" rspace="0em" xref="S4.SS1.p3.7.m3.2.3.2.2.1.cmml">det</mo><mrow id="S4.SS1.p3.7.m3.2.3.2.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.cmml"><msub id="S4.SS1.p3.7.m3.2.3.2.2.2.1" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.cmml"><mo id="S4.SS1.p3.7.m3.2.3.2.2.2.1.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.2.cmml">∑</mo><mrow id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.cmml"><mi id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.2.cmml">i</mi><mo id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.1" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.1.cmml">∈</mo><mi id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.3.cmml">S</mi></mrow></msub><mrow id="S4.SS1.p3.7.m3.2.3.2.2.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.cmml"><msub id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.cmml"><mn id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.2.cmml">𝟏</mn><mi id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.3.cmml">Y</mi></msub><mo id="S4.SS1.p3.7.m3.2.3.2.2.2.2.1" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.1.cmml">⁢</mo><msubsup id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.cmml"><mn id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.2.cmml">𝟏</mn><msup id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.cmml"><mi id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.2" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.2.cmml">Y</mi><mo id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.3.cmml">′</mo></msup><mi id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.3" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.3.cmml">T</mi></msubsup></mrow></mrow></mrow></mrow><mo id="S4.SS1.p3.7.m3.2.3.1" xref="S4.SS1.p3.7.m3.2.3.1.cmml">=</mo><mrow id="S4.SS1.p3.7.m3.2.3.3" xref="S4.SS1.p3.7.m3.2.3.3.cmml"><mi id="S4.SS1.p3.7.m3.2.3.3.2" xref="S4.SS1.p3.7.m3.2.3.3.2.cmml">α</mi><mo id="S4.SS1.p3.7.m3.2.3.3.1" lspace="0.167em" xref="S4.SS1.p3.7.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.7.m3.2.3.3.3" xref="S4.SS1.p3.7.m3.2.3.3.3.cmml"><mo id="S4.SS1.p3.7.m3.2.3.3.3.1" rspace="0.167em" xref="S4.SS1.p3.7.m3.2.3.3.3.1.cmml">det</mo><msub id="S4.SS1.p3.7.m3.2.3.3.3.2" xref="S4.SS1.p3.7.m3.2.3.3.3.2.cmml"><mi id="S4.SS1.p3.7.m3.2.3.3.3.2.2" xref="S4.SS1.p3.7.m3.2.3.3.3.2.2.cmml">U</mi><mrow id="S4.SS1.p3.7.m3.2.2.2.2" xref="S4.SS1.p3.7.m3.2.2.2.3.cmml"><mi id="S4.SS1.p3.7.m3.1.1.1.1" xref="S4.SS1.p3.7.m3.1.1.1.1.cmml">Y</mi><mo id="S4.SS1.p3.7.m3.2.2.2.2.2" xref="S4.SS1.p3.7.m3.2.2.2.3.cmml">,</mo><msup id="S4.SS1.p3.7.m3.2.2.2.2.1" xref="S4.SS1.p3.7.m3.2.2.2.2.1.cmml"><mi id="S4.SS1.p3.7.m3.2.2.2.2.1.2" xref="S4.SS1.p3.7.m3.2.2.2.2.1.2.cmml">Y</mi><mo id="S4.SS1.p3.7.m3.2.2.2.2.1.3" xref="S4.SS1.p3.7.m3.2.2.2.2.1.3.cmml">′</mo></msup></mrow></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m3.2b"><apply id="S4.SS1.p3.7.m3.2.3.cmml" xref="S4.SS1.p3.7.m3.2.3"><eq id="S4.SS1.p3.7.m3.2.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.1"></eq><apply id="S4.SS1.p3.7.m3.2.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2"><ci id="S4.SS1.p3.7.m3.2.3.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.1">𝔼</ci><apply id="S4.SS1.p3.7.m3.2.3.2.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2"><determinant id="S4.SS1.p3.7.m3.2.3.2.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.1"></determinant><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2"><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.2.2.2.1.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1">subscript</csymbol><sum id="S4.SS1.p3.7.m3.2.3.2.2.2.1.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.2"></sum><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3"><in id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.1"></in><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.2">𝑖</ci><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.1.3.3">𝑆</ci></apply></apply><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2"><times id="S4.SS1.p3.7.m3.2.3.2.2.2.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.1"></times><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2">subscript</csymbol><cn id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.2.cmml" type="integer" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.2">1</cn><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.2.3">𝑌</ci></apply><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3">superscript</csymbol><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3">subscript</csymbol><cn id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.2.cmml" type="integer" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.2">1</cn><apply id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3">superscript</csymbol><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.2">𝑌</ci><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.2.3.3">′</ci></apply></apply><ci id="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.3.cmml" xref="S4.SS1.p3.7.m3.2.3.2.2.2.2.3.3">𝑇</ci></apply></apply></apply></apply></apply><apply id="S4.SS1.p3.7.m3.2.3.3.cmml" xref="S4.SS1.p3.7.m3.2.3.3"><times id="S4.SS1.p3.7.m3.2.3.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.3.1"></times><ci id="S4.SS1.p3.7.m3.2.3.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.3.2">𝛼</ci><apply id="S4.SS1.p3.7.m3.2.3.3.3.cmml" xref="S4.SS1.p3.7.m3.2.3.3.3"><determinant id="S4.SS1.p3.7.m3.2.3.3.3.1.cmml" xref="S4.SS1.p3.7.m3.2.3.3.3.1"></determinant><apply id="S4.SS1.p3.7.m3.2.3.3.3.2.cmml" xref="S4.SS1.p3.7.m3.2.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.3.3.3.2.1.cmml" xref="S4.SS1.p3.7.m3.2.3.3.3.2">subscript</csymbol><ci id="S4.SS1.p3.7.m3.2.3.3.3.2.2.cmml" xref="S4.SS1.p3.7.m3.2.3.3.3.2.2">𝑈</ci><list id="S4.SS1.p3.7.m3.2.2.2.3.cmml" xref="S4.SS1.p3.7.m3.2.2.2.2"><ci id="S4.SS1.p3.7.m3.1.1.1.1.cmml" xref="S4.SS1.p3.7.m3.1.1.1.1">𝑌</ci><apply id="S4.SS1.p3.7.m3.2.2.2.2.1.cmml" xref="S4.SS1.p3.7.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m3.2.2.2.2.1.1.cmml" xref="S4.SS1.p3.7.m3.2.2.2.2.1">superscript</csymbol><ci id="S4.SS1.p3.7.m3.2.2.2.2.1.2.cmml" xref="S4.SS1.p3.7.m3.2.2.2.2.1.2">𝑌</ci><ci id="S4.SS1.p3.7.m3.2.2.2.2.1.3.cmml" xref="S4.SS1.p3.7.m3.2.2.2.2.1.3">′</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m3.2c">\mathop{\mathbb{E}}\det\sum_{i\in S}\mathbf{1}_{Y}\mathbf{1}_{Y^{\prime}}^{T}=% \alpha\det U_{Y,Y^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m3.2d">blackboard_E roman_det ∑ start_POSTSUBSCRIPT italic_i ∈ italic_S end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = italic_α roman_det italic_U start_POSTSUBSCRIPT italic_Y , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.5">Back to our real-valued setting, recall that <math alttext="(X_{i},X_{i}^{\prime})\sim\mathcal{D}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.2"><semantics id="S4.SS1.p4.1.m1.2a"><mrow id="S4.SS1.p4.1.m1.2.2" xref="S4.SS1.p4.1.m1.2.2.cmml"><mrow id="S4.SS1.p4.1.m1.2.2.2.2" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml"><mo id="S4.SS1.p4.1.m1.2.2.2.2.3" stretchy="false" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml">(</mo><msub id="S4.SS1.p4.1.m1.1.1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS1.p4.1.m1.1.1.1.1.1.2" xref="S4.SS1.p4.1.m1.1.1.1.1.1.2.cmml">X</mi><mi id="S4.SS1.p4.1.m1.1.1.1.1.1.3" xref="S4.SS1.p4.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS1.p4.1.m1.2.2.2.2.4" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml">,</mo><msubsup id="S4.SS1.p4.1.m1.2.2.2.2.2" xref="S4.SS1.p4.1.m1.2.2.2.2.2.cmml"><mi id="S4.SS1.p4.1.m1.2.2.2.2.2.2.2" xref="S4.SS1.p4.1.m1.2.2.2.2.2.2.2.cmml">X</mi><mi id="S4.SS1.p4.1.m1.2.2.2.2.2.2.3" xref="S4.SS1.p4.1.m1.2.2.2.2.2.2.3.cmml">i</mi><mo id="S4.SS1.p4.1.m1.2.2.2.2.2.3" xref="S4.SS1.p4.1.m1.2.2.2.2.2.3.cmml">′</mo></msubsup><mo id="S4.SS1.p4.1.m1.2.2.2.2.5" stretchy="false" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml">)</mo></mrow><mo id="S4.SS1.p4.1.m1.2.2.3" xref="S4.SS1.p4.1.m1.2.2.3.cmml">∼</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p4.1.m1.2.2.4" xref="S4.SS1.p4.1.m1.2.2.4.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.2b"><apply id="S4.SS1.p4.1.m1.2.2.cmml" xref="S4.SS1.p4.1.m1.2.2"><csymbol cd="latexml" id="S4.SS1.p4.1.m1.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.3">similar-to</csymbol><interval closure="open" id="S4.SS1.p4.1.m1.2.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2"><apply id="S4.SS1.p4.1.m1.1.1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p4.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1.1.2">𝑋</ci><ci id="S4.SS1.p4.1.m1.1.1.1.1.1.3.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS1.p4.1.m1.2.2.2.2.2.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.2.2.2.2.1.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2">superscript</csymbol><apply id="S4.SS1.p4.1.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.2.2.2.2.2.1.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p4.1.m1.2.2.2.2.2.2.2.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2.2.2">𝑋</ci><ci id="S4.SS1.p4.1.m1.2.2.2.2.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2.2.3">𝑖</ci></apply><ci id="S4.SS1.p4.1.m1.2.2.2.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.2.2.2.3">′</ci></apply></interval><ci id="S4.SS1.p4.1.m1.2.2.4.cmml" xref="S4.SS1.p4.1.m1.2.2.4">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.2c">(X_{i},X_{i}^{\prime})\sim\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.2d">( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∼ caligraphic_D</annotation></semantics></math> independently. Given strategies <math alttext="\sigma,\sigma^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.2"><semantics id="S4.SS1.p4.2.m2.2a"><mrow id="S4.SS1.p4.2.m2.2.2.1" xref="S4.SS1.p4.2.m2.2.2.2.cmml"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">σ</mi><mo id="S4.SS1.p4.2.m2.2.2.1.2" xref="S4.SS1.p4.2.m2.2.2.2.cmml">,</mo><msup id="S4.SS1.p4.2.m2.2.2.1.1" xref="S4.SS1.p4.2.m2.2.2.1.1.cmml"><mi id="S4.SS1.p4.2.m2.2.2.1.1.2" xref="S4.SS1.p4.2.m2.2.2.1.1.2.cmml">σ</mi><mo id="S4.SS1.p4.2.m2.2.2.1.1.3" xref="S4.SS1.p4.2.m2.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.2b"><list id="S4.SS1.p4.2.m2.2.2.2.cmml" xref="S4.SS1.p4.2.m2.2.2.1"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝜎</ci><apply id="S4.SS1.p4.2.m2.2.2.1.1.cmml" xref="S4.SS1.p4.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.2.2.1.1.1.cmml" xref="S4.SS1.p4.2.m2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.p4.2.m2.2.2.1.1.2.cmml" xref="S4.SS1.p4.2.m2.2.2.1.1.2">𝜎</ci><ci id="S4.SS1.p4.2.m2.2.2.1.1.3.cmml" xref="S4.SS1.p4.2.m2.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.2c">\sigma,\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.2d">italic_σ , italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with thresholds <math alttext="\tau,\tau^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.2"><semantics id="S4.SS1.p4.3.m3.2a"><mrow id="S4.SS1.p4.3.m3.2.2.1" xref="S4.SS1.p4.3.m3.2.2.2.cmml"><mi id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml">τ</mi><mo id="S4.SS1.p4.3.m3.2.2.1.2" xref="S4.SS1.p4.3.m3.2.2.2.cmml">,</mo><msup id="S4.SS1.p4.3.m3.2.2.1.1" xref="S4.SS1.p4.3.m3.2.2.1.1.cmml"><mi id="S4.SS1.p4.3.m3.2.2.1.1.2" xref="S4.SS1.p4.3.m3.2.2.1.1.2.cmml">τ</mi><mo id="S4.SS1.p4.3.m3.2.2.1.1.3" xref="S4.SS1.p4.3.m3.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.2b"><list id="S4.SS1.p4.3.m3.2.2.2.cmml" xref="S4.SS1.p4.3.m3.2.2.1"><ci id="S4.SS1.p4.3.m3.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1">𝜏</ci><apply id="S4.SS1.p4.3.m3.2.2.1.1.cmml" xref="S4.SS1.p4.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m3.2.2.1.1.1.cmml" xref="S4.SS1.p4.3.m3.2.2.1.1">superscript</csymbol><ci id="S4.SS1.p4.3.m3.2.2.1.1.2.cmml" xref="S4.SS1.p4.3.m3.2.2.1.1.2">𝜏</ci><ci id="S4.SS1.p4.3.m3.2.2.1.1.3.cmml" xref="S4.SS1.p4.3.m3.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.2c">\tau,\tau^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.2d">italic_τ , italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, for some <math alttext="\alpha^{\prime}>0" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m4.1"><semantics id="S4.SS1.p4.4.m4.1a"><mrow id="S4.SS1.p4.4.m4.1.1" xref="S4.SS1.p4.4.m4.1.1.cmml"><msup id="S4.SS1.p4.4.m4.1.1.2" xref="S4.SS1.p4.4.m4.1.1.2.cmml"><mi id="S4.SS1.p4.4.m4.1.1.2.2" xref="S4.SS1.p4.4.m4.1.1.2.2.cmml">α</mi><mo id="S4.SS1.p4.4.m4.1.1.2.3" xref="S4.SS1.p4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS1.p4.4.m4.1.1.1" xref="S4.SS1.p4.4.m4.1.1.1.cmml">></mo><mn id="S4.SS1.p4.4.m4.1.1.3" xref="S4.SS1.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m4.1b"><apply id="S4.SS1.p4.4.m4.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1"><gt id="S4.SS1.p4.4.m4.1.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1.1"></gt><apply id="S4.SS1.p4.4.m4.1.1.2.cmml" xref="S4.SS1.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m4.1.1.2.1.cmml" xref="S4.SS1.p4.4.m4.1.1.2">superscript</csymbol><ci id="S4.SS1.p4.4.m4.1.1.2.2.cmml" xref="S4.SS1.p4.4.m4.1.1.2.2">𝛼</ci><ci id="S4.SS1.p4.4.m4.1.1.2.3.cmml" xref="S4.SS1.p4.4.m4.1.1.2.3">′</ci></apply><cn id="S4.SS1.p4.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m4.1c">\alpha^{\prime}>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m4.1d">italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > 0</annotation></semantics></math> we can rewrite Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4.E14" title="In 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">14</span></a>) as <math alttext="U_{i}(\sigma,\sigma^{\prime})=\alpha^{\prime}h(\tau,\tau^{\prime})^{2}" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m5.4"><semantics id="S4.SS1.p4.5.m5.4a"><mrow id="S4.SS1.p4.5.m5.4.4" xref="S4.SS1.p4.5.m5.4.4.cmml"><mrow id="S4.SS1.p4.5.m5.3.3.1" xref="S4.SS1.p4.5.m5.3.3.1.cmml"><msub 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id="S4.SS1.p4.5.m5.4.4.3" xref="S4.SS1.p4.5.m5.4.4.3.cmml">=</mo><mrow id="S4.SS1.p4.5.m5.4.4.2" xref="S4.SS1.p4.5.m5.4.4.2.cmml"><msup id="S4.SS1.p4.5.m5.4.4.2.3" xref="S4.SS1.p4.5.m5.4.4.2.3.cmml"><mi id="S4.SS1.p4.5.m5.4.4.2.3.2" xref="S4.SS1.p4.5.m5.4.4.2.3.2.cmml">α</mi><mo id="S4.SS1.p4.5.m5.4.4.2.3.3" xref="S4.SS1.p4.5.m5.4.4.2.3.3.cmml">′</mo></msup><mo id="S4.SS1.p4.5.m5.4.4.2.2" xref="S4.SS1.p4.5.m5.4.4.2.2.cmml">⁢</mo><mi id="S4.SS1.p4.5.m5.4.4.2.4" xref="S4.SS1.p4.5.m5.4.4.2.4.cmml">h</mi><mo id="S4.SS1.p4.5.m5.4.4.2.2a" xref="S4.SS1.p4.5.m5.4.4.2.2.cmml">⁢</mo><msup id="S4.SS1.p4.5.m5.4.4.2.1" xref="S4.SS1.p4.5.m5.4.4.2.1.cmml"><mrow id="S4.SS1.p4.5.m5.4.4.2.1.1.1" xref="S4.SS1.p4.5.m5.4.4.2.1.1.2.cmml"><mo id="S4.SS1.p4.5.m5.4.4.2.1.1.1.2" stretchy="false" xref="S4.SS1.p4.5.m5.4.4.2.1.1.2.cmml">(</mo><mi id="S4.SS1.p4.5.m5.2.2" xref="S4.SS1.p4.5.m5.2.2.cmml">τ</mi><mo id="S4.SS1.p4.5.m5.4.4.2.1.1.1.3" xref="S4.SS1.p4.5.m5.4.4.2.1.1.2.cmml">,</mo><msup 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xref="S4.SS1.p4.5.m5.4.4.2.3">superscript</csymbol><ci id="S4.SS1.p4.5.m5.4.4.2.3.2.cmml" xref="S4.SS1.p4.5.m5.4.4.2.3.2">𝛼</ci><ci id="S4.SS1.p4.5.m5.4.4.2.3.3.cmml" xref="S4.SS1.p4.5.m5.4.4.2.3.3">′</ci></apply><ci id="S4.SS1.p4.5.m5.4.4.2.4.cmml" xref="S4.SS1.p4.5.m5.4.4.2.4">ℎ</ci><apply id="S4.SS1.p4.5.m5.4.4.2.1.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m5.4.4.2.1.2.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1">superscript</csymbol><interval closure="open" id="S4.SS1.p4.5.m5.4.4.2.1.1.2.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1.1.1"><ci id="S4.SS1.p4.5.m5.2.2.cmml" xref="S4.SS1.p4.5.m5.2.2">𝜏</ci><apply id="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.1.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1.1.1.1">superscript</csymbol><ci id="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.2.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.2">𝜏</ci><ci id="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.3.cmml" xref="S4.SS1.p4.5.m5.4.4.2.1.1.1.1.3">′</ci></apply></interval><cn id="S4.SS1.p4.5.m5.4.4.2.1.3.cmml" type="integer" xref="S4.SS1.p4.5.m5.4.4.2.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m5.4c">U_{i}(\sigma,\sigma^{\prime})=\alpha^{\prime}h(\tau,\tau^{\prime})^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m5.4d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_σ , italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h ( italic_τ , italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> for</p> <table class="ltx_equation ltx_eqn_table" id="S4.E15"> <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 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end_POSTSUPERSCRIPT ] roman_Pr [ italic_X ≤ italic_τ , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] - roman_Pr [ italic_X > italic_τ , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] roman_Pr [ italic_X ≤ italic_τ , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.6">where <math alttext="(X,X^{\prime})\sim\mathcal{D}" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m1.2"><semantics id="S4.SS1.p4.6.m1.2a"><mrow id="S4.SS1.p4.6.m1.2.2" xref="S4.SS1.p4.6.m1.2.2.cmml"><mrow id="S4.SS1.p4.6.m1.2.2.1.1" 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xref="S4.SS1.p4.6.m1.2.2.2">similar-to</csymbol><interval closure="open" id="S4.SS1.p4.6.m1.2.2.1.2.cmml" xref="S4.SS1.p4.6.m1.2.2.1.1"><ci id="S4.SS1.p4.6.m1.1.1.cmml" xref="S4.SS1.p4.6.m1.1.1">𝑋</ci><apply id="S4.SS1.p4.6.m1.2.2.1.1.1.cmml" xref="S4.SS1.p4.6.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m1.2.2.1.1.1.1.cmml" xref="S4.SS1.p4.6.m1.2.2.1.1.1">superscript</csymbol><ci id="S4.SS1.p4.6.m1.2.2.1.1.1.2.cmml" xref="S4.SS1.p4.6.m1.2.2.1.1.1.2">𝑋</ci><ci id="S4.SS1.p4.6.m1.2.2.1.1.1.3.cmml" xref="S4.SS1.p4.6.m1.2.2.1.1.1.3">′</ci></apply></interval><ci id="S4.SS1.p4.6.m1.2.2.3.cmml" xref="S4.SS1.p4.6.m1.2.2.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m1.2c">(X,X^{\prime})\sim\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m1.2d">( italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∼ caligraphic_D</annotation></semantics></math>. We are now in a position to analyze equilibria and dynamics.</p> </div> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Equilibrium results.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p1.3">Given the definition of <math alttext="h" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px1.p1.1.m1.1a"><mi id="S4.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.1.m1.1b"><ci id="S4.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.1.m1.1d">italic_h</annotation></semantics></math> from Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4.E15" title="In 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">15</span></a>), we can show that uninformative strategies <math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px1.p1.2.m2.1a"><mrow id="S4.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.cmml"><msup id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.2" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.3" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml"><mo id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3a" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml">±</mo><mi id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.2" mathvariant="normal" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1"><eq id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1"></eq><apply id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.2.3"></times></apply><apply id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3">plus-or-minus</csymbol><infinity id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.2.m2.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.2.m2.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> remain equilibria in the DMI mechanism. The proof is immediate, as in both cases one of the two probabilities in each term of Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4.E15" title="In 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">15</span></a>) is zero independent of <math alttext="\tau" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S4.SS1.SSS0.Px1.p1.3.m3.1a"><mi id="S4.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p1.3.m3.1b"><ci id="S4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.3.m3.1d">italic_τ</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition5.1.1.1">Proposition 5</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition5.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition5.p1"> <p class="ltx_p" id="Thmproposition5.p1.1"><math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="Thmproposition5.p1.1.m1.1"><semantics id="Thmproposition5.p1.1.m1.1a"><mrow id="Thmproposition5.p1.1.m1.1.1" xref="Thmproposition5.p1.1.m1.1.1.cmml"><msup id="Thmproposition5.p1.1.m1.1.1.2" xref="Thmproposition5.p1.1.m1.1.1.2.cmml"><mi id="Thmproposition5.p1.1.m1.1.1.2.2" xref="Thmproposition5.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="Thmproposition5.p1.1.m1.1.1.2.3" xref="Thmproposition5.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition5.p1.1.m1.1.1.1" xref="Thmproposition5.p1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmproposition5.p1.1.m1.1.1.3" xref="Thmproposition5.p1.1.m1.1.1.3.cmml"><mo id="Thmproposition5.p1.1.m1.1.1.3a" xref="Thmproposition5.p1.1.m1.1.1.3.cmml">±</mo><mi id="Thmproposition5.p1.1.m1.1.1.3.2" mathvariant="normal" xref="Thmproposition5.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition5.p1.1.m1.1b"><apply id="Thmproposition5.p1.1.m1.1.1.cmml" xref="Thmproposition5.p1.1.m1.1.1"><eq id="Thmproposition5.p1.1.m1.1.1.1.cmml" xref="Thmproposition5.p1.1.m1.1.1.1"></eq><apply id="Thmproposition5.p1.1.m1.1.1.2.cmml" xref="Thmproposition5.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition5.p1.1.m1.1.1.2.1.cmml" xref="Thmproposition5.p1.1.m1.1.1.2">superscript</csymbol><ci id="Thmproposition5.p1.1.m1.1.1.2.2.cmml" xref="Thmproposition5.p1.1.m1.1.1.2.2">𝜏</ci><times id="Thmproposition5.p1.1.m1.1.1.2.3.cmml" xref="Thmproposition5.p1.1.m1.1.1.2.3"></times></apply><apply id="Thmproposition5.p1.1.m1.1.1.3.cmml" xref="Thmproposition5.p1.1.m1.1.1.3"><csymbol cd="latexml" id="Thmproposition5.p1.1.m1.1.1.3.1.cmml" xref="Thmproposition5.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="Thmproposition5.p1.1.m1.1.1.3.2.cmml" xref="Thmproposition5.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition5.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition5.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are both always equilibria restricted to threshold strategies under DG.</p> </div> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p2.1">We can also provide necessary and sufficient conditions for a finite threshold equilibrium for DMI. The theorem relies on several regularity conditions that rule out corner cases where the first order condition is trivially satisfied for spurious reasons.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem5.1.1.1">Theorem 5</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem5.p1"> <p class="ltx_p" id="Thmtheorem5.p1.13">Let finite threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem5.p1.1.m1.1"><semantics id="Thmtheorem5.p1.1.m1.1a"><mi id="Thmtheorem5.p1.1.m1.1.1" xref="Thmtheorem5.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.1.m1.1b"><ci id="Thmtheorem5.p1.1.m1.1.1.cmml" xref="Thmtheorem5.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.1.m1.1d">italic_τ</annotation></semantics></math> be given, <math alttext="h(x,\tau)" class="ltx_Math" display="inline" id="Thmtheorem5.p1.2.m2.2"><semantics id="Thmtheorem5.p1.2.m2.2a"><mrow id="Thmtheorem5.p1.2.m2.2.3" xref="Thmtheorem5.p1.2.m2.2.3.cmml"><mi id="Thmtheorem5.p1.2.m2.2.3.2" xref="Thmtheorem5.p1.2.m2.2.3.2.cmml">h</mi><mo id="Thmtheorem5.p1.2.m2.2.3.1" xref="Thmtheorem5.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.2.m2.2.3.3.2" xref="Thmtheorem5.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem5.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem5.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem5.p1.2.m2.1.1" xref="Thmtheorem5.p1.2.m2.1.1.cmml">x</mi><mo id="Thmtheorem5.p1.2.m2.2.3.3.2.2" xref="Thmtheorem5.p1.2.m2.2.3.3.1.cmml">,</mo><mi id="Thmtheorem5.p1.2.m2.2.2" xref="Thmtheorem5.p1.2.m2.2.2.cmml">τ</mi><mo id="Thmtheorem5.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem5.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.2.m2.2b"><apply id="Thmtheorem5.p1.2.m2.2.3.cmml" xref="Thmtheorem5.p1.2.m2.2.3"><times id="Thmtheorem5.p1.2.m2.2.3.1.cmml" xref="Thmtheorem5.p1.2.m2.2.3.1"></times><ci id="Thmtheorem5.p1.2.m2.2.3.2.cmml" xref="Thmtheorem5.p1.2.m2.2.3.2">ℎ</ci><interval closure="open" id="Thmtheorem5.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem5.p1.2.m2.2.3.3.2"><ci id="Thmtheorem5.p1.2.m2.1.1.cmml" xref="Thmtheorem5.p1.2.m2.1.1">𝑥</ci><ci id="Thmtheorem5.p1.2.m2.2.2.cmml" xref="Thmtheorem5.p1.2.m2.2.2">𝜏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.2.m2.2c">h(x,\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.2.m2.2d">italic_h ( italic_x , italic_τ )</annotation></semantics></math> be differentiable in <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem5.p1.3.m3.1"><semantics id="Thmtheorem5.p1.3.m3.1a"><mi id="Thmtheorem5.p1.3.m3.1.1" xref="Thmtheorem5.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.3.m3.1b"><ci id="Thmtheorem5.p1.3.m3.1.1.cmml" xref="Thmtheorem5.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.3.m3.1d">italic_x</annotation></semantics></math> at <math alttext="x=\tau" class="ltx_Math" display="inline" id="Thmtheorem5.p1.4.m4.1"><semantics id="Thmtheorem5.p1.4.m4.1a"><mrow id="Thmtheorem5.p1.4.m4.1.1" xref="Thmtheorem5.p1.4.m4.1.1.cmml"><mi id="Thmtheorem5.p1.4.m4.1.1.2" xref="Thmtheorem5.p1.4.m4.1.1.2.cmml">x</mi><mo id="Thmtheorem5.p1.4.m4.1.1.1" xref="Thmtheorem5.p1.4.m4.1.1.1.cmml">=</mo><mi id="Thmtheorem5.p1.4.m4.1.1.3" xref="Thmtheorem5.p1.4.m4.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.4.m4.1b"><apply id="Thmtheorem5.p1.4.m4.1.1.cmml" xref="Thmtheorem5.p1.4.m4.1.1"><eq id="Thmtheorem5.p1.4.m4.1.1.1.cmml" xref="Thmtheorem5.p1.4.m4.1.1.1"></eq><ci id="Thmtheorem5.p1.4.m4.1.1.2.cmml" xref="Thmtheorem5.p1.4.m4.1.1.2">𝑥</ci><ci id="Thmtheorem5.p1.4.m4.1.1.3.cmml" xref="Thmtheorem5.p1.4.m4.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.4.m4.1c">x=\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.4.m4.1d">italic_x = italic_τ</annotation></semantics></math>, <math alttext="h(\tau,\tau)\neq 0" class="ltx_Math" display="inline" id="Thmtheorem5.p1.5.m5.2"><semantics id="Thmtheorem5.p1.5.m5.2a"><mrow id="Thmtheorem5.p1.5.m5.2.3" xref="Thmtheorem5.p1.5.m5.2.3.cmml"><mrow id="Thmtheorem5.p1.5.m5.2.3.2" xref="Thmtheorem5.p1.5.m5.2.3.2.cmml"><mi id="Thmtheorem5.p1.5.m5.2.3.2.2" xref="Thmtheorem5.p1.5.m5.2.3.2.2.cmml">h</mi><mo id="Thmtheorem5.p1.5.m5.2.3.2.1" xref="Thmtheorem5.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.5.m5.2.3.2.3.2" xref="Thmtheorem5.p1.5.m5.2.3.2.3.1.cmml"><mo id="Thmtheorem5.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem5.p1.5.m5.2.3.2.3.1.cmml">(</mo><mi id="Thmtheorem5.p1.5.m5.1.1" xref="Thmtheorem5.p1.5.m5.1.1.cmml">τ</mi><mo id="Thmtheorem5.p1.5.m5.2.3.2.3.2.2" xref="Thmtheorem5.p1.5.m5.2.3.2.3.1.cmml">,</mo><mi id="Thmtheorem5.p1.5.m5.2.2" xref="Thmtheorem5.p1.5.m5.2.2.cmml">τ</mi><mo id="Thmtheorem5.p1.5.m5.2.3.2.3.2.3" stretchy="false" xref="Thmtheorem5.p1.5.m5.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.5.m5.2.3.1" xref="Thmtheorem5.p1.5.m5.2.3.1.cmml">≠</mo><mn id="Thmtheorem5.p1.5.m5.2.3.3" xref="Thmtheorem5.p1.5.m5.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.5.m5.2b"><apply id="Thmtheorem5.p1.5.m5.2.3.cmml" xref="Thmtheorem5.p1.5.m5.2.3"><neq id="Thmtheorem5.p1.5.m5.2.3.1.cmml" xref="Thmtheorem5.p1.5.m5.2.3.1"></neq><apply id="Thmtheorem5.p1.5.m5.2.3.2.cmml" xref="Thmtheorem5.p1.5.m5.2.3.2"><times id="Thmtheorem5.p1.5.m5.2.3.2.1.cmml" xref="Thmtheorem5.p1.5.m5.2.3.2.1"></times><ci id="Thmtheorem5.p1.5.m5.2.3.2.2.cmml" xref="Thmtheorem5.p1.5.m5.2.3.2.2">ℎ</ci><interval closure="open" id="Thmtheorem5.p1.5.m5.2.3.2.3.1.cmml" xref="Thmtheorem5.p1.5.m5.2.3.2.3.2"><ci id="Thmtheorem5.p1.5.m5.1.1.cmml" xref="Thmtheorem5.p1.5.m5.1.1">𝜏</ci><ci id="Thmtheorem5.p1.5.m5.2.2.cmml" xref="Thmtheorem5.p1.5.m5.2.2">𝜏</ci></interval></apply><cn id="Thmtheorem5.p1.5.m5.2.3.3.cmml" type="integer" xref="Thmtheorem5.p1.5.m5.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.5.m5.2c">h(\tau,\tau)\neq 0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.5.m5.2d">italic_h ( italic_τ , italic_τ ) ≠ 0</annotation></semantics></math>, and <math alttext="f(\tau)>0" class="ltx_Math" display="inline" id="Thmtheorem5.p1.6.m6.1"><semantics id="Thmtheorem5.p1.6.m6.1a"><mrow id="Thmtheorem5.p1.6.m6.1.2" xref="Thmtheorem5.p1.6.m6.1.2.cmml"><mrow id="Thmtheorem5.p1.6.m6.1.2.2" xref="Thmtheorem5.p1.6.m6.1.2.2.cmml"><mi id="Thmtheorem5.p1.6.m6.1.2.2.2" xref="Thmtheorem5.p1.6.m6.1.2.2.2.cmml">f</mi><mo id="Thmtheorem5.p1.6.m6.1.2.2.1" xref="Thmtheorem5.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.6.m6.1.2.2.3.2" xref="Thmtheorem5.p1.6.m6.1.2.2.cmml"><mo id="Thmtheorem5.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="Thmtheorem5.p1.6.m6.1.2.2.cmml">(</mo><mi id="Thmtheorem5.p1.6.m6.1.1" xref="Thmtheorem5.p1.6.m6.1.1.cmml">τ</mi><mo id="Thmtheorem5.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="Thmtheorem5.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.6.m6.1.2.1" xref="Thmtheorem5.p1.6.m6.1.2.1.cmml">></mo><mn id="Thmtheorem5.p1.6.m6.1.2.3" xref="Thmtheorem5.p1.6.m6.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.6.m6.1b"><apply id="Thmtheorem5.p1.6.m6.1.2.cmml" xref="Thmtheorem5.p1.6.m6.1.2"><gt id="Thmtheorem5.p1.6.m6.1.2.1.cmml" xref="Thmtheorem5.p1.6.m6.1.2.1"></gt><apply id="Thmtheorem5.p1.6.m6.1.2.2.cmml" xref="Thmtheorem5.p1.6.m6.1.2.2"><times id="Thmtheorem5.p1.6.m6.1.2.2.1.cmml" xref="Thmtheorem5.p1.6.m6.1.2.2.1"></times><ci id="Thmtheorem5.p1.6.m6.1.2.2.2.cmml" xref="Thmtheorem5.p1.6.m6.1.2.2.2">𝑓</ci><ci id="Thmtheorem5.p1.6.m6.1.1.cmml" xref="Thmtheorem5.p1.6.m6.1.1">𝜏</ci></apply><cn id="Thmtheorem5.p1.6.m6.1.2.3.cmml" type="integer" xref="Thmtheorem5.p1.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.6.m6.1c">f(\tau)>0</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.6.m6.1d">italic_f ( italic_τ ) > 0</annotation></semantics></math>, where <math alttext="f" class="ltx_Math" display="inline" id="Thmtheorem5.p1.7.m7.1"><semantics id="Thmtheorem5.p1.7.m7.1a"><mi id="Thmtheorem5.p1.7.m7.1.1" xref="Thmtheorem5.p1.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.7.m7.1b"><ci id="Thmtheorem5.p1.7.m7.1.1.cmml" xref="Thmtheorem5.p1.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.7.m7.1d">italic_f</annotation></semantics></math> is the continuous density of <math alttext="F" class="ltx_Math" display="inline" id="Thmtheorem5.p1.8.m8.1"><semantics id="Thmtheorem5.p1.8.m8.1a"><mi id="Thmtheorem5.p1.8.m8.1.1" xref="Thmtheorem5.p1.8.m8.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.8.m8.1b"><ci id="Thmtheorem5.p1.8.m8.1.1.cmml" xref="Thmtheorem5.p1.8.m8.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.8.m8.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.8.m8.1d">italic_F</annotation></semantics></math>. If <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem5.p1.9.m9.1"><semantics id="Thmtheorem5.p1.9.m9.1a"><mi id="Thmtheorem5.p1.9.m9.1.1" xref="Thmtheorem5.p1.9.m9.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.9.m9.1b"><ci id="Thmtheorem5.p1.9.m9.1.1.cmml" xref="Thmtheorem5.p1.9.m9.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.9.m9.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.9.m9.1d">italic_τ</annotation></semantics></math> is an equilibrium restricted to threshold strategies under the DMI mechanism then <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="Thmtheorem5.p1.10.m10.2"><semantics id="Thmtheorem5.p1.10.m10.2a"><mrow id="Thmtheorem5.p1.10.m10.2.3" xref="Thmtheorem5.p1.10.m10.2.3.cmml"><mrow id="Thmtheorem5.p1.10.m10.2.3.2" xref="Thmtheorem5.p1.10.m10.2.3.2.cmml"><mi id="Thmtheorem5.p1.10.m10.2.3.2.2" xref="Thmtheorem5.p1.10.m10.2.3.2.2.cmml">G</mi><mo id="Thmtheorem5.p1.10.m10.2.3.2.1" xref="Thmtheorem5.p1.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.10.m10.2.3.2.3.2" xref="Thmtheorem5.p1.10.m10.2.3.2.cmml"><mo id="Thmtheorem5.p1.10.m10.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem5.p1.10.m10.2.3.2.cmml">(</mo><mi id="Thmtheorem5.p1.10.m10.1.1" xref="Thmtheorem5.p1.10.m10.1.1.cmml">τ</mi><mo id="Thmtheorem5.p1.10.m10.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem5.p1.10.m10.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.10.m10.2.3.1" xref="Thmtheorem5.p1.10.m10.2.3.1.cmml">=</mo><mrow id="Thmtheorem5.p1.10.m10.2.3.3" xref="Thmtheorem5.p1.10.m10.2.3.3.cmml"><mi id="Thmtheorem5.p1.10.m10.2.3.3.2" xref="Thmtheorem5.p1.10.m10.2.3.3.2.cmml">F</mi><mo id="Thmtheorem5.p1.10.m10.2.3.3.1" xref="Thmtheorem5.p1.10.m10.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.10.m10.2.3.3.3.2" xref="Thmtheorem5.p1.10.m10.2.3.3.cmml"><mo id="Thmtheorem5.p1.10.m10.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem5.p1.10.m10.2.3.3.cmml">(</mo><mi id="Thmtheorem5.p1.10.m10.2.2" xref="Thmtheorem5.p1.10.m10.2.2.cmml">τ</mi><mo id="Thmtheorem5.p1.10.m10.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem5.p1.10.m10.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.10.m10.2b"><apply id="Thmtheorem5.p1.10.m10.2.3.cmml" xref="Thmtheorem5.p1.10.m10.2.3"><eq id="Thmtheorem5.p1.10.m10.2.3.1.cmml" xref="Thmtheorem5.p1.10.m10.2.3.1"></eq><apply id="Thmtheorem5.p1.10.m10.2.3.2.cmml" xref="Thmtheorem5.p1.10.m10.2.3.2"><times id="Thmtheorem5.p1.10.m10.2.3.2.1.cmml" xref="Thmtheorem5.p1.10.m10.2.3.2.1"></times><ci id="Thmtheorem5.p1.10.m10.2.3.2.2.cmml" xref="Thmtheorem5.p1.10.m10.2.3.2.2">𝐺</ci><ci id="Thmtheorem5.p1.10.m10.1.1.cmml" xref="Thmtheorem5.p1.10.m10.1.1">𝜏</ci></apply><apply id="Thmtheorem5.p1.10.m10.2.3.3.cmml" xref="Thmtheorem5.p1.10.m10.2.3.3"><times id="Thmtheorem5.p1.10.m10.2.3.3.1.cmml" xref="Thmtheorem5.p1.10.m10.2.3.3.1"></times><ci id="Thmtheorem5.p1.10.m10.2.3.3.2.cmml" xref="Thmtheorem5.p1.10.m10.2.3.3.2">𝐹</ci><ci id="Thmtheorem5.p1.10.m10.2.2.cmml" xref="Thmtheorem5.p1.10.m10.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.10.m10.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.10.m10.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>. Conversely, if <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="Thmtheorem5.p1.11.m11.2"><semantics id="Thmtheorem5.p1.11.m11.2a"><mrow id="Thmtheorem5.p1.11.m11.2.3" xref="Thmtheorem5.p1.11.m11.2.3.cmml"><mrow id="Thmtheorem5.p1.11.m11.2.3.2" xref="Thmtheorem5.p1.11.m11.2.3.2.cmml"><mi id="Thmtheorem5.p1.11.m11.2.3.2.2" xref="Thmtheorem5.p1.11.m11.2.3.2.2.cmml">G</mi><mo id="Thmtheorem5.p1.11.m11.2.3.2.1" xref="Thmtheorem5.p1.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.11.m11.2.3.2.3.2" xref="Thmtheorem5.p1.11.m11.2.3.2.cmml"><mo id="Thmtheorem5.p1.11.m11.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem5.p1.11.m11.2.3.2.cmml">(</mo><mi id="Thmtheorem5.p1.11.m11.1.1" xref="Thmtheorem5.p1.11.m11.1.1.cmml">τ</mi><mo id="Thmtheorem5.p1.11.m11.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem5.p1.11.m11.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem5.p1.11.m11.2.3.1" xref="Thmtheorem5.p1.11.m11.2.3.1.cmml">=</mo><mrow id="Thmtheorem5.p1.11.m11.2.3.3" xref="Thmtheorem5.p1.11.m11.2.3.3.cmml"><mi id="Thmtheorem5.p1.11.m11.2.3.3.2" xref="Thmtheorem5.p1.11.m11.2.3.3.2.cmml">F</mi><mo id="Thmtheorem5.p1.11.m11.2.3.3.1" xref="Thmtheorem5.p1.11.m11.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.11.m11.2.3.3.3.2" xref="Thmtheorem5.p1.11.m11.2.3.3.cmml"><mo id="Thmtheorem5.p1.11.m11.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem5.p1.11.m11.2.3.3.cmml">(</mo><mi id="Thmtheorem5.p1.11.m11.2.2" xref="Thmtheorem5.p1.11.m11.2.2.cmml">τ</mi><mo id="Thmtheorem5.p1.11.m11.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem5.p1.11.m11.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.11.m11.2b"><apply id="Thmtheorem5.p1.11.m11.2.3.cmml" xref="Thmtheorem5.p1.11.m11.2.3"><eq id="Thmtheorem5.p1.11.m11.2.3.1.cmml" xref="Thmtheorem5.p1.11.m11.2.3.1"></eq><apply id="Thmtheorem5.p1.11.m11.2.3.2.cmml" xref="Thmtheorem5.p1.11.m11.2.3.2"><times id="Thmtheorem5.p1.11.m11.2.3.2.1.cmml" xref="Thmtheorem5.p1.11.m11.2.3.2.1"></times><ci id="Thmtheorem5.p1.11.m11.2.3.2.2.cmml" xref="Thmtheorem5.p1.11.m11.2.3.2.2">𝐺</ci><ci id="Thmtheorem5.p1.11.m11.1.1.cmml" xref="Thmtheorem5.p1.11.m11.1.1">𝜏</ci></apply><apply id="Thmtheorem5.p1.11.m11.2.3.3.cmml" xref="Thmtheorem5.p1.11.m11.2.3.3"><times id="Thmtheorem5.p1.11.m11.2.3.3.1.cmml" xref="Thmtheorem5.p1.11.m11.2.3.3.1"></times><ci id="Thmtheorem5.p1.11.m11.2.3.3.2.cmml" xref="Thmtheorem5.p1.11.m11.2.3.3.2">𝐹</ci><ci id="Thmtheorem5.p1.11.m11.2.2.cmml" xref="Thmtheorem5.p1.11.m11.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.11.m11.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.11.m11.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math> and <math alttext="h(x,\tau)" class="ltx_Math" display="inline" id="Thmtheorem5.p1.12.m12.2"><semantics id="Thmtheorem5.p1.12.m12.2a"><mrow id="Thmtheorem5.p1.12.m12.2.3" xref="Thmtheorem5.p1.12.m12.2.3.cmml"><mi id="Thmtheorem5.p1.12.m12.2.3.2" xref="Thmtheorem5.p1.12.m12.2.3.2.cmml">h</mi><mo id="Thmtheorem5.p1.12.m12.2.3.1" xref="Thmtheorem5.p1.12.m12.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem5.p1.12.m12.2.3.3.2" xref="Thmtheorem5.p1.12.m12.2.3.3.1.cmml"><mo id="Thmtheorem5.p1.12.m12.2.3.3.2.1" stretchy="false" xref="Thmtheorem5.p1.12.m12.2.3.3.1.cmml">(</mo><mi id="Thmtheorem5.p1.12.m12.1.1" xref="Thmtheorem5.p1.12.m12.1.1.cmml">x</mi><mo id="Thmtheorem5.p1.12.m12.2.3.3.2.2" xref="Thmtheorem5.p1.12.m12.2.3.3.1.cmml">,</mo><mi id="Thmtheorem5.p1.12.m12.2.2" xref="Thmtheorem5.p1.12.m12.2.2.cmml">τ</mi><mo id="Thmtheorem5.p1.12.m12.2.3.3.2.3" stretchy="false" xref="Thmtheorem5.p1.12.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.12.m12.2b"><apply id="Thmtheorem5.p1.12.m12.2.3.cmml" xref="Thmtheorem5.p1.12.m12.2.3"><times id="Thmtheorem5.p1.12.m12.2.3.1.cmml" xref="Thmtheorem5.p1.12.m12.2.3.1"></times><ci id="Thmtheorem5.p1.12.m12.2.3.2.cmml" xref="Thmtheorem5.p1.12.m12.2.3.2">ℎ</ci><interval closure="open" id="Thmtheorem5.p1.12.m12.2.3.3.1.cmml" xref="Thmtheorem5.p1.12.m12.2.3.3.2"><ci id="Thmtheorem5.p1.12.m12.1.1.cmml" xref="Thmtheorem5.p1.12.m12.1.1">𝑥</ci><ci id="Thmtheorem5.p1.12.m12.2.2.cmml" xref="Thmtheorem5.p1.12.m12.2.2">𝜏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.12.m12.2c">h(x,\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.12.m12.2d">italic_h ( italic_x , italic_τ )</annotation></semantics></math> is non-negative and strictly single-peaked then <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem5.p1.13.m13.1"><semantics id="Thmtheorem5.p1.13.m13.1a"><mi id="Thmtheorem5.p1.13.m13.1.1" xref="Thmtheorem5.p1.13.m13.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem5.p1.13.m13.1b"><ci id="Thmtheorem5.p1.13.m13.1.1.cmml" xref="Thmtheorem5.p1.13.m13.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem5.p1.13.m13.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem5.p1.13.m13.1d">italic_τ</annotation></semantics></math> is an equilibrium restricted to threshold strategies under the DMI mechanism.</p> </div> </div> <div class="ltx_proof" id="S4.SS1.SSS0.Px1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.SSS0.Px1.1.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.1.p1.3">For necessity, assume that <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px1.1.p1.1.m1.1a"><msup id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.1.p1.1.m1.1b"><apply id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.1.m1.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an equilibrium restricted to threshold strategies, so <math alttext="f(\tau,\tau^{*})" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2"><semantics id="S4.SS1.SSS0.Px1.1.p1.2.m2.2a"><mrow id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.3" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.3.cmml">f</mi><mo id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.2" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.2.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.1.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.1.1.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.2.cmml">,</mo><msup id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.4" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2b"><apply id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2"><times id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.2"></times><ci id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.3">𝑓</ci><interval closure="open" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1"><ci id="S4.SS1.SSS0.Px1.1.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.1.1">𝜏</ci><apply id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.2.m2.2.2.1.1.1.3"></times></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2c">f(\tau,\tau^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.2.m2.2d">italic_f ( italic_τ , italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> is maximized at <math alttext="\tau=\tau^{*}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.3.m3.1"><semantics id="S4.SS1.SSS0.Px1.1.p1.3.m3.1a"><mrow id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.1.cmml">=</mo><msup id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.2" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.3" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.1.p1.3.m3.1b"><apply id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1"><eq id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.1"></eq><ci id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.2">𝜏</ci><apply id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.3.m3.1c">\tau=\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.3.m3.1d">italic_τ = italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. The first order condition for optimality is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx7"> <tbody id="S4.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 0" class="ltx_Math" display="inline" id="S4.E16.m1.1"><semantics id="S4.E16.m1.1a"><mn id="S4.E16.m1.1.1" xref="S4.E16.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.E16.m1.1b"><cn id="S4.E16.m1.1.1.cmml" type="integer" xref="S4.E16.m1.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.1c">\displaystyle 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{d}{d\tau}\alpha^{\prime}h(\tau,\tau^{*})^{2}=2h(\tau,\tau^% {*})\alpha^{\prime}f(\tau)\left(\Pr[X^{\prime}>\tau^{*}]-\Pr[X^{\prime}>\tau^{% *}\mid X=\tau]\right)~{}" class="ltx_Math" display="inline" id="S4.E16.m2.8"><semantics id="S4.E16.m2.8a"><mrow id="S4.E16.m2.8.8" xref="S4.E16.m2.8.8.cmml"><mi id="S4.E16.m2.8.8.5" xref="S4.E16.m2.8.8.5.cmml"></mi><mo id="S4.E16.m2.8.8.6" xref="S4.E16.m2.8.8.6.cmml">=</mo><mrow id="S4.E16.m2.6.6.1" xref="S4.E16.m2.6.6.1.cmml"><mstyle displaystyle="true" id="S4.E16.m2.6.6.1.3" xref="S4.E16.m2.6.6.1.3.cmml"><mfrac id="S4.E16.m2.6.6.1.3a" xref="S4.E16.m2.6.6.1.3.cmml"><mi id="S4.E16.m2.6.6.1.3.2" xref="S4.E16.m2.6.6.1.3.2.cmml">d</mi><mrow id="S4.E16.m2.6.6.1.3.3" xref="S4.E16.m2.6.6.1.3.3.cmml"><mi id="S4.E16.m2.6.6.1.3.3.2" xref="S4.E16.m2.6.6.1.3.3.2.cmml">d</mi><mo id="S4.E16.m2.6.6.1.3.3.1" xref="S4.E16.m2.6.6.1.3.3.1.cmml">⁢</mo><mi id="S4.E16.m2.6.6.1.3.3.3" xref="S4.E16.m2.6.6.1.3.3.3.cmml">τ</mi></mrow></mfrac></mstyle><mo id="S4.E16.m2.6.6.1.2" xref="S4.E16.m2.6.6.1.2.cmml">⁢</mo><msup id="S4.E16.m2.6.6.1.4" xref="S4.E16.m2.6.6.1.4.cmml"><mi id="S4.E16.m2.6.6.1.4.2" xref="S4.E16.m2.6.6.1.4.2.cmml">α</mi><mo id="S4.E16.m2.6.6.1.4.3" xref="S4.E16.m2.6.6.1.4.3.cmml">′</mo></msup><mo id="S4.E16.m2.6.6.1.2a" xref="S4.E16.m2.6.6.1.2.cmml">⁢</mo><mi id="S4.E16.m2.6.6.1.5" xref="S4.E16.m2.6.6.1.5.cmml">h</mi><mo id="S4.E16.m2.6.6.1.2b" xref="S4.E16.m2.6.6.1.2.cmml">⁢</mo><msup id="S4.E16.m2.6.6.1.1" xref="S4.E16.m2.6.6.1.1.cmml"><mrow id="S4.E16.m2.6.6.1.1.1.1" xref="S4.E16.m2.6.6.1.1.1.2.cmml"><mo id="S4.E16.m2.6.6.1.1.1.1.2" stretchy="false" xref="S4.E16.m2.6.6.1.1.1.2.cmml">(</mo><mi id="S4.E16.m2.1.1" xref="S4.E16.m2.1.1.cmml">τ</mi><mo id="S4.E16.m2.6.6.1.1.1.1.3" xref="S4.E16.m2.6.6.1.1.1.2.cmml">,</mo><msup id="S4.E16.m2.6.6.1.1.1.1.1" xref="S4.E16.m2.6.6.1.1.1.1.1.cmml"><mi id="S4.E16.m2.6.6.1.1.1.1.1.2" xref="S4.E16.m2.6.6.1.1.1.1.1.2.cmml">τ</mi><mo id="S4.E16.m2.6.6.1.1.1.1.1.3" xref="S4.E16.m2.6.6.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.E16.m2.6.6.1.1.1.1.4" stretchy="false" xref="S4.E16.m2.6.6.1.1.1.2.cmml">)</mo></mrow><mn id="S4.E16.m2.6.6.1.1.3" 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xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.2.2">𝑋</ci><ci id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.2.3.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.2.3">′</ci></apply><apply id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4"><csymbol cd="latexml" id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.1.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.1">conditional</csymbol><apply id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2"><csymbol cd="ambiguous" id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.1.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2">superscript</csymbol><ci id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.2.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.2">𝜏</ci><times id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.3.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.2.3"></times></apply><ci id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.3.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1c.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1"><eq id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.5.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S4.E16.m2.8.8.3.2.1.1.2.1.1.1.4.cmml" id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1d.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1"></share><ci id="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.6.cmml" xref="S4.E16.m2.8.8.3.2.1.1.2.1.1.1.6">𝜏</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m2.8c">\displaystyle=\frac{d}{d\tau}\alpha^{\prime}h(\tau,\tau^{*})^{2}=2h(\tau,\tau^% {*})\alpha^{\prime}f(\tau)\left(\Pr[X^{\prime}>\tau^{*}]-\Pr[X^{\prime}>\tau^{% *}\mid X=\tau]\right)~{}</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m2.8d">= divide start_ARG italic_d end_ARG start_ARG italic_d italic_τ end_ARG italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_h ( italic_τ , italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 2 italic_h ( italic_τ , italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_f ( italic_τ ) ( roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ] - roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∣ italic_X = italic_τ ] )</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="S4.SS1.SSS0.Px1.1.p1.7">By assumption <math alttext="h(\tau^{*},\tau^{*})\neq 0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.4.m1.2"><semantics id="S4.SS1.SSS0.Px1.1.p1.4.m1.2a"><mrow id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.cmml"><mrow 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xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.3"></times><ci id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.4.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.4">ℎ</ci><interval closure="open" id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2"><apply id="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.1.1.1.1.1.1.3"></times></apply><apply id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.2.2.2.2.3"></times></apply></interval></apply><cn id="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.4.cmml" type="integer" xref="S4.SS1.SSS0.Px1.1.p1.4.m1.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.4.m1.2c">h(\tau^{*},\tau^{*})\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.4.m1.2d">italic_h ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≠ 0</annotation></semantics></math> and <math alttext="f(\tau^{*})>0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.5.m2.1"><semantics id="S4.SS1.SSS0.Px1.1.p1.5.m2.1a"><mrow id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.cmml"><mrow id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.3.cmml">f</mi><mo id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.2.cmml">></mo><mn id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.1.p1.5.m2.1b"><apply id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1"><gt id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.2"></gt><apply id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1"><times id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.2"></times><ci id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.3">𝑓</ci><apply id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.1.1.1.1.3"></times></apply></apply><cn id="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px1.1.p1.5.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.5.m2.1c">f(\tau^{*})>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.5.m2.1d">italic_f ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) > 0</annotation></semantics></math>. Thus <math alttext="\Pr[X^{\prime}>\tau^{*}]-\Pr[X^{\prime}>\tau^{*}\mid X=\tau^{*}]=0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4"><semantics id="S4.SS1.SSS0.Px1.1.p1.6.m3.4a"><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.cmml"><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.cmml"><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.6.m3.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.1.1.cmml">Pr</mi><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1a" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.2.cmml">⁡</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.2.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.2.cmml">[</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.cmml"><msup 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xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.3.cmml">−</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.6.m3.2.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.2.2.cmml">Pr</mi><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1a" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.2.cmml">⁡</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.2.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.2.cmml">[</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.cmml"><msup id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.2.cmml">X</mi><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.3" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.3.cmml">></mo><mrow id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.cmml"><msup id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.2" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.3" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.1" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.1.cmml">∣</mo><mi id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.3" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.3.cmml">X</mi></mrow><mo id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.5" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.5.cmml">=</mo><msup 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xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2"><minus id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.3"></minus><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1"><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.1.1">Pr</ci><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1"><gt id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.1"></gt><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.2">𝑋</ci><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.2.3">′</ci></apply><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.3.3.1.1.1.1.1.3.3"></times></apply></apply></apply><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1"><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.2.2">Pr</ci><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1"><and id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1a.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1"></and><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1b.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1"><gt id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.3"></gt><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.2">𝑋</ci><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.2.3">′</ci></apply><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4"><csymbol cd="latexml" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.1">conditional</csymbol><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.2.3"></times></apply><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1c.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1"><eq id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.5.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.4.cmml" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1d.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1"></share><apply id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.2.2.1.1.1.6.3"></times></apply></apply></apply></apply></apply><cn id="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.4.cmml" type="integer" xref="S4.SS1.SSS0.Px1.1.p1.6.m3.4.4.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4c">\Pr[X^{\prime}>\tau^{*}]-\Pr[X^{\prime}>\tau^{*}\mid X=\tau^{*}]=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.6.m3.4d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ] - roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∣ italic_X = italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ] = 0</annotation></semantics></math>, or equivalently <math alttext="G(\tau^{*})-F(\tau^{*})=0" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.1.p1.7.m4.2"><semantics id="S4.SS1.SSS0.Px1.1.p1.7.m4.2a"><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.cmml"><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.cmml"><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.3.cmml">G</mi><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.3.cmml">−</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.3.cmml">F</mi><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.cmml">(</mo><msup id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.2" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.3" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.3.cmml">=</mo><mn id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.4" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.1.p1.7.m4.2b"><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2"><eq id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.3"></eq><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2"><minus id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.3"></minus><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1"><times id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.2"></times><ci id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.3">𝐺</ci><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2"><times id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.2"></times><ci id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.3">𝐹</ci><apply id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.2">𝜏</ci><times id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.2.2.1.1.1.3"></times></apply></apply></apply><cn id="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.4.cmml" type="integer" xref="S4.SS1.SSS0.Px1.1.p1.7.m4.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.1.p1.7.m4.2c">G(\tau^{*})-F(\tau^{*})=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.1.p1.7.m4.2d">italic_G ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_F ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math></p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px1.2.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.2.p2.6">For sufficiency, assume <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.1.m1.2"><semantics id="S4.SS1.SSS0.Px1.2.p2.1.m1.2a"><mrow id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.cmml"><mrow id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.cmml"><mi id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.2.cmml">G</mi><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.1" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.3.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.cmml"><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.2.p2.1.m1.1.1" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.1.1.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.1" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.cmml"><mi id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.2.cmml">F</mi><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.1" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.3.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.cmml"><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.2" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.1.m1.2b"><apply id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3"><eq id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.1"></eq><apply id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2"><times id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.1"></times><ci id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.2.2">𝐺</ci><ci id="S4.SS1.SSS0.Px1.2.p2.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.1.1">𝜏</ci></apply><apply id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3"><times id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.1"></times><ci id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.3.3.2">𝐹</ci><ci id="S4.SS1.SSS0.Px1.2.p2.1.m1.2.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.1.m1.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.1.m1.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>. By the above, <math alttext="\tau" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.2.m2.1"><semantics id="S4.SS1.SSS0.Px1.2.p2.2.m2.1a"><mi id="S4.SS1.SSS0.Px1.2.p2.2.m2.1.1" xref="S4.SS1.SSS0.Px1.2.p2.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.2.m2.1b"><ci id="S4.SS1.SSS0.Px1.2.p2.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.2.m2.1d">italic_τ</annotation></semantics></math> satisfies the first order condition for equilibrium. If <math alttext="h(x,\tau)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.3.m3.2"><semantics id="S4.SS1.SSS0.Px1.2.p2.3.m3.2a"><mrow id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.cmml"><mi id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.2" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.2.cmml">h</mi><mo id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.1" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.2" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.1.cmml"><mo id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.2.p2.3.m3.1.1" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.1.1.cmml">x</mi><mo id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.2.2" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.2" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.3.m3.2b"><apply id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3"><times id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.1"></times><ci id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.2">ℎ</ci><interval closure="open" id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.3.3.2"><ci id="S4.SS1.SSS0.Px1.2.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.1.1">𝑥</ci><ci id="S4.SS1.SSS0.Px1.2.p2.3.m3.2.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.3.m3.2.2">𝜏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.3.m3.2c">h(x,\tau)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.3.m3.2d">italic_h ( italic_x , italic_τ )</annotation></semantics></math> is non-negative and strictly single-peaked as a function of <math alttext="x" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.4.m4.1"><semantics id="S4.SS1.SSS0.Px1.2.p2.4.m4.1a"><mi id="S4.SS1.SSS0.Px1.2.p2.4.m4.1.1" xref="S4.SS1.SSS0.Px1.2.p2.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.4.m4.1b"><ci id="S4.SS1.SSS0.Px1.2.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.4.m4.1d">italic_x</annotation></semantics></math> then <math alttext="U_{i}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.5.m5.1"><semantics id="S4.SS1.SSS0.Px1.2.p2.5.m5.1a"><msub id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.2" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.2.cmml">U</mi><mi id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.3" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.5.m5.1b"><apply id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.2">𝑈</ci><ci id="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.2.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.5.m5.1c">U_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.5.m5.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is also strictly single-peaked as a function of <math alttext="x" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.2.p2.6.m6.1"><semantics id="S4.SS1.SSS0.Px1.2.p2.6.m6.1a"><mi id="S4.SS1.SSS0.Px1.2.p2.6.m6.1.1" xref="S4.SS1.SSS0.Px1.2.p2.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.2.p2.6.m6.1b"><ci id="S4.SS1.SSS0.Px1.2.p2.6.m6.1.1.cmml" xref="S4.SS1.SSS0.Px1.2.p2.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.2.p2.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.2.p2.6.m6.1d">italic_x</annotation></semantics></math> (i.e. it is strictly increasing and then strictly decreasing), so the unique point satisfying the FOC is the global maximum. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p3"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p3.1">Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem5" title="Theorem 5. ‣ Equilibrium results. ‣ 4.1 Equilibrium Characterization ‣ 4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">5</span></a> shows that DMI has the same necessary condition for equilibrium as DG. So, modulo the slightly different sufficient condition for equilibrium, its equilibria can be found at the same thresholds: those where <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.1.m1.2"><semantics id="S4.SS1.SSS0.Px1.p3.1.m1.2a"><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.3" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.cmml"><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.cmml"><mi id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.2.cmml">G</mi><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.cmml"><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.p3.1.m1.1.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.1.1.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.cmml"><mi id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.2.cmml">F</mi><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.cmml"><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.cmml">(</mo><mi id="S4.SS1.SSS0.Px1.p3.1.m1.2.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2.cmml">τ</mi><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.1.m1.2b"><apply id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3"><eq id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.1"></eq><apply id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2"><times id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.1"></times><ci id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.2.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.2.2">𝐺</ci><ci id="S4.SS1.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.1.1">𝜏</ci></apply><apply id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3"><times id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.1"></times><ci id="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.2.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.3.3.2">𝐹</ci><ci id="S4.SS1.SSS0.Px1.p3.1.m1.2.2.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.1.m1.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.1.m1.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>. In particular our result about the dynamics (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem4" title="Theorem 4. ‣ 3.2 Dynamics ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>) immediately applies with the appropriate sufficient condition, and the Gaussian case has the same equilibra with the same stability.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Beyond consistent threshold strategies.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px2.p1.2">The above analysis is restricted to our weaker notion of equilibrium in the space of consistent threshold strategies. It is unclear whether these remain equilibria if agents can use any consistent strategy, or if agents can use any four threshold strategies. Even more broadly, one could consider <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS0.Px2.p1.2.1">joint-task</em> strategies <math alttext="\sigma:\mathbb{R}^{4}\to\{H,L\}^{4}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.1.m1.2"><semantics id="S4.SS1.SSS0.Px2.p1.1.m1.2a"><mrow id="S4.SS1.SSS0.Px2.p1.1.m1.2.3" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.cmml"><mi id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.2.cmml">σ</mi><mo id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.cmml"><msup id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.cmml"><mi id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.2.cmml">ℝ</mi><mn id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.3" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.3.cmml">4</mn></msup><mo id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.1" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.1.cmml">→</mo><msup id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.cmml"><mrow id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.1.cmml"><mo id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.1.cmml">{</mo><mi id="S4.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">H</mi><mo id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.2.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.1.cmml">,</mo><mi id="S4.SS1.SSS0.Px2.p1.1.m1.2.2" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.2.cmml">L</mi><mo id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.2.3" stretchy="false" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.1.cmml">}</mo></mrow><mn id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.3" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.3.cmml">4</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.1.m1.2b"><apply id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3"><ci id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.1">:</ci><ci id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.2.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.2">𝜎</ci><apply id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3"><ci id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.1">→</ci><apply id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.2">ℝ</ci><cn id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.2.3">4</cn></apply><apply id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3">superscript</csymbol><set id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.2.2"><ci id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1">𝐻</ci><ci id="S4.SS1.SSS0.Px2.p1.1.m1.2.2.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.2">𝐿</ci></set><cn id="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.3.cmml" type="integer" xref="S4.SS1.SSS0.Px2.p1.1.m1.2.3.3.3.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.1.m1.2c">\sigma:\mathbb{R}^{4}\to\{H,L\}^{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.1.m1.2d">italic_σ : blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT → { italic_H , italic_L } start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, which map all four signals simultaneously to all four reports. While not shown in <cite class="ltx_cite ltx_citemacro_citet">Kong [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib9" title="">2020</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib10" title="">2024</a>]</cite>, in the binary report setting the DMI mechanism turns out to satisfy the even stronger property that truthfulness is an equilibrium among all joint-task strategies. We prove this claim in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A3" title="Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">C</span></a>. We then show that truthfulness is <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS0.Px2.p1.2.2">not</em> an equilibrium for real-valued reports: in many cases the best response flips one or two of the truthful reports to increase the chance of a nonzero payoff in <math alttext="M_{\textrm{DMI}}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.2.m2.1a"><msub id="S4.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">M</mi><mtext id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3a.cmml">DMI</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝑀</ci><ci id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3a.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3"><mtext id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.3">DMI</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.2.m2.1c">M_{\textrm{DMI}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.2.m2.1d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Robust Bayesian Truth Serum</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We now study the Robust Bayesian Truth Serum (RBTS) mechanism proposed by <cite class="ltx_cite ltx_citemacro_citet">Witkowski and Parkes [<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib19" title="">2012</a>]</cite>. RBTS is inspired by the Bayesian Truth Serum (BTS) <cite class="ltx_cite ltx_citemacro_citep">[Prelec, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib13" title="">2004</a>]</cite>. However, while incentive alignment under BTS requires an arbitrarily large number of agents, RBTS guarantees truthfulness for <math alttext="n\geq 3" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">n</mi><mo id="S5.p1.1.m1.1.1.1" xref="S5.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><geq id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1"></geq><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝑛</ci><cn id="S5.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">n\geq 3</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_n ≥ 3</annotation></semantics></math> agents.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.9">RBTS is a non-minimal mechanism: each agent <math alttext="i" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_i</annotation></semantics></math> submits both an information report <math alttext="r_{i}\in\{L,H\}" class="ltx_Math" display="inline" id="S5.p2.2.m2.2"><semantics id="S5.p2.2.m2.2a"><mrow id="S5.p2.2.m2.2.3" xref="S5.p2.2.m2.2.3.cmml"><msub id="S5.p2.2.m2.2.3.2" xref="S5.p2.2.m2.2.3.2.cmml"><mi id="S5.p2.2.m2.2.3.2.2" xref="S5.p2.2.m2.2.3.2.2.cmml">r</mi><mi id="S5.p2.2.m2.2.3.2.3" xref="S5.p2.2.m2.2.3.2.3.cmml">i</mi></msub><mo id="S5.p2.2.m2.2.3.1" xref="S5.p2.2.m2.2.3.1.cmml">∈</mo><mrow id="S5.p2.2.m2.2.3.3.2" xref="S5.p2.2.m2.2.3.3.1.cmml"><mo id="S5.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S5.p2.2.m2.2.3.3.1.cmml">{</mo><mi id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">L</mi><mo id="S5.p2.2.m2.2.3.3.2.2" xref="S5.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S5.p2.2.m2.2.2" xref="S5.p2.2.m2.2.2.cmml">H</mi><mo id="S5.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S5.p2.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.2b"><apply id="S5.p2.2.m2.2.3.cmml" xref="S5.p2.2.m2.2.3"><in id="S5.p2.2.m2.2.3.1.cmml" xref="S5.p2.2.m2.2.3.1"></in><apply id="S5.p2.2.m2.2.3.2.cmml" xref="S5.p2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S5.p2.2.m2.2.3.2.1.cmml" xref="S5.p2.2.m2.2.3.2">subscript</csymbol><ci id="S5.p2.2.m2.2.3.2.2.cmml" xref="S5.p2.2.m2.2.3.2.2">𝑟</ci><ci id="S5.p2.2.m2.2.3.2.3.cmml" xref="S5.p2.2.m2.2.3.2.3">𝑖</ci></apply><set id="S5.p2.2.m2.2.3.3.1.cmml" xref="S5.p2.2.m2.2.3.3.2"><ci id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1">𝐿</ci><ci id="S5.p2.2.m2.2.2.cmml" xref="S5.p2.2.m2.2.2">𝐻</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.2c">r_{i}\in\{L,H\}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.2d">italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { italic_L , italic_H }</annotation></semantics></math> and also a <em class="ltx_emph ltx_font_italic" id="S5.p2.9.1">prediction</em> report <math alttext="p_{i}\in[0,1]" class="ltx_Math" display="inline" id="S5.p2.3.m3.2"><semantics id="S5.p2.3.m3.2a"><mrow id="S5.p2.3.m3.2.3" xref="S5.p2.3.m3.2.3.cmml"><msub id="S5.p2.3.m3.2.3.2" xref="S5.p2.3.m3.2.3.2.cmml"><mi id="S5.p2.3.m3.2.3.2.2" xref="S5.p2.3.m3.2.3.2.2.cmml">p</mi><mi id="S5.p2.3.m3.2.3.2.3" xref="S5.p2.3.m3.2.3.2.3.cmml">i</mi></msub><mo id="S5.p2.3.m3.2.3.1" xref="S5.p2.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.p2.3.m3.2.3.3.2" xref="S5.p2.3.m3.2.3.3.1.cmml"><mo id="S5.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S5.p2.3.m3.2.3.3.1.cmml">[</mo><mn id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml">0</mn><mo id="S5.p2.3.m3.2.3.3.2.2" xref="S5.p2.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.p2.3.m3.2.2" xref="S5.p2.3.m3.2.2.cmml">1</mn><mo id="S5.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S5.p2.3.m3.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.2b"><apply id="S5.p2.3.m3.2.3.cmml" xref="S5.p2.3.m3.2.3"><in id="S5.p2.3.m3.2.3.1.cmml" xref="S5.p2.3.m3.2.3.1"></in><apply id="S5.p2.3.m3.2.3.2.cmml" xref="S5.p2.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.p2.3.m3.2.3.2.1.cmml" xref="S5.p2.3.m3.2.3.2">subscript</csymbol><ci id="S5.p2.3.m3.2.3.2.2.cmml" xref="S5.p2.3.m3.2.3.2.2">𝑝</ci><ci id="S5.p2.3.m3.2.3.2.3.cmml" xref="S5.p2.3.m3.2.3.2.3">𝑖</ci></apply><interval closure="closed" id="S5.p2.3.m3.2.3.3.1.cmml" xref="S5.p2.3.m3.2.3.3.2"><cn id="S5.p2.3.m3.1.1.cmml" type="integer" xref="S5.p2.3.m3.1.1">0</cn><cn id="S5.p2.3.m3.2.2.cmml" type="integer" xref="S5.p2.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.2c">p_{i}\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ [ 0 , 1 ]</annotation></semantics></math>. The prediction report represents agent <math alttext="i" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mi id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><ci id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">italic_i</annotation></semantics></math>’s belief about the frequency of <math alttext="H" class="ltx_Math" display="inline" id="S5.p2.5.m5.1"><semantics id="S5.p2.5.m5.1a"><mi id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.1b"><ci id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.1d">italic_H</annotation></semantics></math> reports from all agents. The mechanism incentivizes prediction reports using the <em class="ltx_emph ltx_font_italic" id="S5.p2.9.2">Brier score</em> <math alttext="S(p,r)=2p\mathbf{1}[r=H]-p^{2}" class="ltx_Math" display="inline" id="S5.p2.6.m6.3"><semantics id="S5.p2.6.m6.3a"><mrow id="S5.p2.6.m6.3.3" xref="S5.p2.6.m6.3.3.cmml"><mrow id="S5.p2.6.m6.3.3.3" xref="S5.p2.6.m6.3.3.3.cmml"><mi id="S5.p2.6.m6.3.3.3.2" xref="S5.p2.6.m6.3.3.3.2.cmml">S</mi><mo id="S5.p2.6.m6.3.3.3.1" xref="S5.p2.6.m6.3.3.3.1.cmml">⁢</mo><mrow id="S5.p2.6.m6.3.3.3.3.2" xref="S5.p2.6.m6.3.3.3.3.1.cmml"><mo id="S5.p2.6.m6.3.3.3.3.2.1" stretchy="false" xref="S5.p2.6.m6.3.3.3.3.1.cmml">(</mo><mi id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml">p</mi><mo id="S5.p2.6.m6.3.3.3.3.2.2" xref="S5.p2.6.m6.3.3.3.3.1.cmml">,</mo><mi id="S5.p2.6.m6.2.2" xref="S5.p2.6.m6.2.2.cmml">r</mi><mo id="S5.p2.6.m6.3.3.3.3.2.3" stretchy="false" xref="S5.p2.6.m6.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S5.p2.6.m6.3.3.2" xref="S5.p2.6.m6.3.3.2.cmml">=</mo><mrow id="S5.p2.6.m6.3.3.1" xref="S5.p2.6.m6.3.3.1.cmml"><mrow id="S5.p2.6.m6.3.3.1.1" xref="S5.p2.6.m6.3.3.1.1.cmml"><mn id="S5.p2.6.m6.3.3.1.1.3" xref="S5.p2.6.m6.3.3.1.1.3.cmml">2</mn><mo id="S5.p2.6.m6.3.3.1.1.2" xref="S5.p2.6.m6.3.3.1.1.2.cmml">⁢</mo><mi id="S5.p2.6.m6.3.3.1.1.4" xref="S5.p2.6.m6.3.3.1.1.4.cmml">p</mi><mo id="S5.p2.6.m6.3.3.1.1.2a" xref="S5.p2.6.m6.3.3.1.1.2.cmml">⁢</mo><mn id="S5.p2.6.m6.3.3.1.1.5" xref="S5.p2.6.m6.3.3.1.1.5.cmml">𝟏</mn><mo id="S5.p2.6.m6.3.3.1.1.2b" xref="S5.p2.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.p2.6.m6.3.3.1.1.1.1" xref="S5.p2.6.m6.3.3.1.1.1.2.cmml"><mo id="S5.p2.6.m6.3.3.1.1.1.1.2" stretchy="false" xref="S5.p2.6.m6.3.3.1.1.1.2.1.cmml">[</mo><mrow id="S5.p2.6.m6.3.3.1.1.1.1.1" xref="S5.p2.6.m6.3.3.1.1.1.1.1.cmml"><mi id="S5.p2.6.m6.3.3.1.1.1.1.1.2" xref="S5.p2.6.m6.3.3.1.1.1.1.1.2.cmml">r</mi><mo id="S5.p2.6.m6.3.3.1.1.1.1.1.1" xref="S5.p2.6.m6.3.3.1.1.1.1.1.1.cmml">=</mo><mi id="S5.p2.6.m6.3.3.1.1.1.1.1.3" xref="S5.p2.6.m6.3.3.1.1.1.1.1.3.cmml">H</mi></mrow><mo id="S5.p2.6.m6.3.3.1.1.1.1.3" stretchy="false" xref="S5.p2.6.m6.3.3.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.p2.6.m6.3.3.1.2" xref="S5.p2.6.m6.3.3.1.2.cmml">−</mo><msup id="S5.p2.6.m6.3.3.1.3" xref="S5.p2.6.m6.3.3.1.3.cmml"><mi id="S5.p2.6.m6.3.3.1.3.2" xref="S5.p2.6.m6.3.3.1.3.2.cmml">p</mi><mn id="S5.p2.6.m6.3.3.1.3.3" xref="S5.p2.6.m6.3.3.1.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.3b"><apply id="S5.p2.6.m6.3.3.cmml" xref="S5.p2.6.m6.3.3"><eq id="S5.p2.6.m6.3.3.2.cmml" xref="S5.p2.6.m6.3.3.2"></eq><apply id="S5.p2.6.m6.3.3.3.cmml" xref="S5.p2.6.m6.3.3.3"><times id="S5.p2.6.m6.3.3.3.1.cmml" xref="S5.p2.6.m6.3.3.3.1"></times><ci id="S5.p2.6.m6.3.3.3.2.cmml" xref="S5.p2.6.m6.3.3.3.2">𝑆</ci><interval closure="open" id="S5.p2.6.m6.3.3.3.3.1.cmml" xref="S5.p2.6.m6.3.3.3.3.2"><ci id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1">𝑝</ci><ci id="S5.p2.6.m6.2.2.cmml" xref="S5.p2.6.m6.2.2">𝑟</ci></interval></apply><apply id="S5.p2.6.m6.3.3.1.cmml" xref="S5.p2.6.m6.3.3.1"><minus id="S5.p2.6.m6.3.3.1.2.cmml" xref="S5.p2.6.m6.3.3.1.2"></minus><apply id="S5.p2.6.m6.3.3.1.1.cmml" xref="S5.p2.6.m6.3.3.1.1"><times id="S5.p2.6.m6.3.3.1.1.2.cmml" xref="S5.p2.6.m6.3.3.1.1.2"></times><cn id="S5.p2.6.m6.3.3.1.1.3.cmml" type="integer" xref="S5.p2.6.m6.3.3.1.1.3">2</cn><ci id="S5.p2.6.m6.3.3.1.1.4.cmml" xref="S5.p2.6.m6.3.3.1.1.4">𝑝</ci><cn id="S5.p2.6.m6.3.3.1.1.5.cmml" type="integer" xref="S5.p2.6.m6.3.3.1.1.5">1</cn><apply id="S5.p2.6.m6.3.3.1.1.1.2.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1"><csymbol cd="latexml" id="S5.p2.6.m6.3.3.1.1.1.2.1.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1.2">delimited-[]</csymbol><apply id="S5.p2.6.m6.3.3.1.1.1.1.1.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1.1"><eq id="S5.p2.6.m6.3.3.1.1.1.1.1.1.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1.1.1"></eq><ci id="S5.p2.6.m6.3.3.1.1.1.1.1.2.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1.1.2">𝑟</ci><ci id="S5.p2.6.m6.3.3.1.1.1.1.1.3.cmml" xref="S5.p2.6.m6.3.3.1.1.1.1.1.3">𝐻</ci></apply></apply></apply><apply id="S5.p2.6.m6.3.3.1.3.cmml" xref="S5.p2.6.m6.3.3.1.3"><csymbol cd="ambiguous" id="S5.p2.6.m6.3.3.1.3.1.cmml" xref="S5.p2.6.m6.3.3.1.3">superscript</csymbol><ci id="S5.p2.6.m6.3.3.1.3.2.cmml" xref="S5.p2.6.m6.3.3.1.3.2">𝑝</ci><cn id="S5.p2.6.m6.3.3.1.3.3.cmml" type="integer" xref="S5.p2.6.m6.3.3.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.3c">S(p,r)=2p\mathbf{1}[r=H]-p^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.3d">italic_S ( italic_p , italic_r ) = 2 italic_p bold_1 [ italic_r = italic_H ] - italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (also known as the quadratic score), which is an example of a strictly proper scoring rule <cite class="ltx_cite ltx_citemacro_citep">[Brier, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib3" title="">1950</a>, Gneiting and Raftery, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib8" title="">2007</a>]</cite>. RBTS randomly picks a reference agent <math alttext="j" class="ltx_Math" display="inline" id="S5.p2.7.m7.1"><semantics id="S5.p2.7.m7.1a"><mi id="S5.p2.7.m7.1.1" xref="S5.p2.7.m7.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S5.p2.7.m7.1b"><ci id="S5.p2.7.m7.1.1.cmml" xref="S5.p2.7.m7.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m7.1c">j</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m7.1d">italic_j</annotation></semantics></math> and peer agent <math alttext="k" class="ltx_Math" display="inline" id="S5.p2.8.m8.1"><semantics id="S5.p2.8.m8.1a"><mi id="S5.p2.8.m8.1.1" xref="S5.p2.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p2.8.m8.1b"><ci id="S5.p2.8.m8.1.1.cmml" xref="S5.p2.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m8.1d">italic_k</annotation></semantics></math>, and pays agent <math alttext="i" class="ltx_Math" display="inline" id="S5.p2.9.m9.1"><semantics id="S5.p2.9.m9.1a"><mi id="S5.p2.9.m9.1.1" xref="S5.p2.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.p2.9.m9.1b"><ci id="S5.p2.9.m9.1.1.cmml" xref="S5.p2.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.p2.9.m9.1d">italic_i</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S5.E17"> <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="M_{\textrm{RBTS}}((r_{i},p_{i}),(r_{j},p_{j}),(r_{k},p_{k}))=S(p_{j}+\delta t(% r_{i}),r_{k})+S(p_{i},r_{k})," class="ltx_Math" display="block" id="S5.E17.m1.1"><semantics id="S5.E17.m1.1a"><mrow id="S5.E17.m1.1.1.1" xref="S5.E17.m1.1.1.1.1.cmml"><mrow id="S5.E17.m1.1.1.1.1" 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italic_i end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) = italic_S ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_δ italic_t ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_S ( italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_k 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">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p2.12">where <math alttext="t(L)=-1" class="ltx_Math" display="inline" id="S5.p2.10.m1.1"><semantics id="S5.p2.10.m1.1a"><mrow id="S5.p2.10.m1.1.2" xref="S5.p2.10.m1.1.2.cmml"><mrow id="S5.p2.10.m1.1.2.2" xref="S5.p2.10.m1.1.2.2.cmml"><mi id="S5.p2.10.m1.1.2.2.2" xref="S5.p2.10.m1.1.2.2.2.cmml">t</mi><mo id="S5.p2.10.m1.1.2.2.1" xref="S5.p2.10.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.p2.10.m1.1.2.2.3.2" xref="S5.p2.10.m1.1.2.2.cmml"><mo id="S5.p2.10.m1.1.2.2.3.2.1" stretchy="false" xref="S5.p2.10.m1.1.2.2.cmml">(</mo><mi id="S5.p2.10.m1.1.1" xref="S5.p2.10.m1.1.1.cmml">L</mi><mo id="S5.p2.10.m1.1.2.2.3.2.2" stretchy="false" xref="S5.p2.10.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.p2.10.m1.1.2.1" xref="S5.p2.10.m1.1.2.1.cmml">=</mo><mrow id="S5.p2.10.m1.1.2.3" xref="S5.p2.10.m1.1.2.3.cmml"><mo id="S5.p2.10.m1.1.2.3a" xref="S5.p2.10.m1.1.2.3.cmml">−</mo><mn id="S5.p2.10.m1.1.2.3.2" xref="S5.p2.10.m1.1.2.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.10.m1.1b"><apply id="S5.p2.10.m1.1.2.cmml" xref="S5.p2.10.m1.1.2"><eq 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id="S5.p2.11.m2.1.1.cmml" xref="S5.p2.11.m2.1.1">𝐻</ci></apply><cn id="S5.p2.11.m2.1.2.3.cmml" type="integer" xref="S5.p2.11.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.11.m2.1c">t(H)=1</annotation><annotation encoding="application/x-llamapun" id="S5.p2.11.m2.1d">italic_t ( italic_H ) = 1</annotation></semantics></math>, and <math alttext="\delta>0" class="ltx_Math" display="inline" id="S5.p2.12.m3.1"><semantics id="S5.p2.12.m3.1a"><mrow id="S5.p2.12.m3.1.1" xref="S5.p2.12.m3.1.1.cmml"><mi id="S5.p2.12.m3.1.1.2" xref="S5.p2.12.m3.1.1.2.cmml">δ</mi><mo id="S5.p2.12.m3.1.1.1" xref="S5.p2.12.m3.1.1.1.cmml">></mo><mn id="S5.p2.12.m3.1.1.3" xref="S5.p2.12.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.12.m3.1b"><apply id="S5.p2.12.m3.1.1.cmml" xref="S5.p2.12.m3.1.1"><gt id="S5.p2.12.m3.1.1.1.cmml" xref="S5.p2.12.m3.1.1.1"></gt><ci id="S5.p2.12.m3.1.1.2.cmml" xref="S5.p2.12.m3.1.1.2">𝛿</ci><cn 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Under binary signals and reports, truthfulness over both the information and prediction report forms a strict Bayes-Nash equilibrium.</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>Equilibrium Characterization</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.10">If we consider a threshold strategy <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1a"><mi id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1b"><ci id="S5.SS1.p1.1.m1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1d">italic_τ</annotation></semantics></math> in our real-valued signal model, an agent with signal 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id="S5.SS1.p1.3.m3.5.5.3.1.3.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.3">𝑃</ci><list id="S5.SS1.p1.3.m3.5.5.3.1.1.2.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.1.1"><ci id="S5.SS1.p1.3.m3.2.2.cmml" xref="S5.SS1.p1.3.m3.2.2">𝜏</ci><apply id="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.1.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.2.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.2">𝑥</ci><ci id="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.3.cmml" xref="S5.SS1.p1.3.m3.5.5.3.1.1.1.1.3">𝑖</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.3.m3.5c">p_{i}(x_{i})=\Pr[X^{\prime}>\tau\mid X=x_{i}]=1-P(\tau;x_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.3.m3.5d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ ∣ italic_X = italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] = 1 - italic_P ( italic_τ ; italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Since the Brier score <math alttext="S" class="ltx_Math" display="inline" id="S5.SS1.p1.4.m4.1"><semantics id="S5.SS1.p1.4.m4.1a"><mi id="S5.SS1.p1.4.m4.1.1" xref="S5.SS1.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.4.m4.1b"><ci id="S5.SS1.p1.4.m4.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.4.m4.1d">italic_S</annotation></semantics></math> is proper, it immediately follows that agents will report <math alttext="p_{i}" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m5.1"><semantics id="S5.SS1.p1.5.m5.1a"><msub id="S5.SS1.p1.5.m5.1.1" xref="S5.SS1.p1.5.m5.1.1.cmml"><mi id="S5.SS1.p1.5.m5.1.1.2" xref="S5.SS1.p1.5.m5.1.1.2.cmml">p</mi><mi id="S5.SS1.p1.5.m5.1.1.3" xref="S5.SS1.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m5.1b"><apply id="S5.SS1.p1.5.m5.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S5.SS1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.p1.5.m5.1.1.2">𝑝</ci><ci id="S5.SS1.p1.5.m5.1.1.3.cmml" xref="S5.SS1.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m5.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m5.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> truthfully in a symmetric threshold Bayes-Nash equilibrium. Thus it is WLOG that we can study incentives in threshold equilibria under only the information score, <math alttext="M_{\textrm{RBTS}}(r_{i},p_{j},r_{k})=S(p_{j}+\delta t(r_{i}),r_{k})" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m6.5"><semantics id="S5.SS1.p1.6.m6.5a"><mrow id="S5.SS1.p1.6.m6.5.5" xref="S5.SS1.p1.6.m6.5.5.cmml"><mrow id="S5.SS1.p1.6.m6.3.3.3" xref="S5.SS1.p1.6.m6.3.3.3.cmml"><msub id="S5.SS1.p1.6.m6.3.3.3.5" xref="S5.SS1.p1.6.m6.3.3.3.5.cmml"><mi id="S5.SS1.p1.6.m6.3.3.3.5.2" xref="S5.SS1.p1.6.m6.3.3.3.5.2.cmml">M</mi><mtext id="S5.SS1.p1.6.m6.3.3.3.5.3" xref="S5.SS1.p1.6.m6.3.3.3.5.3a.cmml">RBTS</mtext></msub><mo id="S5.SS1.p1.6.m6.3.3.3.4" xref="S5.SS1.p1.6.m6.3.3.3.4.cmml">⁢</mo><mrow id="S5.SS1.p1.6.m6.3.3.3.3.3" xref="S5.SS1.p1.6.m6.3.3.3.3.4.cmml"><mo id="S5.SS1.p1.6.m6.3.3.3.3.3.4" stretchy="false" xref="S5.SS1.p1.6.m6.3.3.3.3.4.cmml">(</mo><msub id="S5.SS1.p1.6.m6.1.1.1.1.1.1" xref="S5.SS1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.SS1.p1.6.m6.1.1.1.1.1.1.2" 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xref="S5.SS1.p1.6.m6.3.3.3.5.3"><mtext id="S5.SS1.p1.6.m6.3.3.3.5.3.cmml" mathsize="70%" xref="S5.SS1.p1.6.m6.3.3.3.5.3">RBTS</mtext></ci></apply><vector id="S5.SS1.p1.6.m6.3.3.3.3.4.cmml" xref="S5.SS1.p1.6.m6.3.3.3.3.3"><apply id="S5.SS1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.6.m6.1.1.1.1.1.1.2">𝑟</ci><ci id="S5.SS1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.6.m6.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S5.SS1.p1.6.m6.2.2.2.2.2.2.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S5.SS1.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.2.2.2">𝑝</ci><ci id="S5.SS1.p1.6.m6.2.2.2.2.2.2.3.cmml" xref="S5.SS1.p1.6.m6.2.2.2.2.2.2.3">𝑗</ci></apply><apply 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id="S5.SS1.p1.6.m6.4.4.4.1.1.1.3.1.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.3">subscript</csymbol><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.3.2.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.3.2">𝑝</ci><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.3.3.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.3.3">𝑗</ci></apply><apply id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1"><times id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.2.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.2"></times><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.3.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.3">𝛿</ci><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.4.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.4">𝑡</ci><apply id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.2">𝑟</ci><ci id="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.6.m6.4.4.4.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S5.SS1.p1.6.m6.5.5.5.2.2.2.cmml" xref="S5.SS1.p1.6.m6.5.5.5.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m6.5.5.5.2.2.2.1.cmml" xref="S5.SS1.p1.6.m6.5.5.5.2.2.2">subscript</csymbol><ci id="S5.SS1.p1.6.m6.5.5.5.2.2.2.2.cmml" xref="S5.SS1.p1.6.m6.5.5.5.2.2.2.2">𝑟</ci><ci id="S5.SS1.p1.6.m6.5.5.5.2.2.2.3.cmml" xref="S5.SS1.p1.6.m6.5.5.5.2.2.2.3">𝑘</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m6.5c">M_{\textrm{RBTS}}(r_{i},p_{j},r_{k})=S(p_{j}+\delta t(r_{i}),r_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m6.5d">italic_M start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_S ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_δ italic_t ( italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. Moreover, we can operate under the assumption that the prediction report <math alttext="p_{j}" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m7.1"><semantics id="S5.SS1.p1.7.m7.1a"><msub id="S5.SS1.p1.7.m7.1.1" xref="S5.SS1.p1.7.m7.1.1.cmml"><mi id="S5.SS1.p1.7.m7.1.1.2" xref="S5.SS1.p1.7.m7.1.1.2.cmml">p</mi><mi id="S5.SS1.p1.7.m7.1.1.3" xref="S5.SS1.p1.7.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m7.1b"><apply id="S5.SS1.p1.7.m7.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.7.m7.1.1.1.cmml" xref="S5.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S5.SS1.p1.7.m7.1.1.2.cmml" xref="S5.SS1.p1.7.m7.1.1.2">𝑝</ci><ci id="S5.SS1.p1.7.m7.1.1.3.cmml" xref="S5.SS1.p1.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m7.1c">p_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m7.1d">italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is truthful, i.e. <math alttext="p_{j}=p_{j}(x_{j})=1-P(\tau;x_{j})" class="ltx_Math" display="inline" id="S5.SS1.p1.8.m8.3"><semantics id="S5.SS1.p1.8.m8.3a"><mrow id="S5.SS1.p1.8.m8.3.3" xref="S5.SS1.p1.8.m8.3.3.cmml"><msub id="S5.SS1.p1.8.m8.3.3.4" xref="S5.SS1.p1.8.m8.3.3.4.cmml"><mi id="S5.SS1.p1.8.m8.3.3.4.2" xref="S5.SS1.p1.8.m8.3.3.4.2.cmml">p</mi><mi id="S5.SS1.p1.8.m8.3.3.4.3" xref="S5.SS1.p1.8.m8.3.3.4.3.cmml">j</mi></msub><mo id="S5.SS1.p1.8.m8.3.3.5" xref="S5.SS1.p1.8.m8.3.3.5.cmml">=</mo><mrow id="S5.SS1.p1.8.m8.2.2.1" xref="S5.SS1.p1.8.m8.2.2.1.cmml"><msub id="S5.SS1.p1.8.m8.2.2.1.3" xref="S5.SS1.p1.8.m8.2.2.1.3.cmml"><mi id="S5.SS1.p1.8.m8.2.2.1.3.2" xref="S5.SS1.p1.8.m8.2.2.1.3.2.cmml">p</mi><mi id="S5.SS1.p1.8.m8.2.2.1.3.3" xref="S5.SS1.p1.8.m8.2.2.1.3.3.cmml">j</mi></msub><mo id="S5.SS1.p1.8.m8.2.2.1.2" xref="S5.SS1.p1.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.p1.8.m8.2.2.1.1.1" 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xref="S5.SS1.p1.8.m8.3.3.2.1.3">𝑃</ci><list id="S5.SS1.p1.8.m8.3.3.2.1.1.2.cmml" xref="S5.SS1.p1.8.m8.3.3.2.1.1.1"><ci id="S5.SS1.p1.8.m8.1.1.cmml" xref="S5.SS1.p1.8.m8.1.1">𝜏</ci><apply id="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.cmml" xref="S5.SS1.p1.8.m8.3.3.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.1.cmml" xref="S5.SS1.p1.8.m8.3.3.2.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.2.cmml" xref="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.2">𝑥</ci><ci id="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.3.cmml" xref="S5.SS1.p1.8.m8.3.3.2.1.1.1.1.3">𝑗</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.8.m8.3c">p_{j}=p_{j}(x_{j})=1-P(\tau;x_{j})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.8.m8.3d">italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = 1 - italic_P ( italic_τ ; italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> for agent <math alttext="j" class="ltx_Math" display="inline" id="S5.SS1.p1.9.m9.1"><semantics id="S5.SS1.p1.9.m9.1a"><mi id="S5.SS1.p1.9.m9.1.1" xref="S5.SS1.p1.9.m9.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.9.m9.1b"><ci id="S5.SS1.p1.9.m9.1.1.cmml" xref="S5.SS1.p1.9.m9.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.9.m9.1c">j</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.9.m9.1d">italic_j</annotation></semantics></math>’s true signal <math alttext="x_{j}" class="ltx_Math" display="inline" id="S5.SS1.p1.10.m10.1"><semantics id="S5.SS1.p1.10.m10.1a"><msub id="S5.SS1.p1.10.m10.1.1" xref="S5.SS1.p1.10.m10.1.1.cmml"><mi id="S5.SS1.p1.10.m10.1.1.2" xref="S5.SS1.p1.10.m10.1.1.2.cmml">x</mi><mi id="S5.SS1.p1.10.m10.1.1.3" xref="S5.SS1.p1.10.m10.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.10.m10.1b"><apply id="S5.SS1.p1.10.m10.1.1.cmml" xref="S5.SS1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.10.m10.1.1.1.cmml" xref="S5.SS1.p1.10.m10.1.1">subscript</csymbol><ci id="S5.SS1.p1.10.m10.1.1.2.cmml" xref="S5.SS1.p1.10.m10.1.1.2">𝑥</ci><ci id="S5.SS1.p1.10.m10.1.1.3.cmml" xref="S5.SS1.p1.10.m10.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.10.m10.1c">x_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.10.m10.1d">italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.3">Assume an agent’s signal <math alttext="x" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><mi id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><ci id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">italic_x</annotation></semantics></math> satisfies <math alttext="x\leq\tau" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><mrow id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml"><mi id="S5.SS1.p2.2.m2.1.1.2" xref="S5.SS1.p2.2.m2.1.1.2.cmml">x</mi><mo id="S5.SS1.p2.2.m2.1.1.1" xref="S5.SS1.p2.2.m2.1.1.1.cmml">≤</mo><mi id="S5.SS1.p2.2.m2.1.1.3" xref="S5.SS1.p2.2.m2.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><apply id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1"><leq id="S5.SS1.p2.2.m2.1.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1.1"></leq><ci id="S5.SS1.p2.2.m2.1.1.2.cmml" xref="S5.SS1.p2.2.m2.1.1.2">𝑥</ci><ci id="S5.SS1.p2.2.m2.1.1.3.cmml" xref="S5.SS1.p2.2.m2.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_x ≤ italic_τ</annotation></semantics></math>. Then a threshold equilibrium <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS1.p2.3.m3.1"><semantics id="S5.SS1.p2.3.m3.1a"><mi id="S5.SS1.p2.3.m3.1.1" xref="S5.SS1.p2.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.3.m3.1b"><ci id="S5.SS1.p2.3.m3.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.3.m3.1d">italic_τ</annotation></semantics></math> must satisfy</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx8"> <tbody id="S5.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathop{\mathbb{E}}_{x_{j},x_{k}\sim\beta(x)}\left[S\left(p_{j}(x% _{j})-\delta,\sigma(x_{k})\right)\right]" class="ltx_Math" display="inline" 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italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT [ italic_S ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - italic_δ , italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\mathop{\mathbb{E}}_{x_{j},x_{k}\sim\beta(x)}\left[S\left(p_{% j}(x_{j})+\delta,\sigma(x_{k})\right)\right]" class="ltx_Math" display="inline" id="S5.Ex13.m2.4"><semantics id="S5.Ex13.m2.4a"><mrow id="S5.Ex13.m2.4.4" xref="S5.Ex13.m2.4.4.cmml"><mi id="S5.Ex13.m2.4.4.3" xref="S5.Ex13.m2.4.4.3.cmml"></mi><mo id="S5.Ex13.m2.4.4.2" rspace="0.1389em" xref="S5.Ex13.m2.4.4.2.cmml">≥</mo><mrow id="S5.Ex13.m2.4.4.1" xref="S5.Ex13.m2.4.4.1.cmml"><munder id="S5.Ex13.m2.4.4.1.2" xref="S5.Ex13.m2.4.4.1.2.cmml"><mo id="S5.Ex13.m2.4.4.1.2.2" lspace="0.1389em" 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id="S5.Ex13.m2.4c">\displaystyle\geq\mathop{\mathbb{E}}_{x_{j},x_{k}\sim\beta(x)}\left[S\left(p_{% j}(x_{j})+\delta,\sigma(x_{k})\right)\right]</annotation><annotation encoding="application/x-llamapun" id="S5.Ex13.m2.4d">≥ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT [ italic_S ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) + italic_δ , italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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id="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.1.1.1.1.3">𝑗</ci></apply></apply><ci id="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.3.cmml" xref="S5.Ex14.m1.1.1.1.1.1.3.1.1.1.3">𝛿</ci></apply><cn id="S5.Ex14.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex14.m1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex14.m1.1c">\displaystyle\mathop{\mathbb{E}}\left[2(p_{j}(x_{j})-\delta)\mathbf{1}[\sigma(% x_{k})=H]-(p_{j}(x_{j})-\delta)^{2}\right]</annotation><annotation encoding="application/x-llamapun" id="S5.Ex14.m1.1d">blackboard_E [ 2 ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - italic_δ ) bold_1 [ italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_H ] - ( italic_p start_POSTSUBSCRIPT italic_j 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) + italic_δ ) bold_1 [ italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_H ] - ( italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) + italic_δ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 4\delta\mathop{\mathbb{E}}\left[p_{j}(x_{j})-\mathbf{1}[\sigma(x% _{k})=H]\right]" class="ltx_Math" display="inline" id="S5.Ex15.m1.1"><semantics id="S5.Ex15.m1.1a"><mrow id="S5.Ex15.m1.1.1" xref="S5.Ex15.m1.1.1.cmml"><mn id="S5.Ex15.m1.1.1.3" xref="S5.Ex15.m1.1.1.3.cmml">4</mn><mo id="S5.Ex15.m1.1.1.2" xref="S5.Ex15.m1.1.1.2.cmml">⁢</mo><mi id="S5.Ex15.m1.1.1.4" 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_{k})=H]\right]</annotation><annotation encoding="application/x-llamapun" id="S5.Ex15.m1.1d">4 italic_δ blackboard_E [ italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - bold_1 [ italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_H ] ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq 0" class="ltx_Math" display="inline" id="S5.Ex15.m2.1"><semantics id="S5.Ex15.m2.1a"><mrow id="S5.Ex15.m2.1.1" xref="S5.Ex15.m2.1.1.cmml"><mi id="S5.Ex15.m2.1.1.2" xref="S5.Ex15.m2.1.1.2.cmml"></mi><mo id="S5.Ex15.m2.1.1.1" xref="S5.Ex15.m2.1.1.1.cmml">≥</mo><mn id="S5.Ex15.m2.1.1.3" xref="S5.Ex15.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex15.m2.1b"><apply id="S5.Ex15.m2.1.1.cmml" xref="S5.Ex15.m2.1.1"><geq id="S5.Ex15.m2.1.1.1.cmml" xref="S5.Ex15.m2.1.1.1"></geq><csymbol cd="latexml" id="S5.Ex15.m2.1.1.2.cmml" 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xref="S5.Ex16.m1.2.2.1.1.1.4.2">𝜏</ci><ci id="S5.Ex16.m1.2.2.1.1.1.4.3.cmml" xref="S5.Ex16.m1.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S5.Ex16.m1.2.2.1.1.1c.cmml" xref="S5.Ex16.m1.2.2.1.1.1"><eq id="S5.Ex16.m1.2.2.1.1.1.5.cmml" xref="S5.Ex16.m1.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S5.Ex16.m1.2.2.1.1.1.4.cmml" id="S5.Ex16.m1.2.2.1.1.1d.cmml" xref="S5.Ex16.m1.2.2.1.1.1"></share><ci id="S5.Ex16.m1.2.2.1.1.1.6.cmml" xref="S5.Ex16.m1.2.2.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex16.m1.2c">\displaystyle\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="S5.Ex16.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\mathop{\mathbb{E}}_{x_{j}\sim\beta(x)}\left[\Pr[X^{\prime}% \leq\tau\mid X_{j}=x_{j}]\right]" class="ltx_Math" display="inline" id="S5.Ex16.m2.3"><semantics id="S5.Ex16.m2.3a"><mrow id="S5.Ex16.m2.3.3" xref="S5.Ex16.m2.3.3.cmml"><mi id="S5.Ex16.m2.3.3.3" xref="S5.Ex16.m2.3.3.3.cmml"></mi><mo id="S5.Ex16.m2.3.3.2" rspace="0.1389em" xref="S5.Ex16.m2.3.3.2.cmml">≥</mo><mrow id="S5.Ex16.m2.3.3.1" xref="S5.Ex16.m2.3.3.1.cmml"><munder id="S5.Ex16.m2.3.3.1.2" xref="S5.Ex16.m2.3.3.1.2.cmml"><mo id="S5.Ex16.m2.3.3.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0em" xref="S5.Ex16.m2.3.3.1.2.2.cmml">𝔼</mo><mrow id="S5.Ex16.m2.1.1.1" xref="S5.Ex16.m2.1.1.1.cmml"><msub id="S5.Ex16.m2.1.1.1.3" xref="S5.Ex16.m2.1.1.1.3.cmml"><mi id="S5.Ex16.m2.1.1.1.3.2" xref="S5.Ex16.m2.1.1.1.3.2.cmml">x</mi><mi id="S5.Ex16.m2.1.1.1.3.3" xref="S5.Ex16.m2.1.1.1.3.3.cmml">j</mi></msub><mo id="S5.Ex16.m2.1.1.1.2" xref="S5.Ex16.m2.1.1.1.2.cmml">∼</mo><mrow id="S5.Ex16.m2.1.1.1.4" xref="S5.Ex16.m2.1.1.1.4.cmml"><mi id="S5.Ex16.m2.1.1.1.4.2" xref="S5.Ex16.m2.1.1.1.4.2.cmml">β</mi><mo 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id="S5.Ex16.m2.3.3.1.1.1.1.1.1.1d.cmml" xref="S5.Ex16.m2.3.3.1.1.1.1.1.1.1"></share><apply id="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.cmml" xref="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6"><csymbol cd="ambiguous" id="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.1.cmml" xref="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6">subscript</csymbol><ci id="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.2.cmml" xref="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.2">𝑥</ci><ci id="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.3.cmml" xref="S5.Ex16.m2.3.3.1.1.1.1.1.1.1.6.3">𝑗</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex16.m2.3c">\displaystyle\geq\mathop{\mathbb{E}}_{x_{j}\sim\beta(x)}\left[\Pr[X^{\prime}% \leq\tau\mid X_{j}=x_{j}]\right]</annotation><annotation encoding="application/x-llamapun" id="S5.Ex16.m2.3d">≥ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT [ roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ] ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS1.p2.8">where the last line follows from plugging in <math alttext="p_{j}(x_{j})=1-\Pr[X^{\prime}\leq\tau\mid X_{j}=x_{j}]" class="ltx_Math" display="inline" id="S5.SS1.p2.4.m1.3"><semantics id="S5.SS1.p2.4.m1.3a"><mrow id="S5.SS1.p2.4.m1.3.3" xref="S5.SS1.p2.4.m1.3.3.cmml"><mrow id="S5.SS1.p2.4.m1.2.2.1" xref="S5.SS1.p2.4.m1.2.2.1.cmml"><msub id="S5.SS1.p2.4.m1.2.2.1.3" xref="S5.SS1.p2.4.m1.2.2.1.3.cmml"><mi id="S5.SS1.p2.4.m1.2.2.1.3.2" xref="S5.SS1.p2.4.m1.2.2.1.3.2.cmml">p</mi><mi id="S5.SS1.p2.4.m1.2.2.1.3.3" xref="S5.SS1.p2.4.m1.2.2.1.3.3.cmml">j</mi></msub><mo id="S5.SS1.p2.4.m1.2.2.1.2" xref="S5.SS1.p2.4.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.p2.4.m1.2.2.1.1.1" xref="S5.SS1.p2.4.m1.2.2.1.1.1.1.cmml"><mo 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]</annotation></semantics></math>, and noting that <math alttext="\mathop{\mathbb{E}}_{x_{j},x_{k}\sim\beta(x)}\mathbf{1}[\sigma(x_{k})=H]=1-\Pr% [X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="S5.SS1.p2.5.m2.6"><semantics id="S5.SS1.p2.5.m2.6a"><mrow id="S5.SS1.p2.5.m2.6.6" xref="S5.SS1.p2.5.m2.6.6.cmml"><mrow id="S5.SS1.p2.5.m2.5.5.1" xref="S5.SS1.p2.5.m2.5.5.1.cmml"><msub id="S5.SS1.p2.5.m2.5.5.1.2" xref="S5.SS1.p2.5.m2.5.5.1.2.cmml"><mo id="S5.SS1.p2.5.m2.5.5.1.2.2" xref="S5.SS1.p2.5.m2.5.5.1.2.2.cmml">𝔼</mo><mrow id="S5.SS1.p2.5.m2.3.3.3" xref="S5.SS1.p2.5.m2.3.3.3.cmml"><mrow id="S5.SS1.p2.5.m2.3.3.3.3.2" xref="S5.SS1.p2.5.m2.3.3.3.3.3.cmml"><msub id="S5.SS1.p2.5.m2.2.2.2.2.1.1" xref="S5.SS1.p2.5.m2.2.2.2.2.1.1.cmml"><mi id="S5.SS1.p2.5.m2.2.2.2.2.1.1.2" xref="S5.SS1.p2.5.m2.2.2.2.2.1.1.2.cmml">x</mi><mi id="S5.SS1.p2.5.m2.2.2.2.2.1.1.3" xref="S5.SS1.p2.5.m2.2.2.2.2.1.1.3.cmml">j</mi></msub><mo id="S5.SS1.p2.5.m2.3.3.3.3.2.3" 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id="S5.SS1.p2.5.m2.6c">\mathop{\mathbb{E}}_{x_{j},x_{k}\sim\beta(x)}\mathbf{1}[\sigma(x_{k})=H]=1-\Pr% [X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.5.m2.6d">blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT bold_1 [ italic_σ ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_H ] = 1 - roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math>. One can show a symmetric condition if <math alttext="i" class="ltx_Math" display="inline" id="S5.SS1.p2.6.m3.1"><semantics id="S5.SS1.p2.6.m3.1a"><mi id="S5.SS1.p2.6.m3.1.1" xref="S5.SS1.p2.6.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.6.m3.1b"><ci id="S5.SS1.p2.6.m3.1.1.cmml" xref="S5.SS1.p2.6.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.6.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.6.m3.1d">italic_i</annotation></semantics></math>’s signal satisfies <math alttext="x>\tau" class="ltx_Math" display="inline" id="S5.SS1.p2.7.m4.1"><semantics id="S5.SS1.p2.7.m4.1a"><mrow id="S5.SS1.p2.7.m4.1.1" xref="S5.SS1.p2.7.m4.1.1.cmml"><mi id="S5.SS1.p2.7.m4.1.1.2" xref="S5.SS1.p2.7.m4.1.1.2.cmml">x</mi><mo id="S5.SS1.p2.7.m4.1.1.1" xref="S5.SS1.p2.7.m4.1.1.1.cmml">></mo><mi id="S5.SS1.p2.7.m4.1.1.3" xref="S5.SS1.p2.7.m4.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.7.m4.1b"><apply id="S5.SS1.p2.7.m4.1.1.cmml" xref="S5.SS1.p2.7.m4.1.1"><gt id="S5.SS1.p2.7.m4.1.1.1.cmml" xref="S5.SS1.p2.7.m4.1.1.1"></gt><ci id="S5.SS1.p2.7.m4.1.1.2.cmml" xref="S5.SS1.p2.7.m4.1.1.2">𝑥</ci><ci id="S5.SS1.p2.7.m4.1.1.3.cmml" xref="S5.SS1.p2.7.m4.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.7.m4.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.7.m4.1d">italic_x > italic_τ</annotation></semantics></math>, so that all symmetric Bayes-Nash threshold equilibria <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS1.p2.8.m5.1"><semantics id="S5.SS1.p2.8.m5.1a"><mi id="S5.SS1.p2.8.m5.1.1" xref="S5.SS1.p2.8.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.8.m5.1b"><ci id="S5.SS1.p2.8.m5.1.1.cmml" xref="S5.SS1.p2.8.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.8.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.8.m5.1d">italic_τ</annotation></semantics></math> are characterized by the following conditions:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx9"> <tbody id="S5.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall x\leq\tau,P(\tau;x)" class="ltx_Math" display="inline" id="S5.E18.m1.4"><semantics id="S5.E18.m1.4a"><mrow id="S5.E18.m1.4.4" xref="S5.E18.m1.4.4.cmml"><mrow id="S5.E18.m1.4.4.3" xref="S5.E18.m1.4.4.3.cmml"><mo id="S5.E18.m1.4.4.3.1" rspace="0.167em" xref="S5.E18.m1.4.4.3.1.cmml">∀</mo><mi id="S5.E18.m1.4.4.3.2" xref="S5.E18.m1.4.4.3.2.cmml">x</mi></mrow><mo id="S5.E18.m1.4.4.2" xref="S5.E18.m1.4.4.2.cmml">≤</mo><mrow id="S5.E18.m1.4.4.1.1" xref="S5.E18.m1.4.4.1.2.cmml"><mi id="S5.E18.m1.3.3" xref="S5.E18.m1.3.3.cmml">τ</mi><mo id="S5.E18.m1.4.4.1.1.2" xref="S5.E18.m1.4.4.1.2.cmml">,</mo><mrow id="S5.E18.m1.4.4.1.1.1" xref="S5.E18.m1.4.4.1.1.1.cmml"><mi id="S5.E18.m1.4.4.1.1.1.2" xref="S5.E18.m1.4.4.1.1.1.2.cmml">P</mi><mo id="S5.E18.m1.4.4.1.1.1.1" xref="S5.E18.m1.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S5.E18.m1.4.4.1.1.1.3.2" xref="S5.E18.m1.4.4.1.1.1.3.1.cmml"><mo id="S5.E18.m1.4.4.1.1.1.3.2.1" stretchy="false" xref="S5.E18.m1.4.4.1.1.1.3.1.cmml">(</mo><mi id="S5.E18.m1.1.1" xref="S5.E18.m1.1.1.cmml">τ</mi><mo id="S5.E18.m1.4.4.1.1.1.3.2.2" xref="S5.E18.m1.4.4.1.1.1.3.1.cmml">;</mo><mi id="S5.E18.m1.2.2" xref="S5.E18.m1.2.2.cmml">x</mi><mo id="S5.E18.m1.4.4.1.1.1.3.2.3" stretchy="false" xref="S5.E18.m1.4.4.1.1.1.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E18.m1.4b"><apply id="S5.E18.m1.4.4.cmml" xref="S5.E18.m1.4.4"><leq id="S5.E18.m1.4.4.2.cmml" xref="S5.E18.m1.4.4.2"></leq><apply id="S5.E18.m1.4.4.3.cmml" xref="S5.E18.m1.4.4.3"><csymbol cd="latexml" id="S5.E18.m1.4.4.3.1.cmml" xref="S5.E18.m1.4.4.3.1">for-all</csymbol><ci id="S5.E18.m1.4.4.3.2.cmml" xref="S5.E18.m1.4.4.3.2">𝑥</ci></apply><list id="S5.E18.m1.4.4.1.2.cmml" xref="S5.E18.m1.4.4.1.1"><ci id="S5.E18.m1.3.3.cmml" xref="S5.E18.m1.3.3">𝜏</ci><apply id="S5.E18.m1.4.4.1.1.1.cmml" xref="S5.E18.m1.4.4.1.1.1"><times id="S5.E18.m1.4.4.1.1.1.1.cmml" xref="S5.E18.m1.4.4.1.1.1.1"></times><ci id="S5.E18.m1.4.4.1.1.1.2.cmml" xref="S5.E18.m1.4.4.1.1.1.2">𝑃</ci><list id="S5.E18.m1.4.4.1.1.1.3.1.cmml" xref="S5.E18.m1.4.4.1.1.1.3.2"><ci id="S5.E18.m1.1.1.cmml" xref="S5.E18.m1.1.1">𝜏</ci><ci id="S5.E18.m1.2.2.cmml" xref="S5.E18.m1.2.2">𝑥</ci></list></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E18.m1.4c">\displaystyle\forall x\leq\tau,P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.E18.m1.4d">∀ italic_x ≤ italic_τ , italic_P ( italic_τ ; italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime})," class="ltx_Math" display="inline" id="S5.E18.m2.3"><semantics id="S5.E18.m2.3a"><mrow id="S5.E18.m2.3.3.1" xref="S5.E18.m2.3.3.1.1.cmml"><mrow id="S5.E18.m2.3.3.1.1" xref="S5.E18.m2.3.3.1.1.cmml"><mi id="S5.E18.m2.3.3.1.1.3" xref="S5.E18.m2.3.3.1.1.3.cmml"></mi><mo id="S5.E18.m2.3.3.1.1.2" rspace="0.1389em" xref="S5.E18.m2.3.3.1.1.2.cmml">≥</mo><mrow id="S5.E18.m2.3.3.1.1.1" xref="S5.E18.m2.3.3.1.1.1.cmml"><munder id="S5.E18.m2.3.3.1.1.1.2" xref="S5.E18.m2.3.3.1.1.1.2.cmml"><mo id="S5.E18.m2.3.3.1.1.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S5.E18.m2.3.3.1.1.1.2.2.cmml">𝔼</mo><mrow id="S5.E18.m2.1.1.1" xref="S5.E18.m2.1.1.1.cmml"><msup id="S5.E18.m2.1.1.1.3" xref="S5.E18.m2.1.1.1.3.cmml"><mi id="S5.E18.m2.1.1.1.3.2" xref="S5.E18.m2.1.1.1.3.2.cmml">x</mi><mo id="S5.E18.m2.1.1.1.3.3" xref="S5.E18.m2.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.E18.m2.1.1.1.2" xref="S5.E18.m2.1.1.1.2.cmml">∼</mo><mrow id="S5.E18.m2.1.1.1.4" xref="S5.E18.m2.1.1.1.4.cmml"><mi id="S5.E18.m2.1.1.1.4.2" xref="S5.E18.m2.1.1.1.4.2.cmml">β</mi><mo id="S5.E18.m2.1.1.1.4.1" xref="S5.E18.m2.1.1.1.4.1.cmml">⁢</mo><mrow id="S5.E18.m2.1.1.1.4.3.2" xref="S5.E18.m2.1.1.1.4.cmml"><mo id="S5.E18.m2.1.1.1.4.3.2.1" stretchy="false" xref="S5.E18.m2.1.1.1.4.cmml">(</mo><mi id="S5.E18.m2.1.1.1.1" xref="S5.E18.m2.1.1.1.1.cmml">x</mi><mo id="S5.E18.m2.1.1.1.4.3.2.2" stretchy="false" xref="S5.E18.m2.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.E18.m2.3.3.1.1.1.1" xref="S5.E18.m2.3.3.1.1.1.1.cmml"><mi id="S5.E18.m2.3.3.1.1.1.1.3" xref="S5.E18.m2.3.3.1.1.1.1.3.cmml">P</mi><mo id="S5.E18.m2.3.3.1.1.1.1.2" xref="S5.E18.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E18.m2.3.3.1.1.1.1.1.1" xref="S5.E18.m2.3.3.1.1.1.1.1.2.cmml"><mo id="S5.E18.m2.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S5.E18.m2.3.3.1.1.1.1.1.2.cmml">(</mo><mi id="S5.E18.m2.2.2" xref="S5.E18.m2.2.2.cmml">τ</mi><mo id="S5.E18.m2.3.3.1.1.1.1.1.1.3" xref="S5.E18.m2.3.3.1.1.1.1.1.2.cmml">;</mo><msup id="S5.E18.m2.3.3.1.1.1.1.1.1.1" xref="S5.E18.m2.3.3.1.1.1.1.1.1.1.cmml"><mi id="S5.E18.m2.3.3.1.1.1.1.1.1.1.2" xref="S5.E18.m2.3.3.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.E18.m2.3.3.1.1.1.1.1.1.1.3" xref="S5.E18.m2.3.3.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.E18.m2.3.3.1.1.1.1.1.1.4" stretchy="false" xref="S5.E18.m2.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.E18.m2.3.3.1.2" xref="S5.E18.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E18.m2.3b"><apply id="S5.E18.m2.3.3.1.1.cmml" xref="S5.E18.m2.3.3.1"><geq id="S5.E18.m2.3.3.1.1.2.cmml" xref="S5.E18.m2.3.3.1.1.2"></geq><csymbol cd="latexml" id="S5.E18.m2.3.3.1.1.3.cmml" xref="S5.E18.m2.3.3.1.1.3">absent</csymbol><apply id="S5.E18.m2.3.3.1.1.1.cmml" xref="S5.E18.m2.3.3.1.1.1"><apply id="S5.E18.m2.3.3.1.1.1.2.cmml" xref="S5.E18.m2.3.3.1.1.1.2"><csymbol 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id="S5.E19.m2.3c">\displaystyle\leq\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime}).</annotation><annotation encoding="application/x-llamapun" id="S5.E19.m2.3d">≤ blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS1.p2.9">Therefore, unlike the DG mechanism, in RBTS an agent’s actions in equilibrium depend on the <em class="ltx_emph ltx_font_italic" id="S5.SS1.p2.9.1">expected</em> conditional probability distribution. We can immediately see that the uninformative thresholds <math alttext="\tau=\pm\infty" class="ltx_Math" display="inline" id="S5.SS1.p2.9.m1.1"><semantics id="S5.SS1.p2.9.m1.1a"><mrow id="S5.SS1.p2.9.m1.1.1" xref="S5.SS1.p2.9.m1.1.1.cmml"><mi id="S5.SS1.p2.9.m1.1.1.2" xref="S5.SS1.p2.9.m1.1.1.2.cmml">τ</mi><mo id="S5.SS1.p2.9.m1.1.1.1" xref="S5.SS1.p2.9.m1.1.1.1.cmml">=</mo><mrow id="S5.SS1.p2.9.m1.1.1.3" xref="S5.SS1.p2.9.m1.1.1.3.cmml"><mo id="S5.SS1.p2.9.m1.1.1.3a" xref="S5.SS1.p2.9.m1.1.1.3.cmml">±</mo><mi id="S5.SS1.p2.9.m1.1.1.3.2" mathvariant="normal" xref="S5.SS1.p2.9.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.9.m1.1b"><apply id="S5.SS1.p2.9.m1.1.1.cmml" xref="S5.SS1.p2.9.m1.1.1"><eq id="S5.SS1.p2.9.m1.1.1.1.cmml" xref="S5.SS1.p2.9.m1.1.1.1"></eq><ci id="S5.SS1.p2.9.m1.1.1.2.cmml" xref="S5.SS1.p2.9.m1.1.1.2">𝜏</ci><apply id="S5.SS1.p2.9.m1.1.1.3.cmml" xref="S5.SS1.p2.9.m1.1.1.3"><csymbol cd="latexml" id="S5.SS1.p2.9.m1.1.1.3.1.cmml" xref="S5.SS1.p2.9.m1.1.1.3">plus-or-minus</csymbol><infinity id="S5.SS1.p2.9.m1.1.1.3.2.cmml" xref="S5.SS1.p2.9.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.9.m1.1c">\tau=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.9.m1.1d">italic_τ = ± ∞</annotation></semantics></math> still always appear as equilibria:</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition6.1.1.1">Proposition 6</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition6.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition6.p1"> <p class="ltx_p" id="Thmproposition6.p1.1"><math alttext="\tau^{*}=\pm\infty" class="ltx_Math" display="inline" id="Thmproposition6.p1.1.m1.1"><semantics id="Thmproposition6.p1.1.m1.1a"><mrow id="Thmproposition6.p1.1.m1.1.1" xref="Thmproposition6.p1.1.m1.1.1.cmml"><msup id="Thmproposition6.p1.1.m1.1.1.2" xref="Thmproposition6.p1.1.m1.1.1.2.cmml"><mi id="Thmproposition6.p1.1.m1.1.1.2.2" xref="Thmproposition6.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="Thmproposition6.p1.1.m1.1.1.2.3" xref="Thmproposition6.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition6.p1.1.m1.1.1.1" xref="Thmproposition6.p1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmproposition6.p1.1.m1.1.1.3" xref="Thmproposition6.p1.1.m1.1.1.3.cmml"><mo id="Thmproposition6.p1.1.m1.1.1.3a" xref="Thmproposition6.p1.1.m1.1.1.3.cmml">±</mo><mi id="Thmproposition6.p1.1.m1.1.1.3.2" mathvariant="normal" xref="Thmproposition6.p1.1.m1.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition6.p1.1.m1.1b"><apply id="Thmproposition6.p1.1.m1.1.1.cmml" xref="Thmproposition6.p1.1.m1.1.1"><eq id="Thmproposition6.p1.1.m1.1.1.1.cmml" xref="Thmproposition6.p1.1.m1.1.1.1"></eq><apply id="Thmproposition6.p1.1.m1.1.1.2.cmml" xref="Thmproposition6.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition6.p1.1.m1.1.1.2.1.cmml" xref="Thmproposition6.p1.1.m1.1.1.2">superscript</csymbol><ci id="Thmproposition6.p1.1.m1.1.1.2.2.cmml" xref="Thmproposition6.p1.1.m1.1.1.2.2">𝜏</ci><times id="Thmproposition6.p1.1.m1.1.1.2.3.cmml" xref="Thmproposition6.p1.1.m1.1.1.2.3"></times></apply><apply id="Thmproposition6.p1.1.m1.1.1.3.cmml" xref="Thmproposition6.p1.1.m1.1.1.3"><csymbol cd="latexml" id="Thmproposition6.p1.1.m1.1.1.3.1.cmml" xref="Thmproposition6.p1.1.m1.1.1.3">plus-or-minus</csymbol><infinity id="Thmproposition6.p1.1.m1.1.1.3.2.cmml" xref="Thmproposition6.p1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition6.p1.1.m1.1c">\tau^{*}=\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition6.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ± ∞</annotation></semantics></math> are threshold equilibria under RBTS.</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.p1.6">Let <math alttext="\tau^{*}=\infty" 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"><msup id="S5.SS1.1.p1.1.m1.1.1.2" xref="S5.SS1.1.p1.1.m1.1.1.2.cmml"><mi id="S5.SS1.1.p1.1.m1.1.1.2.2" xref="S5.SS1.1.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="S5.SS1.1.p1.1.m1.1.1.2.3" xref="S5.SS1.1.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.SS1.1.p1.1.m1.1.1.1" xref="S5.SS1.1.p1.1.m1.1.1.1.cmml">=</mo><mi id="S5.SS1.1.p1.1.m1.1.1.3" mathvariant="normal" xref="S5.SS1.1.p1.1.m1.1.1.3.cmml">∞</mi></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><apply id="S5.SS1.1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.1.p1.1.m1.1.1.2.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2">superscript</csymbol><ci id="S5.SS1.1.p1.1.m1.1.1.2.2.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2.2">𝜏</ci><times id="S5.SS1.1.p1.1.m1.1.1.2.3.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2.3"></times></apply><infinity id="S5.SS1.1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.1.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.1.m1.1c">\tau^{*}=\infty</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ∞</annotation></semantics></math>. Then for all signals <math alttext="x,x^{\prime}\in\mathbb{R}" 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.m2.2.2" xref="S5.SS1.1.p1.2.m2.2.2.cmml"><mrow id="S5.SS1.1.p1.2.m2.2.2.1.1" xref="S5.SS1.1.p1.2.m2.2.2.1.2.cmml"><mi id="S5.SS1.1.p1.2.m2.1.1" xref="S5.SS1.1.p1.2.m2.1.1.cmml">x</mi><mo id="S5.SS1.1.p1.2.m2.2.2.1.1.2" xref="S5.SS1.1.p1.2.m2.2.2.1.2.cmml">,</mo><msup id="S5.SS1.1.p1.2.m2.2.2.1.1.1" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.cmml"><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><mo id="S5.SS1.1.p1.2.m2.2.2.1.1.1.3" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S5.SS1.1.p1.2.m2.2.2.2" xref="S5.SS1.1.p1.2.m2.2.2.2.cmml">∈</mo><mi id="S5.SS1.1.p1.2.m2.2.2.3" xref="S5.SS1.1.p1.2.m2.2.2.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.2.m2.2b"><apply id="S5.SS1.1.p1.2.m2.2.2.cmml" xref="S5.SS1.1.p1.2.m2.2.2"><in id="S5.SS1.1.p1.2.m2.2.2.2.cmml" xref="S5.SS1.1.p1.2.m2.2.2.2"></in><list id="S5.SS1.1.p1.2.m2.2.2.1.2.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1"><ci id="S5.SS1.1.p1.2.m2.1.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1">𝑥</ci><apply id="S5.SS1.1.p1.2.m2.2.2.1.1.1.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1">superscript</csymbol><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><ci id="S5.SS1.1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S5.SS1.1.p1.2.m2.2.2.1.1.1.3">′</ci></apply></list><ci id="S5.SS1.1.p1.2.m2.2.2.3.cmml" xref="S5.SS1.1.p1.2.m2.2.2.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.2.m2.2c">x,x^{\prime}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.2.m2.2d">italic_x , italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R</annotation></semantics></math>, <math alttext="P(\tau^{*};x)=1" class="ltx_Math" display="inline" id="S5.SS1.1.p1.3.m3.2"><semantics id="S5.SS1.1.p1.3.m3.2a"><mrow id="S5.SS1.1.p1.3.m3.2.2" xref="S5.SS1.1.p1.3.m3.2.2.cmml"><mrow id="S5.SS1.1.p1.3.m3.2.2.1" xref="S5.SS1.1.p1.3.m3.2.2.1.cmml"><mi id="S5.SS1.1.p1.3.m3.2.2.1.3" xref="S5.SS1.1.p1.3.m3.2.2.1.3.cmml">P</mi><mo id="S5.SS1.1.p1.3.m3.2.2.1.2" xref="S5.SS1.1.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.1.p1.3.m3.2.2.1.1.1" xref="S5.SS1.1.p1.3.m3.2.2.1.1.2.cmml"><mo id="S5.SS1.1.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S5.SS1.1.p1.3.m3.2.2.1.1.2.cmml">(</mo><msup id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.cmml"><mi id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.2" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.2.cmml">τ</mi><mo id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.3" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS1.1.p1.3.m3.2.2.1.1.1.3" xref="S5.SS1.1.p1.3.m3.2.2.1.1.2.cmml">;</mo><mi id="S5.SS1.1.p1.3.m3.1.1" xref="S5.SS1.1.p1.3.m3.1.1.cmml">x</mi><mo id="S5.SS1.1.p1.3.m3.2.2.1.1.1.4" stretchy="false" xref="S5.SS1.1.p1.3.m3.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.SS1.1.p1.3.m3.2.2.2" xref="S5.SS1.1.p1.3.m3.2.2.2.cmml">=</mo><mn id="S5.SS1.1.p1.3.m3.2.2.3" xref="S5.SS1.1.p1.3.m3.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.3.m3.2b"><apply id="S5.SS1.1.p1.3.m3.2.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2"><eq id="S5.SS1.1.p1.3.m3.2.2.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2.2"></eq><apply id="S5.SS1.1.p1.3.m3.2.2.1.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1"><times id="S5.SS1.1.p1.3.m3.2.2.1.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.2"></times><ci id="S5.SS1.1.p1.3.m3.2.2.1.3.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.3">𝑃</ci><list id="S5.SS1.1.p1.3.m3.2.2.1.1.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1"><apply id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1">superscript</csymbol><ci id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.2">𝜏</ci><times id="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.3.m3.2.2.1.1.1.1.3"></times></apply><ci id="S5.SS1.1.p1.3.m3.1.1.cmml" xref="S5.SS1.1.p1.3.m3.1.1">𝑥</ci></list></apply><cn id="S5.SS1.1.p1.3.m3.2.2.3.cmml" type="integer" xref="S5.SS1.1.p1.3.m3.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.3.m3.2c">P(\tau^{*};x)=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.3.m3.2d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x ) = 1</annotation></semantics></math> and <math alttext="P(\tau^{*};x^{\prime})=1" class="ltx_Math" display="inline" id="S5.SS1.1.p1.4.m4.2"><semantics id="S5.SS1.1.p1.4.m4.2a"><mrow id="S5.SS1.1.p1.4.m4.2.2" xref="S5.SS1.1.p1.4.m4.2.2.cmml"><mrow id="S5.SS1.1.p1.4.m4.2.2.2" xref="S5.SS1.1.p1.4.m4.2.2.2.cmml"><mi id="S5.SS1.1.p1.4.m4.2.2.2.4" xref="S5.SS1.1.p1.4.m4.2.2.2.4.cmml">P</mi><mo id="S5.SS1.1.p1.4.m4.2.2.2.3" xref="S5.SS1.1.p1.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS1.1.p1.4.m4.2.2.2.2.2" xref="S5.SS1.1.p1.4.m4.2.2.2.2.3.cmml"><mo id="S5.SS1.1.p1.4.m4.2.2.2.2.2.3" stretchy="false" xref="S5.SS1.1.p1.4.m4.2.2.2.2.3.cmml">(</mo><msup id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.2" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.2.cmml">τ</mi><mo id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.3" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS1.1.p1.4.m4.2.2.2.2.2.4" xref="S5.SS1.1.p1.4.m4.2.2.2.2.3.cmml">;</mo><msup id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.cmml"><mi id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.2" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.2.cmml">x</mi><mo id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.3" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S5.SS1.1.p1.4.m4.2.2.2.2.2.5" stretchy="false" xref="S5.SS1.1.p1.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.SS1.1.p1.4.m4.2.2.3" xref="S5.SS1.1.p1.4.m4.2.2.3.cmml">=</mo><mn id="S5.SS1.1.p1.4.m4.2.2.4" xref="S5.SS1.1.p1.4.m4.2.2.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.4.m4.2b"><apply id="S5.SS1.1.p1.4.m4.2.2.cmml" xref="S5.SS1.1.p1.4.m4.2.2"><eq id="S5.SS1.1.p1.4.m4.2.2.3.cmml" xref="S5.SS1.1.p1.4.m4.2.2.3"></eq><apply id="S5.SS1.1.p1.4.m4.2.2.2.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2"><times id="S5.SS1.1.p1.4.m4.2.2.2.3.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.3"></times><ci id="S5.SS1.1.p1.4.m4.2.2.2.4.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.4">𝑃</ci><list id="S5.SS1.1.p1.4.m4.2.2.2.2.3.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2"><apply id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.2">𝜏</ci><times id="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.4.m4.1.1.1.1.1.1.3"></times></apply><apply id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2">superscript</csymbol><ci id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.2">𝑥</ci><ci id="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S5.SS1.1.p1.4.m4.2.2.2.2.2.2.3">′</ci></apply></list></apply><cn id="S5.SS1.1.p1.4.m4.2.2.4.cmml" type="integer" xref="S5.SS1.1.p1.4.m4.2.2.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.4.m4.2c">P(\tau^{*};x^{\prime})=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.4.m4.2d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = 1</annotation></semantics></math>, so that each side of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E18" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">18</span></a> always evaluates to 1 and the condition for <math alttext="x<\tau^{*}" 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"><mi id="S5.SS1.1.p1.5.m5.1.1.2" xref="S5.SS1.1.p1.5.m5.1.1.2.cmml">x</mi><mo id="S5.SS1.1.p1.5.m5.1.1.1" xref="S5.SS1.1.p1.5.m5.1.1.1.cmml"><</mo><msup id="S5.SS1.1.p1.5.m5.1.1.3" xref="S5.SS1.1.p1.5.m5.1.1.3.cmml"><mi id="S5.SS1.1.p1.5.m5.1.1.3.2" xref="S5.SS1.1.p1.5.m5.1.1.3.2.cmml">τ</mi><mo id="S5.SS1.1.p1.5.m5.1.1.3.3" xref="S5.SS1.1.p1.5.m5.1.1.3.3.cmml">∗</mo></msup></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"><lt id="S5.SS1.1.p1.5.m5.1.1.1.cmml" xref="S5.SS1.1.p1.5.m5.1.1.1"></lt><ci id="S5.SS1.1.p1.5.m5.1.1.2.cmml" xref="S5.SS1.1.p1.5.m5.1.1.2">𝑥</ci><apply id="S5.SS1.1.p1.5.m5.1.1.3.cmml" xref="S5.SS1.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.1.p1.5.m5.1.1.3.1.cmml" xref="S5.SS1.1.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.SS1.1.p1.5.m5.1.1.3.2.cmml" xref="S5.SS1.1.p1.5.m5.1.1.3.2">𝜏</ci><times id="S5.SS1.1.p1.5.m5.1.1.3.3.cmml" xref="S5.SS1.1.p1.5.m5.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.5.m5.1c">x<\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.5.m5.1d">italic_x < italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is satisfied. The condition of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E19" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">19</span></a> is vacuous. An analagous argument holds for <math alttext="\tau^{*}=-\infty" class="ltx_Math" display="inline" id="S5.SS1.1.p1.6.m6.1"><semantics id="S5.SS1.1.p1.6.m6.1a"><mrow id="S5.SS1.1.p1.6.m6.1.1" xref="S5.SS1.1.p1.6.m6.1.1.cmml"><msup id="S5.SS1.1.p1.6.m6.1.1.2" xref="S5.SS1.1.p1.6.m6.1.1.2.cmml"><mi id="S5.SS1.1.p1.6.m6.1.1.2.2" xref="S5.SS1.1.p1.6.m6.1.1.2.2.cmml">τ</mi><mo id="S5.SS1.1.p1.6.m6.1.1.2.3" xref="S5.SS1.1.p1.6.m6.1.1.2.3.cmml">∗</mo></msup><mo id="S5.SS1.1.p1.6.m6.1.1.1" xref="S5.SS1.1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S5.SS1.1.p1.6.m6.1.1.3" xref="S5.SS1.1.p1.6.m6.1.1.3.cmml"><mo id="S5.SS1.1.p1.6.m6.1.1.3a" xref="S5.SS1.1.p1.6.m6.1.1.3.cmml">−</mo><mi id="S5.SS1.1.p1.6.m6.1.1.3.2" mathvariant="normal" xref="S5.SS1.1.p1.6.m6.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.6.m6.1b"><apply id="S5.SS1.1.p1.6.m6.1.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1"><eq id="S5.SS1.1.p1.6.m6.1.1.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1.1"></eq><apply id="S5.SS1.1.p1.6.m6.1.1.2.cmml" xref="S5.SS1.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.1.p1.6.m6.1.1.2.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1.2">superscript</csymbol><ci id="S5.SS1.1.p1.6.m6.1.1.2.2.cmml" xref="S5.SS1.1.p1.6.m6.1.1.2.2">𝜏</ci><times id="S5.SS1.1.p1.6.m6.1.1.2.3.cmml" xref="S5.SS1.1.p1.6.m6.1.1.2.3"></times></apply><apply id="S5.SS1.1.p1.6.m6.1.1.3.cmml" xref="S5.SS1.1.p1.6.m6.1.1.3"><minus id="S5.SS1.1.p1.6.m6.1.1.3.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1.3"></minus><infinity id="S5.SS1.1.p1.6.m6.1.1.3.2.cmml" xref="S5.SS1.1.p1.6.m6.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.6.m6.1c">\tau^{*}=-\infty</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.6.m6.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = - ∞</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.6">We also observe necessary and sufficient conditions for non-infinite equilibria <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS1.p3.1.m1.1"><semantics id="S5.SS1.p3.1.m1.1a"><mi id="S5.SS1.p3.1.m1.1.1" xref="S5.SS1.p3.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.1.m1.1b"><ci id="S5.SS1.p3.1.m1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.1.m1.1d">italic_τ</annotation></semantics></math>, relative to the function <math alttext="G" class="ltx_Math" display="inline" id="S5.SS1.p3.2.m2.1"><semantics id="S5.SS1.p3.2.m2.1a"><mi id="S5.SS1.p3.2.m2.1.1" xref="S5.SS1.p3.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.2.m2.1b"><ci id="S5.SS1.p3.2.m2.1.1.cmml" xref="S5.SS1.p3.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.2.m2.1d">italic_G</annotation></semantics></math>. Let <math alttext="Q(x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(x;x^{\prime})" class="ltx_Math" display="inline" id="S5.SS1.p3.3.m3.4"><semantics id="S5.SS1.p3.3.m3.4a"><mrow id="S5.SS1.p3.3.m3.4.4" xref="S5.SS1.p3.3.m3.4.4.cmml"><mrow id="S5.SS1.p3.3.m3.4.4.3" xref="S5.SS1.p3.3.m3.4.4.3.cmml"><mi id="S5.SS1.p3.3.m3.4.4.3.2" xref="S5.SS1.p3.3.m3.4.4.3.2.cmml">Q</mi><mo id="S5.SS1.p3.3.m3.4.4.3.1" xref="S5.SS1.p3.3.m3.4.4.3.1.cmml">⁢</mo><mrow id="S5.SS1.p3.3.m3.4.4.3.3.2" xref="S5.SS1.p3.3.m3.4.4.3.cmml"><mo id="S5.SS1.p3.3.m3.4.4.3.3.2.1" stretchy="false" xref="S5.SS1.p3.3.m3.4.4.3.cmml">(</mo><mi id="S5.SS1.p3.3.m3.2.2" xref="S5.SS1.p3.3.m3.2.2.cmml">x</mi><mo id="S5.SS1.p3.3.m3.4.4.3.3.2.2" stretchy="false" xref="S5.SS1.p3.3.m3.4.4.3.cmml">)</mo></mrow></mrow><mo id="S5.SS1.p3.3.m3.4.4.2" rspace="0.1389em" xref="S5.SS1.p3.3.m3.4.4.2.cmml">=</mo><mrow id="S5.SS1.p3.3.m3.4.4.1" xref="S5.SS1.p3.3.m3.4.4.1.cmml"><msub id="S5.SS1.p3.3.m3.4.4.1.2" xref="S5.SS1.p3.3.m3.4.4.1.2.cmml"><mo id="S5.SS1.p3.3.m3.4.4.1.2.2" lspace="0.1389em" rspace="0.167em" xref="S5.SS1.p3.3.m3.4.4.1.2.2.cmml">𝔼</mo><mrow id="S5.SS1.p3.3.m3.1.1.1" xref="S5.SS1.p3.3.m3.1.1.1.cmml"><msup id="S5.SS1.p3.3.m3.1.1.1.3" xref="S5.SS1.p3.3.m3.1.1.1.3.cmml"><mi id="S5.SS1.p3.3.m3.1.1.1.3.2" xref="S5.SS1.p3.3.m3.1.1.1.3.2.cmml">x</mi><mo id="S5.SS1.p3.3.m3.1.1.1.3.3" xref="S5.SS1.p3.3.m3.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.SS1.p3.3.m3.1.1.1.2" xref="S5.SS1.p3.3.m3.1.1.1.2.cmml">∼</mo><mrow id="S5.SS1.p3.3.m3.1.1.1.4" xref="S5.SS1.p3.3.m3.1.1.1.4.cmml"><mi id="S5.SS1.p3.3.m3.1.1.1.4.2" xref="S5.SS1.p3.3.m3.1.1.1.4.2.cmml">β</mi><mo id="S5.SS1.p3.3.m3.1.1.1.4.1" xref="S5.SS1.p3.3.m3.1.1.1.4.1.cmml">⁢</mo><mrow id="S5.SS1.p3.3.m3.1.1.1.4.3.2" xref="S5.SS1.p3.3.m3.1.1.1.4.cmml"><mo id="S5.SS1.p3.3.m3.1.1.1.4.3.2.1" stretchy="false" xref="S5.SS1.p3.3.m3.1.1.1.4.cmml">(</mo><mi id="S5.SS1.p3.3.m3.1.1.1.1" xref="S5.SS1.p3.3.m3.1.1.1.1.cmml">x</mi><mo id="S5.SS1.p3.3.m3.1.1.1.4.3.2.2" stretchy="false" xref="S5.SS1.p3.3.m3.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S5.SS1.p3.3.m3.4.4.1.1" xref="S5.SS1.p3.3.m3.4.4.1.1.cmml"><mi id="S5.SS1.p3.3.m3.4.4.1.1.3" xref="S5.SS1.p3.3.m3.4.4.1.1.3.cmml">P</mi><mo id="S5.SS1.p3.3.m3.4.4.1.1.2" xref="S5.SS1.p3.3.m3.4.4.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.p3.3.m3.4.4.1.1.1.1" xref="S5.SS1.p3.3.m3.4.4.1.1.1.2.cmml"><mo id="S5.SS1.p3.3.m3.4.4.1.1.1.1.2" stretchy="false" xref="S5.SS1.p3.3.m3.4.4.1.1.1.2.cmml">(</mo><mi id="S5.SS1.p3.3.m3.3.3" xref="S5.SS1.p3.3.m3.3.3.cmml">x</mi><mo id="S5.SS1.p3.3.m3.4.4.1.1.1.1.3" xref="S5.SS1.p3.3.m3.4.4.1.1.1.2.cmml">;</mo><msup id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.cmml"><mi id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.2" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.2.cmml">x</mi><mo id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.3" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.SS1.p3.3.m3.4.4.1.1.1.1.4" stretchy="false" xref="S5.SS1.p3.3.m3.4.4.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.3.m3.4b"><apply id="S5.SS1.p3.3.m3.4.4.cmml" xref="S5.SS1.p3.3.m3.4.4"><eq id="S5.SS1.p3.3.m3.4.4.2.cmml" xref="S5.SS1.p3.3.m3.4.4.2"></eq><apply id="S5.SS1.p3.3.m3.4.4.3.cmml" xref="S5.SS1.p3.3.m3.4.4.3"><times id="S5.SS1.p3.3.m3.4.4.3.1.cmml" xref="S5.SS1.p3.3.m3.4.4.3.1"></times><ci id="S5.SS1.p3.3.m3.4.4.3.2.cmml" xref="S5.SS1.p3.3.m3.4.4.3.2">𝑄</ci><ci id="S5.SS1.p3.3.m3.2.2.cmml" xref="S5.SS1.p3.3.m3.2.2">𝑥</ci></apply><apply id="S5.SS1.p3.3.m3.4.4.1.cmml" xref="S5.SS1.p3.3.m3.4.4.1"><apply id="S5.SS1.p3.3.m3.4.4.1.2.cmml" xref="S5.SS1.p3.3.m3.4.4.1.2"><csymbol cd="ambiguous" id="S5.SS1.p3.3.m3.4.4.1.2.1.cmml" xref="S5.SS1.p3.3.m3.4.4.1.2">subscript</csymbol><ci id="S5.SS1.p3.3.m3.4.4.1.2.2.cmml" xref="S5.SS1.p3.3.m3.4.4.1.2.2">𝔼</ci><apply id="S5.SS1.p3.3.m3.1.1.1.cmml" xref="S5.SS1.p3.3.m3.1.1.1"><csymbol cd="latexml" id="S5.SS1.p3.3.m3.1.1.1.2.cmml" xref="S5.SS1.p3.3.m3.1.1.1.2">similar-to</csymbol><apply id="S5.SS1.p3.3.m3.1.1.1.3.cmml" xref="S5.SS1.p3.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p3.3.m3.1.1.1.3.1.cmml" xref="S5.SS1.p3.3.m3.1.1.1.3">superscript</csymbol><ci id="S5.SS1.p3.3.m3.1.1.1.3.2.cmml" xref="S5.SS1.p3.3.m3.1.1.1.3.2">𝑥</ci><ci id="S5.SS1.p3.3.m3.1.1.1.3.3.cmml" xref="S5.SS1.p3.3.m3.1.1.1.3.3">′</ci></apply><apply id="S5.SS1.p3.3.m3.1.1.1.4.cmml" xref="S5.SS1.p3.3.m3.1.1.1.4"><times id="S5.SS1.p3.3.m3.1.1.1.4.1.cmml" xref="S5.SS1.p3.3.m3.1.1.1.4.1"></times><ci id="S5.SS1.p3.3.m3.1.1.1.4.2.cmml" xref="S5.SS1.p3.3.m3.1.1.1.4.2">𝛽</ci><ci id="S5.SS1.p3.3.m3.1.1.1.1.cmml" xref="S5.SS1.p3.3.m3.1.1.1.1">𝑥</ci></apply></apply></apply><apply id="S5.SS1.p3.3.m3.4.4.1.1.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1"><times id="S5.SS1.p3.3.m3.4.4.1.1.2.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.2"></times><ci id="S5.SS1.p3.3.m3.4.4.1.1.3.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.3">𝑃</ci><list id="S5.SS1.p3.3.m3.4.4.1.1.1.2.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1"><ci id="S5.SS1.p3.3.m3.3.3.cmml" xref="S5.SS1.p3.3.m3.3.3">𝑥</ci><apply id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.1.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1">superscript</csymbol><ci id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.2.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.2">𝑥</ci><ci id="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.3.cmml" xref="S5.SS1.p3.3.m3.4.4.1.1.1.1.1.3">′</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.3.m3.4c">Q(x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(x;x^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.3.m3.4d">italic_Q ( italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT italic_P ( italic_x ; italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, and let <math alttext="\bar{P}(\tau;x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime})" class="ltx_Math" display="inline" id="S5.SS1.p3.4.m4.5"><semantics id="S5.SS1.p3.4.m4.5a"><mrow id="S5.SS1.p3.4.m4.5.5" xref="S5.SS1.p3.4.m4.5.5.cmml"><mrow id="S5.SS1.p3.4.m4.5.5.3" xref="S5.SS1.p3.4.m4.5.5.3.cmml"><mover accent="true" id="S5.SS1.p3.4.m4.5.5.3.2" xref="S5.SS1.p3.4.m4.5.5.3.2.cmml"><mi id="S5.SS1.p3.4.m4.5.5.3.2.2" xref="S5.SS1.p3.4.m4.5.5.3.2.2.cmml">P</mi><mo id="S5.SS1.p3.4.m4.5.5.3.2.1" xref="S5.SS1.p3.4.m4.5.5.3.2.1.cmml">¯</mo></mover><mo id="S5.SS1.p3.4.m4.5.5.3.1" xref="S5.SS1.p3.4.m4.5.5.3.1.cmml">⁢</mo><mrow id="S5.SS1.p3.4.m4.5.5.3.3.2" xref="S5.SS1.p3.4.m4.5.5.3.3.1.cmml"><mo id="S5.SS1.p3.4.m4.5.5.3.3.2.1" stretchy="false" xref="S5.SS1.p3.4.m4.5.5.3.3.1.cmml">(</mo><mi id="S5.SS1.p3.4.m4.2.2" xref="S5.SS1.p3.4.m4.2.2.cmml">τ</mi><mo id="S5.SS1.p3.4.m4.5.5.3.3.2.2" xref="S5.SS1.p3.4.m4.5.5.3.3.1.cmml">;</mo><mi id="S5.SS1.p3.4.m4.3.3" xref="S5.SS1.p3.4.m4.3.3.cmml">x</mi><mo id="S5.SS1.p3.4.m4.5.5.3.3.2.3" stretchy="false" xref="S5.SS1.p3.4.m4.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.p3.4.m4.5.5.2" rspace="0.1389em" xref="S5.SS1.p3.4.m4.5.5.2.cmml">=</mo><mrow id="S5.SS1.p3.4.m4.5.5.1" xref="S5.SS1.p3.4.m4.5.5.1.cmml"><msub id="S5.SS1.p3.4.m4.5.5.1.2" xref="S5.SS1.p3.4.m4.5.5.1.2.cmml"><mo id="S5.SS1.p3.4.m4.5.5.1.2.2" lspace="0.1389em" rspace="0.167em" xref="S5.SS1.p3.4.m4.5.5.1.2.2.cmml">𝔼</mo><mrow id="S5.SS1.p3.4.m4.1.1.1" xref="S5.SS1.p3.4.m4.1.1.1.cmml"><msup id="S5.SS1.p3.4.m4.1.1.1.3" xref="S5.SS1.p3.4.m4.1.1.1.3.cmml"><mi id="S5.SS1.p3.4.m4.1.1.1.3.2" xref="S5.SS1.p3.4.m4.1.1.1.3.2.cmml">x</mi><mo id="S5.SS1.p3.4.m4.1.1.1.3.3" xref="S5.SS1.p3.4.m4.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.SS1.p3.4.m4.1.1.1.2" xref="S5.SS1.p3.4.m4.1.1.1.2.cmml">∼</mo><mrow id="S5.SS1.p3.4.m4.1.1.1.4" xref="S5.SS1.p3.4.m4.1.1.1.4.cmml"><mi id="S5.SS1.p3.4.m4.1.1.1.4.2" xref="S5.SS1.p3.4.m4.1.1.1.4.2.cmml">β</mi><mo id="S5.SS1.p3.4.m4.1.1.1.4.1" xref="S5.SS1.p3.4.m4.1.1.1.4.1.cmml">⁢</mo><mrow id="S5.SS1.p3.4.m4.1.1.1.4.3.2" xref="S5.SS1.p3.4.m4.1.1.1.4.cmml"><mo id="S5.SS1.p3.4.m4.1.1.1.4.3.2.1" stretchy="false" xref="S5.SS1.p3.4.m4.1.1.1.4.cmml">(</mo><mi id="S5.SS1.p3.4.m4.1.1.1.1" xref="S5.SS1.p3.4.m4.1.1.1.1.cmml">x</mi><mo id="S5.SS1.p3.4.m4.1.1.1.4.3.2.2" stretchy="false" xref="S5.SS1.p3.4.m4.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S5.SS1.p3.4.m4.5.5.1.1" xref="S5.SS1.p3.4.m4.5.5.1.1.cmml"><mi id="S5.SS1.p3.4.m4.5.5.1.1.3" xref="S5.SS1.p3.4.m4.5.5.1.1.3.cmml">P</mi><mo id="S5.SS1.p3.4.m4.5.5.1.1.2" xref="S5.SS1.p3.4.m4.5.5.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.p3.4.m4.5.5.1.1.1.1" xref="S5.SS1.p3.4.m4.5.5.1.1.1.2.cmml"><mo id="S5.SS1.p3.4.m4.5.5.1.1.1.1.2" stretchy="false" xref="S5.SS1.p3.4.m4.5.5.1.1.1.2.cmml">(</mo><mi id="S5.SS1.p3.4.m4.4.4" xref="S5.SS1.p3.4.m4.4.4.cmml">τ</mi><mo id="S5.SS1.p3.4.m4.5.5.1.1.1.1.3" xref="S5.SS1.p3.4.m4.5.5.1.1.1.2.cmml">;</mo><msup id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.cmml"><mi id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.2" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.2.cmml">x</mi><mo id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.3" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.SS1.p3.4.m4.5.5.1.1.1.1.4" stretchy="false" xref="S5.SS1.p3.4.m4.5.5.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.4.m4.5b"><apply id="S5.SS1.p3.4.m4.5.5.cmml" xref="S5.SS1.p3.4.m4.5.5"><eq id="S5.SS1.p3.4.m4.5.5.2.cmml" xref="S5.SS1.p3.4.m4.5.5.2"></eq><apply id="S5.SS1.p3.4.m4.5.5.3.cmml" xref="S5.SS1.p3.4.m4.5.5.3"><times id="S5.SS1.p3.4.m4.5.5.3.1.cmml" xref="S5.SS1.p3.4.m4.5.5.3.1"></times><apply id="S5.SS1.p3.4.m4.5.5.3.2.cmml" xref="S5.SS1.p3.4.m4.5.5.3.2"><ci id="S5.SS1.p3.4.m4.5.5.3.2.1.cmml" xref="S5.SS1.p3.4.m4.5.5.3.2.1">¯</ci><ci id="S5.SS1.p3.4.m4.5.5.3.2.2.cmml" xref="S5.SS1.p3.4.m4.5.5.3.2.2">𝑃</ci></apply><list id="S5.SS1.p3.4.m4.5.5.3.3.1.cmml" xref="S5.SS1.p3.4.m4.5.5.3.3.2"><ci id="S5.SS1.p3.4.m4.2.2.cmml" xref="S5.SS1.p3.4.m4.2.2">𝜏</ci><ci id="S5.SS1.p3.4.m4.3.3.cmml" xref="S5.SS1.p3.4.m4.3.3">𝑥</ci></list></apply><apply id="S5.SS1.p3.4.m4.5.5.1.cmml" xref="S5.SS1.p3.4.m4.5.5.1"><apply id="S5.SS1.p3.4.m4.5.5.1.2.cmml" xref="S5.SS1.p3.4.m4.5.5.1.2"><csymbol cd="ambiguous" id="S5.SS1.p3.4.m4.5.5.1.2.1.cmml" xref="S5.SS1.p3.4.m4.5.5.1.2">subscript</csymbol><ci id="S5.SS1.p3.4.m4.5.5.1.2.2.cmml" xref="S5.SS1.p3.4.m4.5.5.1.2.2">𝔼</ci><apply id="S5.SS1.p3.4.m4.1.1.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1"><csymbol cd="latexml" id="S5.SS1.p3.4.m4.1.1.1.2.cmml" xref="S5.SS1.p3.4.m4.1.1.1.2">similar-to</csymbol><apply id="S5.SS1.p3.4.m4.1.1.1.3.cmml" xref="S5.SS1.p3.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p3.4.m4.1.1.1.3.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1.3">superscript</csymbol><ci id="S5.SS1.p3.4.m4.1.1.1.3.2.cmml" xref="S5.SS1.p3.4.m4.1.1.1.3.2">𝑥</ci><ci id="S5.SS1.p3.4.m4.1.1.1.3.3.cmml" xref="S5.SS1.p3.4.m4.1.1.1.3.3">′</ci></apply><apply id="S5.SS1.p3.4.m4.1.1.1.4.cmml" xref="S5.SS1.p3.4.m4.1.1.1.4"><times id="S5.SS1.p3.4.m4.1.1.1.4.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1.4.1"></times><ci id="S5.SS1.p3.4.m4.1.1.1.4.2.cmml" xref="S5.SS1.p3.4.m4.1.1.1.4.2">𝛽</ci><ci id="S5.SS1.p3.4.m4.1.1.1.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1.1">𝑥</ci></apply></apply></apply><apply id="S5.SS1.p3.4.m4.5.5.1.1.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1"><times id="S5.SS1.p3.4.m4.5.5.1.1.2.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.2"></times><ci id="S5.SS1.p3.4.m4.5.5.1.1.3.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.3">𝑃</ci><list id="S5.SS1.p3.4.m4.5.5.1.1.1.2.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1"><ci id="S5.SS1.p3.4.m4.4.4.cmml" xref="S5.SS1.p3.4.m4.4.4">𝜏</ci><apply id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.1.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1">superscript</csymbol><ci id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.2.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.2">𝑥</ci><ci id="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.3.cmml" xref="S5.SS1.p3.4.m4.5.5.1.1.1.1.1.3">′</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.4.m4.5c">\bar{P}(\tau;x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.4.m4.5d">over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> be the expected conditional probability of another agent reporting <math alttext="L" class="ltx_Math" display="inline" id="S5.SS1.p3.5.m5.1"><semantics id="S5.SS1.p3.5.m5.1a"><mi id="S5.SS1.p3.5.m5.1.1" xref="S5.SS1.p3.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.5.m5.1b"><ci id="S5.SS1.p3.5.m5.1.1.cmml" xref="S5.SS1.p3.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.5.m5.1d">italic_L</annotation></semantics></math> over signal <math alttext="x" class="ltx_Math" display="inline" id="S5.SS1.p3.6.m6.1"><semantics id="S5.SS1.p3.6.m6.1a"><mi id="S5.SS1.p3.6.m6.1.1" xref="S5.SS1.p3.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.6.m6.1b"><ci id="S5.SS1.p3.6.m6.1.1.cmml" xref="S5.SS1.p3.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.6.m6.1d">italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="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="Thmtheorem6.1.1.1">Theorem 6</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem6.p1"> <p class="ltx_p" id="Thmtheorem6.p1.11">Let a finite threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem6.p1.1.m1.1"><semantics id="Thmtheorem6.p1.1.m1.1a"><mi id="Thmtheorem6.p1.1.m1.1.1" xref="Thmtheorem6.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.1.m1.1b"><ci id="Thmtheorem6.p1.1.m1.1.1.cmml" xref="Thmtheorem6.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.1.m1.1d">italic_τ</annotation></semantics></math> be given and <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem6.p1.2.m2.2"><semantics id="Thmtheorem6.p1.2.m2.2a"><mrow id="Thmtheorem6.p1.2.m2.2.3" xref="Thmtheorem6.p1.2.m2.2.3.cmml"><mi id="Thmtheorem6.p1.2.m2.2.3.2" xref="Thmtheorem6.p1.2.m2.2.3.2.cmml">P</mi><mo id="Thmtheorem6.p1.2.m2.2.3.1" xref="Thmtheorem6.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.2.m2.2.3.3.2" xref="Thmtheorem6.p1.2.m2.2.3.3.1.cmml"><mo id="Thmtheorem6.p1.2.m2.2.3.3.2.1" stretchy="false" xref="Thmtheorem6.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="Thmtheorem6.p1.2.m2.1.1" xref="Thmtheorem6.p1.2.m2.1.1.cmml">τ</mi><mo id="Thmtheorem6.p1.2.m2.2.3.3.2.2" xref="Thmtheorem6.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="Thmtheorem6.p1.2.m2.2.2" xref="Thmtheorem6.p1.2.m2.2.2.cmml">x</mi><mo id="Thmtheorem6.p1.2.m2.2.3.3.2.3" stretchy="false" xref="Thmtheorem6.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.2.m2.2b"><apply id="Thmtheorem6.p1.2.m2.2.3.cmml" xref="Thmtheorem6.p1.2.m2.2.3"><times id="Thmtheorem6.p1.2.m2.2.3.1.cmml" xref="Thmtheorem6.p1.2.m2.2.3.1"></times><ci id="Thmtheorem6.p1.2.m2.2.3.2.cmml" xref="Thmtheorem6.p1.2.m2.2.3.2">𝑃</ci><list id="Thmtheorem6.p1.2.m2.2.3.3.1.cmml" xref="Thmtheorem6.p1.2.m2.2.3.3.2"><ci id="Thmtheorem6.p1.2.m2.1.1.cmml" xref="Thmtheorem6.p1.2.m2.1.1">𝜏</ci><ci id="Thmtheorem6.p1.2.m2.2.2.cmml" xref="Thmtheorem6.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> be continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem6.p1.3.m3.1"><semantics id="Thmtheorem6.p1.3.m3.1a"><mi id="Thmtheorem6.p1.3.m3.1.1" xref="Thmtheorem6.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.3.m3.1b"><ci id="Thmtheorem6.p1.3.m3.1.1.cmml" xref="Thmtheorem6.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.3.m3.1d">italic_x</annotation></semantics></math>; also assume <math alttext="\beta(x)" class="ltx_Math" display="inline" id="Thmtheorem6.p1.4.m4.1"><semantics id="Thmtheorem6.p1.4.m4.1a"><mrow id="Thmtheorem6.p1.4.m4.1.2" xref="Thmtheorem6.p1.4.m4.1.2.cmml"><mi id="Thmtheorem6.p1.4.m4.1.2.2" xref="Thmtheorem6.p1.4.m4.1.2.2.cmml">β</mi><mo id="Thmtheorem6.p1.4.m4.1.2.1" xref="Thmtheorem6.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.4.m4.1.2.3.2" xref="Thmtheorem6.p1.4.m4.1.2.cmml"><mo id="Thmtheorem6.p1.4.m4.1.2.3.2.1" stretchy="false" xref="Thmtheorem6.p1.4.m4.1.2.cmml">(</mo><mi id="Thmtheorem6.p1.4.m4.1.1" xref="Thmtheorem6.p1.4.m4.1.1.cmml">x</mi><mo id="Thmtheorem6.p1.4.m4.1.2.3.2.2" stretchy="false" xref="Thmtheorem6.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.4.m4.1b"><apply id="Thmtheorem6.p1.4.m4.1.2.cmml" xref="Thmtheorem6.p1.4.m4.1.2"><times id="Thmtheorem6.p1.4.m4.1.2.1.cmml" xref="Thmtheorem6.p1.4.m4.1.2.1"></times><ci id="Thmtheorem6.p1.4.m4.1.2.2.cmml" xref="Thmtheorem6.p1.4.m4.1.2.2">𝛽</ci><ci id="Thmtheorem6.p1.4.m4.1.1.cmml" xref="Thmtheorem6.p1.4.m4.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.4.m4.1c">\beta(x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.4.m4.1d">italic_β ( italic_x )</annotation></semantics></math> is continuous over <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem6.p1.5.m5.1"><semantics id="Thmtheorem6.p1.5.m5.1a"><mi id="Thmtheorem6.p1.5.m5.1.1" xref="Thmtheorem6.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.5.m5.1b"><ci id="Thmtheorem6.p1.5.m5.1.1.cmml" xref="Thmtheorem6.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.5.m5.1d">italic_x</annotation></semantics></math>. If <math alttext="\tau\in\mathbb{R}" class="ltx_Math" display="inline" id="Thmtheorem6.p1.6.m6.1"><semantics id="Thmtheorem6.p1.6.m6.1a"><mrow id="Thmtheorem6.p1.6.m6.1.1" xref="Thmtheorem6.p1.6.m6.1.1.cmml"><mi id="Thmtheorem6.p1.6.m6.1.1.2" xref="Thmtheorem6.p1.6.m6.1.1.2.cmml">τ</mi><mo id="Thmtheorem6.p1.6.m6.1.1.1" xref="Thmtheorem6.p1.6.m6.1.1.1.cmml">∈</mo><mi id="Thmtheorem6.p1.6.m6.1.1.3" xref="Thmtheorem6.p1.6.m6.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.6.m6.1b"><apply id="Thmtheorem6.p1.6.m6.1.1.cmml" xref="Thmtheorem6.p1.6.m6.1.1"><in id="Thmtheorem6.p1.6.m6.1.1.1.cmml" xref="Thmtheorem6.p1.6.m6.1.1.1"></in><ci id="Thmtheorem6.p1.6.m6.1.1.2.cmml" xref="Thmtheorem6.p1.6.m6.1.1.2">𝜏</ci><ci id="Thmtheorem6.p1.6.m6.1.1.3.cmml" xref="Thmtheorem6.p1.6.m6.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.6.m6.1c">\tau\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.6.m6.1d">italic_τ ∈ blackboard_R</annotation></semantics></math> is threshold equilibrium under the RBTS mechanism then <math alttext="Q(\tau)=G(\tau)" class="ltx_Math" display="inline" id="Thmtheorem6.p1.7.m7.2"><semantics id="Thmtheorem6.p1.7.m7.2a"><mrow id="Thmtheorem6.p1.7.m7.2.3" xref="Thmtheorem6.p1.7.m7.2.3.cmml"><mrow id="Thmtheorem6.p1.7.m7.2.3.2" xref="Thmtheorem6.p1.7.m7.2.3.2.cmml"><mi id="Thmtheorem6.p1.7.m7.2.3.2.2" xref="Thmtheorem6.p1.7.m7.2.3.2.2.cmml">Q</mi><mo id="Thmtheorem6.p1.7.m7.2.3.2.1" xref="Thmtheorem6.p1.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.7.m7.2.3.2.3.2" xref="Thmtheorem6.p1.7.m7.2.3.2.cmml"><mo id="Thmtheorem6.p1.7.m7.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem6.p1.7.m7.2.3.2.cmml">(</mo><mi id="Thmtheorem6.p1.7.m7.1.1" xref="Thmtheorem6.p1.7.m7.1.1.cmml">τ</mi><mo id="Thmtheorem6.p1.7.m7.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem6.p1.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem6.p1.7.m7.2.3.1" xref="Thmtheorem6.p1.7.m7.2.3.1.cmml">=</mo><mrow id="Thmtheorem6.p1.7.m7.2.3.3" xref="Thmtheorem6.p1.7.m7.2.3.3.cmml"><mi id="Thmtheorem6.p1.7.m7.2.3.3.2" xref="Thmtheorem6.p1.7.m7.2.3.3.2.cmml">G</mi><mo id="Thmtheorem6.p1.7.m7.2.3.3.1" xref="Thmtheorem6.p1.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.7.m7.2.3.3.3.2" xref="Thmtheorem6.p1.7.m7.2.3.3.cmml"><mo id="Thmtheorem6.p1.7.m7.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem6.p1.7.m7.2.3.3.cmml">(</mo><mi id="Thmtheorem6.p1.7.m7.2.2" xref="Thmtheorem6.p1.7.m7.2.2.cmml">τ</mi><mo id="Thmtheorem6.p1.7.m7.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem6.p1.7.m7.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.7.m7.2b"><apply id="Thmtheorem6.p1.7.m7.2.3.cmml" xref="Thmtheorem6.p1.7.m7.2.3"><eq id="Thmtheorem6.p1.7.m7.2.3.1.cmml" xref="Thmtheorem6.p1.7.m7.2.3.1"></eq><apply id="Thmtheorem6.p1.7.m7.2.3.2.cmml" xref="Thmtheorem6.p1.7.m7.2.3.2"><times id="Thmtheorem6.p1.7.m7.2.3.2.1.cmml" xref="Thmtheorem6.p1.7.m7.2.3.2.1"></times><ci id="Thmtheorem6.p1.7.m7.2.3.2.2.cmml" xref="Thmtheorem6.p1.7.m7.2.3.2.2">𝑄</ci><ci id="Thmtheorem6.p1.7.m7.1.1.cmml" xref="Thmtheorem6.p1.7.m7.1.1">𝜏</ci></apply><apply id="Thmtheorem6.p1.7.m7.2.3.3.cmml" xref="Thmtheorem6.p1.7.m7.2.3.3"><times id="Thmtheorem6.p1.7.m7.2.3.3.1.cmml" xref="Thmtheorem6.p1.7.m7.2.3.3.1"></times><ci id="Thmtheorem6.p1.7.m7.2.3.3.2.cmml" xref="Thmtheorem6.p1.7.m7.2.3.3.2">𝐺</ci><ci id="Thmtheorem6.p1.7.m7.2.2.cmml" xref="Thmtheorem6.p1.7.m7.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.7.m7.2c">Q(\tau)=G(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.7.m7.2d">italic_Q ( italic_τ ) = italic_G ( italic_τ )</annotation></semantics></math>. Conversely, if <math alttext="Q(\tau)=G(\tau)" class="ltx_Math" display="inline" id="Thmtheorem6.p1.8.m8.2"><semantics id="Thmtheorem6.p1.8.m8.2a"><mrow id="Thmtheorem6.p1.8.m8.2.3" xref="Thmtheorem6.p1.8.m8.2.3.cmml"><mrow id="Thmtheorem6.p1.8.m8.2.3.2" xref="Thmtheorem6.p1.8.m8.2.3.2.cmml"><mi id="Thmtheorem6.p1.8.m8.2.3.2.2" xref="Thmtheorem6.p1.8.m8.2.3.2.2.cmml">Q</mi><mo id="Thmtheorem6.p1.8.m8.2.3.2.1" xref="Thmtheorem6.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.8.m8.2.3.2.3.2" xref="Thmtheorem6.p1.8.m8.2.3.2.cmml"><mo id="Thmtheorem6.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="Thmtheorem6.p1.8.m8.2.3.2.cmml">(</mo><mi id="Thmtheorem6.p1.8.m8.1.1" xref="Thmtheorem6.p1.8.m8.1.1.cmml">τ</mi><mo id="Thmtheorem6.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="Thmtheorem6.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="Thmtheorem6.p1.8.m8.2.3.1" xref="Thmtheorem6.p1.8.m8.2.3.1.cmml">=</mo><mrow id="Thmtheorem6.p1.8.m8.2.3.3" xref="Thmtheorem6.p1.8.m8.2.3.3.cmml"><mi id="Thmtheorem6.p1.8.m8.2.3.3.2" xref="Thmtheorem6.p1.8.m8.2.3.3.2.cmml">G</mi><mo id="Thmtheorem6.p1.8.m8.2.3.3.1" xref="Thmtheorem6.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.8.m8.2.3.3.3.2" xref="Thmtheorem6.p1.8.m8.2.3.3.cmml"><mo id="Thmtheorem6.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="Thmtheorem6.p1.8.m8.2.3.3.cmml">(</mo><mi id="Thmtheorem6.p1.8.m8.2.2" xref="Thmtheorem6.p1.8.m8.2.2.cmml">τ</mi><mo id="Thmtheorem6.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="Thmtheorem6.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.8.m8.2b"><apply id="Thmtheorem6.p1.8.m8.2.3.cmml" xref="Thmtheorem6.p1.8.m8.2.3"><eq id="Thmtheorem6.p1.8.m8.2.3.1.cmml" xref="Thmtheorem6.p1.8.m8.2.3.1"></eq><apply id="Thmtheorem6.p1.8.m8.2.3.2.cmml" xref="Thmtheorem6.p1.8.m8.2.3.2"><times id="Thmtheorem6.p1.8.m8.2.3.2.1.cmml" xref="Thmtheorem6.p1.8.m8.2.3.2.1"></times><ci id="Thmtheorem6.p1.8.m8.2.3.2.2.cmml" xref="Thmtheorem6.p1.8.m8.2.3.2.2">𝑄</ci><ci id="Thmtheorem6.p1.8.m8.1.1.cmml" xref="Thmtheorem6.p1.8.m8.1.1">𝜏</ci></apply><apply id="Thmtheorem6.p1.8.m8.2.3.3.cmml" xref="Thmtheorem6.p1.8.m8.2.3.3"><times id="Thmtheorem6.p1.8.m8.2.3.3.1.cmml" xref="Thmtheorem6.p1.8.m8.2.3.3.1"></times><ci id="Thmtheorem6.p1.8.m8.2.3.3.2.cmml" xref="Thmtheorem6.p1.8.m8.2.3.3.2">𝐺</ci><ci id="Thmtheorem6.p1.8.m8.2.2.cmml" xref="Thmtheorem6.p1.8.m8.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.8.m8.2c">Q(\tau)=G(\tau)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.8.m8.2d">italic_Q ( italic_τ ) = italic_G ( italic_τ )</annotation></semantics></math> and <math alttext="P(\tau;x)-\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="Thmtheorem6.p1.9.m9.4"><semantics id="Thmtheorem6.p1.9.m9.4a"><mrow id="Thmtheorem6.p1.9.m9.4.5" xref="Thmtheorem6.p1.9.m9.4.5.cmml"><mrow id="Thmtheorem6.p1.9.m9.4.5.2" xref="Thmtheorem6.p1.9.m9.4.5.2.cmml"><mi id="Thmtheorem6.p1.9.m9.4.5.2.2" xref="Thmtheorem6.p1.9.m9.4.5.2.2.cmml">P</mi><mo id="Thmtheorem6.p1.9.m9.4.5.2.1" xref="Thmtheorem6.p1.9.m9.4.5.2.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.9.m9.4.5.2.3.2" xref="Thmtheorem6.p1.9.m9.4.5.2.3.1.cmml"><mo id="Thmtheorem6.p1.9.m9.4.5.2.3.2.1" stretchy="false" xref="Thmtheorem6.p1.9.m9.4.5.2.3.1.cmml">(</mo><mi id="Thmtheorem6.p1.9.m9.1.1" xref="Thmtheorem6.p1.9.m9.1.1.cmml">τ</mi><mo id="Thmtheorem6.p1.9.m9.4.5.2.3.2.2" xref="Thmtheorem6.p1.9.m9.4.5.2.3.1.cmml">;</mo><mi id="Thmtheorem6.p1.9.m9.2.2" xref="Thmtheorem6.p1.9.m9.2.2.cmml">x</mi><mo id="Thmtheorem6.p1.9.m9.4.5.2.3.2.3" stretchy="false" xref="Thmtheorem6.p1.9.m9.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="Thmtheorem6.p1.9.m9.4.5.1" xref="Thmtheorem6.p1.9.m9.4.5.1.cmml">−</mo><mrow id="Thmtheorem6.p1.9.m9.4.5.3" xref="Thmtheorem6.p1.9.m9.4.5.3.cmml"><mover accent="true" id="Thmtheorem6.p1.9.m9.4.5.3.2" xref="Thmtheorem6.p1.9.m9.4.5.3.2.cmml"><mi id="Thmtheorem6.p1.9.m9.4.5.3.2.2" xref="Thmtheorem6.p1.9.m9.4.5.3.2.2.cmml">P</mi><mo id="Thmtheorem6.p1.9.m9.4.5.3.2.1" xref="Thmtheorem6.p1.9.m9.4.5.3.2.1.cmml">¯</mo></mover><mo id="Thmtheorem6.p1.9.m9.4.5.3.1" xref="Thmtheorem6.p1.9.m9.4.5.3.1.cmml">⁢</mo><mrow id="Thmtheorem6.p1.9.m9.4.5.3.3.2" xref="Thmtheorem6.p1.9.m9.4.5.3.3.1.cmml"><mo id="Thmtheorem6.p1.9.m9.4.5.3.3.2.1" stretchy="false" xref="Thmtheorem6.p1.9.m9.4.5.3.3.1.cmml">(</mo><mi id="Thmtheorem6.p1.9.m9.3.3" xref="Thmtheorem6.p1.9.m9.3.3.cmml">τ</mi><mo id="Thmtheorem6.p1.9.m9.4.5.3.3.2.2" xref="Thmtheorem6.p1.9.m9.4.5.3.3.1.cmml">;</mo><mi id="Thmtheorem6.p1.9.m9.4.4" xref="Thmtheorem6.p1.9.m9.4.4.cmml">x</mi><mo id="Thmtheorem6.p1.9.m9.4.5.3.3.2.3" stretchy="false" xref="Thmtheorem6.p1.9.m9.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.9.m9.4b"><apply id="Thmtheorem6.p1.9.m9.4.5.cmml" xref="Thmtheorem6.p1.9.m9.4.5"><minus id="Thmtheorem6.p1.9.m9.4.5.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.1"></minus><apply id="Thmtheorem6.p1.9.m9.4.5.2.cmml" xref="Thmtheorem6.p1.9.m9.4.5.2"><times id="Thmtheorem6.p1.9.m9.4.5.2.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.2.1"></times><ci id="Thmtheorem6.p1.9.m9.4.5.2.2.cmml" xref="Thmtheorem6.p1.9.m9.4.5.2.2">𝑃</ci><list id="Thmtheorem6.p1.9.m9.4.5.2.3.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.2.3.2"><ci id="Thmtheorem6.p1.9.m9.1.1.cmml" xref="Thmtheorem6.p1.9.m9.1.1">𝜏</ci><ci id="Thmtheorem6.p1.9.m9.2.2.cmml" xref="Thmtheorem6.p1.9.m9.2.2">𝑥</ci></list></apply><apply id="Thmtheorem6.p1.9.m9.4.5.3.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3"><times id="Thmtheorem6.p1.9.m9.4.5.3.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3.1"></times><apply id="Thmtheorem6.p1.9.m9.4.5.3.2.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3.2"><ci id="Thmtheorem6.p1.9.m9.4.5.3.2.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3.2.1">¯</ci><ci id="Thmtheorem6.p1.9.m9.4.5.3.2.2.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3.2.2">𝑃</ci></apply><list id="Thmtheorem6.p1.9.m9.4.5.3.3.1.cmml" xref="Thmtheorem6.p1.9.m9.4.5.3.3.2"><ci id="Thmtheorem6.p1.9.m9.3.3.cmml" xref="Thmtheorem6.p1.9.m9.3.3">𝜏</ci><ci id="Thmtheorem6.p1.9.m9.4.4.cmml" xref="Thmtheorem6.p1.9.m9.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.9.m9.4c">P(\tau;x)-\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.9.m9.4d">italic_P ( italic_τ ; italic_x ) - over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math> has a single crossing of <math alttext="0" class="ltx_Math" display="inline" id="Thmtheorem6.p1.10.m10.1"><semantics id="Thmtheorem6.p1.10.m10.1a"><mn id="Thmtheorem6.p1.10.m10.1.1" xref="Thmtheorem6.p1.10.m10.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.10.m10.1b"><cn id="Thmtheorem6.p1.10.m10.1.1.cmml" type="integer" xref="Thmtheorem6.p1.10.m10.1.1">0</cn></annotation-xml></semantics></math> from positive to negative then <math alttext="\tau" class="ltx_Math" display="inline" id="Thmtheorem6.p1.11.m11.1"><semantics id="Thmtheorem6.p1.11.m11.1a"><mi id="Thmtheorem6.p1.11.m11.1.1" xref="Thmtheorem6.p1.11.m11.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem6.p1.11.m11.1b"><ci id="Thmtheorem6.p1.11.m11.1.1.cmml" xref="Thmtheorem6.p1.11.m11.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem6.p1.11.m11.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem6.p1.11.m11.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium under the RBTS mechanism.</p> </div> </div> <div class="ltx_proof" id="S5.SS1.3"> <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.p1.11">For necessity, first note that since <math alttext="\beta(x)" 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.m1.1.2" xref="S5.SS1.2.p1.1.m1.1.2.cmml"><mi id="S5.SS1.2.p1.1.m1.1.2.2" xref="S5.SS1.2.p1.1.m1.1.2.2.cmml">β</mi><mo id="S5.SS1.2.p1.1.m1.1.2.1" xref="S5.SS1.2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.1.m1.1.2.3.2" xref="S5.SS1.2.p1.1.m1.1.2.cmml"><mo id="S5.SS1.2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.1.m1.1.2.cmml">(</mo><mi id="S5.SS1.2.p1.1.m1.1.1" xref="S5.SS1.2.p1.1.m1.1.1.cmml">x</mi><mo id="S5.SS1.2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS1.2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.1.m1.1b"><apply id="S5.SS1.2.p1.1.m1.1.2.cmml" xref="S5.SS1.2.p1.1.m1.1.2"><times id="S5.SS1.2.p1.1.m1.1.2.1.cmml" xref="S5.SS1.2.p1.1.m1.1.2.1"></times><ci id="S5.SS1.2.p1.1.m1.1.2.2.cmml" xref="S5.SS1.2.p1.1.m1.1.2.2">𝛽</ci><ci id="S5.SS1.2.p1.1.m1.1.1.cmml" xref="S5.SS1.2.p1.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.1.m1.1c">\beta(x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.1.m1.1d">italic_β ( italic_x )</annotation></semantics></math> and <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="S5.SS1.2.p1.2.m2.2"><semantics id="S5.SS1.2.p1.2.m2.2a"><mrow id="S5.SS1.2.p1.2.m2.2.3" xref="S5.SS1.2.p1.2.m2.2.3.cmml"><mi id="S5.SS1.2.p1.2.m2.2.3.2" xref="S5.SS1.2.p1.2.m2.2.3.2.cmml">P</mi><mo id="S5.SS1.2.p1.2.m2.2.3.1" xref="S5.SS1.2.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.2.m2.2.3.3.2" xref="S5.SS1.2.p1.2.m2.2.3.3.1.cmml"><mo id="S5.SS1.2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S5.SS1.2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.2.m2.1.1" xref="S5.SS1.2.p1.2.m2.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.2.m2.2.3.3.2.2" xref="S5.SS1.2.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.2.m2.2.2" xref="S5.SS1.2.p1.2.m2.2.2.cmml">x</mi><mo id="S5.SS1.2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S5.SS1.2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.2.m2.2b"><apply id="S5.SS1.2.p1.2.m2.2.3.cmml" xref="S5.SS1.2.p1.2.m2.2.3"><times id="S5.SS1.2.p1.2.m2.2.3.1.cmml" xref="S5.SS1.2.p1.2.m2.2.3.1"></times><ci id="S5.SS1.2.p1.2.m2.2.3.2.cmml" xref="S5.SS1.2.p1.2.m2.2.3.2">𝑃</ci><list id="S5.SS1.2.p1.2.m2.2.3.3.1.cmml" xref="S5.SS1.2.p1.2.m2.2.3.3.2"><ci id="S5.SS1.2.p1.2.m2.1.1.cmml" xref="S5.SS1.2.p1.2.m2.1.1">𝜏</ci><ci id="S5.SS1.2.p1.2.m2.2.2.cmml" xref="S5.SS1.2.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> are continuous over <math alttext="x" class="ltx_Math" display="inline" id="S5.SS1.2.p1.3.m3.1"><semantics id="S5.SS1.2.p1.3.m3.1a"><mi id="S5.SS1.2.p1.3.m3.1.1" xref="S5.SS1.2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.3.m3.1b"><ci id="S5.SS1.2.p1.3.m3.1.1.cmml" xref="S5.SS1.2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.3.m3.1d">italic_x</annotation></semantics></math>, <math alttext="\bar{P}(\tau;x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime})" class="ltx_Math" display="inline" id="S5.SS1.2.p1.4.m4.5"><semantics id="S5.SS1.2.p1.4.m4.5a"><mrow id="S5.SS1.2.p1.4.m4.5.5" xref="S5.SS1.2.p1.4.m4.5.5.cmml"><mrow id="S5.SS1.2.p1.4.m4.5.5.3" xref="S5.SS1.2.p1.4.m4.5.5.3.cmml"><mover accent="true" id="S5.SS1.2.p1.4.m4.5.5.3.2" xref="S5.SS1.2.p1.4.m4.5.5.3.2.cmml"><mi id="S5.SS1.2.p1.4.m4.5.5.3.2.2" xref="S5.SS1.2.p1.4.m4.5.5.3.2.2.cmml">P</mi><mo id="S5.SS1.2.p1.4.m4.5.5.3.2.1" xref="S5.SS1.2.p1.4.m4.5.5.3.2.1.cmml">¯</mo></mover><mo 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encoding="MathML-Content" id="S5.SS1.2.p1.4.m4.5b"><apply id="S5.SS1.2.p1.4.m4.5.5.cmml" xref="S5.SS1.2.p1.4.m4.5.5"><eq id="S5.SS1.2.p1.4.m4.5.5.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.2"></eq><apply id="S5.SS1.2.p1.4.m4.5.5.3.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3"><times id="S5.SS1.2.p1.4.m4.5.5.3.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3.1"></times><apply id="S5.SS1.2.p1.4.m4.5.5.3.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3.2"><ci id="S5.SS1.2.p1.4.m4.5.5.3.2.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3.2.1">¯</ci><ci id="S5.SS1.2.p1.4.m4.5.5.3.2.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3.2.2">𝑃</ci></apply><list id="S5.SS1.2.p1.4.m4.5.5.3.3.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.3.3.2"><ci id="S5.SS1.2.p1.4.m4.2.2.cmml" xref="S5.SS1.2.p1.4.m4.2.2">𝜏</ci><ci id="S5.SS1.2.p1.4.m4.3.3.cmml" xref="S5.SS1.2.p1.4.m4.3.3">𝑥</ci></list></apply><apply id="S5.SS1.2.p1.4.m4.5.5.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1"><apply id="S5.SS1.2.p1.4.m4.5.5.1.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.2"><csymbol cd="ambiguous" 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xref="S5.SS1.2.p1.4.m4.1.1.1.1">𝑥</ci></apply></apply></apply><apply id="S5.SS1.2.p1.4.m4.5.5.1.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1"><times id="S5.SS1.2.p1.4.m4.5.5.1.1.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.2"></times><ci id="S5.SS1.2.p1.4.m4.5.5.1.1.3.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.3">𝑃</ci><list id="S5.SS1.2.p1.4.m4.5.5.1.1.1.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.1.1"><ci id="S5.SS1.2.p1.4.m4.4.4.cmml" xref="S5.SS1.2.p1.4.m4.4.4">𝜏</ci><apply id="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.1.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1">superscript</csymbol><ci id="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.2.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.2">𝑥</ci><ci id="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.3.cmml" xref="S5.SS1.2.p1.4.m4.5.5.1.1.1.1.1.3">′</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.4.m4.5c">\bar{P}(\tau;x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\beta(x)}P(\tau;x^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.4.m4.5d">over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_β ( italic_x ) end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is also continuous over <math alttext="x" class="ltx_Math" display="inline" id="S5.SS1.2.p1.5.m5.1"><semantics id="S5.SS1.2.p1.5.m5.1a"><mi id="S5.SS1.2.p1.5.m5.1.1" xref="S5.SS1.2.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.5.m5.1b"><ci id="S5.SS1.2.p1.5.m5.1.1.cmml" xref="S5.SS1.2.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.5.m5.1d">italic_x</annotation></semantics></math> by an application of the Dominated Convergence Theorem. Let <math alttext="f(\tau;x)=P(\tau;x)-\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="S5.SS1.2.p1.6.m6.6"><semantics id="S5.SS1.2.p1.6.m6.6a"><mrow id="S5.SS1.2.p1.6.m6.6.7" xref="S5.SS1.2.p1.6.m6.6.7.cmml"><mrow id="S5.SS1.2.p1.6.m6.6.7.2" xref="S5.SS1.2.p1.6.m6.6.7.2.cmml"><mi id="S5.SS1.2.p1.6.m6.6.7.2.2" xref="S5.SS1.2.p1.6.m6.6.7.2.2.cmml">f</mi><mo id="S5.SS1.2.p1.6.m6.6.7.2.1" xref="S5.SS1.2.p1.6.m6.6.7.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.6.m6.6.7.2.3.2" xref="S5.SS1.2.p1.6.m6.6.7.2.3.1.cmml"><mo id="S5.SS1.2.p1.6.m6.6.7.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.2.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.6.m6.1.1" xref="S5.SS1.2.p1.6.m6.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.6.m6.6.7.2.3.2.2" xref="S5.SS1.2.p1.6.m6.6.7.2.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.6.m6.2.2" xref="S5.SS1.2.p1.6.m6.2.2.cmml">x</mi><mo id="S5.SS1.2.p1.6.m6.6.7.2.3.2.3" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.2.p1.6.m6.6.7.1" xref="S5.SS1.2.p1.6.m6.6.7.1.cmml">=</mo><mrow id="S5.SS1.2.p1.6.m6.6.7.3" xref="S5.SS1.2.p1.6.m6.6.7.3.cmml"><mrow id="S5.SS1.2.p1.6.m6.6.7.3.2" xref="S5.SS1.2.p1.6.m6.6.7.3.2.cmml"><mi id="S5.SS1.2.p1.6.m6.6.7.3.2.2" xref="S5.SS1.2.p1.6.m6.6.7.3.2.2.cmml">P</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.2.1" xref="S5.SS1.2.p1.6.m6.6.7.3.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.6.m6.6.7.3.2.3.2" xref="S5.SS1.2.p1.6.m6.6.7.3.2.3.1.cmml"><mo id="S5.SS1.2.p1.6.m6.6.7.3.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.3.2.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.6.m6.3.3" xref="S5.SS1.2.p1.6.m6.3.3.cmml">τ</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.2.3.2.2" xref="S5.SS1.2.p1.6.m6.6.7.3.2.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.6.m6.4.4" xref="S5.SS1.2.p1.6.m6.4.4.cmml">x</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.2.3.2.3" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.2.p1.6.m6.6.7.3.1" xref="S5.SS1.2.p1.6.m6.6.7.3.1.cmml">−</mo><mrow id="S5.SS1.2.p1.6.m6.6.7.3.3" xref="S5.SS1.2.p1.6.m6.6.7.3.3.cmml"><mover accent="true" id="S5.SS1.2.p1.6.m6.6.7.3.3.2" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2.cmml"><mi id="S5.SS1.2.p1.6.m6.6.7.3.3.2.2" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2.2.cmml">P</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.3.2.1" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2.1.cmml">¯</mo></mover><mo id="S5.SS1.2.p1.6.m6.6.7.3.3.1" xref="S5.SS1.2.p1.6.m6.6.7.3.3.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.6.m6.6.7.3.3.3.2" xref="S5.SS1.2.p1.6.m6.6.7.3.3.3.1.cmml"><mo id="S5.SS1.2.p1.6.m6.6.7.3.3.3.2.1" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.3.3.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.6.m6.5.5" xref="S5.SS1.2.p1.6.m6.5.5.cmml">τ</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.3.3.2.2" xref="S5.SS1.2.p1.6.m6.6.7.3.3.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.6.m6.6.6" xref="S5.SS1.2.p1.6.m6.6.6.cmml">x</mi><mo id="S5.SS1.2.p1.6.m6.6.7.3.3.3.2.3" stretchy="false" xref="S5.SS1.2.p1.6.m6.6.7.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.6.m6.6b"><apply id="S5.SS1.2.p1.6.m6.6.7.cmml" xref="S5.SS1.2.p1.6.m6.6.7"><eq id="S5.SS1.2.p1.6.m6.6.7.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.1"></eq><apply id="S5.SS1.2.p1.6.m6.6.7.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.2"><times id="S5.SS1.2.p1.6.m6.6.7.2.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.2.1"></times><ci id="S5.SS1.2.p1.6.m6.6.7.2.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.2.2">𝑓</ci><list id="S5.SS1.2.p1.6.m6.6.7.2.3.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.2.3.2"><ci id="S5.SS1.2.p1.6.m6.1.1.cmml" xref="S5.SS1.2.p1.6.m6.1.1">𝜏</ci><ci id="S5.SS1.2.p1.6.m6.2.2.cmml" xref="S5.SS1.2.p1.6.m6.2.2">𝑥</ci></list></apply><apply id="S5.SS1.2.p1.6.m6.6.7.3.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3"><minus id="S5.SS1.2.p1.6.m6.6.7.3.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.1"></minus><apply id="S5.SS1.2.p1.6.m6.6.7.3.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.2"><times id="S5.SS1.2.p1.6.m6.6.7.3.2.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.2.1"></times><ci id="S5.SS1.2.p1.6.m6.6.7.3.2.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.2.2">𝑃</ci><list id="S5.SS1.2.p1.6.m6.6.7.3.2.3.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.2.3.2"><ci id="S5.SS1.2.p1.6.m6.3.3.cmml" xref="S5.SS1.2.p1.6.m6.3.3">𝜏</ci><ci id="S5.SS1.2.p1.6.m6.4.4.cmml" xref="S5.SS1.2.p1.6.m6.4.4">𝑥</ci></list></apply><apply id="S5.SS1.2.p1.6.m6.6.7.3.3.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3"><times id="S5.SS1.2.p1.6.m6.6.7.3.3.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3.1"></times><apply id="S5.SS1.2.p1.6.m6.6.7.3.3.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2"><ci id="S5.SS1.2.p1.6.m6.6.7.3.3.2.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2.1">¯</ci><ci id="S5.SS1.2.p1.6.m6.6.7.3.3.2.2.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3.2.2">𝑃</ci></apply><list id="S5.SS1.2.p1.6.m6.6.7.3.3.3.1.cmml" xref="S5.SS1.2.p1.6.m6.6.7.3.3.3.2"><ci id="S5.SS1.2.p1.6.m6.5.5.cmml" xref="S5.SS1.2.p1.6.m6.5.5">𝜏</ci><ci id="S5.SS1.2.p1.6.m6.6.6.cmml" xref="S5.SS1.2.p1.6.m6.6.6">𝑥</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.6.m6.6c">f(\tau;x)=P(\tau;x)-\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.6.m6.6d">italic_f ( italic_τ ; italic_x ) = italic_P ( italic_τ ; italic_x ) - over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>. Since <math alttext="f(\tau;x)" class="ltx_Math" display="inline" id="S5.SS1.2.p1.7.m7.2"><semantics id="S5.SS1.2.p1.7.m7.2a"><mrow id="S5.SS1.2.p1.7.m7.2.3" xref="S5.SS1.2.p1.7.m7.2.3.cmml"><mi id="S5.SS1.2.p1.7.m7.2.3.2" xref="S5.SS1.2.p1.7.m7.2.3.2.cmml">f</mi><mo id="S5.SS1.2.p1.7.m7.2.3.1" xref="S5.SS1.2.p1.7.m7.2.3.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.7.m7.2.3.3.2" xref="S5.SS1.2.p1.7.m7.2.3.3.1.cmml"><mo id="S5.SS1.2.p1.7.m7.2.3.3.2.1" stretchy="false" xref="S5.SS1.2.p1.7.m7.2.3.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.7.m7.1.1" xref="S5.SS1.2.p1.7.m7.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.7.m7.2.3.3.2.2" xref="S5.SS1.2.p1.7.m7.2.3.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.7.m7.2.2" xref="S5.SS1.2.p1.7.m7.2.2.cmml">x</mi><mo id="S5.SS1.2.p1.7.m7.2.3.3.2.3" stretchy="false" xref="S5.SS1.2.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.7.m7.2b"><apply id="S5.SS1.2.p1.7.m7.2.3.cmml" xref="S5.SS1.2.p1.7.m7.2.3"><times id="S5.SS1.2.p1.7.m7.2.3.1.cmml" xref="S5.SS1.2.p1.7.m7.2.3.1"></times><ci id="S5.SS1.2.p1.7.m7.2.3.2.cmml" xref="S5.SS1.2.p1.7.m7.2.3.2">𝑓</ci><list id="S5.SS1.2.p1.7.m7.2.3.3.1.cmml" xref="S5.SS1.2.p1.7.m7.2.3.3.2"><ci id="S5.SS1.2.p1.7.m7.1.1.cmml" xref="S5.SS1.2.p1.7.m7.1.1">𝜏</ci><ci id="S5.SS1.2.p1.7.m7.2.2.cmml" xref="S5.SS1.2.p1.7.m7.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.7.m7.2c">f(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.7.m7.2d">italic_f ( italic_τ ; italic_x )</annotation></semantics></math> is continuous, we must have <math alttext="\lim_{x\to\tau^{+}}f(\tau;x)=\lim_{x\to\tau^{-}}f(\tau;x)=f(\tau;\tau)." class="ltx_Math" display="inline" id="S5.SS1.2.p1.8.m8.7"><semantics id="S5.SS1.2.p1.8.m8.7a"><mrow id="S5.SS1.2.p1.8.m8.7.7.1" xref="S5.SS1.2.p1.8.m8.7.7.1.1.cmml"><mrow id="S5.SS1.2.p1.8.m8.7.7.1.1" xref="S5.SS1.2.p1.8.m8.7.7.1.1.cmml"><mrow id="S5.SS1.2.p1.8.m8.7.7.1.1.2" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.cmml"><msub id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.cmml"><mo id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.2" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.2.cmml">lim</mo><mrow id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.cmml"><mi id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.2" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.2.cmml">x</mi><mo id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.1" stretchy="false" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.1.cmml">→</mo><msup id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3.cmml"><mi id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3.2" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3.2.cmml">τ</mi><mo id="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3.3" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.1.3.3.3.cmml">+</mo></msup></mrow></msub><mrow id="S5.SS1.2.p1.8.m8.7.7.1.1.2.2" xref="S5.SS1.2.p1.8.m8.7.7.1.1.2.2.cmml"><mi id="S5.SS1.2.p1.8.m8.7.7.1.1.2.2.2" 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id="S5.SS1.2.p1.8.m8.7.7.1.1.4.2.3.1.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.4.2.3.2"><ci id="S5.SS1.2.p1.8.m8.3.3.cmml" xref="S5.SS1.2.p1.8.m8.3.3">𝜏</ci><ci id="S5.SS1.2.p1.8.m8.4.4.cmml" xref="S5.SS1.2.p1.8.m8.4.4">𝑥</ci></list></apply></apply></apply><apply id="S5.SS1.2.p1.8.m8.7.7.1.1c.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1"><eq id="S5.SS1.2.p1.8.m8.7.7.1.1.5.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S5.SS1.2.p1.8.m8.7.7.1.1.4.cmml" id="S5.SS1.2.p1.8.m8.7.7.1.1d.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1"></share><apply id="S5.SS1.2.p1.8.m8.7.7.1.1.6.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.6"><times id="S5.SS1.2.p1.8.m8.7.7.1.1.6.1.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.6.1"></times><ci id="S5.SS1.2.p1.8.m8.7.7.1.1.6.2.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.6.2">𝑓</ci><list id="S5.SS1.2.p1.8.m8.7.7.1.1.6.3.1.cmml" xref="S5.SS1.2.p1.8.m8.7.7.1.1.6.3.2"><ci id="S5.SS1.2.p1.8.m8.5.5.cmml" xref="S5.SS1.2.p1.8.m8.5.5">𝜏</ci><ci id="S5.SS1.2.p1.8.m8.6.6.cmml" xref="S5.SS1.2.p1.8.m8.6.6">𝜏</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.8.m8.7c">\lim_{x\to\tau^{+}}f(\tau;x)=\lim_{x\to\tau^{-}}f(\tau;x)=f(\tau;\tau).</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.8.m8.7d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_τ ; italic_x ) = roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_τ ; italic_x ) = italic_f ( italic_τ ; italic_τ ) .</annotation></semantics></math> Further, <math alttext="\lim_{x\to\tau^{-}}f(\tau;x)\geq 0" class="ltx_Math" display="inline" id="S5.SS1.2.p1.9.m9.2"><semantics id="S5.SS1.2.p1.9.m9.2a"><mrow id="S5.SS1.2.p1.9.m9.2.3" xref="S5.SS1.2.p1.9.m9.2.3.cmml"><mrow id="S5.SS1.2.p1.9.m9.2.3.2" xref="S5.SS1.2.p1.9.m9.2.3.2.cmml"><msub id="S5.SS1.2.p1.9.m9.2.3.2.1" xref="S5.SS1.2.p1.9.m9.2.3.2.1.cmml"><mo id="S5.SS1.2.p1.9.m9.2.3.2.1.2" xref="S5.SS1.2.p1.9.m9.2.3.2.1.2.cmml">lim</mo><mrow id="S5.SS1.2.p1.9.m9.2.3.2.1.3" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.cmml"><mi id="S5.SS1.2.p1.9.m9.2.3.2.1.3.2" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.2.cmml">x</mi><mo id="S5.SS1.2.p1.9.m9.2.3.2.1.3.1" stretchy="false" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.1.cmml">→</mo><msup id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.cmml"><mi id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.2" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.2.cmml">τ</mi><mo id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.3" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.3.cmml">−</mo></msup></mrow></msub><mrow id="S5.SS1.2.p1.9.m9.2.3.2.2" xref="S5.SS1.2.p1.9.m9.2.3.2.2.cmml"><mi id="S5.SS1.2.p1.9.m9.2.3.2.2.2" xref="S5.SS1.2.p1.9.m9.2.3.2.2.2.cmml">f</mi><mo id="S5.SS1.2.p1.9.m9.2.3.2.2.1" xref="S5.SS1.2.p1.9.m9.2.3.2.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.9.m9.2.3.2.2.3.2" xref="S5.SS1.2.p1.9.m9.2.3.2.2.3.1.cmml"><mo id="S5.SS1.2.p1.9.m9.2.3.2.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.9.m9.2.3.2.2.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.9.m9.1.1" xref="S5.SS1.2.p1.9.m9.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.9.m9.2.3.2.2.3.2.2" xref="S5.SS1.2.p1.9.m9.2.3.2.2.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.9.m9.2.2" xref="S5.SS1.2.p1.9.m9.2.2.cmml">x</mi><mo id="S5.SS1.2.p1.9.m9.2.3.2.2.3.2.3" stretchy="false" xref="S5.SS1.2.p1.9.m9.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.SS1.2.p1.9.m9.2.3.1" xref="S5.SS1.2.p1.9.m9.2.3.1.cmml">≥</mo><mn id="S5.SS1.2.p1.9.m9.2.3.3" xref="S5.SS1.2.p1.9.m9.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.9.m9.2b"><apply id="S5.SS1.2.p1.9.m9.2.3.cmml" xref="S5.SS1.2.p1.9.m9.2.3"><geq id="S5.SS1.2.p1.9.m9.2.3.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.1"></geq><apply id="S5.SS1.2.p1.9.m9.2.3.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2"><apply id="S5.SS1.2.p1.9.m9.2.3.2.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1"><csymbol cd="ambiguous" id="S5.SS1.2.p1.9.m9.2.3.2.1.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1">subscript</csymbol><limit id="S5.SS1.2.p1.9.m9.2.3.2.1.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.2"></limit><apply id="S5.SS1.2.p1.9.m9.2.3.2.1.3.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3"><ci id="S5.SS1.2.p1.9.m9.2.3.2.1.3.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.1">→</ci><ci id="S5.SS1.2.p1.9.m9.2.3.2.1.3.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.2">𝑥</ci><apply id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3">superscript</csymbol><ci id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.2">𝜏</ci><minus id="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.3.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.1.3.3.3"></minus></apply></apply></apply><apply id="S5.SS1.2.p1.9.m9.2.3.2.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.2"><times id="S5.SS1.2.p1.9.m9.2.3.2.2.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.2.1"></times><ci id="S5.SS1.2.p1.9.m9.2.3.2.2.2.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.2.2">𝑓</ci><list id="S5.SS1.2.p1.9.m9.2.3.2.2.3.1.cmml" xref="S5.SS1.2.p1.9.m9.2.3.2.2.3.2"><ci id="S5.SS1.2.p1.9.m9.1.1.cmml" xref="S5.SS1.2.p1.9.m9.1.1">𝜏</ci><ci id="S5.SS1.2.p1.9.m9.2.2.cmml" xref="S5.SS1.2.p1.9.m9.2.2">𝑥</ci></list></apply></apply><cn id="S5.SS1.2.p1.9.m9.2.3.3.cmml" type="integer" xref="S5.SS1.2.p1.9.m9.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.9.m9.2c">\lim_{x\to\tau^{-}}f(\tau;x)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.9.m9.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_τ ; italic_x ) ≥ 0</annotation></semantics></math> and <math alttext="\lim_{x\to\tau^{+}}f(\tau;x)\leq 0" class="ltx_Math" display="inline" id="S5.SS1.2.p1.10.m10.2"><semantics id="S5.SS1.2.p1.10.m10.2a"><mrow id="S5.SS1.2.p1.10.m10.2.3" xref="S5.SS1.2.p1.10.m10.2.3.cmml"><mrow id="S5.SS1.2.p1.10.m10.2.3.2" xref="S5.SS1.2.p1.10.m10.2.3.2.cmml"><msub id="S5.SS1.2.p1.10.m10.2.3.2.1" xref="S5.SS1.2.p1.10.m10.2.3.2.1.cmml"><mo id="S5.SS1.2.p1.10.m10.2.3.2.1.2" xref="S5.SS1.2.p1.10.m10.2.3.2.1.2.cmml">lim</mo><mrow id="S5.SS1.2.p1.10.m10.2.3.2.1.3" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.cmml"><mi id="S5.SS1.2.p1.10.m10.2.3.2.1.3.2" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.2.cmml">x</mi><mo id="S5.SS1.2.p1.10.m10.2.3.2.1.3.1" stretchy="false" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.1.cmml">→</mo><msup id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.cmml"><mi id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.2" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.2.cmml">τ</mi><mo id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.3" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.3.cmml">+</mo></msup></mrow></msub><mrow id="S5.SS1.2.p1.10.m10.2.3.2.2" xref="S5.SS1.2.p1.10.m10.2.3.2.2.cmml"><mi id="S5.SS1.2.p1.10.m10.2.3.2.2.2" xref="S5.SS1.2.p1.10.m10.2.3.2.2.2.cmml">f</mi><mo id="S5.SS1.2.p1.10.m10.2.3.2.2.1" xref="S5.SS1.2.p1.10.m10.2.3.2.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.10.m10.2.3.2.2.3.2" xref="S5.SS1.2.p1.10.m10.2.3.2.2.3.1.cmml"><mo id="S5.SS1.2.p1.10.m10.2.3.2.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.10.m10.2.3.2.2.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.10.m10.1.1" xref="S5.SS1.2.p1.10.m10.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.10.m10.2.3.2.2.3.2.2" xref="S5.SS1.2.p1.10.m10.2.3.2.2.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.10.m10.2.2" xref="S5.SS1.2.p1.10.m10.2.2.cmml">x</mi><mo id="S5.SS1.2.p1.10.m10.2.3.2.2.3.2.3" stretchy="false" xref="S5.SS1.2.p1.10.m10.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.SS1.2.p1.10.m10.2.3.1" xref="S5.SS1.2.p1.10.m10.2.3.1.cmml">≤</mo><mn id="S5.SS1.2.p1.10.m10.2.3.3" xref="S5.SS1.2.p1.10.m10.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.10.m10.2b"><apply id="S5.SS1.2.p1.10.m10.2.3.cmml" xref="S5.SS1.2.p1.10.m10.2.3"><leq id="S5.SS1.2.p1.10.m10.2.3.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.1"></leq><apply id="S5.SS1.2.p1.10.m10.2.3.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2"><apply id="S5.SS1.2.p1.10.m10.2.3.2.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1"><csymbol cd="ambiguous" id="S5.SS1.2.p1.10.m10.2.3.2.1.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1">subscript</csymbol><limit id="S5.SS1.2.p1.10.m10.2.3.2.1.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.2"></limit><apply id="S5.SS1.2.p1.10.m10.2.3.2.1.3.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3"><ci id="S5.SS1.2.p1.10.m10.2.3.2.1.3.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.1">→</ci><ci id="S5.SS1.2.p1.10.m10.2.3.2.1.3.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.2">𝑥</ci><apply id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3">superscript</csymbol><ci id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.2">𝜏</ci><plus id="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.3.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.1.3.3.3"></plus></apply></apply></apply><apply id="S5.SS1.2.p1.10.m10.2.3.2.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.2"><times id="S5.SS1.2.p1.10.m10.2.3.2.2.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.2.1"></times><ci id="S5.SS1.2.p1.10.m10.2.3.2.2.2.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.2.2">𝑓</ci><list id="S5.SS1.2.p1.10.m10.2.3.2.2.3.1.cmml" xref="S5.SS1.2.p1.10.m10.2.3.2.2.3.2"><ci id="S5.SS1.2.p1.10.m10.1.1.cmml" xref="S5.SS1.2.p1.10.m10.1.1">𝜏</ci><ci id="S5.SS1.2.p1.10.m10.2.2.cmml" xref="S5.SS1.2.p1.10.m10.2.2">𝑥</ci></list></apply></apply><cn id="S5.SS1.2.p1.10.m10.2.3.3.cmml" type="integer" xref="S5.SS1.2.p1.10.m10.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.10.m10.2c">\lim_{x\to\tau^{+}}f(\tau;x)\leq 0</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.10.m10.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_τ ; italic_x ) ≤ 0</annotation></semantics></math> by Conditions <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E18" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">18</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E19" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">19</span></a>, so <math alttext="f(\tau;\tau)=0=G(\tau)-Q(\tau)" class="ltx_Math" display="inline" id="S5.SS1.2.p1.11.m11.4"><semantics id="S5.SS1.2.p1.11.m11.4a"><mrow id="S5.SS1.2.p1.11.m11.4.5" xref="S5.SS1.2.p1.11.m11.4.5.cmml"><mrow id="S5.SS1.2.p1.11.m11.4.5.2" xref="S5.SS1.2.p1.11.m11.4.5.2.cmml"><mi id="S5.SS1.2.p1.11.m11.4.5.2.2" xref="S5.SS1.2.p1.11.m11.4.5.2.2.cmml">f</mi><mo id="S5.SS1.2.p1.11.m11.4.5.2.1" xref="S5.SS1.2.p1.11.m11.4.5.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.11.m11.4.5.2.3.2" xref="S5.SS1.2.p1.11.m11.4.5.2.3.1.cmml"><mo id="S5.SS1.2.p1.11.m11.4.5.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.2.3.1.cmml">(</mo><mi id="S5.SS1.2.p1.11.m11.1.1" xref="S5.SS1.2.p1.11.m11.1.1.cmml">τ</mi><mo id="S5.SS1.2.p1.11.m11.4.5.2.3.2.2" xref="S5.SS1.2.p1.11.m11.4.5.2.3.1.cmml">;</mo><mi id="S5.SS1.2.p1.11.m11.2.2" xref="S5.SS1.2.p1.11.m11.2.2.cmml">τ</mi><mo id="S5.SS1.2.p1.11.m11.4.5.2.3.2.3" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.2.p1.11.m11.4.5.3" xref="S5.SS1.2.p1.11.m11.4.5.3.cmml">=</mo><mn id="S5.SS1.2.p1.11.m11.4.5.4" xref="S5.SS1.2.p1.11.m11.4.5.4.cmml">0</mn><mo id="S5.SS1.2.p1.11.m11.4.5.5" xref="S5.SS1.2.p1.11.m11.4.5.5.cmml">=</mo><mrow id="S5.SS1.2.p1.11.m11.4.5.6" xref="S5.SS1.2.p1.11.m11.4.5.6.cmml"><mrow id="S5.SS1.2.p1.11.m11.4.5.6.2" xref="S5.SS1.2.p1.11.m11.4.5.6.2.cmml"><mi id="S5.SS1.2.p1.11.m11.4.5.6.2.2" xref="S5.SS1.2.p1.11.m11.4.5.6.2.2.cmml">G</mi><mo id="S5.SS1.2.p1.11.m11.4.5.6.2.1" xref="S5.SS1.2.p1.11.m11.4.5.6.2.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.11.m11.4.5.6.2.3.2" xref="S5.SS1.2.p1.11.m11.4.5.6.2.cmml"><mo id="S5.SS1.2.p1.11.m11.4.5.6.2.3.2.1" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.6.2.cmml">(</mo><mi id="S5.SS1.2.p1.11.m11.3.3" xref="S5.SS1.2.p1.11.m11.3.3.cmml">τ</mi><mo id="S5.SS1.2.p1.11.m11.4.5.6.2.3.2.2" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.6.2.cmml">)</mo></mrow></mrow><mo id="S5.SS1.2.p1.11.m11.4.5.6.1" xref="S5.SS1.2.p1.11.m11.4.5.6.1.cmml">−</mo><mrow id="S5.SS1.2.p1.11.m11.4.5.6.3" xref="S5.SS1.2.p1.11.m11.4.5.6.3.cmml"><mi id="S5.SS1.2.p1.11.m11.4.5.6.3.2" xref="S5.SS1.2.p1.11.m11.4.5.6.3.2.cmml">Q</mi><mo id="S5.SS1.2.p1.11.m11.4.5.6.3.1" xref="S5.SS1.2.p1.11.m11.4.5.6.3.1.cmml">⁢</mo><mrow id="S5.SS1.2.p1.11.m11.4.5.6.3.3.2" xref="S5.SS1.2.p1.11.m11.4.5.6.3.cmml"><mo id="S5.SS1.2.p1.11.m11.4.5.6.3.3.2.1" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.6.3.cmml">(</mo><mi id="S5.SS1.2.p1.11.m11.4.4" xref="S5.SS1.2.p1.11.m11.4.4.cmml">τ</mi><mo id="S5.SS1.2.p1.11.m11.4.5.6.3.3.2.2" stretchy="false" xref="S5.SS1.2.p1.11.m11.4.5.6.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p1.11.m11.4b"><apply id="S5.SS1.2.p1.11.m11.4.5.cmml" xref="S5.SS1.2.p1.11.m11.4.5"><and id="S5.SS1.2.p1.11.m11.4.5a.cmml" xref="S5.SS1.2.p1.11.m11.4.5"></and><apply id="S5.SS1.2.p1.11.m11.4.5b.cmml" xref="S5.SS1.2.p1.11.m11.4.5"><eq id="S5.SS1.2.p1.11.m11.4.5.3.cmml" xref="S5.SS1.2.p1.11.m11.4.5.3"></eq><apply id="S5.SS1.2.p1.11.m11.4.5.2.cmml" xref="S5.SS1.2.p1.11.m11.4.5.2"><times id="S5.SS1.2.p1.11.m11.4.5.2.1.cmml" xref="S5.SS1.2.p1.11.m11.4.5.2.1"></times><ci id="S5.SS1.2.p1.11.m11.4.5.2.2.cmml" xref="S5.SS1.2.p1.11.m11.4.5.2.2">𝑓</ci><list id="S5.SS1.2.p1.11.m11.4.5.2.3.1.cmml" xref="S5.SS1.2.p1.11.m11.4.5.2.3.2"><ci id="S5.SS1.2.p1.11.m11.1.1.cmml" xref="S5.SS1.2.p1.11.m11.1.1">𝜏</ci><ci id="S5.SS1.2.p1.11.m11.2.2.cmml" xref="S5.SS1.2.p1.11.m11.2.2">𝜏</ci></list></apply><cn id="S5.SS1.2.p1.11.m11.4.5.4.cmml" type="integer" xref="S5.SS1.2.p1.11.m11.4.5.4">0</cn></apply><apply id="S5.SS1.2.p1.11.m11.4.5c.cmml" xref="S5.SS1.2.p1.11.m11.4.5"><eq id="S5.SS1.2.p1.11.m11.4.5.5.cmml" xref="S5.SS1.2.p1.11.m11.4.5.5"></eq><share href="https://arxiv.org/html/2503.16280v1#S5.SS1.2.p1.11.m11.4.5.4.cmml" id="S5.SS1.2.p1.11.m11.4.5d.cmml" xref="S5.SS1.2.p1.11.m11.4.5"></share><apply id="S5.SS1.2.p1.11.m11.4.5.6.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6"><minus id="S5.SS1.2.p1.11.m11.4.5.6.1.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.1"></minus><apply id="S5.SS1.2.p1.11.m11.4.5.6.2.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.2"><times id="S5.SS1.2.p1.11.m11.4.5.6.2.1.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.2.1"></times><ci id="S5.SS1.2.p1.11.m11.4.5.6.2.2.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.2.2">𝐺</ci><ci id="S5.SS1.2.p1.11.m11.3.3.cmml" xref="S5.SS1.2.p1.11.m11.3.3">𝜏</ci></apply><apply id="S5.SS1.2.p1.11.m11.4.5.6.3.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.3"><times id="S5.SS1.2.p1.11.m11.4.5.6.3.1.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.3.1"></times><ci id="S5.SS1.2.p1.11.m11.4.5.6.3.2.cmml" xref="S5.SS1.2.p1.11.m11.4.5.6.3.2">𝑄</ci><ci id="S5.SS1.2.p1.11.m11.4.4.cmml" xref="S5.SS1.2.p1.11.m11.4.4">𝜏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p1.11.m11.4c">f(\tau;\tau)=0=G(\tau)-Q(\tau)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p1.11.m11.4d">italic_f ( italic_τ ; italic_τ ) = 0 = italic_G ( italic_τ ) - italic_Q ( italic_τ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS1.3.p2"> <p class="ltx_p" id="S5.SS1.3.p2.5">For sufficiency, take some <math alttext="x>\tau" class="ltx_Math" display="inline" id="S5.SS1.3.p2.1.m1.1"><semantics id="S5.SS1.3.p2.1.m1.1a"><mrow id="S5.SS1.3.p2.1.m1.1.1" xref="S5.SS1.3.p2.1.m1.1.1.cmml"><mi id="S5.SS1.3.p2.1.m1.1.1.2" xref="S5.SS1.3.p2.1.m1.1.1.2.cmml">x</mi><mo id="S5.SS1.3.p2.1.m1.1.1.1" xref="S5.SS1.3.p2.1.m1.1.1.1.cmml">></mo><mi id="S5.SS1.3.p2.1.m1.1.1.3" xref="S5.SS1.3.p2.1.m1.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.3.p2.1.m1.1b"><apply id="S5.SS1.3.p2.1.m1.1.1.cmml" xref="S5.SS1.3.p2.1.m1.1.1"><gt id="S5.SS1.3.p2.1.m1.1.1.1.cmml" xref="S5.SS1.3.p2.1.m1.1.1.1"></gt><ci id="S5.SS1.3.p2.1.m1.1.1.2.cmml" xref="S5.SS1.3.p2.1.m1.1.1.2">𝑥</ci><ci id="S5.SS1.3.p2.1.m1.1.1.3.cmml" xref="S5.SS1.3.p2.1.m1.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.3.p2.1.m1.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.3.p2.1.m1.1d">italic_x > italic_τ</annotation></semantics></math>. By single crossing, <math alttext="P(\tau;x)\leq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="S5.SS1.3.p2.2.m2.4"><semantics id="S5.SS1.3.p2.2.m2.4a"><mrow id="S5.SS1.3.p2.2.m2.4.5" xref="S5.SS1.3.p2.2.m2.4.5.cmml"><mrow id="S5.SS1.3.p2.2.m2.4.5.2" xref="S5.SS1.3.p2.2.m2.4.5.2.cmml"><mi id="S5.SS1.3.p2.2.m2.4.5.2.2" xref="S5.SS1.3.p2.2.m2.4.5.2.2.cmml">P</mi><mo id="S5.SS1.3.p2.2.m2.4.5.2.1" xref="S5.SS1.3.p2.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="S5.SS1.3.p2.2.m2.4.5.2.3.2" xref="S5.SS1.3.p2.2.m2.4.5.2.3.1.cmml"><mo id="S5.SS1.3.p2.2.m2.4.5.2.3.2.1" stretchy="false" xref="S5.SS1.3.p2.2.m2.4.5.2.3.1.cmml">(</mo><mi id="S5.SS1.3.p2.2.m2.1.1" xref="S5.SS1.3.p2.2.m2.1.1.cmml">τ</mi><mo id="S5.SS1.3.p2.2.m2.4.5.2.3.2.2" xref="S5.SS1.3.p2.2.m2.4.5.2.3.1.cmml">;</mo><mi id="S5.SS1.3.p2.2.m2.2.2" xref="S5.SS1.3.p2.2.m2.2.2.cmml">x</mi><mo id="S5.SS1.3.p2.2.m2.4.5.2.3.2.3" stretchy="false" xref="S5.SS1.3.p2.2.m2.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.3.p2.2.m2.4.5.1" xref="S5.SS1.3.p2.2.m2.4.5.1.cmml">≤</mo><mrow id="S5.SS1.3.p2.2.m2.4.5.3" xref="S5.SS1.3.p2.2.m2.4.5.3.cmml"><mover accent="true" id="S5.SS1.3.p2.2.m2.4.5.3.2" xref="S5.SS1.3.p2.2.m2.4.5.3.2.cmml"><mi id="S5.SS1.3.p2.2.m2.4.5.3.2.2" xref="S5.SS1.3.p2.2.m2.4.5.3.2.2.cmml">P</mi><mo id="S5.SS1.3.p2.2.m2.4.5.3.2.1" xref="S5.SS1.3.p2.2.m2.4.5.3.2.1.cmml">¯</mo></mover><mo id="S5.SS1.3.p2.2.m2.4.5.3.1" xref="S5.SS1.3.p2.2.m2.4.5.3.1.cmml">⁢</mo><mrow id="S5.SS1.3.p2.2.m2.4.5.3.3.2" xref="S5.SS1.3.p2.2.m2.4.5.3.3.1.cmml"><mo id="S5.SS1.3.p2.2.m2.4.5.3.3.2.1" stretchy="false" xref="S5.SS1.3.p2.2.m2.4.5.3.3.1.cmml">(</mo><mi id="S5.SS1.3.p2.2.m2.3.3" xref="S5.SS1.3.p2.2.m2.3.3.cmml">τ</mi><mo id="S5.SS1.3.p2.2.m2.4.5.3.3.2.2" xref="S5.SS1.3.p2.2.m2.4.5.3.3.1.cmml">;</mo><mi id="S5.SS1.3.p2.2.m2.4.4" xref="S5.SS1.3.p2.2.m2.4.4.cmml">x</mi><mo id="S5.SS1.3.p2.2.m2.4.5.3.3.2.3" stretchy="false" xref="S5.SS1.3.p2.2.m2.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.3.p2.2.m2.4b"><apply id="S5.SS1.3.p2.2.m2.4.5.cmml" xref="S5.SS1.3.p2.2.m2.4.5"><leq id="S5.SS1.3.p2.2.m2.4.5.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.1"></leq><apply id="S5.SS1.3.p2.2.m2.4.5.2.cmml" xref="S5.SS1.3.p2.2.m2.4.5.2"><times id="S5.SS1.3.p2.2.m2.4.5.2.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.2.1"></times><ci id="S5.SS1.3.p2.2.m2.4.5.2.2.cmml" xref="S5.SS1.3.p2.2.m2.4.5.2.2">𝑃</ci><list id="S5.SS1.3.p2.2.m2.4.5.2.3.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.2.3.2"><ci id="S5.SS1.3.p2.2.m2.1.1.cmml" xref="S5.SS1.3.p2.2.m2.1.1">𝜏</ci><ci id="S5.SS1.3.p2.2.m2.2.2.cmml" xref="S5.SS1.3.p2.2.m2.2.2">𝑥</ci></list></apply><apply id="S5.SS1.3.p2.2.m2.4.5.3.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3"><times id="S5.SS1.3.p2.2.m2.4.5.3.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3.1"></times><apply id="S5.SS1.3.p2.2.m2.4.5.3.2.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3.2"><ci id="S5.SS1.3.p2.2.m2.4.5.3.2.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3.2.1">¯</ci><ci id="S5.SS1.3.p2.2.m2.4.5.3.2.2.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3.2.2">𝑃</ci></apply><list id="S5.SS1.3.p2.2.m2.4.5.3.3.1.cmml" xref="S5.SS1.3.p2.2.m2.4.5.3.3.2"><ci id="S5.SS1.3.p2.2.m2.3.3.cmml" xref="S5.SS1.3.p2.2.m2.3.3">𝜏</ci><ci id="S5.SS1.3.p2.2.m2.4.4.cmml" xref="S5.SS1.3.p2.2.m2.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.3.p2.2.m2.4c">P(\tau;x)\leq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.3.p2.2.m2.4d">italic_P ( italic_τ ; italic_x ) ≤ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>. Similarly, if <math alttext="x\leq\tau" class="ltx_Math" display="inline" id="S5.SS1.3.p2.3.m3.1"><semantics id="S5.SS1.3.p2.3.m3.1a"><mrow id="S5.SS1.3.p2.3.m3.1.1" xref="S5.SS1.3.p2.3.m3.1.1.cmml"><mi id="S5.SS1.3.p2.3.m3.1.1.2" xref="S5.SS1.3.p2.3.m3.1.1.2.cmml">x</mi><mo id="S5.SS1.3.p2.3.m3.1.1.1" xref="S5.SS1.3.p2.3.m3.1.1.1.cmml">≤</mo><mi id="S5.SS1.3.p2.3.m3.1.1.3" xref="S5.SS1.3.p2.3.m3.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.3.p2.3.m3.1b"><apply id="S5.SS1.3.p2.3.m3.1.1.cmml" xref="S5.SS1.3.p2.3.m3.1.1"><leq id="S5.SS1.3.p2.3.m3.1.1.1.cmml" xref="S5.SS1.3.p2.3.m3.1.1.1"></leq><ci id="S5.SS1.3.p2.3.m3.1.1.2.cmml" xref="S5.SS1.3.p2.3.m3.1.1.2">𝑥</ci><ci id="S5.SS1.3.p2.3.m3.1.1.3.cmml" xref="S5.SS1.3.p2.3.m3.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.3.p2.3.m3.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.3.p2.3.m3.1d">italic_x ≤ italic_τ</annotation></semantics></math>, <math alttext="P(\tau;x)\geq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="S5.SS1.3.p2.4.m4.4"><semantics id="S5.SS1.3.p2.4.m4.4a"><mrow id="S5.SS1.3.p2.4.m4.4.5" xref="S5.SS1.3.p2.4.m4.4.5.cmml"><mrow id="S5.SS1.3.p2.4.m4.4.5.2" xref="S5.SS1.3.p2.4.m4.4.5.2.cmml"><mi id="S5.SS1.3.p2.4.m4.4.5.2.2" xref="S5.SS1.3.p2.4.m4.4.5.2.2.cmml">P</mi><mo id="S5.SS1.3.p2.4.m4.4.5.2.1" xref="S5.SS1.3.p2.4.m4.4.5.2.1.cmml">⁢</mo><mrow id="S5.SS1.3.p2.4.m4.4.5.2.3.2" xref="S5.SS1.3.p2.4.m4.4.5.2.3.1.cmml"><mo id="S5.SS1.3.p2.4.m4.4.5.2.3.2.1" stretchy="false" xref="S5.SS1.3.p2.4.m4.4.5.2.3.1.cmml">(</mo><mi id="S5.SS1.3.p2.4.m4.1.1" xref="S5.SS1.3.p2.4.m4.1.1.cmml">τ</mi><mo id="S5.SS1.3.p2.4.m4.4.5.2.3.2.2" xref="S5.SS1.3.p2.4.m4.4.5.2.3.1.cmml">;</mo><mi id="S5.SS1.3.p2.4.m4.2.2" xref="S5.SS1.3.p2.4.m4.2.2.cmml">x</mi><mo id="S5.SS1.3.p2.4.m4.4.5.2.3.2.3" stretchy="false" xref="S5.SS1.3.p2.4.m4.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.3.p2.4.m4.4.5.1" xref="S5.SS1.3.p2.4.m4.4.5.1.cmml">≥</mo><mrow id="S5.SS1.3.p2.4.m4.4.5.3" xref="S5.SS1.3.p2.4.m4.4.5.3.cmml"><mover accent="true" id="S5.SS1.3.p2.4.m4.4.5.3.2" xref="S5.SS1.3.p2.4.m4.4.5.3.2.cmml"><mi id="S5.SS1.3.p2.4.m4.4.5.3.2.2" xref="S5.SS1.3.p2.4.m4.4.5.3.2.2.cmml">P</mi><mo id="S5.SS1.3.p2.4.m4.4.5.3.2.1" xref="S5.SS1.3.p2.4.m4.4.5.3.2.1.cmml">¯</mo></mover><mo id="S5.SS1.3.p2.4.m4.4.5.3.1" xref="S5.SS1.3.p2.4.m4.4.5.3.1.cmml">⁢</mo><mrow id="S5.SS1.3.p2.4.m4.4.5.3.3.2" xref="S5.SS1.3.p2.4.m4.4.5.3.3.1.cmml"><mo id="S5.SS1.3.p2.4.m4.4.5.3.3.2.1" stretchy="false" xref="S5.SS1.3.p2.4.m4.4.5.3.3.1.cmml">(</mo><mi id="S5.SS1.3.p2.4.m4.3.3" xref="S5.SS1.3.p2.4.m4.3.3.cmml">τ</mi><mo id="S5.SS1.3.p2.4.m4.4.5.3.3.2.2" xref="S5.SS1.3.p2.4.m4.4.5.3.3.1.cmml">;</mo><mi id="S5.SS1.3.p2.4.m4.4.4" xref="S5.SS1.3.p2.4.m4.4.4.cmml">x</mi><mo id="S5.SS1.3.p2.4.m4.4.5.3.3.2.3" stretchy="false" xref="S5.SS1.3.p2.4.m4.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.3.p2.4.m4.4b"><apply id="S5.SS1.3.p2.4.m4.4.5.cmml" xref="S5.SS1.3.p2.4.m4.4.5"><geq id="S5.SS1.3.p2.4.m4.4.5.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.1"></geq><apply id="S5.SS1.3.p2.4.m4.4.5.2.cmml" xref="S5.SS1.3.p2.4.m4.4.5.2"><times id="S5.SS1.3.p2.4.m4.4.5.2.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.2.1"></times><ci id="S5.SS1.3.p2.4.m4.4.5.2.2.cmml" xref="S5.SS1.3.p2.4.m4.4.5.2.2">𝑃</ci><list id="S5.SS1.3.p2.4.m4.4.5.2.3.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.2.3.2"><ci id="S5.SS1.3.p2.4.m4.1.1.cmml" xref="S5.SS1.3.p2.4.m4.1.1">𝜏</ci><ci id="S5.SS1.3.p2.4.m4.2.2.cmml" xref="S5.SS1.3.p2.4.m4.2.2">𝑥</ci></list></apply><apply id="S5.SS1.3.p2.4.m4.4.5.3.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3"><times id="S5.SS1.3.p2.4.m4.4.5.3.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3.1"></times><apply id="S5.SS1.3.p2.4.m4.4.5.3.2.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3.2"><ci id="S5.SS1.3.p2.4.m4.4.5.3.2.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3.2.1">¯</ci><ci id="S5.SS1.3.p2.4.m4.4.5.3.2.2.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3.2.2">𝑃</ci></apply><list id="S5.SS1.3.p2.4.m4.4.5.3.3.1.cmml" xref="S5.SS1.3.p2.4.m4.4.5.3.3.2"><ci id="S5.SS1.3.p2.4.m4.3.3.cmml" xref="S5.SS1.3.p2.4.m4.3.3">𝜏</ci><ci id="S5.SS1.3.p2.4.m4.4.4.cmml" xref="S5.SS1.3.p2.4.m4.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.3.p2.4.m4.4c">P(\tau;x)\geq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.3.p2.4.m4.4d">italic_P ( italic_τ ; italic_x ) ≥ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>. This establishes Equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E18" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">18</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E19" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">19</span></a>), so <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS1.3.p2.5.m5.1"><semantics id="S5.SS1.3.p2.5.m5.1a"><mi id="S5.SS1.3.p2.5.m5.1.1" xref="S5.SS1.3.p2.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.3.p2.5.m5.1b"><ci id="S5.SS1.3.p2.5.m5.1.1.cmml" xref="S5.SS1.3.p2.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.3.p2.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.3.p2.5.m5.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.2">While the exact crossing condition has changed from <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.2"><semantics id="S5.SS1.p4.1.m1.2a"><mrow id="S5.SS1.p4.1.m1.2.3" xref="S5.SS1.p4.1.m1.2.3.cmml"><mrow id="S5.SS1.p4.1.m1.2.3.2" xref="S5.SS1.p4.1.m1.2.3.2.cmml"><mi id="S5.SS1.p4.1.m1.2.3.2.2" xref="S5.SS1.p4.1.m1.2.3.2.2.cmml">G</mi><mo id="S5.SS1.p4.1.m1.2.3.2.1" xref="S5.SS1.p4.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S5.SS1.p4.1.m1.2.3.2.3.2" xref="S5.SS1.p4.1.m1.2.3.2.cmml"><mo id="S5.SS1.p4.1.m1.2.3.2.3.2.1" stretchy="false" xref="S5.SS1.p4.1.m1.2.3.2.cmml">(</mo><mi id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml">τ</mi><mo id="S5.SS1.p4.1.m1.2.3.2.3.2.2" stretchy="false" xref="S5.SS1.p4.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.SS1.p4.1.m1.2.3.1" xref="S5.SS1.p4.1.m1.2.3.1.cmml">=</mo><mrow id="S5.SS1.p4.1.m1.2.3.3" xref="S5.SS1.p4.1.m1.2.3.3.cmml"><mi id="S5.SS1.p4.1.m1.2.3.3.2" xref="S5.SS1.p4.1.m1.2.3.3.2.cmml">F</mi><mo id="S5.SS1.p4.1.m1.2.3.3.1" xref="S5.SS1.p4.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS1.p4.1.m1.2.3.3.3.2" xref="S5.SS1.p4.1.m1.2.3.3.cmml"><mo id="S5.SS1.p4.1.m1.2.3.3.3.2.1" stretchy="false" xref="S5.SS1.p4.1.m1.2.3.3.cmml">(</mo><mi id="S5.SS1.p4.1.m1.2.2" xref="S5.SS1.p4.1.m1.2.2.cmml">τ</mi><mo id="S5.SS1.p4.1.m1.2.3.3.3.2.2" stretchy="false" xref="S5.SS1.p4.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.2b"><apply id="S5.SS1.p4.1.m1.2.3.cmml" xref="S5.SS1.p4.1.m1.2.3"><eq id="S5.SS1.p4.1.m1.2.3.1.cmml" xref="S5.SS1.p4.1.m1.2.3.1"></eq><apply id="S5.SS1.p4.1.m1.2.3.2.cmml" xref="S5.SS1.p4.1.m1.2.3.2"><times id="S5.SS1.p4.1.m1.2.3.2.1.cmml" xref="S5.SS1.p4.1.m1.2.3.2.1"></times><ci id="S5.SS1.p4.1.m1.2.3.2.2.cmml" xref="S5.SS1.p4.1.m1.2.3.2.2">𝐺</ci><ci id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1">𝜏</ci></apply><apply id="S5.SS1.p4.1.m1.2.3.3.cmml" xref="S5.SS1.p4.1.m1.2.3.3"><times id="S5.SS1.p4.1.m1.2.3.3.1.cmml" xref="S5.SS1.p4.1.m1.2.3.3.1"></times><ci id="S5.SS1.p4.1.m1.2.3.3.2.cmml" xref="S5.SS1.p4.1.m1.2.3.3.2">𝐹</ci><ci id="S5.SS1.p4.1.m1.2.2.cmml" xref="S5.SS1.p4.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math> to <math alttext="G(\tau)=Q(\tau)" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.2"><semantics id="S5.SS1.p4.2.m2.2a"><mrow id="S5.SS1.p4.2.m2.2.3" xref="S5.SS1.p4.2.m2.2.3.cmml"><mrow id="S5.SS1.p4.2.m2.2.3.2" xref="S5.SS1.p4.2.m2.2.3.2.cmml"><mi 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id="S5.SS1.p4.2.m2.2.3.3.3.2.2" stretchy="false" xref="S5.SS1.p4.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.2.m2.2b"><apply id="S5.SS1.p4.2.m2.2.3.cmml" xref="S5.SS1.p4.2.m2.2.3"><eq id="S5.SS1.p4.2.m2.2.3.1.cmml" xref="S5.SS1.p4.2.m2.2.3.1"></eq><apply id="S5.SS1.p4.2.m2.2.3.2.cmml" xref="S5.SS1.p4.2.m2.2.3.2"><times id="S5.SS1.p4.2.m2.2.3.2.1.cmml" xref="S5.SS1.p4.2.m2.2.3.2.1"></times><ci id="S5.SS1.p4.2.m2.2.3.2.2.cmml" xref="S5.SS1.p4.2.m2.2.3.2.2">𝐺</ci><ci id="S5.SS1.p4.2.m2.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1">𝜏</ci></apply><apply id="S5.SS1.p4.2.m2.2.3.3.cmml" xref="S5.SS1.p4.2.m2.2.3.3"><times id="S5.SS1.p4.2.m2.2.3.3.1.cmml" xref="S5.SS1.p4.2.m2.2.3.3.1"></times><ci id="S5.SS1.p4.2.m2.2.3.3.2.cmml" xref="S5.SS1.p4.2.m2.2.3.3.2">𝑄</ci><ci id="S5.SS1.p4.2.m2.2.2.cmml" xref="S5.SS1.p4.2.m2.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.2.m2.2c">G(\tau)=Q(\tau)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.2.m2.2d">italic_G ( italic_τ ) = italic_Q ( italic_τ )</annotation></semantics></math>, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem6" title="Theorem 6. ‣ 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a> paints a similar picture to what we saw for DG. Best responses arew now more complex to compute, so we do not treat the general dynamics formally, but we can provide a comparison of their behavior in the Gaussian case.</p> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Gaussian Model</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.1">We return to our Normal example, where each agent receives a noisy version of a Gaussian signal. Leaving the technical details to § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A4" title="Appendix D Omitted Proofs for RBTS ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">D</span></a>, we first note that the same equilibria occur as the previous mechanisms.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary5.1.1.1">Corollary 5</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary5.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary5.p1"> <p class="ltx_p" id="Thmcorollary5.p1.2">In the Gaussian model under RBTS, there are three equilibria at <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmcorollary5.p1.1.m1.1"><semantics id="Thmcorollary5.p1.1.m1.1a"><mrow id="Thmcorollary5.p1.1.m1.1.1" xref="Thmcorollary5.p1.1.m1.1.1.cmml"><mi id="Thmcorollary5.p1.1.m1.1.1.2" xref="Thmcorollary5.p1.1.m1.1.1.2.cmml">τ</mi><mo id="Thmcorollary5.p1.1.m1.1.1.1" xref="Thmcorollary5.p1.1.m1.1.1.1.cmml">=</mo><mn id="Thmcorollary5.p1.1.m1.1.1.3" xref="Thmcorollary5.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary5.p1.1.m1.1b"><apply id="Thmcorollary5.p1.1.m1.1.1.cmml" xref="Thmcorollary5.p1.1.m1.1.1"><eq id="Thmcorollary5.p1.1.m1.1.1.1.cmml" xref="Thmcorollary5.p1.1.m1.1.1.1"></eq><ci id="Thmcorollary5.p1.1.m1.1.1.2.cmml" xref="Thmcorollary5.p1.1.m1.1.1.2">𝜏</ci><cn id="Thmcorollary5.p1.1.m1.1.1.3.cmml" type="integer" xref="Thmcorollary5.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary5.p1.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary5.p1.1.m1.1d">italic_τ = 0</annotation></semantics></math> and <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmcorollary5.p1.2.m2.1"><semantics id="Thmcorollary5.p1.2.m2.1a"><mrow id="Thmcorollary5.p1.2.m2.1.1" xref="Thmcorollary5.p1.2.m2.1.1.cmml"><mo id="Thmcorollary5.p1.2.m2.1.1a" xref="Thmcorollary5.p1.2.m2.1.1.cmml">±</mo><mi id="Thmcorollary5.p1.2.m2.1.1.2" mathvariant="normal" xref="Thmcorollary5.p1.2.m2.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcorollary5.p1.2.m2.1b"><apply id="Thmcorollary5.p1.2.m2.1.1.cmml" xref="Thmcorollary5.p1.2.m2.1.1"><csymbol cd="latexml" id="Thmcorollary5.p1.2.m2.1.1.1.cmml" xref="Thmcorollary5.p1.2.m2.1.1">plus-or-minus</csymbol><infinity id="Thmcorollary5.p1.2.m2.1.1.2.cmml" xref="Thmcorollary5.p1.2.m2.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary5.p1.2.m2.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary5.p1.2.m2.1d">± ∞</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.10">We also derive the exact form of the best response <math alttext="\hat{\tau}=c(\rho)\tau" 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"><mover accent="true" id="S5.SS2.p2.1.m1.1.2.2" xref="S5.SS2.p2.1.m1.1.2.2.cmml"><mi id="S5.SS2.p2.1.m1.1.2.2.2" xref="S5.SS2.p2.1.m1.1.2.2.2.cmml">τ</mi><mo id="S5.SS2.p2.1.m1.1.2.2.1" xref="S5.SS2.p2.1.m1.1.2.2.1.cmml">^</mo></mover><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" xref="S5.SS2.p2.1.m1.1.2.3.cmml"><mi id="S5.SS2.p2.1.m1.1.2.3.2" xref="S5.SS2.p2.1.m1.1.2.3.2.cmml">c</mi><mo id="S5.SS2.p2.1.m1.1.2.3.1" xref="S5.SS2.p2.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p2.1.m1.1.2.3.3.2" xref="S5.SS2.p2.1.m1.1.2.3.cmml"><mo id="S5.SS2.p2.1.m1.1.2.3.3.2.1" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.3.cmml">(</mo><mi id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml">ρ</mi><mo id="S5.SS2.p2.1.m1.1.2.3.3.2.2" stretchy="false" xref="S5.SS2.p2.1.m1.1.2.3.cmml">)</mo></mrow><mo id="S5.SS2.p2.1.m1.1.2.3.1a" xref="S5.SS2.p2.1.m1.1.2.3.1.cmml">⁢</mo><mi id="S5.SS2.p2.1.m1.1.2.3.4" xref="S5.SS2.p2.1.m1.1.2.3.4.cmml">τ</mi></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"><eq id="S5.SS2.p2.1.m1.1.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.1"></eq><apply id="S5.SS2.p2.1.m1.1.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2"><ci id="S5.SS2.p2.1.m1.1.2.2.1.cmml" xref="S5.SS2.p2.1.m1.1.2.2.1">^</ci><ci id="S5.SS2.p2.1.m1.1.2.2.2.cmml" xref="S5.SS2.p2.1.m1.1.2.2.2">𝜏</ci></apply><apply id="S5.SS2.p2.1.m1.1.2.3.cmml" xref="S5.SS2.p2.1.m1.1.2.3"><times id="S5.SS2.p2.1.m1.1.2.3.1.cmml" xref="S5.SS2.p2.1.m1.1.2.3.1"></times><ci id="S5.SS2.p2.1.m1.1.2.3.2.cmml" xref="S5.SS2.p2.1.m1.1.2.3.2">𝑐</ci><ci id="S5.SS2.p2.1.m1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1">𝜌</ci><ci id="S5.SS2.p2.1.m1.1.2.3.4.cmml" xref="S5.SS2.p2.1.m1.1.2.3.4">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">\hat{\tau}=c(\rho)\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">over^ start_ARG italic_τ end_ARG = italic_c ( italic_ρ ) italic_τ</annotation></semantics></math> in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A4" title="Appendix D Omitted Proofs for RBTS ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">D</span></a>. Now, note that for any <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.1"><semantics id="S5.SS2.p2.2.m2.1a"><mi id="S5.SS2.p2.2.m2.1.1" xref="S5.SS2.p2.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.1b"><ci id="S5.SS2.p2.2.m2.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.1d">italic_τ</annotation></semantics></math>, <math alttext="\hat{\tau}=c(\rho)\tau" class="ltx_Math" display="inline" id="S5.SS2.p2.3.m3.1"><semantics id="S5.SS2.p2.3.m3.1a"><mrow id="S5.SS2.p2.3.m3.1.2" xref="S5.SS2.p2.3.m3.1.2.cmml"><mover accent="true" id="S5.SS2.p2.3.m3.1.2.2" xref="S5.SS2.p2.3.m3.1.2.2.cmml"><mi id="S5.SS2.p2.3.m3.1.2.2.2" xref="S5.SS2.p2.3.m3.1.2.2.2.cmml">τ</mi><mo id="S5.SS2.p2.3.m3.1.2.2.1" xref="S5.SS2.p2.3.m3.1.2.2.1.cmml">^</mo></mover><mo id="S5.SS2.p2.3.m3.1.2.1" xref="S5.SS2.p2.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS2.p2.3.m3.1.2.3" xref="S5.SS2.p2.3.m3.1.2.3.cmml"><mi id="S5.SS2.p2.3.m3.1.2.3.2" xref="S5.SS2.p2.3.m3.1.2.3.2.cmml">c</mi><mo id="S5.SS2.p2.3.m3.1.2.3.1" xref="S5.SS2.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p2.3.m3.1.2.3.3.2" xref="S5.SS2.p2.3.m3.1.2.3.cmml"><mo id="S5.SS2.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.SS2.p2.3.m3.1.2.3.cmml">(</mo><mi id="S5.SS2.p2.3.m3.1.1" xref="S5.SS2.p2.3.m3.1.1.cmml">ρ</mi><mo id="S5.SS2.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.SS2.p2.3.m3.1.2.3.cmml">)</mo></mrow><mo id="S5.SS2.p2.3.m3.1.2.3.1a" xref="S5.SS2.p2.3.m3.1.2.3.1.cmml">⁢</mo><mi id="S5.SS2.p2.3.m3.1.2.3.4" xref="S5.SS2.p2.3.m3.1.2.3.4.cmml">τ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.3.m3.1b"><apply id="S5.SS2.p2.3.m3.1.2.cmml" xref="S5.SS2.p2.3.m3.1.2"><eq id="S5.SS2.p2.3.m3.1.2.1.cmml" xref="S5.SS2.p2.3.m3.1.2.1"></eq><apply id="S5.SS2.p2.3.m3.1.2.2.cmml" xref="S5.SS2.p2.3.m3.1.2.2"><ci id="S5.SS2.p2.3.m3.1.2.2.1.cmml" xref="S5.SS2.p2.3.m3.1.2.2.1">^</ci><ci id="S5.SS2.p2.3.m3.1.2.2.2.cmml" xref="S5.SS2.p2.3.m3.1.2.2.2">𝜏</ci></apply><apply id="S5.SS2.p2.3.m3.1.2.3.cmml" xref="S5.SS2.p2.3.m3.1.2.3"><times id="S5.SS2.p2.3.m3.1.2.3.1.cmml" xref="S5.SS2.p2.3.m3.1.2.3.1"></times><ci id="S5.SS2.p2.3.m3.1.2.3.2.cmml" xref="S5.SS2.p2.3.m3.1.2.3.2">𝑐</ci><ci id="S5.SS2.p2.3.m3.1.1.cmml" xref="S5.SS2.p2.3.m3.1.1">𝜌</ci><ci id="S5.SS2.p2.3.m3.1.2.3.4.cmml" xref="S5.SS2.p2.3.m3.1.2.3.4">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.3.m3.1c">\hat{\tau}=c(\rho)\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.3.m3.1d">over^ start_ARG italic_τ end_ARG = italic_c ( italic_ρ ) italic_τ</annotation></semantics></math> for <math alttext="c(\rho)\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.p2.4.m4.3"><semantics id="S5.SS2.p2.4.m4.3a"><mrow id="S5.SS2.p2.4.m4.3.4" xref="S5.SS2.p2.4.m4.3.4.cmml"><mrow id="S5.SS2.p2.4.m4.3.4.2" xref="S5.SS2.p2.4.m4.3.4.2.cmml"><mi id="S5.SS2.p2.4.m4.3.4.2.2" xref="S5.SS2.p2.4.m4.3.4.2.2.cmml">c</mi><mo id="S5.SS2.p2.4.m4.3.4.2.1" xref="S5.SS2.p2.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="S5.SS2.p2.4.m4.3.4.2.3.2" xref="S5.SS2.p2.4.m4.3.4.2.cmml"><mo id="S5.SS2.p2.4.m4.3.4.2.3.2.1" stretchy="false" xref="S5.SS2.p2.4.m4.3.4.2.cmml">(</mo><mi id="S5.SS2.p2.4.m4.1.1" xref="S5.SS2.p2.4.m4.1.1.cmml">ρ</mi><mo id="S5.SS2.p2.4.m4.3.4.2.3.2.2" stretchy="false" xref="S5.SS2.p2.4.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p2.4.m4.3.4.1" xref="S5.SS2.p2.4.m4.3.4.1.cmml">∈</mo><mrow id="S5.SS2.p2.4.m4.3.4.3.2" xref="S5.SS2.p2.4.m4.3.4.3.1.cmml"><mo id="S5.SS2.p2.4.m4.3.4.3.2.1" stretchy="false" xref="S5.SS2.p2.4.m4.3.4.3.1.cmml">(</mo><mn id="S5.SS2.p2.4.m4.2.2" xref="S5.SS2.p2.4.m4.2.2.cmml">0</mn><mo id="S5.SS2.p2.4.m4.3.4.3.2.2" xref="S5.SS2.p2.4.m4.3.4.3.1.cmml">,</mo><mn id="S5.SS2.p2.4.m4.3.3" xref="S5.SS2.p2.4.m4.3.3.cmml">1</mn><mo id="S5.SS2.p2.4.m4.3.4.3.2.3" stretchy="false" xref="S5.SS2.p2.4.m4.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.4.m4.3b"><apply id="S5.SS2.p2.4.m4.3.4.cmml" xref="S5.SS2.p2.4.m4.3.4"><in id="S5.SS2.p2.4.m4.3.4.1.cmml" xref="S5.SS2.p2.4.m4.3.4.1"></in><apply id="S5.SS2.p2.4.m4.3.4.2.cmml" xref="S5.SS2.p2.4.m4.3.4.2"><times id="S5.SS2.p2.4.m4.3.4.2.1.cmml" xref="S5.SS2.p2.4.m4.3.4.2.1"></times><ci id="S5.SS2.p2.4.m4.3.4.2.2.cmml" xref="S5.SS2.p2.4.m4.3.4.2.2">𝑐</ci><ci id="S5.SS2.p2.4.m4.1.1.cmml" xref="S5.SS2.p2.4.m4.1.1">𝜌</ci></apply><interval closure="open" id="S5.SS2.p2.4.m4.3.4.3.1.cmml" xref="S5.SS2.p2.4.m4.3.4.3.2"><cn id="S5.SS2.p2.4.m4.2.2.cmml" type="integer" xref="S5.SS2.p2.4.m4.2.2">0</cn><cn id="S5.SS2.p2.4.m4.3.3.cmml" type="integer" xref="S5.SS2.p2.4.m4.3.3">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.4.m4.3c">c(\rho)\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.4.m4.3d">italic_c ( italic_ρ ) ∈ ( 0 , 1 )</annotation></semantics></math>. Under our dynamics, <math alttext="\dot{\tau}=\hat{\tau}-\tau." class="ltx_Math" display="inline" id="S5.SS2.p2.5.m5.1"><semantics id="S5.SS2.p2.5.m5.1a"><mrow id="S5.SS2.p2.5.m5.1.1.1" xref="S5.SS2.p2.5.m5.1.1.1.1.cmml"><mrow id="S5.SS2.p2.5.m5.1.1.1.1" xref="S5.SS2.p2.5.m5.1.1.1.1.cmml"><mover accent="true" id="S5.SS2.p2.5.m5.1.1.1.1.2" xref="S5.SS2.p2.5.m5.1.1.1.1.2.cmml"><mi id="S5.SS2.p2.5.m5.1.1.1.1.2.2" xref="S5.SS2.p2.5.m5.1.1.1.1.2.2.cmml">τ</mi><mo id="S5.SS2.p2.5.m5.1.1.1.1.2.1" xref="S5.SS2.p2.5.m5.1.1.1.1.2.1.cmml">˙</mo></mover><mo id="S5.SS2.p2.5.m5.1.1.1.1.1" xref="S5.SS2.p2.5.m5.1.1.1.1.1.cmml">=</mo><mrow id="S5.SS2.p2.5.m5.1.1.1.1.3" xref="S5.SS2.p2.5.m5.1.1.1.1.3.cmml"><mover accent="true" id="S5.SS2.p2.5.m5.1.1.1.1.3.2" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2.cmml"><mi id="S5.SS2.p2.5.m5.1.1.1.1.3.2.2" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2.2.cmml">τ</mi><mo id="S5.SS2.p2.5.m5.1.1.1.1.3.2.1" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2.1.cmml">^</mo></mover><mo id="S5.SS2.p2.5.m5.1.1.1.1.3.1" xref="S5.SS2.p2.5.m5.1.1.1.1.3.1.cmml">−</mo><mi id="S5.SS2.p2.5.m5.1.1.1.1.3.3" xref="S5.SS2.p2.5.m5.1.1.1.1.3.3.cmml">τ</mi></mrow></mrow><mo id="S5.SS2.p2.5.m5.1.1.1.2" lspace="0em" xref="S5.SS2.p2.5.m5.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.5.m5.1b"><apply id="S5.SS2.p2.5.m5.1.1.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1"><eq id="S5.SS2.p2.5.m5.1.1.1.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.1"></eq><apply id="S5.SS2.p2.5.m5.1.1.1.1.2.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.2"><ci id="S5.SS2.p2.5.m5.1.1.1.1.2.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.2.1">˙</ci><ci id="S5.SS2.p2.5.m5.1.1.1.1.2.2.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.2.2">𝜏</ci></apply><apply id="S5.SS2.p2.5.m5.1.1.1.1.3.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3"><minus id="S5.SS2.p2.5.m5.1.1.1.1.3.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3.1"></minus><apply id="S5.SS2.p2.5.m5.1.1.1.1.3.2.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2"><ci id="S5.SS2.p2.5.m5.1.1.1.1.3.2.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2.1">^</ci><ci id="S5.SS2.p2.5.m5.1.1.1.1.3.2.2.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3.2.2">𝜏</ci></apply><ci id="S5.SS2.p2.5.m5.1.1.1.1.3.3.cmml" xref="S5.SS2.p2.5.m5.1.1.1.1.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.5.m5.1c">\dot{\tau}=\hat{\tau}-\tau.</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.5.m5.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ .</annotation></semantics></math> If <math alttext="\tau>0" class="ltx_Math" display="inline" id="S5.SS2.p2.6.m6.1"><semantics id="S5.SS2.p2.6.m6.1a"><mrow id="S5.SS2.p2.6.m6.1.1" xref="S5.SS2.p2.6.m6.1.1.cmml"><mi id="S5.SS2.p2.6.m6.1.1.2" xref="S5.SS2.p2.6.m6.1.1.2.cmml">τ</mi><mo id="S5.SS2.p2.6.m6.1.1.1" xref="S5.SS2.p2.6.m6.1.1.1.cmml">></mo><mn id="S5.SS2.p2.6.m6.1.1.3" xref="S5.SS2.p2.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.6.m6.1b"><apply id="S5.SS2.p2.6.m6.1.1.cmml" xref="S5.SS2.p2.6.m6.1.1"><gt id="S5.SS2.p2.6.m6.1.1.1.cmml" xref="S5.SS2.p2.6.m6.1.1.1"></gt><ci id="S5.SS2.p2.6.m6.1.1.2.cmml" xref="S5.SS2.p2.6.m6.1.1.2">𝜏</ci><cn id="S5.SS2.p2.6.m6.1.1.3.cmml" type="integer" xref="S5.SS2.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.6.m6.1c">\tau>0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.6.m6.1d">italic_τ > 0</annotation></semantics></math>, then <math alttext="\hat{\tau}<\tau" class="ltx_Math" display="inline" id="S5.SS2.p2.7.m7.1"><semantics id="S5.SS2.p2.7.m7.1a"><mrow id="S5.SS2.p2.7.m7.1.1" xref="S5.SS2.p2.7.m7.1.1.cmml"><mover accent="true" id="S5.SS2.p2.7.m7.1.1.2" xref="S5.SS2.p2.7.m7.1.1.2.cmml"><mi id="S5.SS2.p2.7.m7.1.1.2.2" xref="S5.SS2.p2.7.m7.1.1.2.2.cmml">τ</mi><mo id="S5.SS2.p2.7.m7.1.1.2.1" xref="S5.SS2.p2.7.m7.1.1.2.1.cmml">^</mo></mover><mo id="S5.SS2.p2.7.m7.1.1.1" xref="S5.SS2.p2.7.m7.1.1.1.cmml"><</mo><mi id="S5.SS2.p2.7.m7.1.1.3" xref="S5.SS2.p2.7.m7.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.7.m7.1b"><apply id="S5.SS2.p2.7.m7.1.1.cmml" xref="S5.SS2.p2.7.m7.1.1"><lt id="S5.SS2.p2.7.m7.1.1.1.cmml" xref="S5.SS2.p2.7.m7.1.1.1"></lt><apply id="S5.SS2.p2.7.m7.1.1.2.cmml" xref="S5.SS2.p2.7.m7.1.1.2"><ci id="S5.SS2.p2.7.m7.1.1.2.1.cmml" xref="S5.SS2.p2.7.m7.1.1.2.1">^</ci><ci id="S5.SS2.p2.7.m7.1.1.2.2.cmml" xref="S5.SS2.p2.7.m7.1.1.2.2">𝜏</ci></apply><ci id="S5.SS2.p2.7.m7.1.1.3.cmml" xref="S5.SS2.p2.7.m7.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.7.m7.1c">\hat{\tau}<\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.7.m7.1d">over^ start_ARG italic_τ end_ARG < italic_τ</annotation></semantics></math>, while if <math alttext="\tau<0" class="ltx_Math" display="inline" id="S5.SS2.p2.8.m8.1"><semantics id="S5.SS2.p2.8.m8.1a"><mrow id="S5.SS2.p2.8.m8.1.1" xref="S5.SS2.p2.8.m8.1.1.cmml"><mi id="S5.SS2.p2.8.m8.1.1.2" xref="S5.SS2.p2.8.m8.1.1.2.cmml">τ</mi><mo id="S5.SS2.p2.8.m8.1.1.1" xref="S5.SS2.p2.8.m8.1.1.1.cmml"><</mo><mn id="S5.SS2.p2.8.m8.1.1.3" xref="S5.SS2.p2.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.8.m8.1b"><apply id="S5.SS2.p2.8.m8.1.1.cmml" xref="S5.SS2.p2.8.m8.1.1"><lt id="S5.SS2.p2.8.m8.1.1.1.cmml" xref="S5.SS2.p2.8.m8.1.1.1"></lt><ci id="S5.SS2.p2.8.m8.1.1.2.cmml" xref="S5.SS2.p2.8.m8.1.1.2">𝜏</ci><cn id="S5.SS2.p2.8.m8.1.1.3.cmml" type="integer" xref="S5.SS2.p2.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.8.m8.1c">\tau<0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.8.m8.1d">italic_τ < 0</annotation></semantics></math>, <math alttext="\hat{\tau}>\tau" class="ltx_Math" display="inline" id="S5.SS2.p2.9.m9.1"><semantics id="S5.SS2.p2.9.m9.1a"><mrow id="S5.SS2.p2.9.m9.1.1" xref="S5.SS2.p2.9.m9.1.1.cmml"><mover accent="true" id="S5.SS2.p2.9.m9.1.1.2" xref="S5.SS2.p2.9.m9.1.1.2.cmml"><mi id="S5.SS2.p2.9.m9.1.1.2.2" xref="S5.SS2.p2.9.m9.1.1.2.2.cmml">τ</mi><mo id="S5.SS2.p2.9.m9.1.1.2.1" xref="S5.SS2.p2.9.m9.1.1.2.1.cmml">^</mo></mover><mo id="S5.SS2.p2.9.m9.1.1.1" xref="S5.SS2.p2.9.m9.1.1.1.cmml">></mo><mi id="S5.SS2.p2.9.m9.1.1.3" xref="S5.SS2.p2.9.m9.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.9.m9.1b"><apply id="S5.SS2.p2.9.m9.1.1.cmml" xref="S5.SS2.p2.9.m9.1.1"><gt id="S5.SS2.p2.9.m9.1.1.1.cmml" xref="S5.SS2.p2.9.m9.1.1.1"></gt><apply id="S5.SS2.p2.9.m9.1.1.2.cmml" xref="S5.SS2.p2.9.m9.1.1.2"><ci id="S5.SS2.p2.9.m9.1.1.2.1.cmml" xref="S5.SS2.p2.9.m9.1.1.2.1">^</ci><ci id="S5.SS2.p2.9.m9.1.1.2.2.cmml" xref="S5.SS2.p2.9.m9.1.1.2.2">𝜏</ci></apply><ci id="S5.SS2.p2.9.m9.1.1.3.cmml" xref="S5.SS2.p2.9.m9.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.9.m9.1c">\hat{\tau}>\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.9.m9.1d">over^ start_ARG italic_τ end_ARG > italic_τ</annotation></semantics></math>. It follows that <math alttext="\tau=0" class="ltx_Math" display="inline" id="S5.SS2.p2.10.m10.1"><semantics id="S5.SS2.p2.10.m10.1a"><mrow id="S5.SS2.p2.10.m10.1.1" xref="S5.SS2.p2.10.m10.1.1.cmml"><mi id="S5.SS2.p2.10.m10.1.1.2" xref="S5.SS2.p2.10.m10.1.1.2.cmml">τ</mi><mo id="S5.SS2.p2.10.m10.1.1.1" xref="S5.SS2.p2.10.m10.1.1.1.cmml">=</mo><mn id="S5.SS2.p2.10.m10.1.1.3" xref="S5.SS2.p2.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.10.m10.1b"><apply id="S5.SS2.p2.10.m10.1.1.cmml" xref="S5.SS2.p2.10.m10.1.1"><eq id="S5.SS2.p2.10.m10.1.1.1.cmml" xref="S5.SS2.p2.10.m10.1.1.1"></eq><ci id="S5.SS2.p2.10.m10.1.1.2.cmml" xref="S5.SS2.p2.10.m10.1.1.2">𝜏</ci><cn id="S5.SS2.p2.10.m10.1.1.3.cmml" type="integer" xref="S5.SS2.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.10.m10.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.10.m10.1d">italic_τ = 0</annotation></semantics></math> is stable, while by topological necessity the uninformative equilibria are unstable.</p> </div> <div class="ltx_para" id="S5.SS2.p3"> <p class="ltx_p" id="S5.SS2.p3.9">While the equilibria and dynamics behave the same in RBTS as in DG for the Gaussian model, we can ask if there are differences in the convergence rate to stable point <math alttext="\tau=0" class="ltx_Math" display="inline" id="S5.SS2.p3.1.m1.1"><semantics id="S5.SS2.p3.1.m1.1a"><mrow id="S5.SS2.p3.1.m1.1.1" xref="S5.SS2.p3.1.m1.1.1.cmml"><mi id="S5.SS2.p3.1.m1.1.1.2" xref="S5.SS2.p3.1.m1.1.1.2.cmml">τ</mi><mo id="S5.SS2.p3.1.m1.1.1.1" xref="S5.SS2.p3.1.m1.1.1.1.cmml">=</mo><mn id="S5.SS2.p3.1.m1.1.1.3" xref="S5.SS2.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.1.m1.1b"><apply id="S5.SS2.p3.1.m1.1.1.cmml" xref="S5.SS2.p3.1.m1.1.1"><eq id="S5.SS2.p3.1.m1.1.1.1.cmml" xref="S5.SS2.p3.1.m1.1.1.1"></eq><ci id="S5.SS2.p3.1.m1.1.1.2.cmml" xref="S5.SS2.p3.1.m1.1.1.2">𝜏</ci><cn id="S5.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.1.m1.1d">italic_τ = 0</annotation></semantics></math>. In fact, we can observe that the best response in both settings is a linear function of <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS2.p3.2.m2.1"><semantics id="S5.SS2.p3.2.m2.1a"><mi id="S5.SS2.p3.2.m2.1.1" xref="S5.SS2.p3.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.2.m2.1b"><ci id="S5.SS2.p3.2.m2.1.1.cmml" xref="S5.SS2.p3.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.2.m2.1d">italic_τ</annotation></semantics></math> with coefficients <math alttext="m_{\textrm{DG}}" class="ltx_Math" display="inline" id="S5.SS2.p3.3.m3.1"><semantics id="S5.SS2.p3.3.m3.1a"><msub id="S5.SS2.p3.3.m3.1.1" xref="S5.SS2.p3.3.m3.1.1.cmml"><mi id="S5.SS2.p3.3.m3.1.1.2" xref="S5.SS2.p3.3.m3.1.1.2.cmml">m</mi><mtext id="S5.SS2.p3.3.m3.1.1.3" xref="S5.SS2.p3.3.m3.1.1.3a.cmml">DG</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.3.m3.1b"><apply id="S5.SS2.p3.3.m3.1.1.cmml" xref="S5.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.3.m3.1.1.1.cmml" xref="S5.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S5.SS2.p3.3.m3.1.1.2.cmml" xref="S5.SS2.p3.3.m3.1.1.2">𝑚</ci><ci id="S5.SS2.p3.3.m3.1.1.3a.cmml" xref="S5.SS2.p3.3.m3.1.1.3"><mtext id="S5.SS2.p3.3.m3.1.1.3.cmml" mathsize="70%" xref="S5.SS2.p3.3.m3.1.1.3">DG</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.3.m3.1c">m_{\textrm{DG}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.3.m3.1d">italic_m start_POSTSUBSCRIPT DG end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="m_{\textrm{RBTS}}" class="ltx_Math" display="inline" id="S5.SS2.p3.4.m4.1"><semantics id="S5.SS2.p3.4.m4.1a"><msub id="S5.SS2.p3.4.m4.1.1" xref="S5.SS2.p3.4.m4.1.1.cmml"><mi id="S5.SS2.p3.4.m4.1.1.2" xref="S5.SS2.p3.4.m4.1.1.2.cmml">m</mi><mtext id="S5.SS2.p3.4.m4.1.1.3" xref="S5.SS2.p3.4.m4.1.1.3a.cmml">RBTS</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.4.m4.1b"><apply id="S5.SS2.p3.4.m4.1.1.cmml" xref="S5.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.4.m4.1.1.1.cmml" xref="S5.SS2.p3.4.m4.1.1">subscript</csymbol><ci id="S5.SS2.p3.4.m4.1.1.2.cmml" xref="S5.SS2.p3.4.m4.1.1.2">𝑚</ci><ci id="S5.SS2.p3.4.m4.1.1.3a.cmml" xref="S5.SS2.p3.4.m4.1.1.3"><mtext id="S5.SS2.p3.4.m4.1.1.3.cmml" mathsize="70%" xref="S5.SS2.p3.4.m4.1.1.3">RBTS</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.4.m4.1c">m_{\textrm{RBTS}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.4.m4.1d">italic_m start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT</annotation></semantics></math> respectively (see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A4" title="Appendix D Omitted Proofs for RBTS ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">D</span></a> for a formal derivation). It turns out that in any setting of parameters <math alttext="a" class="ltx_Math" display="inline" id="S5.SS2.p3.5.m5.1"><semantics id="S5.SS2.p3.5.m5.1a"><mi id="S5.SS2.p3.5.m5.1.1" xref="S5.SS2.p3.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.5.m5.1b"><ci id="S5.SS2.p3.5.m5.1.1.cmml" xref="S5.SS2.p3.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.5.m5.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S5.SS2.p3.6.m6.1"><semantics id="S5.SS2.p3.6.m6.1a"><mi id="S5.SS2.p3.6.m6.1.1" xref="S5.SS2.p3.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.6.m6.1b"><ci id="S5.SS2.p3.6.m6.1.1.cmml" xref="S5.SS2.p3.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.6.m6.1c">b</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.6.m6.1d">italic_b</annotation></semantics></math>, <math alttext="m_{\textrm{RBTS}}>m_{\textrm{DG}}" class="ltx_Math" display="inline" id="S5.SS2.p3.7.m7.1"><semantics id="S5.SS2.p3.7.m7.1a"><mrow id="S5.SS2.p3.7.m7.1.1" xref="S5.SS2.p3.7.m7.1.1.cmml"><msub id="S5.SS2.p3.7.m7.1.1.2" xref="S5.SS2.p3.7.m7.1.1.2.cmml"><mi id="S5.SS2.p3.7.m7.1.1.2.2" xref="S5.SS2.p3.7.m7.1.1.2.2.cmml">m</mi><mtext id="S5.SS2.p3.7.m7.1.1.2.3" xref="S5.SS2.p3.7.m7.1.1.2.3a.cmml">RBTS</mtext></msub><mo id="S5.SS2.p3.7.m7.1.1.1" xref="S5.SS2.p3.7.m7.1.1.1.cmml">></mo><msub id="S5.SS2.p3.7.m7.1.1.3" xref="S5.SS2.p3.7.m7.1.1.3.cmml"><mi id="S5.SS2.p3.7.m7.1.1.3.2" xref="S5.SS2.p3.7.m7.1.1.3.2.cmml">m</mi><mtext id="S5.SS2.p3.7.m7.1.1.3.3" xref="S5.SS2.p3.7.m7.1.1.3.3a.cmml">DG</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.7.m7.1b"><apply id="S5.SS2.p3.7.m7.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1"><gt id="S5.SS2.p3.7.m7.1.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1.1"></gt><apply id="S5.SS2.p3.7.m7.1.1.2.cmml" xref="S5.SS2.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p3.7.m7.1.1.2.1.cmml" xref="S5.SS2.p3.7.m7.1.1.2">subscript</csymbol><ci id="S5.SS2.p3.7.m7.1.1.2.2.cmml" xref="S5.SS2.p3.7.m7.1.1.2.2">𝑚</ci><ci id="S5.SS2.p3.7.m7.1.1.2.3a.cmml" xref="S5.SS2.p3.7.m7.1.1.2.3"><mtext id="S5.SS2.p3.7.m7.1.1.2.3.cmml" mathsize="70%" xref="S5.SS2.p3.7.m7.1.1.2.3">RBTS</mtext></ci></apply><apply id="S5.SS2.p3.7.m7.1.1.3.cmml" xref="S5.SS2.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p3.7.m7.1.1.3.1.cmml" xref="S5.SS2.p3.7.m7.1.1.3">subscript</csymbol><ci id="S5.SS2.p3.7.m7.1.1.3.2.cmml" xref="S5.SS2.p3.7.m7.1.1.3.2">𝑚</ci><ci id="S5.SS2.p3.7.m7.1.1.3.3a.cmml" xref="S5.SS2.p3.7.m7.1.1.3.3"><mtext id="S5.SS2.p3.7.m7.1.1.3.3.cmml" mathsize="70%" xref="S5.SS2.p3.7.m7.1.1.3.3">DG</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.7.m7.1c">m_{\textrm{RBTS}}>m_{\textrm{DG}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.7.m7.1d">italic_m start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT > italic_m start_POSTSUBSCRIPT DG end_POSTSUBSCRIPT</annotation></semantics></math>, so that if both mechanisms begin with the same starting threshold <math alttext="\tau(0)" class="ltx_Math" display="inline" id="S5.SS2.p3.8.m8.1"><semantics id="S5.SS2.p3.8.m8.1a"><mrow id="S5.SS2.p3.8.m8.1.2" xref="S5.SS2.p3.8.m8.1.2.cmml"><mi id="S5.SS2.p3.8.m8.1.2.2" xref="S5.SS2.p3.8.m8.1.2.2.cmml">τ</mi><mo id="S5.SS2.p3.8.m8.1.2.1" xref="S5.SS2.p3.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p3.8.m8.1.2.3.2" xref="S5.SS2.p3.8.m8.1.2.cmml"><mo id="S5.SS2.p3.8.m8.1.2.3.2.1" stretchy="false" xref="S5.SS2.p3.8.m8.1.2.cmml">(</mo><mn id="S5.SS2.p3.8.m8.1.1" xref="S5.SS2.p3.8.m8.1.1.cmml">0</mn><mo id="S5.SS2.p3.8.m8.1.2.3.2.2" stretchy="false" xref="S5.SS2.p3.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.8.m8.1b"><apply id="S5.SS2.p3.8.m8.1.2.cmml" xref="S5.SS2.p3.8.m8.1.2"><times id="S5.SS2.p3.8.m8.1.2.1.cmml" xref="S5.SS2.p3.8.m8.1.2.1"></times><ci id="S5.SS2.p3.8.m8.1.2.2.cmml" xref="S5.SS2.p3.8.m8.1.2.2">𝜏</ci><cn id="S5.SS2.p3.8.m8.1.1.cmml" type="integer" xref="S5.SS2.p3.8.m8.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.8.m8.1c">\tau(0)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.8.m8.1d">italic_τ ( 0 )</annotation></semantics></math>, the stable equilibrium at <math alttext="0" class="ltx_Math" display="inline" id="S5.SS2.p3.9.m9.1"><semantics id="S5.SS2.p3.9.m9.1a"><mn id="S5.SS2.p3.9.m9.1.1" xref="S5.SS2.p3.9.m9.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.9.m9.1b"><cn id="S5.SS2.p3.9.m9.1.1.cmml" type="integer" xref="S5.SS2.p3.9.m9.1.1">0</cn></annotation-xml></semantics></math> is reached faster in DG than RBTS. In a peer grading setting, then, graders will more quickly converge to a half-good half-bad quality consensus than RBTS graders, so that the designer may have more time to react and reset expectations in the latter case.</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>Experiments</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">We now aim to numerically explore existence and dynamics of equilibria across mechanisms in settings other than the noisy Gaussian model. We consider two natural departures from the “nice” properties of Gaussians, skewness and multimodality.</p> </div> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.1 </span>Skewness</h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.11">Skewed models of information capture settings where there is some asymmetry in the tasks. To study such settings, we adapt the same noisy signal model as our running Gaussian example, but with a base distribution which is skewed. Formally, each agent <math alttext="i" class="ltx_Math" display="inline" id="S6.SS1.p1.1.m1.1"><semantics id="S6.SS1.p1.1.m1.1a"><mi id="S6.SS1.p1.1.m1.1.1" xref="S6.SS1.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.1.m1.1b"><ci id="S6.SS1.p1.1.m1.1.1.cmml" xref="S6.SS1.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.1.m1.1d">italic_i</annotation></semantics></math> receives a signal <math alttext="X_{i}" class="ltx_Math" display="inline" id="S6.SS1.p1.2.m2.1"><semantics id="S6.SS1.p1.2.m2.1a"><msub id="S6.SS1.p1.2.m2.1.1" xref="S6.SS1.p1.2.m2.1.1.cmml"><mi id="S6.SS1.p1.2.m2.1.1.2" xref="S6.SS1.p1.2.m2.1.1.2.cmml">X</mi><mi id="S6.SS1.p1.2.m2.1.1.3" xref="S6.SS1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.2.m2.1b"><apply id="S6.SS1.p1.2.m2.1.1.cmml" xref="S6.SS1.p1.2.m2.1.1"><csymbol 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id="S6.SS1.p1.4.m4.1a"><mi id="S6.SS1.p1.4.m4.1.1" xref="S6.SS1.p1.4.m4.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.4.m4.1b"><ci id="S6.SS1.p1.4.m4.1.1.cmml" xref="S6.SS1.p1.4.m4.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.4.m4.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.4.m4.1d">italic_Z</annotation></semantics></math> is the skewed Normal distribution with mean 0 and variance 1, and <math alttext="Z_{i}\sim N(0,1)" class="ltx_Math" display="inline" id="S6.SS1.p1.5.m5.2"><semantics id="S6.SS1.p1.5.m5.2a"><mrow id="S6.SS1.p1.5.m5.2.3" xref="S6.SS1.p1.5.m5.2.3.cmml"><msub id="S6.SS1.p1.5.m5.2.3.2" xref="S6.SS1.p1.5.m5.2.3.2.cmml"><mi id="S6.SS1.p1.5.m5.2.3.2.2" xref="S6.SS1.p1.5.m5.2.3.2.2.cmml">Z</mi><mi id="S6.SS1.p1.5.m5.2.3.2.3" xref="S6.SS1.p1.5.m5.2.3.2.3.cmml">i</mi></msub><mo id="S6.SS1.p1.5.m5.2.3.1" xref="S6.SS1.p1.5.m5.2.3.1.cmml">∼</mo><mrow id="S6.SS1.p1.5.m5.2.3.3" 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)</annotation></semantics></math>. <math alttext="Z" class="ltx_Math" display="inline" id="S6.SS1.p1.6.m6.1"><semantics id="S6.SS1.p1.6.m6.1a"><mi id="S6.SS1.p1.6.m6.1.1" xref="S6.SS1.p1.6.m6.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.6.m6.1b"><ci id="S6.SS1.p1.6.m6.1.1.cmml" xref="S6.SS1.p1.6.m6.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.6.m6.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.6.m6.1d">italic_Z</annotation></semantics></math> has another parameter, <math alttext="\alpha" class="ltx_Math" display="inline" id="S6.SS1.p1.7.m7.1"><semantics id="S6.SS1.p1.7.m7.1a"><mi id="S6.SS1.p1.7.m7.1.1" xref="S6.SS1.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.7.m7.1b"><ci id="S6.SS1.p1.7.m7.1.1.cmml" xref="S6.SS1.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.7.m7.1d">italic_α</annotation></semantics></math>, which controls the skewness of the distribution. The magnitude of <math alttext="\alpha" class="ltx_Math" display="inline" id="S6.SS1.p1.8.m8.1"><semantics id="S6.SS1.p1.8.m8.1a"><mi id="S6.SS1.p1.8.m8.1.1" xref="S6.SS1.p1.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.8.m8.1b"><ci id="S6.SS1.p1.8.m8.1.1.cmml" xref="S6.SS1.p1.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.8.m8.1d">italic_α</annotation></semantics></math> increases the magnitude of skewness; moreover, <math alttext="\alpha<0" class="ltx_Math" display="inline" id="S6.SS1.p1.9.m9.1"><semantics id="S6.SS1.p1.9.m9.1a"><mrow id="S6.SS1.p1.9.m9.1.1" xref="S6.SS1.p1.9.m9.1.1.cmml"><mi id="S6.SS1.p1.9.m9.1.1.2" xref="S6.SS1.p1.9.m9.1.1.2.cmml">α</mi><mo id="S6.SS1.p1.9.m9.1.1.1" xref="S6.SS1.p1.9.m9.1.1.1.cmml"><</mo><mn id="S6.SS1.p1.9.m9.1.1.3" xref="S6.SS1.p1.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.9.m9.1b"><apply id="S6.SS1.p1.9.m9.1.1.cmml" xref="S6.SS1.p1.9.m9.1.1"><lt id="S6.SS1.p1.9.m9.1.1.1.cmml" xref="S6.SS1.p1.9.m9.1.1.1"></lt><ci id="S6.SS1.p1.9.m9.1.1.2.cmml" xref="S6.SS1.p1.9.m9.1.1.2">𝛼</ci><cn id="S6.SS1.p1.9.m9.1.1.3.cmml" type="integer" xref="S6.SS1.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.9.m9.1c">\alpha<0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.9.m9.1d">italic_α < 0</annotation></semantics></math> leads to a left-skewed distribution, and <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S6.SS1.p1.10.m10.1"><semantics id="S6.SS1.p1.10.m10.1a"><mrow id="S6.SS1.p1.10.m10.1.1" xref="S6.SS1.p1.10.m10.1.1.cmml"><mi id="S6.SS1.p1.10.m10.1.1.2" xref="S6.SS1.p1.10.m10.1.1.2.cmml">α</mi><mo id="S6.SS1.p1.10.m10.1.1.1" xref="S6.SS1.p1.10.m10.1.1.1.cmml">></mo><mn id="S6.SS1.p1.10.m10.1.1.3" xref="S6.SS1.p1.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.10.m10.1b"><apply id="S6.SS1.p1.10.m10.1.1.cmml" xref="S6.SS1.p1.10.m10.1.1"><gt id="S6.SS1.p1.10.m10.1.1.1.cmml" xref="S6.SS1.p1.10.m10.1.1.1"></gt><ci id="S6.SS1.p1.10.m10.1.1.2.cmml" xref="S6.SS1.p1.10.m10.1.1.2">𝛼</ci><cn id="S6.SS1.p1.10.m10.1.1.3.cmml" type="integer" xref="S6.SS1.p1.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.10.m10.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.10.m10.1d">italic_α > 0</annotation></semantics></math> a right-skewed distribution. We study how threshold equilibria move as a function of the parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S6.SS1.p1.11.m11.1"><semantics id="S6.SS1.p1.11.m11.1a"><mi id="S6.SS1.p1.11.m11.1.1" xref="S6.SS1.p1.11.m11.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.11.m11.1b"><ci id="S6.SS1.p1.11.m11.1.1.cmml" xref="S6.SS1.p1.11.m11.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.11.m11.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.11.m11.1d">italic_α</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.2">As depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6.F4" title="Figure 4 ‣ 6.2 Multimodality ‣ 6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>, we find a unique equilibrium under both OA and DG which shifts toward the longer tail, i.e. the thresholds move from negative to positive as the skewness moves from left to right (and we recover the <math alttext="\tau=0" class="ltx_Math" display="inline" id="S6.SS1.p2.1.m1.1"><semantics id="S6.SS1.p2.1.m1.1a"><mrow id="S6.SS1.p2.1.m1.1.1" xref="S6.SS1.p2.1.m1.1.1.cmml"><mi id="S6.SS1.p2.1.m1.1.1.2" xref="S6.SS1.p2.1.m1.1.1.2.cmml">τ</mi><mo id="S6.SS1.p2.1.m1.1.1.1" xref="S6.SS1.p2.1.m1.1.1.1.cmml">=</mo><mn id="S6.SS1.p2.1.m1.1.1.3" xref="S6.SS1.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.1b"><apply id="S6.SS1.p2.1.m1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1"><eq id="S6.SS1.p2.1.m1.1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1.1"></eq><ci id="S6.SS1.p2.1.m1.1.1.2.cmml" xref="S6.SS1.p2.1.m1.1.1.2">𝜏</ci><cn id="S6.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.SS1.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.1.m1.1d">italic_τ = 0</annotation></semantics></math> equilibrium in the original Gaussian model when <math alttext="\alpha=0" class="ltx_Math" display="inline" id="S6.SS1.p2.2.m2.1"><semantics id="S6.SS1.p2.2.m2.1a"><mrow id="S6.SS1.p2.2.m2.1.1" xref="S6.SS1.p2.2.m2.1.1.cmml"><mi id="S6.SS1.p2.2.m2.1.1.2" xref="S6.SS1.p2.2.m2.1.1.2.cmml">α</mi><mo id="S6.SS1.p2.2.m2.1.1.1" xref="S6.SS1.p2.2.m2.1.1.1.cmml">=</mo><mn id="S6.SS1.p2.2.m2.1.1.3" xref="S6.SS1.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.1b"><apply id="S6.SS1.p2.2.m2.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1"><eq id="S6.SS1.p2.2.m2.1.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1.1"></eq><ci id="S6.SS1.p2.2.m2.1.1.2.cmml" xref="S6.SS1.p2.2.m2.1.1.2">𝛼</ci><cn id="S6.SS1.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.SS1.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.1c">\alpha=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.2.m2.1d">italic_α = 0</annotation></semantics></math>). Thus we expect when there is some asymmetry in the distribution of signals, agents in DG will settle at a threshold that balances out the asymmetry in signal mass. While we observe the same high-level dynamics for OA, these equilibria are unstable and so we still expect agents will move toward an uninformative consensus. For DMI, meanwhile, we note that the same equilibria will occur as DG by our theoretical results. Moreover, given the behavior observed under the Gaussian, we expect the stable equilibria in RBTS to share the same pattern as DG in tracking the skewness of the distribution, but at a rate in between OA and DG.</p> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.2 </span>Multimodality</h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.14">We now explore the effects that a distribution with multiple peaks has on the equilibria that occur under our signal model. Consider a classic mixed Gaussian setting where each agent’s signal <math alttext="X_{i}" class="ltx_Math" display="inline" id="S6.SS2.p1.1.m1.1"><semantics id="S6.SS2.p1.1.m1.1a"><msub id="S6.SS2.p1.1.m1.1.1" xref="S6.SS2.p1.1.m1.1.1.cmml"><mi id="S6.SS2.p1.1.m1.1.1.2" xref="S6.SS2.p1.1.m1.1.1.2.cmml">X</mi><mi id="S6.SS2.p1.1.m1.1.1.3" xref="S6.SS2.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.1.m1.1b"><apply id="S6.SS2.p1.1.m1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.1.m1.1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.p1.1.m1.1.1.2.cmml" xref="S6.SS2.p1.1.m1.1.1.2">𝑋</ci><ci id="S6.SS2.p1.1.m1.1.1.3.cmml" xref="S6.SS2.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.1.m1.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is received i.i.d. from one of <math alttext="K" class="ltx_Math" display="inline" id="S6.SS2.p1.2.m2.1"><semantics id="S6.SS2.p1.2.m2.1a"><mi id="S6.SS2.p1.2.m2.1.1" xref="S6.SS2.p1.2.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.2.m2.1b"><ci id="S6.SS2.p1.2.m2.1.1.cmml" xref="S6.SS2.p1.2.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.2.m2.1c">K</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.2.m2.1d">italic_K</annotation></semantics></math> Gaussian components, each with mean <math alttext="\mu_{k}" class="ltx_Math" display="inline" id="S6.SS2.p1.3.m3.1"><semantics id="S6.SS2.p1.3.m3.1a"><msub id="S6.SS2.p1.3.m3.1.1" xref="S6.SS2.p1.3.m3.1.1.cmml"><mi id="S6.SS2.p1.3.m3.1.1.2" xref="S6.SS2.p1.3.m3.1.1.2.cmml">μ</mi><mi id="S6.SS2.p1.3.m3.1.1.3" xref="S6.SS2.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.3.m3.1b"><apply id="S6.SS2.p1.3.m3.1.1.cmml" xref="S6.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.3.m3.1.1.1.cmml" xref="S6.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.p1.3.m3.1.1.2.cmml" xref="S6.SS2.p1.3.m3.1.1.2">𝜇</ci><ci id="S6.SS2.p1.3.m3.1.1.3.cmml" xref="S6.SS2.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.3.m3.1c">\mu_{k}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.3.m3.1d">italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and variance <math alttext="\sigma_{k}^{2}" class="ltx_Math" display="inline" id="S6.SS2.p1.4.m4.1"><semantics id="S6.SS2.p1.4.m4.1a"><msubsup id="S6.SS2.p1.4.m4.1.1" xref="S6.SS2.p1.4.m4.1.1.cmml"><mi id="S6.SS2.p1.4.m4.1.1.2.2" xref="S6.SS2.p1.4.m4.1.1.2.2.cmml">σ</mi><mi id="S6.SS2.p1.4.m4.1.1.2.3" xref="S6.SS2.p1.4.m4.1.1.2.3.cmml">k</mi><mn id="S6.SS2.p1.4.m4.1.1.3" xref="S6.SS2.p1.4.m4.1.1.3.cmml">2</mn></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.4.m4.1b"><apply id="S6.SS2.p1.4.m4.1.1.cmml" xref="S6.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.4.m4.1.1.1.cmml" xref="S6.SS2.p1.4.m4.1.1">superscript</csymbol><apply id="S6.SS2.p1.4.m4.1.1.2.cmml" xref="S6.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.4.m4.1.1.2.1.cmml" xref="S6.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S6.SS2.p1.4.m4.1.1.2.2.cmml" xref="S6.SS2.p1.4.m4.1.1.2.2">𝜎</ci><ci id="S6.SS2.p1.4.m4.1.1.2.3.cmml" xref="S6.SS2.p1.4.m4.1.1.2.3">𝑘</ci></apply><cn id="S6.SS2.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.SS2.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.4.m4.1c">\sigma_{k}^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. There is an underlying parameter <math alttext="z\sim\text{Categorical}(\pi)" class="ltx_Math" display="inline" id="S6.SS2.p1.5.m5.1"><semantics id="S6.SS2.p1.5.m5.1a"><mrow id="S6.SS2.p1.5.m5.1.2" xref="S6.SS2.p1.5.m5.1.2.cmml"><mi id="S6.SS2.p1.5.m5.1.2.2" xref="S6.SS2.p1.5.m5.1.2.2.cmml">z</mi><mo id="S6.SS2.p1.5.m5.1.2.1" xref="S6.SS2.p1.5.m5.1.2.1.cmml">∼</mo><mrow id="S6.SS2.p1.5.m5.1.2.3" xref="S6.SS2.p1.5.m5.1.2.3.cmml"><mtext id="S6.SS2.p1.5.m5.1.2.3.2" xref="S6.SS2.p1.5.m5.1.2.3.2a.cmml">Categorical</mtext><mo id="S6.SS2.p1.5.m5.1.2.3.1" xref="S6.SS2.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.p1.5.m5.1.2.3.3.2" xref="S6.SS2.p1.5.m5.1.2.3.cmml"><mo id="S6.SS2.p1.5.m5.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.p1.5.m5.1.2.3.cmml">(</mo><mi id="S6.SS2.p1.5.m5.1.1" xref="S6.SS2.p1.5.m5.1.1.cmml">π</mi><mo id="S6.SS2.p1.5.m5.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.p1.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.5.m5.1b"><apply id="S6.SS2.p1.5.m5.1.2.cmml" xref="S6.SS2.p1.5.m5.1.2"><csymbol cd="latexml" id="S6.SS2.p1.5.m5.1.2.1.cmml" xref="S6.SS2.p1.5.m5.1.2.1">similar-to</csymbol><ci id="S6.SS2.p1.5.m5.1.2.2.cmml" xref="S6.SS2.p1.5.m5.1.2.2">𝑧</ci><apply id="S6.SS2.p1.5.m5.1.2.3.cmml" xref="S6.SS2.p1.5.m5.1.2.3"><times id="S6.SS2.p1.5.m5.1.2.3.1.cmml" xref="S6.SS2.p1.5.m5.1.2.3.1"></times><ci id="S6.SS2.p1.5.m5.1.2.3.2a.cmml" xref="S6.SS2.p1.5.m5.1.2.3.2"><mtext id="S6.SS2.p1.5.m5.1.2.3.2.cmml" xref="S6.SS2.p1.5.m5.1.2.3.2">Categorical</mtext></ci><ci id="S6.SS2.p1.5.m5.1.1.cmml" xref="S6.SS2.p1.5.m5.1.1">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.5.m5.1c">z\sim\text{Categorical}(\pi)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.5.m5.1d">italic_z ∼ Categorical ( italic_π )</annotation></semantics></math> for <math alttext="\pi\in\Delta^{K}" class="ltx_Math" display="inline" id="S6.SS2.p1.6.m6.1"><semantics id="S6.SS2.p1.6.m6.1a"><mrow id="S6.SS2.p1.6.m6.1.1" xref="S6.SS2.p1.6.m6.1.1.cmml"><mi id="S6.SS2.p1.6.m6.1.1.2" xref="S6.SS2.p1.6.m6.1.1.2.cmml">π</mi><mo id="S6.SS2.p1.6.m6.1.1.1" xref="S6.SS2.p1.6.m6.1.1.1.cmml">∈</mo><msup id="S6.SS2.p1.6.m6.1.1.3" xref="S6.SS2.p1.6.m6.1.1.3.cmml"><mi id="S6.SS2.p1.6.m6.1.1.3.2" mathvariant="normal" xref="S6.SS2.p1.6.m6.1.1.3.2.cmml">Δ</mi><mi id="S6.SS2.p1.6.m6.1.1.3.3" xref="S6.SS2.p1.6.m6.1.1.3.3.cmml">K</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.6.m6.1b"><apply id="S6.SS2.p1.6.m6.1.1.cmml" xref="S6.SS2.p1.6.m6.1.1"><in id="S6.SS2.p1.6.m6.1.1.1.cmml" xref="S6.SS2.p1.6.m6.1.1.1"></in><ci id="S6.SS2.p1.6.m6.1.1.2.cmml" xref="S6.SS2.p1.6.m6.1.1.2">𝜋</ci><apply id="S6.SS2.p1.6.m6.1.1.3.cmml" xref="S6.SS2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p1.6.m6.1.1.3.1.cmml" xref="S6.SS2.p1.6.m6.1.1.3">superscript</csymbol><ci id="S6.SS2.p1.6.m6.1.1.3.2.cmml" xref="S6.SS2.p1.6.m6.1.1.3.2">Δ</ci><ci id="S6.SS2.p1.6.m6.1.1.3.3.cmml" xref="S6.SS2.p1.6.m6.1.1.3.3">𝐾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.6.m6.1c">\pi\in\Delta^{K}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.6.m6.1d">italic_π ∈ roman_Δ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math> that determines which Gaussian component all the signals come from. For the purposes of our experiments, we fix <math alttext="\pi" class="ltx_Math" display="inline" id="S6.SS2.p1.7.m7.1"><semantics id="S6.SS2.p1.7.m7.1a"><mi id="S6.SS2.p1.7.m7.1.1" xref="S6.SS2.p1.7.m7.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.7.m7.1b"><ci id="S6.SS2.p1.7.m7.1.1.cmml" xref="S6.SS2.p1.7.m7.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.7.m7.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.7.m7.1d">italic_π</annotation></semantics></math> to be the uniform distribution over the <math alttext="K" class="ltx_Math" display="inline" id="S6.SS2.p1.8.m8.1"><semantics id="S6.SS2.p1.8.m8.1a"><mi id="S6.SS2.p1.8.m8.1.1" xref="S6.SS2.p1.8.m8.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.8.m8.1b"><ci id="S6.SS2.p1.8.m8.1.1.cmml" xref="S6.SS2.p1.8.m8.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.8.m8.1c">K</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.8.m8.1d">italic_K</annotation></semantics></math> components. We numerically calculate the functions <math alttext="F" class="ltx_Math" display="inline" id="S6.SS2.p1.9.m9.1"><semantics id="S6.SS2.p1.9.m9.1a"><mi id="S6.SS2.p1.9.m9.1.1" xref="S6.SS2.p1.9.m9.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.9.m9.1b"><ci id="S6.SS2.p1.9.m9.1.1.cmml" xref="S6.SS2.p1.9.m9.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.9.m9.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.9.m9.1d">italic_F</annotation></semantics></math>, <math alttext="G" class="ltx_Math" display="inline" id="S6.SS2.p1.10.m10.1"><semantics id="S6.SS2.p1.10.m10.1a"><mi id="S6.SS2.p1.10.m10.1.1" xref="S6.SS2.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.10.m10.1b"><ci id="S6.SS2.p1.10.m10.1.1.cmml" xref="S6.SS2.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.10.m10.1d">italic_G</annotation></semantics></math>, and <math alttext="Q" class="ltx_Math" display="inline" id="S6.SS2.p1.11.m11.1"><semantics id="S6.SS2.p1.11.m11.1a"><mi id="S6.SS2.p1.11.m11.1.1" xref="S6.SS2.p1.11.m11.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.11.m11.1b"><ci id="S6.SS2.p1.11.m11.1.1.cmml" xref="S6.SS2.p1.11.m11.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.11.m11.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.11.m11.1d">italic_Q</annotation></semantics></math>, and check for the sufficient and necessary conditions of our theorems, to find equilibria and their stability in this setting. We set the precision of each component to be the same, and then vary this precision parameter to generate bifurcation graphs and study how these equilibria evolve for <math alttext="K=2" class="ltx_Math" display="inline" id="S6.SS2.p1.12.m12.1"><semantics id="S6.SS2.p1.12.m12.1a"><mrow id="S6.SS2.p1.12.m12.1.1" xref="S6.SS2.p1.12.m12.1.1.cmml"><mi id="S6.SS2.p1.12.m12.1.1.2" xref="S6.SS2.p1.12.m12.1.1.2.cmml">K</mi><mo id="S6.SS2.p1.12.m12.1.1.1" xref="S6.SS2.p1.12.m12.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.12.m12.1.1.3" xref="S6.SS2.p1.12.m12.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.12.m12.1b"><apply id="S6.SS2.p1.12.m12.1.1.cmml" xref="S6.SS2.p1.12.m12.1.1"><eq id="S6.SS2.p1.12.m12.1.1.1.cmml" xref="S6.SS2.p1.12.m12.1.1.1"></eq><ci id="S6.SS2.p1.12.m12.1.1.2.cmml" xref="S6.SS2.p1.12.m12.1.1.2">𝐾</ci><cn id="S6.SS2.p1.12.m12.1.1.3.cmml" type="integer" xref="S6.SS2.p1.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.12.m12.1c">K=2</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.12.m12.1d">italic_K = 2</annotation></semantics></math> and <math alttext="3" class="ltx_Math" display="inline" id="S6.SS2.p1.13.m13.1"><semantics id="S6.SS2.p1.13.m13.1a"><mn id="S6.SS2.p1.13.m13.1.1" xref="S6.SS2.p1.13.m13.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.13.m13.1b"><cn id="S6.SS2.p1.13.m13.1.1.cmml" type="integer" xref="S6.SS2.p1.13.m13.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.13.m13.1c">3</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.13.m13.1d">3</annotation></semantics></math> (see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A5" title="Appendix E Omitted Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">E</span></a> for <math alttext="K=4" class="ltx_Math" display="inline" id="S6.SS2.p1.14.m14.1"><semantics id="S6.SS2.p1.14.m14.1a"><mrow id="S6.SS2.p1.14.m14.1.1" xref="S6.SS2.p1.14.m14.1.1.cmml"><mi id="S6.SS2.p1.14.m14.1.1.2" xref="S6.SS2.p1.14.m14.1.1.2.cmml">K</mi><mo id="S6.SS2.p1.14.m14.1.1.1" xref="S6.SS2.p1.14.m14.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.14.m14.1.1.3" xref="S6.SS2.p1.14.m14.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.14.m14.1b"><apply id="S6.SS2.p1.14.m14.1.1.cmml" xref="S6.SS2.p1.14.m14.1.1"><eq id="S6.SS2.p1.14.m14.1.1.1.cmml" xref="S6.SS2.p1.14.m14.1.1.1"></eq><ci id="S6.SS2.p1.14.m14.1.1.2.cmml" xref="S6.SS2.p1.14.m14.1.1.2">𝐾</ci><cn id="S6.SS2.p1.14.m14.1.1.3.cmml" type="integer" xref="S6.SS2.p1.14.m14.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.14.m14.1c">K=4</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.14.m14.1d">italic_K = 4</annotation></semantics></math>).</p> </div> <figure class="ltx_figure" id="S6.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top" id="S6.F4.3" style="width:195.1pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="519" id="S6.F4.1.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F4.3.3.2.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S6.F4.3.2.1" style="font-size:90%;">Equilibria in OA and DG when signals are noisy versions of a skewed Normal, across different values of the skewness parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S6.F4.3.2.1.m1.1"><semantics id="S6.F4.3.2.1.m1.1b"><mi id="S6.F4.3.2.1.m1.1.1" xref="S6.F4.3.2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S6.F4.3.2.1.m1.1c"><ci id="S6.F4.3.2.1.m1.1.1.cmml" xref="S6.F4.3.2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.3.2.1.m1.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S6.F4.3.2.1.m1.1e">italic_α</annotation></semantics></math>. </span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top" id="S6.F4.14" style="width:195.1pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="499" id="S6.F4.4.g1" src="x4.png" width="747"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F4.14.11.5.1" style="font-size:90%;">Figure 4</span>: </span><math alttext="F" class="ltx_Math" display="inline" id="S6.F4.10.6.m1.1"><semantics id="S6.F4.10.6.m1.1b"><mi id="S6.F4.10.6.m1.1.1" mathsize="90%" xref="S6.F4.10.6.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.F4.10.6.m1.1c"><ci id="S6.F4.10.6.m1.1.1.cmml" xref="S6.F4.10.6.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.10.6.m1.1d">F</annotation><annotation encoding="application/x-llamapun" id="S6.F4.10.6.m1.1e">italic_F</annotation></semantics></math><span class="ltx_text" id="S6.F4.14.10.4" style="font-size:90%;">, <math alttext="G" class="ltx_Math" display="inline" id="S6.F4.11.7.1.m1.1"><semantics id="S6.F4.11.7.1.m1.1b"><mi id="S6.F4.11.7.1.m1.1.1" xref="S6.F4.11.7.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.F4.11.7.1.m1.1c"><ci id="S6.F4.11.7.1.m1.1.1.cmml" xref="S6.F4.11.7.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.11.7.1.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="S6.F4.11.7.1.m1.1e">italic_G</annotation></semantics></math>, and <math alttext="Q" class="ltx_Math" display="inline" id="S6.F4.12.8.2.m2.1"><semantics id="S6.F4.12.8.2.m2.1b"><mi id="S6.F4.12.8.2.m2.1.1" xref="S6.F4.12.8.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S6.F4.12.8.2.m2.1c"><ci id="S6.F4.12.8.2.m2.1.1.cmml" xref="S6.F4.12.8.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.12.8.2.m2.1d">Q</annotation><annotation encoding="application/x-llamapun" id="S6.F4.12.8.2.m2.1e">italic_Q</annotation></semantics></math> over signals <math alttext="x" class="ltx_Math" display="inline" id="S6.F4.13.9.3.m3.1"><semantics id="S6.F4.13.9.3.m3.1b"><mi id="S6.F4.13.9.3.m3.1.1" xref="S6.F4.13.9.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S6.F4.13.9.3.m3.1c"><ci id="S6.F4.13.9.3.m3.1.1.cmml" xref="S6.F4.13.9.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.13.9.3.m3.1d">x</annotation><annotation encoding="application/x-llamapun" id="S6.F4.13.9.3.m3.1e">italic_x</annotation></semantics></math> in the two-component Gaussian mixture setting with means <math alttext="-2,2" class="ltx_Math" display="inline" id="S6.F4.14.10.4.m4.2"><semantics id="S6.F4.14.10.4.m4.2b"><mrow id="S6.F4.14.10.4.m4.2.2.1" xref="S6.F4.14.10.4.m4.2.2.2.cmml"><mrow id="S6.F4.14.10.4.m4.2.2.1.1" xref="S6.F4.14.10.4.m4.2.2.1.1.cmml"><mo id="S6.F4.14.10.4.m4.2.2.1.1b" xref="S6.F4.14.10.4.m4.2.2.1.1.cmml">−</mo><mn id="S6.F4.14.10.4.m4.2.2.1.1.2" xref="S6.F4.14.10.4.m4.2.2.1.1.2.cmml">2</mn></mrow><mo id="S6.F4.14.10.4.m4.2.2.1.2" xref="S6.F4.14.10.4.m4.2.2.2.cmml">,</mo><mn id="S6.F4.14.10.4.m4.1.1" xref="S6.F4.14.10.4.m4.1.1.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F4.14.10.4.m4.2c"><list id="S6.F4.14.10.4.m4.2.2.2.cmml" xref="S6.F4.14.10.4.m4.2.2.1"><apply id="S6.F4.14.10.4.m4.2.2.1.1.cmml" xref="S6.F4.14.10.4.m4.2.2.1.1"><minus id="S6.F4.14.10.4.m4.2.2.1.1.1.cmml" xref="S6.F4.14.10.4.m4.2.2.1.1"></minus><cn id="S6.F4.14.10.4.m4.2.2.1.1.2.cmml" type="integer" xref="S6.F4.14.10.4.m4.2.2.1.1.2">2</cn></apply><cn id="S6.F4.14.10.4.m4.1.1.cmml" type="integer" xref="S6.F4.14.10.4.m4.1.1">2</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S6.F4.14.10.4.m4.2d">-2,2</annotation><annotation encoding="application/x-llamapun" id="S6.F4.14.10.4.m4.2e">- 2 , 2</annotation></semantics></math> and precision 2. </span></figcaption> </figure> </div> </div> </figure> <figure class="ltx_figure" id="S6.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S6.F5.1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S6.F5.1.g1" src="x5.png" width="830"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S6.F5.2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S6.F5.2.g1" src="x6.png" width="830"/> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S6.F5.3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S6.F5.3.g1" src="x7.png" width="830"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S6.F5.4"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="S6.F5.4.g1" src="x8.png" width="830"/> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F5.10.3.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S6.F5.8.2" style="font-size:90%;">Bifurcation diagrams for two- and three-component Gaussian mixtures. (Top) Two-component case: OA (left) exhibits three equilibria at high precision, while DG (right) maintains a single stable equilibrium at <math alttext="\tau=0" class="ltx_Math" display="inline" id="S6.F5.7.1.m1.1"><semantics id="S6.F5.7.1.m1.1b"><mrow id="S6.F5.7.1.m1.1.1" xref="S6.F5.7.1.m1.1.1.cmml"><mi id="S6.F5.7.1.m1.1.1.2" xref="S6.F5.7.1.m1.1.1.2.cmml">τ</mi><mo id="S6.F5.7.1.m1.1.1.1" xref="S6.F5.7.1.m1.1.1.1.cmml">=</mo><mn id="S6.F5.7.1.m1.1.1.3" xref="S6.F5.7.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F5.7.1.m1.1c"><apply id="S6.F5.7.1.m1.1.1.cmml" xref="S6.F5.7.1.m1.1.1"><eq id="S6.F5.7.1.m1.1.1.1.cmml" xref="S6.F5.7.1.m1.1.1.1"></eq><ci id="S6.F5.7.1.m1.1.1.2.cmml" xref="S6.F5.7.1.m1.1.1.2">𝜏</ci><cn id="S6.F5.7.1.m1.1.1.3.cmml" type="integer" xref="S6.F5.7.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.7.1.m1.1d">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S6.F5.7.1.m1.1e">italic_τ = 0</annotation></semantics></math>. (Bottom) Three-component case: OA (left) has five equilibria, with two stable at <math alttext="\tau=\pm 1" class="ltx_Math" display="inline" id="S6.F5.8.2.m2.1"><semantics id="S6.F5.8.2.m2.1b"><mrow id="S6.F5.8.2.m2.1.1" xref="S6.F5.8.2.m2.1.1.cmml"><mi id="S6.F5.8.2.m2.1.1.2" xref="S6.F5.8.2.m2.1.1.2.cmml">τ</mi><mo id="S6.F5.8.2.m2.1.1.1" xref="S6.F5.8.2.m2.1.1.1.cmml">=</mo><mrow id="S6.F5.8.2.m2.1.1.3" xref="S6.F5.8.2.m2.1.1.3.cmml"><mo id="S6.F5.8.2.m2.1.1.3b" xref="S6.F5.8.2.m2.1.1.3.cmml">±</mo><mn id="S6.F5.8.2.m2.1.1.3.2" xref="S6.F5.8.2.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.F5.8.2.m2.1c"><apply id="S6.F5.8.2.m2.1.1.cmml" xref="S6.F5.8.2.m2.1.1"><eq id="S6.F5.8.2.m2.1.1.1.cmml" xref="S6.F5.8.2.m2.1.1.1"></eq><ci id="S6.F5.8.2.m2.1.1.2.cmml" xref="S6.F5.8.2.m2.1.1.2">𝜏</ci><apply id="S6.F5.8.2.m2.1.1.3.cmml" xref="S6.F5.8.2.m2.1.1.3"><csymbol cd="latexml" id="S6.F5.8.2.m2.1.1.3.1.cmml" xref="S6.F5.8.2.m2.1.1.3">plus-or-minus</csymbol><cn id="S6.F5.8.2.m2.1.1.3.2.cmml" type="integer" xref="S6.F5.8.2.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F5.8.2.m2.1d">\tau=\pm 1</annotation><annotation encoding="application/x-llamapun" id="S6.F5.8.2.m2.1e">italic_τ = ± 1</annotation></semantics></math>; DG (right) has three equilibria, with stability reversed from the two-component OA case.</span></figcaption> </figure> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.1">We first observe that based on our experiments, the number of equilibria increases by two each time we add a Gaussian component (Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6.F5" title="Figure 5 ‣ 6.2 Multimodality ‣ 6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">5</span></a>). But OA begins with three equilibria for large enough precision under the two-component model, while DG begins with one. Since a unique equilibrium in DG is stable, this is a testament to DG being robust to uninformative equilibria. However, for a high enough separation between the two modes, stable equilibria emerge (at <math alttext="\tau=0" class="ltx_Math" display="inline" id="S6.SS2.p2.1.m1.1"><semantics id="S6.SS2.p2.1.m1.1a"><mrow id="S6.SS2.p2.1.m1.1.1" xref="S6.SS2.p2.1.m1.1.1.cmml"><mi id="S6.SS2.p2.1.m1.1.1.2" xref="S6.SS2.p2.1.m1.1.1.2.cmml">τ</mi><mo id="S6.SS2.p2.1.m1.1.1.1" xref="S6.SS2.p2.1.m1.1.1.1.cmml">=</mo><mn id="S6.SS2.p2.1.m1.1.1.3" xref="S6.SS2.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.1.m1.1b"><apply id="S6.SS2.p2.1.m1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1"><eq id="S6.SS2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1.1"></eq><ci id="S6.SS2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.p2.1.m1.1.1.2">𝜏</ci><cn id="S6.SS2.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.1.m1.1d">italic_τ = 0</annotation></semantics></math> in the three component case) in OA, and the area of starting conditions that lead to this equilibrium increases. One can view this phenomenon as a reflection of the original binary signal model: with enough separation between the two modes of the richer signal space, we essentially return to a setting where Output Agreement is useful because the “signal” is effectively which mode has been chosen.</p> </div> <div class="ltx_para" id="S6.SS2.p3"> <p class="ltx_p" id="S6.SS2.p3.5">We can view equilibria in the snapshot of a specific precision value in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6.F4" title="Figure 4 ‣ 6.2 Multimodality ‣ 6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>. Here, the intersection of <math alttext="G" class="ltx_Math" display="inline" id="S6.SS2.p3.1.m1.1"><semantics id="S6.SS2.p3.1.m1.1a"><mi id="S6.SS2.p3.1.m1.1.1" xref="S6.SS2.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.1.m1.1b"><ci id="S6.SS2.p3.1.m1.1.1.cmml" xref="S6.SS2.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.1.m1.1d">italic_G</annotation></semantics></math> with 1/2 corresponds to equilibria in OA, with <math alttext="F" class="ltx_Math" display="inline" id="S6.SS2.p3.2.m2.1"><semantics id="S6.SS2.p3.2.m2.1a"><mi id="S6.SS2.p3.2.m2.1.1" xref="S6.SS2.p3.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.2.m2.1b"><ci id="S6.SS2.p3.2.m2.1.1.cmml" xref="S6.SS2.p3.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.2.m2.1d">italic_F</annotation></semantics></math> corresponds to equilibria in DG (and DMI), and with <math alttext="Q" class="ltx_Math" display="inline" id="S6.SS2.p3.3.m3.1"><semantics id="S6.SS2.p3.3.m3.1a"><mi id="S6.SS2.p3.3.m3.1.1" xref="S6.SS2.p3.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.3.m3.1b"><ci id="S6.SS2.p3.3.m3.1.1.cmml" xref="S6.SS2.p3.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.3.m3.1d">italic_Q</annotation></semantics></math> corresponds to equilibria in RBTS. The function <math alttext="Q" class="ltx_Math" display="inline" id="S6.SS2.p3.4.m4.1"><semantics id="S6.SS2.p3.4.m4.1a"><mi id="S6.SS2.p3.4.m4.1.1" xref="S6.SS2.p3.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.4.m4.1b"><ci id="S6.SS2.p3.4.m4.1.1.cmml" xref="S6.SS2.p3.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.4.m4.1d">italic_Q</annotation></semantics></math> closely tracks <math alttext="G" class="ltx_Math" display="inline" id="S6.SS2.p3.5.m5.1"><semantics id="S6.SS2.p3.5.m5.1a"><mi id="S6.SS2.p3.5.m5.1.1" xref="S6.SS2.p3.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.5.m5.1b"><ci id="S6.SS2.p3.5.m5.1.1.cmml" xref="S6.SS2.p3.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.5.m5.1d">italic_G</annotation></semantics></math> for high enough precision values. We therefore find that the same stability and number of equilibria occur in RBTS as in DG. However, we note that bifurcation (the increase in equilibria) occurs at lower precision values for RBTS (see § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#A5" title="Appendix E Omitted Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">E</span></a> for visualization). From a design perspective, a lower bifurcation point in RBTS is nice because there is a wider set of signal distributions allowing for some choice over stable thresholds. Meanwhile, our theoretical results show that equilibria restricted to the threshold space in DMI correspond to those in DG, so that we would expect the trends we observe over precision to transfer directly from DG to DMI.</p> </div> <div class="ltx_para" id="S6.SS2.p4"> <p class="ltx_p" id="S6.SS2.p4.1">In general, then, our simulations show that multiple threshold equilibria occur across mechanisms with numbers increasing as modes separate, but similar patterns of stability emerge. Moreover, it is even more clear how different behavior emerges under a richer signal model. One could imagine two different real-valued distributions, the first unimodal as in a peer grading setting and the second bimodal as in a data labeling setting, with the same collapsed binary signal model. While previous results on the collapsed space concludes the same behavior would emerge in both settings, our results show that the number and stability of equilibria will be significantly different.</p> </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>Discussion</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">We study behavior in several peer prediction mechanisms when signals are real-valued and reports are binary. While all these mechanisms have various truthfulness guarantees in the binary signal setting, in our model we find that the notion of truthfulness breaks. Specifically, when agents play according to thresholds which partition the signal space into “high” or “low” regions, under dynamics we find equilibria rarely stay at the initial threshold a mechanism designer sets. Instead, for several mechanisms (Dasgupta–Ghosh, Determinant Mutual Information, and the Robust Bayesian Truth Serum) agents will often stabilize at a different threshold according to the specific payment scheme. Moreover, in Output Agreement we find that uninformative equilibria are often stable, so that in many information structure settings agents will <em class="ltx_emph ltx_font_italic" id="S7.p1.1.1">always</em> shift toward an uninformative consensus. We show how such equilibria play out in a natural, noisy Gaussian signal setting and extend our analysis to experiments with multimodal and skewed distributions.</p> </div> <div class="ltx_para" id="S7.p2"> <p class="ltx_p" id="S7.p2.1">Our results imply that the standard modeling assumptions in peer prediction literature miss important nuances, and these holes can propagate to real changes in how we expect agents to behave in practice. In particular, models of peer prediction have often taken as given an arbitrary joint distribution <math alttext="P" class="ltx_Math" display="inline" id="S7.p2.1.m1.1"><semantics id="S7.p2.1.m1.1a"><mi id="S7.p2.1.m1.1.1" xref="S7.p2.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S7.p2.1.m1.1b"><ci id="S7.p2.1.m1.1.1.cmml" xref="S7.p2.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S7.p2.1.m1.1d">italic_P</annotation></semantics></math> over pairs of signals drawn from a finite set. This suggests that we should be able to choose an arbitrary meaning for each signal and then peer prediction should “just work” for the resulting distribution. In contrast, our results show that peer prediction is substantially more inflexible in practice. Whenever the meaning of signals has endogeneity to it there will typically be only a small number of plausible equilibria in this richer space.</p> </div> <section class="ltx_paragraph" id="S7.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Adding flexibility.</h4> <div class="ltx_para" id="S7.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S7.SS0.SSS0.Px1.p1.3">We saw in the Gaussian case that a 50/50 report distribution was a natural outcome, which might be unsuitable for some applications. For example in peer grading or norm enforcement <cite class="ltx_cite ltx_citemacro_citep">[Alechina et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib2" title="">2017</a>]</cite>, one might want the final decision for most tasks to be “accept”. So what should a mechanism designer do? Since peer prediction naturally relies on multiple reports about each task, one option is to use a decision rule, which combines multiple reports on a task in a way that is aware of this limitation. In peer grading an assignment might be satisfactory if <span class="ltx_text ltx_font_italic" id="S7.SS0.SSS0.Px1.p1.3.1">any</span> peer grader reports <math alttext="H" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S7.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="S7.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S7.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S7.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S7.SS0.SSS0.Px1.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px1.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px1.p1.1.m1.1d">italic_H</annotation></semantics></math>. Doing so would allow more flexibility on the final decisions, e.g. increasing the fraction of satisfactory assignments, even under fixed semantics of <math alttext="H" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px1.p1.2.m2.1"><semantics id="S7.SS0.SSS0.Px1.p1.2.m2.1a"><mi id="S7.SS0.SSS0.Px1.p1.2.m2.1.1" xref="S7.SS0.SSS0.Px1.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px1.p1.2.m2.1b"><ci id="S7.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S7.SS0.SSS0.Px1.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px1.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px1.p1.2.m2.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="S7.SS0.SSS0.Px1.p1.3.m3.1a"><mi id="S7.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S7.SS0.SSS0.Px1.p1.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px1.p1.3.m3.1b"><ci id="S7.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S7.SS0.SSS0.Px1.p1.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px1.p1.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px1.p1.3.m3.1d">italic_L</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S7.SS0.SSS0.Px1.p2.1">An alternative is to seek to design new families of peer prediction mechanisms that allow control over where the equilibrium thresholds occur. The fact that our characterizations yielded different conditions among our four mechanisms suggests at least some scope for this, although the the fact that in the Gaussiance case these conditions coincided also points to potential limitations. Nevertheless, this seems an interesting direction for future work.</p> </div> </section> <section class="ltx_paragraph" id="S7.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Beyond binary reports.</h4> <div class="ltx_para" id="S7.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S7.SS0.SSS0.Px2.p1.4">Another direction for future work is to extend our characterizations of what we should expect from various peer prediction mechanisms beyond the binary case. For example, in peer grading it may make sense to have reports <math alttext="\{A,B,C,D,F\}" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px2.p1.1.m1.5"><semantics id="S7.SS0.SSS0.Px2.p1.1.m1.5a"><mrow id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml"><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.1" stretchy="false" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">{</mo><mi id="S7.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S7.SS0.SSS0.Px2.p1.1.m1.1.1.cmml">A</mi><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.2" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">,</mo><mi id="S7.SS0.SSS0.Px2.p1.1.m1.2.2" xref="S7.SS0.SSS0.Px2.p1.1.m1.2.2.cmml">B</mi><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.3" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">,</mo><mi id="S7.SS0.SSS0.Px2.p1.1.m1.3.3" xref="S7.SS0.SSS0.Px2.p1.1.m1.3.3.cmml">C</mi><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.4" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">,</mo><mi id="S7.SS0.SSS0.Px2.p1.1.m1.4.4" xref="S7.SS0.SSS0.Px2.p1.1.m1.4.4.cmml">D</mi><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.5" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">,</mo><mi id="S7.SS0.SSS0.Px2.p1.1.m1.5.5" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.5.cmml">F</mi><mo id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2.6" stretchy="false" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px2.p1.1.m1.5b"><set id="S7.SS0.SSS0.Px2.p1.1.m1.5.6.1.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.6.2"><ci id="S7.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.1.1">𝐴</ci><ci id="S7.SS0.SSS0.Px2.p1.1.m1.2.2.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.2.2">𝐵</ci><ci id="S7.SS0.SSS0.Px2.p1.1.m1.3.3.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.3.3">𝐶</ci><ci id="S7.SS0.SSS0.Px2.p1.1.m1.4.4.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.4.4">𝐷</ci><ci id="S7.SS0.SSS0.Px2.p1.1.m1.5.5.cmml" xref="S7.SS0.SSS0.Px2.p1.1.m1.5.5">𝐹</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px2.p1.1.m1.5c">\{A,B,C,D,F\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px2.p1.1.m1.5d">{ italic_A , italic_B , italic_C , italic_D , italic_F }</annotation></semantics></math>, while for peer review between five and ten signals is common when rating papers <cite class="ltx_cite ltx_citemacro_citep">[Srinivasan and Morgenstern, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib15" title="">2021</a>, Ugarov, <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib16" title="">2023</a>]</cite>. It would be natural to extend our model from strategies with a single threshold to multiple thresholds for such settings. While in principle our approach of analyzing best responses is applicable, deriving these becomes more complex, as does stating necessary and sufficient conditions that are easy to apply to natural classes of distributions. For other settings like labeling an image from <math alttext="\{" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px2.p1.2.m2.1"><semantics id="S7.SS0.SSS0.Px2.p1.2.m2.1a"><mo id="S7.SS0.SSS0.Px2.p1.2.m2.1.1" stretchy="false" xref="S7.SS0.SSS0.Px2.p1.2.m2.1.1.cmml">{</mo><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px2.p1.2.m2.1b"><ci id="S7.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S7.SS0.SSS0.Px2.p1.2.m2.1.1">{</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px2.p1.2.m2.1c">\{</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px2.p1.2.m2.1d">{</annotation></semantics></math>cat,dog,fish<math alttext="\}" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px2.p1.3.m3.1"><semantics id="S7.SS0.SSS0.Px2.p1.3.m3.1a"><mo id="S7.SS0.SSS0.Px2.p1.3.m3.1.1" stretchy="false" xref="S7.SS0.SSS0.Px2.p1.3.m3.1.1.cmml">}</mo><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px2.p1.3.m3.1b"><ci id="S7.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S7.SS0.SSS0.Px2.p1.3.m3.1.1">}</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px2.p1.3.m3.1c">\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px2.p1.3.m3.1d">}</annotation></semantics></math>, one would not expect a real-valued signal model to be appropriate, with perhaps <math alttext="\mathbb{R}^{3}" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px2.p1.4.m4.1"><semantics id="S7.SS0.SSS0.Px2.p1.4.m4.1a"><msup id="S7.SS0.SSS0.Px2.p1.4.m4.1.1" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1.cmml"><mi id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.2" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml">ℝ</mi><mn id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.3" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml">3</mn></msup><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px2.p1.4.m4.1b"><apply id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1">superscript</csymbol><ci id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1.2">ℝ</ci><cn id="S7.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml" type="integer" xref="S7.SS0.SSS0.Px2.p1.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px2.p1.4.m4.1c">\mathbb{R}^{3}</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px2.p1.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math> being a better choice.</p> </div> </section> <section class="ltx_paragraph" id="S7.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Effort.</h4> <div class="ltx_para" id="S7.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S7.SS0.SSS0.Px3.p1.2">Studying effort would be natural in our model. For example, one could model effort in the Gaussian model as giving the agent <math alttext="aZ+bZ_{i}" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px3.p1.1.m1.1"><semantics id="S7.SS0.SSS0.Px3.p1.1.m1.1a"><mrow id="S7.SS0.SSS0.Px3.p1.1.m1.1.1" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.cmml"><mrow id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml"><mi id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.2" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.2.cmml">a</mi><mo id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.1" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.3" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.3.cmml">Z</mi></mrow><mo id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.1" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.1.cmml">+</mo><mrow id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml"><mi id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.2" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.2.cmml">b</mi><mo id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.1" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.cmml"><mi id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.2" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.2.cmml">Z</mi><mi id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.3" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px3.p1.1.m1.1b"><apply id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1"><plus id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.1"></plus><apply id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2"><times id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.1.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.1"></times><ci id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.2.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.2">𝑎</ci><ci id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.3.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.2.3">𝑍</ci></apply><apply id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3"><times id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.1.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.1"></times><ci id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.2.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.2">𝑏</ci><apply id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.1.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.2.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.2">𝑍</ci><ci id="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.3.cmml" xref="S7.SS0.SSS0.Px3.p1.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px3.p1.1.m1.1c">aZ+bZ_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px3.p1.1.m1.1d">italic_a italic_Z + italic_b italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="b" class="ltx_Math" display="inline" id="S7.SS0.SSS0.Px3.p1.2.m2.1"><semantics id="S7.SS0.SSS0.Px3.p1.2.m2.1a"><mi id="S7.SS0.SSS0.Px3.p1.2.m2.1.1" xref="S7.SS0.SSS0.Px3.p1.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.Px3.p1.2.m2.1b"><ci id="S7.SS0.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S7.SS0.SSS0.Px3.p1.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.Px3.p1.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.Px3.p1.2.m2.1d">italic_b</annotation></semantics></math> is decreasing in their exerted effort; in the limit, they obtain the “ground truth” and need only contend with the noisy reports of others. One interesting interplay with our dynamics is that low-effort agents seem most likely to move the threshold, as high-effort agents will have more reason to respect the current one; e.g. when all agents are high-effort, they all coordinate perfectly on the correct report.</p> </div> </section> <section class="ltx_paragraph" id="S7.SS0.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Further broadening the model.</h4> <div class="ltx_para" id="S7.SS0.SSS0.Px4.p1"> <p class="ltx_p" id="S7.SS0.SSS0.Px4.p1.1">We have peeled back the peer prediction abstraction by one layer, from binary signals to real-valued ones. This already substantially changed the behavior of mechanisms. Yet even this is quite a stylized abstraction of the information available. Another missing piece is the fact that in many settings some deterministic <em class="ltx_emph ltx_font_italic" id="S7.SS0.SSS0.Px4.p1.1.1">context</em>, like the content of the essay being graded, is shared by all looking at the same task. From this “shared signal” comes the well-known problem of spurious correlations where in principle participants may choose strategies that depend on inessential details of the context, like the first word of the essay, to correlate their reports without correlating with the ground truth, all while getting optimal payoffs in multi-task mechansisms. While there is work seeking to deal with heterogeneity among agent behaviors <cite class="ltx_cite ltx_citemacro_citep">[Agarwal et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#bib.bib1" title="">2020</a>]</cite>, such behavior seems impossible to fully rule out. But even if such pathological equilibra seem unlikely in systems with many well-meaning participants, there is often still ample metadata about tasks such as categories or difficulties that naturally influence participants’ process of translating their actual experience into a “signal.” Designing mechanisms that are robust in the face of such richness is a further challenge.</p> </div> </section> <section class="ltx_subsection" id="S7.SSx1"> <h3 class="ltx_title ltx_title_subsection">Acknowledgments</h3> <div class="ltx_para" id="S7.SSx1.p1"> <p class="ltx_p" id="S7.SSx1.p1.1">This paper is based on initial results when the first two authors were working with the Oinc cryptocurrency team. We thank Tim Roughgarden, and members of Oinc and Kleros, for useful discussions.</p> </div> </section> </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">Agarwal et al. [2020]</span> <span class="ltx_bibblock"> A. Agarwal, D. Mandal, D. C. Parkes, and N. Shah. </span> <span class="ltx_bibblock">Peer prediction with heterogeneous users. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">ACM Transactions on Economics and Computation (TEAC)</em>, 8(1):1–34, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Alechina et al. [2017]</span> <span class="ltx_bibblock"> N. Alechina, J. Halpern, I. Kash, and B. 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Parkes. </span> <span class="ltx_bibblock">A robust Bayesian truth serum for small populations. </span> <span class="ltx_bibblock"><em class="ltx_emph ltx_font_italic" id="bib.bib19.1.1">Proceedings of the AAAI Conference on Artificial Intelligence</em>, 26(1):1492–1498, 2012. </span> </li> </ul> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Output Agreement</h2> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.1 </span>Equilibrium Characterization Generalization</h3> <div class="ltx_para" id="A1.SS1.p1"> <p class="ltx_p" id="A1.SS1.p1.1">In general, we can identify <em class="ltx_emph ltx_font_italic" id="A1.SS1.p1.1.1">where</em> finite equilibria exist in OA based on the extreme behavior of the function <math alttext="G" class="ltx_Math" display="inline" id="A1.SS1.p1.1.m1.1"><semantics id="A1.SS1.p1.1.m1.1a"><mi id="A1.SS1.p1.1.m1.1.1" xref="A1.SS1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.1.m1.1b"><ci id="A1.SS1.p1.1.m1.1.1.cmml" xref="A1.SS1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p1.1.m1.1d">italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_condition" id="Thmcondition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcondition1.1.1.1">Condition 1</span></span><span class="ltx_text ltx_font_bold" id="Thmcondition1.2.2">.</span> </h6> <div class="ltx_para" id="Thmcondition1.p1"> <p class="ltx_p" id="Thmcondition1.p1.7"><span class="ltx_text ltx_font_italic" id="Thmcondition1.p1.7.7">Let <math alttext="I=[a,b]\subset\mathbb{R}" class="ltx_Math" display="inline" id="Thmcondition1.p1.1.1.m1.2"><semantics id="Thmcondition1.p1.1.1.m1.2a"><mrow id="Thmcondition1.p1.1.1.m1.2.3" xref="Thmcondition1.p1.1.1.m1.2.3.cmml"><mi id="Thmcondition1.p1.1.1.m1.2.3.2" xref="Thmcondition1.p1.1.1.m1.2.3.2.cmml">I</mi><mo id="Thmcondition1.p1.1.1.m1.2.3.3" xref="Thmcondition1.p1.1.1.m1.2.3.3.cmml">=</mo><mrow id="Thmcondition1.p1.1.1.m1.2.3.4.2" xref="Thmcondition1.p1.1.1.m1.2.3.4.1.cmml"><mo id="Thmcondition1.p1.1.1.m1.2.3.4.2.1" stretchy="false" xref="Thmcondition1.p1.1.1.m1.2.3.4.1.cmml">[</mo><mi id="Thmcondition1.p1.1.1.m1.1.1" xref="Thmcondition1.p1.1.1.m1.1.1.cmml">a</mi><mo id="Thmcondition1.p1.1.1.m1.2.3.4.2.2" xref="Thmcondition1.p1.1.1.m1.2.3.4.1.cmml">,</mo><mi id="Thmcondition1.p1.1.1.m1.2.2" xref="Thmcondition1.p1.1.1.m1.2.2.cmml">b</mi><mo id="Thmcondition1.p1.1.1.m1.2.3.4.2.3" stretchy="false" xref="Thmcondition1.p1.1.1.m1.2.3.4.1.cmml">]</mo></mrow><mo id="Thmcondition1.p1.1.1.m1.2.3.5" xref="Thmcondition1.p1.1.1.m1.2.3.5.cmml">⊂</mo><mi id="Thmcondition1.p1.1.1.m1.2.3.6" xref="Thmcondition1.p1.1.1.m1.2.3.6.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.1.1.m1.2b"><apply id="Thmcondition1.p1.1.1.m1.2.3.cmml" xref="Thmcondition1.p1.1.1.m1.2.3"><and id="Thmcondition1.p1.1.1.m1.2.3a.cmml" xref="Thmcondition1.p1.1.1.m1.2.3"></and><apply id="Thmcondition1.p1.1.1.m1.2.3b.cmml" xref="Thmcondition1.p1.1.1.m1.2.3"><eq id="Thmcondition1.p1.1.1.m1.2.3.3.cmml" xref="Thmcondition1.p1.1.1.m1.2.3.3"></eq><ci id="Thmcondition1.p1.1.1.m1.2.3.2.cmml" xref="Thmcondition1.p1.1.1.m1.2.3.2">𝐼</ci><interval closure="closed" id="Thmcondition1.p1.1.1.m1.2.3.4.1.cmml" xref="Thmcondition1.p1.1.1.m1.2.3.4.2"><ci id="Thmcondition1.p1.1.1.m1.1.1.cmml" xref="Thmcondition1.p1.1.1.m1.1.1">𝑎</ci><ci id="Thmcondition1.p1.1.1.m1.2.2.cmml" xref="Thmcondition1.p1.1.1.m1.2.2">𝑏</ci></interval></apply><apply id="Thmcondition1.p1.1.1.m1.2.3c.cmml" xref="Thmcondition1.p1.1.1.m1.2.3"><subset id="Thmcondition1.p1.1.1.m1.2.3.5.cmml" xref="Thmcondition1.p1.1.1.m1.2.3.5"></subset><share href="https://arxiv.org/html/2503.16280v1#Thmcondition1.p1.1.1.m1.2.3.4.cmml" id="Thmcondition1.p1.1.1.m1.2.3d.cmml" xref="Thmcondition1.p1.1.1.m1.2.3"></share><ci id="Thmcondition1.p1.1.1.m1.2.3.6.cmml" xref="Thmcondition1.p1.1.1.m1.2.3.6">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.1.1.m1.2c">I=[a,b]\subset\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.1.1.m1.2d">italic_I = [ italic_a , italic_b ] ⊂ blackboard_R</annotation></semantics></math> be an interval. Upon seeing signal <math alttext="x\leq a" class="ltx_Math" display="inline" id="Thmcondition1.p1.2.2.m2.1"><semantics id="Thmcondition1.p1.2.2.m2.1a"><mrow id="Thmcondition1.p1.2.2.m2.1.1" xref="Thmcondition1.p1.2.2.m2.1.1.cmml"><mi id="Thmcondition1.p1.2.2.m2.1.1.2" xref="Thmcondition1.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmcondition1.p1.2.2.m2.1.1.1" xref="Thmcondition1.p1.2.2.m2.1.1.1.cmml">≤</mo><mi id="Thmcondition1.p1.2.2.m2.1.1.3" xref="Thmcondition1.p1.2.2.m2.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.2.2.m2.1b"><apply id="Thmcondition1.p1.2.2.m2.1.1.cmml" xref="Thmcondition1.p1.2.2.m2.1.1"><leq id="Thmcondition1.p1.2.2.m2.1.1.1.cmml" xref="Thmcondition1.p1.2.2.m2.1.1.1"></leq><ci id="Thmcondition1.p1.2.2.m2.1.1.2.cmml" xref="Thmcondition1.p1.2.2.m2.1.1.2">𝑥</ci><ci id="Thmcondition1.p1.2.2.m2.1.1.3.cmml" xref="Thmcondition1.p1.2.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.2.2.m2.1c">x\leq a</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.2.2.m2.1d">italic_x ≤ italic_a</annotation></semantics></math>, then an agent believes with probability less than 1/2 that another signal will be less than <math alttext="x" class="ltx_Math" display="inline" id="Thmcondition1.p1.3.3.m3.1"><semantics id="Thmcondition1.p1.3.3.m3.1a"><mi id="Thmcondition1.p1.3.3.m3.1.1" xref="Thmcondition1.p1.3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.3.3.m3.1b"><ci id="Thmcondition1.p1.3.3.m3.1.1.cmml" xref="Thmcondition1.p1.3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.3.3.m3.1d">italic_x</annotation></semantics></math>. Meanwhile, upon seeing signal <math alttext="x\geq b" class="ltx_Math" display="inline" id="Thmcondition1.p1.4.4.m4.1"><semantics id="Thmcondition1.p1.4.4.m4.1a"><mrow id="Thmcondition1.p1.4.4.m4.1.1" xref="Thmcondition1.p1.4.4.m4.1.1.cmml"><mi id="Thmcondition1.p1.4.4.m4.1.1.2" xref="Thmcondition1.p1.4.4.m4.1.1.2.cmml">x</mi><mo id="Thmcondition1.p1.4.4.m4.1.1.1" xref="Thmcondition1.p1.4.4.m4.1.1.1.cmml">≥</mo><mi id="Thmcondition1.p1.4.4.m4.1.1.3" xref="Thmcondition1.p1.4.4.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.4.4.m4.1b"><apply id="Thmcondition1.p1.4.4.m4.1.1.cmml" xref="Thmcondition1.p1.4.4.m4.1.1"><geq id="Thmcondition1.p1.4.4.m4.1.1.1.cmml" xref="Thmcondition1.p1.4.4.m4.1.1.1"></geq><ci id="Thmcondition1.p1.4.4.m4.1.1.2.cmml" xref="Thmcondition1.p1.4.4.m4.1.1.2">𝑥</ci><ci id="Thmcondition1.p1.4.4.m4.1.1.3.cmml" xref="Thmcondition1.p1.4.4.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.4.4.m4.1c">x\geq b</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.4.4.m4.1d">italic_x ≥ italic_b</annotation></semantics></math>, then an agent believes with probability greater than 1/2 that another signal will be less than <math alttext="x" class="ltx_Math" display="inline" id="Thmcondition1.p1.5.5.m5.1"><semantics id="Thmcondition1.p1.5.5.m5.1a"><mi id="Thmcondition1.p1.5.5.m5.1.1" xref="Thmcondition1.p1.5.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.5.5.m5.1b"><ci id="Thmcondition1.p1.5.5.m5.1.1.cmml" xref="Thmcondition1.p1.5.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.5.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.5.5.m5.1d">italic_x</annotation></semantics></math>. Formally, <math alttext="x\leq a\implies P(x^{\prime}\leq x\mid x)<1/2" class="ltx_Math" display="inline" id="Thmcondition1.p1.6.6.m6.1"><semantics id="Thmcondition1.p1.6.6.m6.1a"><mrow id="Thmcondition1.p1.6.6.m6.1.1" xref="Thmcondition1.p1.6.6.m6.1.1.cmml"><mi id="Thmcondition1.p1.6.6.m6.1.1.3" xref="Thmcondition1.p1.6.6.m6.1.1.3.cmml">x</mi><mo id="Thmcondition1.p1.6.6.m6.1.1.4" xref="Thmcondition1.p1.6.6.m6.1.1.4.cmml">≤</mo><mi id="Thmcondition1.p1.6.6.m6.1.1.5" xref="Thmcondition1.p1.6.6.m6.1.1.5.cmml">a</mi><mo id="Thmcondition1.p1.6.6.m6.1.1.6" stretchy="false" xref="Thmcondition1.p1.6.6.m6.1.1.6.cmml">⟹</mo><mrow id="Thmcondition1.p1.6.6.m6.1.1.1" xref="Thmcondition1.p1.6.6.m6.1.1.1.cmml"><mi id="Thmcondition1.p1.6.6.m6.1.1.1.3" xref="Thmcondition1.p1.6.6.m6.1.1.1.3.cmml">P</mi><mo id="Thmcondition1.p1.6.6.m6.1.1.1.2" xref="Thmcondition1.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmcondition1.p1.6.6.m6.1.1.1.1.1" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.cmml"><msup id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.cmml"><mi id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.2" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.2.cmml">x</mi><mo id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.3" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.1" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.1.cmml">≤</mo><mrow id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.cmml"><mi id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.2" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.2.cmml">x</mi><mo id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.1" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.3" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.3.cmml">x</mi></mrow></mrow><mo id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmcondition1.p1.6.6.m6.1.1.7" xref="Thmcondition1.p1.6.6.m6.1.1.7.cmml"><</mo><mrow id="Thmcondition1.p1.6.6.m6.1.1.8" xref="Thmcondition1.p1.6.6.m6.1.1.8.cmml"><mn id="Thmcondition1.p1.6.6.m6.1.1.8.2" xref="Thmcondition1.p1.6.6.m6.1.1.8.2.cmml">1</mn><mo id="Thmcondition1.p1.6.6.m6.1.1.8.1" xref="Thmcondition1.p1.6.6.m6.1.1.8.1.cmml">/</mo><mn id="Thmcondition1.p1.6.6.m6.1.1.8.3" xref="Thmcondition1.p1.6.6.m6.1.1.8.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition1.p1.6.6.m6.1b"><apply id="Thmcondition1.p1.6.6.m6.1.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1"><and id="Thmcondition1.p1.6.6.m6.1.1a.cmml" xref="Thmcondition1.p1.6.6.m6.1.1"></and><apply id="Thmcondition1.p1.6.6.m6.1.1b.cmml" xref="Thmcondition1.p1.6.6.m6.1.1"><leq id="Thmcondition1.p1.6.6.m6.1.1.4.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.4"></leq><ci id="Thmcondition1.p1.6.6.m6.1.1.3.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.3">𝑥</ci><ci id="Thmcondition1.p1.6.6.m6.1.1.5.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.5">𝑎</ci></apply><apply id="Thmcondition1.p1.6.6.m6.1.1c.cmml" xref="Thmcondition1.p1.6.6.m6.1.1"><implies id="Thmcondition1.p1.6.6.m6.1.1.6.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.6"></implies><share href="https://arxiv.org/html/2503.16280v1#Thmcondition1.p1.6.6.m6.1.1.5.cmml" id="Thmcondition1.p1.6.6.m6.1.1d.cmml" xref="Thmcondition1.p1.6.6.m6.1.1"></share><apply id="Thmcondition1.p1.6.6.m6.1.1.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1"><times id="Thmcondition1.p1.6.6.m6.1.1.1.2.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.2"></times><ci id="Thmcondition1.p1.6.6.m6.1.1.1.3.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.3">𝑃</ci><apply id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1"><leq id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.1"></leq><apply id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2">superscript</csymbol><ci id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.2.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.2">𝑥</ci><ci id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.3.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.2.3">′</ci></apply><apply id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3"><csymbol cd="latexml" id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.1.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.1">conditional</csymbol><ci id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.2.cmml" xref="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.2">𝑥</ci><ci id="Thmcondition1.p1.6.6.m6.1.1.1.1.1.1.3.3.cmml" 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xref="Thmcondition1.p1.7.7.m7.1.1.1.2"></times><ci id="Thmcondition1.p1.7.7.m7.1.1.1.3.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.3">𝑃</ci><apply id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1"><leq id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.1"></leq><apply id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.1.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2">superscript</csymbol><ci id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.2.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.2">𝑥</ci><ci id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.3.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.2.3">′</ci></apply><apply id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3"><csymbol cd="latexml" id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.1.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.1">conditional</csymbol><ci id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.2.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.2">𝑥</ci><ci id="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.3.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.1.1.1.1.3.3">𝑥</ci></apply></apply></apply></apply><apply id="Thmcondition1.p1.7.7.m7.1.1e.cmml" xref="Thmcondition1.p1.7.7.m7.1.1"><gt id="Thmcondition1.p1.7.7.m7.1.1.7.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.7"></gt><share href="https://arxiv.org/html/2503.16280v1#Thmcondition1.p1.7.7.m7.1.1.1.cmml" id="Thmcondition1.p1.7.7.m7.1.1f.cmml" xref="Thmcondition1.p1.7.7.m7.1.1"></share><apply id="Thmcondition1.p1.7.7.m7.1.1.8.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.8"><divide id="Thmcondition1.p1.7.7.m7.1.1.8.1.cmml" xref="Thmcondition1.p1.7.7.m7.1.1.8.1"></divide><cn id="Thmcondition1.p1.7.7.m7.1.1.8.2.cmml" type="integer" xref="Thmcondition1.p1.7.7.m7.1.1.8.2">1</cn><cn id="Thmcondition1.p1.7.7.m7.1.1.8.3.cmml" type="integer" xref="Thmcondition1.p1.7.7.m7.1.1.8.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition1.p1.7.7.m7.1c">x\geq b\implies P(x^{\prime}\leq x\mid x)>1/2</annotation><annotation encoding="application/x-llamapun" id="Thmcondition1.p1.7.7.m7.1d">italic_x ≥ italic_b ⟹ italic_P ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_x ∣ italic_x ) > 1 / 2</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem7.1.1.1">Theorem 7</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem7.p1"> <p class="ltx_p" id="Thmtheorem7.p1.4">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>, with <math alttext="\Pr[X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="Thmtheorem7.p1.1.m1.2"><semantics id="Thmtheorem7.p1.1.m1.2a"><mrow id="Thmtheorem7.p1.1.m1.2.2.1" xref="Thmtheorem7.p1.1.m1.2.2.2.cmml"><mi id="Thmtheorem7.p1.1.m1.1.1" xref="Thmtheorem7.p1.1.m1.1.1.cmml">Pr</mi><mo id="Thmtheorem7.p1.1.m1.2.2.1a" xref="Thmtheorem7.p1.1.m1.2.2.2.cmml">⁡</mo><mrow id="Thmtheorem7.p1.1.m1.2.2.1.1" xref="Thmtheorem7.p1.1.m1.2.2.2.cmml"><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.2" stretchy="false" xref="Thmtheorem7.p1.1.m1.2.2.2.cmml">[</mo><mrow id="Thmtheorem7.p1.1.m1.2.2.1.1.1" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.cmml"><msup id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.cmml"><mi id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.2" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.2.cmml">X</mi><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.3" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.1.3" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.3.cmml">≤</mo><mrow id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.cmml"><mi id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.2" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.2.cmml">τ</mi><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.1" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.1.cmml">∣</mo><mi id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.3" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.3.cmml">X</mi></mrow><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.1.5" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.5.cmml">=</mo><mi id="Thmtheorem7.p1.1.m1.2.2.1.1.1.6" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.6.cmml">x</mi></mrow><mo id="Thmtheorem7.p1.1.m1.2.2.1.1.3" stretchy="false" xref="Thmtheorem7.p1.1.m1.2.2.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem7.p1.1.m1.2b"><apply id="Thmtheorem7.p1.1.m1.2.2.2.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1"><ci id="Thmtheorem7.p1.1.m1.1.1.cmml" xref="Thmtheorem7.p1.1.m1.1.1">Pr</ci><apply id="Thmtheorem7.p1.1.m1.2.2.1.1.1.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1"><and id="Thmtheorem7.p1.1.m1.2.2.1.1.1a.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1"></and><apply id="Thmtheorem7.p1.1.m1.2.2.1.1.1b.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1"><leq id="Thmtheorem7.p1.1.m1.2.2.1.1.1.3.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.3"></leq><apply id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.1.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.2">𝑋</ci><ci id="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.3.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.2.3">′</ci></apply><apply id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4"><csymbol cd="latexml" id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.1.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.1">conditional</csymbol><ci id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.2.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.2">𝜏</ci><ci id="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.3.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="Thmtheorem7.p1.1.m1.2.2.1.1.1c.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1"><eq id="Thmtheorem7.p1.1.m1.2.2.1.1.1.5.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#Thmtheorem7.p1.1.m1.2.2.1.1.1.4.cmml" id="Thmtheorem7.p1.1.m1.2.2.1.1.1d.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1"></share><ci id="Thmtheorem7.p1.1.m1.2.2.1.1.1.6.cmml" xref="Thmtheorem7.p1.1.m1.2.2.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem7.p1.1.m1.2c">\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem7.p1.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math> monotone decreasing and continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem7.p1.2.m2.1"><semantics id="Thmtheorem7.p1.2.m2.1a"><mi id="Thmtheorem7.p1.2.m2.1.1" xref="Thmtheorem7.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem7.p1.2.m2.1b"><ci id="Thmtheorem7.p1.2.m2.1.1.cmml" xref="Thmtheorem7.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem7.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem7.p1.2.m2.1d">italic_x</annotation></semantics></math>, and <math alttext="G" class="ltx_Math" display="inline" id="Thmtheorem7.p1.3.m3.1"><semantics id="Thmtheorem7.p1.3.m3.1a"><mi id="Thmtheorem7.p1.3.m3.1.1" xref="Thmtheorem7.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem7.p1.3.m3.1b"><ci id="Thmtheorem7.p1.3.m3.1.1.cmml" xref="Thmtheorem7.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem7.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem7.p1.3.m3.1d">italic_G</annotation></semantics></math> continuous. Then there exists an equilibrium <math alttext="\tau^{*}\in I" class="ltx_Math" display="inline" id="Thmtheorem7.p1.4.m4.1"><semantics id="Thmtheorem7.p1.4.m4.1a"><mrow id="Thmtheorem7.p1.4.m4.1.1" xref="Thmtheorem7.p1.4.m4.1.1.cmml"><msup id="Thmtheorem7.p1.4.m4.1.1.2" xref="Thmtheorem7.p1.4.m4.1.1.2.cmml"><mi id="Thmtheorem7.p1.4.m4.1.1.2.2" xref="Thmtheorem7.p1.4.m4.1.1.2.2.cmml">τ</mi><mo id="Thmtheorem7.p1.4.m4.1.1.2.3" xref="Thmtheorem7.p1.4.m4.1.1.2.3.cmml">∗</mo></msup><mo id="Thmtheorem7.p1.4.m4.1.1.1" xref="Thmtheorem7.p1.4.m4.1.1.1.cmml">∈</mo><mi id="Thmtheorem7.p1.4.m4.1.1.3" xref="Thmtheorem7.p1.4.m4.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem7.p1.4.m4.1b"><apply id="Thmtheorem7.p1.4.m4.1.1.cmml" xref="Thmtheorem7.p1.4.m4.1.1"><in id="Thmtheorem7.p1.4.m4.1.1.1.cmml" xref="Thmtheorem7.p1.4.m4.1.1.1"></in><apply id="Thmtheorem7.p1.4.m4.1.1.2.cmml" xref="Thmtheorem7.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem7.p1.4.m4.1.1.2.1.cmml" xref="Thmtheorem7.p1.4.m4.1.1.2">superscript</csymbol><ci id="Thmtheorem7.p1.4.m4.1.1.2.2.cmml" xref="Thmtheorem7.p1.4.m4.1.1.2.2">𝜏</ci><times id="Thmtheorem7.p1.4.m4.1.1.2.3.cmml" xref="Thmtheorem7.p1.4.m4.1.1.2.3"></times></apply><ci id="Thmtheorem7.p1.4.m4.1.1.3.cmml" xref="Thmtheorem7.p1.4.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem7.p1.4.m4.1c">\tau^{*}\in I</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem7.p1.4.m4.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="A1.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.1.p1"> <p class="ltx_p" id="A1.SS1.1.p1.6">Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> implies <math alttext="G(a)<1/2" class="ltx_Math" display="inline" id="A1.SS1.1.p1.1.m1.1"><semantics id="A1.SS1.1.p1.1.m1.1a"><mrow id="A1.SS1.1.p1.1.m1.1.2" xref="A1.SS1.1.p1.1.m1.1.2.cmml"><mrow id="A1.SS1.1.p1.1.m1.1.2.2" xref="A1.SS1.1.p1.1.m1.1.2.2.cmml"><mi id="A1.SS1.1.p1.1.m1.1.2.2.2" xref="A1.SS1.1.p1.1.m1.1.2.2.2.cmml">G</mi><mo id="A1.SS1.1.p1.1.m1.1.2.2.1" xref="A1.SS1.1.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS1.1.p1.1.m1.1.2.2.3.2" xref="A1.SS1.1.p1.1.m1.1.2.2.cmml"><mo id="A1.SS1.1.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="A1.SS1.1.p1.1.m1.1.2.2.cmml">(</mo><mi id="A1.SS1.1.p1.1.m1.1.1" xref="A1.SS1.1.p1.1.m1.1.1.cmml">a</mi><mo id="A1.SS1.1.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="A1.SS1.1.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.1.p1.1.m1.1.2.1" xref="A1.SS1.1.p1.1.m1.1.2.1.cmml"><</mo><mrow id="A1.SS1.1.p1.1.m1.1.2.3" xref="A1.SS1.1.p1.1.m1.1.2.3.cmml"><mn id="A1.SS1.1.p1.1.m1.1.2.3.2" xref="A1.SS1.1.p1.1.m1.1.2.3.2.cmml">1</mn><mo id="A1.SS1.1.p1.1.m1.1.2.3.1" xref="A1.SS1.1.p1.1.m1.1.2.3.1.cmml">/</mo><mn id="A1.SS1.1.p1.1.m1.1.2.3.3" xref="A1.SS1.1.p1.1.m1.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.1.m1.1b"><apply id="A1.SS1.1.p1.1.m1.1.2.cmml" xref="A1.SS1.1.p1.1.m1.1.2"><lt id="A1.SS1.1.p1.1.m1.1.2.1.cmml" xref="A1.SS1.1.p1.1.m1.1.2.1"></lt><apply id="A1.SS1.1.p1.1.m1.1.2.2.cmml" xref="A1.SS1.1.p1.1.m1.1.2.2"><times id="A1.SS1.1.p1.1.m1.1.2.2.1.cmml" xref="A1.SS1.1.p1.1.m1.1.2.2.1"></times><ci id="A1.SS1.1.p1.1.m1.1.2.2.2.cmml" xref="A1.SS1.1.p1.1.m1.1.2.2.2">𝐺</ci><ci id="A1.SS1.1.p1.1.m1.1.1.cmml" xref="A1.SS1.1.p1.1.m1.1.1">𝑎</ci></apply><apply id="A1.SS1.1.p1.1.m1.1.2.3.cmml" xref="A1.SS1.1.p1.1.m1.1.2.3"><divide id="A1.SS1.1.p1.1.m1.1.2.3.1.cmml" xref="A1.SS1.1.p1.1.m1.1.2.3.1"></divide><cn id="A1.SS1.1.p1.1.m1.1.2.3.2.cmml" type="integer" xref="A1.SS1.1.p1.1.m1.1.2.3.2">1</cn><cn id="A1.SS1.1.p1.1.m1.1.2.3.3.cmml" type="integer" xref="A1.SS1.1.p1.1.m1.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.1.m1.1c">G(a)<1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.1.m1.1d">italic_G ( italic_a ) < 1 / 2</annotation></semantics></math> and <math alttext="G(b)>1/2" class="ltx_Math" display="inline" id="A1.SS1.1.p1.2.m2.1"><semantics id="A1.SS1.1.p1.2.m2.1a"><mrow id="A1.SS1.1.p1.2.m2.1.2" xref="A1.SS1.1.p1.2.m2.1.2.cmml"><mrow id="A1.SS1.1.p1.2.m2.1.2.2" xref="A1.SS1.1.p1.2.m2.1.2.2.cmml"><mi id="A1.SS1.1.p1.2.m2.1.2.2.2" xref="A1.SS1.1.p1.2.m2.1.2.2.2.cmml">G</mi><mo id="A1.SS1.1.p1.2.m2.1.2.2.1" xref="A1.SS1.1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS1.1.p1.2.m2.1.2.2.3.2" xref="A1.SS1.1.p1.2.m2.1.2.2.cmml"><mo id="A1.SS1.1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="A1.SS1.1.p1.2.m2.1.2.2.cmml">(</mo><mi id="A1.SS1.1.p1.2.m2.1.1" xref="A1.SS1.1.p1.2.m2.1.1.cmml">b</mi><mo id="A1.SS1.1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="A1.SS1.1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.1.p1.2.m2.1.2.1" xref="A1.SS1.1.p1.2.m2.1.2.1.cmml">></mo><mrow id="A1.SS1.1.p1.2.m2.1.2.3" xref="A1.SS1.1.p1.2.m2.1.2.3.cmml"><mn id="A1.SS1.1.p1.2.m2.1.2.3.2" xref="A1.SS1.1.p1.2.m2.1.2.3.2.cmml">1</mn><mo id="A1.SS1.1.p1.2.m2.1.2.3.1" xref="A1.SS1.1.p1.2.m2.1.2.3.1.cmml">/</mo><mn id="A1.SS1.1.p1.2.m2.1.2.3.3" xref="A1.SS1.1.p1.2.m2.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.2.m2.1b"><apply id="A1.SS1.1.p1.2.m2.1.2.cmml" xref="A1.SS1.1.p1.2.m2.1.2"><gt id="A1.SS1.1.p1.2.m2.1.2.1.cmml" xref="A1.SS1.1.p1.2.m2.1.2.1"></gt><apply id="A1.SS1.1.p1.2.m2.1.2.2.cmml" xref="A1.SS1.1.p1.2.m2.1.2.2"><times id="A1.SS1.1.p1.2.m2.1.2.2.1.cmml" xref="A1.SS1.1.p1.2.m2.1.2.2.1"></times><ci id="A1.SS1.1.p1.2.m2.1.2.2.2.cmml" xref="A1.SS1.1.p1.2.m2.1.2.2.2">𝐺</ci><ci id="A1.SS1.1.p1.2.m2.1.1.cmml" xref="A1.SS1.1.p1.2.m2.1.1">𝑏</ci></apply><apply id="A1.SS1.1.p1.2.m2.1.2.3.cmml" xref="A1.SS1.1.p1.2.m2.1.2.3"><divide id="A1.SS1.1.p1.2.m2.1.2.3.1.cmml" xref="A1.SS1.1.p1.2.m2.1.2.3.1"></divide><cn id="A1.SS1.1.p1.2.m2.1.2.3.2.cmml" type="integer" xref="A1.SS1.1.p1.2.m2.1.2.3.2">1</cn><cn id="A1.SS1.1.p1.2.m2.1.2.3.3.cmml" type="integer" xref="A1.SS1.1.p1.2.m2.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.2.m2.1c">G(b)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.2.m2.1d">italic_G ( italic_b ) > 1 / 2</annotation></semantics></math>. Since <math alttext="G" class="ltx_Math" display="inline" id="A1.SS1.1.p1.3.m3.1"><semantics id="A1.SS1.1.p1.3.m3.1a"><mi id="A1.SS1.1.p1.3.m3.1.1" xref="A1.SS1.1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.3.m3.1b"><ci id="A1.SS1.1.p1.3.m3.1.1.cmml" xref="A1.SS1.1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.3.m3.1d">italic_G</annotation></semantics></math> is continuous, we can apply the Intermediate Value Theorem to conclude there exists a point <math alttext="\tau^{*}\in[a,b]" class="ltx_Math" display="inline" id="A1.SS1.1.p1.4.m4.2"><semantics id="A1.SS1.1.p1.4.m4.2a"><mrow id="A1.SS1.1.p1.4.m4.2.3" xref="A1.SS1.1.p1.4.m4.2.3.cmml"><msup id="A1.SS1.1.p1.4.m4.2.3.2" xref="A1.SS1.1.p1.4.m4.2.3.2.cmml"><mi id="A1.SS1.1.p1.4.m4.2.3.2.2" xref="A1.SS1.1.p1.4.m4.2.3.2.2.cmml">τ</mi><mo id="A1.SS1.1.p1.4.m4.2.3.2.3" xref="A1.SS1.1.p1.4.m4.2.3.2.3.cmml">∗</mo></msup><mo id="A1.SS1.1.p1.4.m4.2.3.1" xref="A1.SS1.1.p1.4.m4.2.3.1.cmml">∈</mo><mrow id="A1.SS1.1.p1.4.m4.2.3.3.2" xref="A1.SS1.1.p1.4.m4.2.3.3.1.cmml"><mo id="A1.SS1.1.p1.4.m4.2.3.3.2.1" stretchy="false" xref="A1.SS1.1.p1.4.m4.2.3.3.1.cmml">[</mo><mi id="A1.SS1.1.p1.4.m4.1.1" xref="A1.SS1.1.p1.4.m4.1.1.cmml">a</mi><mo id="A1.SS1.1.p1.4.m4.2.3.3.2.2" xref="A1.SS1.1.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="A1.SS1.1.p1.4.m4.2.2" xref="A1.SS1.1.p1.4.m4.2.2.cmml">b</mi><mo id="A1.SS1.1.p1.4.m4.2.3.3.2.3" stretchy="false" xref="A1.SS1.1.p1.4.m4.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.4.m4.2b"><apply id="A1.SS1.1.p1.4.m4.2.3.cmml" xref="A1.SS1.1.p1.4.m4.2.3"><in id="A1.SS1.1.p1.4.m4.2.3.1.cmml" xref="A1.SS1.1.p1.4.m4.2.3.1"></in><apply id="A1.SS1.1.p1.4.m4.2.3.2.cmml" xref="A1.SS1.1.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.1.p1.4.m4.2.3.2.1.cmml" xref="A1.SS1.1.p1.4.m4.2.3.2">superscript</csymbol><ci id="A1.SS1.1.p1.4.m4.2.3.2.2.cmml" xref="A1.SS1.1.p1.4.m4.2.3.2.2">𝜏</ci><times id="A1.SS1.1.p1.4.m4.2.3.2.3.cmml" xref="A1.SS1.1.p1.4.m4.2.3.2.3"></times></apply><interval closure="closed" id="A1.SS1.1.p1.4.m4.2.3.3.1.cmml" xref="A1.SS1.1.p1.4.m4.2.3.3.2"><ci id="A1.SS1.1.p1.4.m4.1.1.cmml" xref="A1.SS1.1.p1.4.m4.1.1">𝑎</ci><ci id="A1.SS1.1.p1.4.m4.2.2.cmml" xref="A1.SS1.1.p1.4.m4.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.4.m4.2c">\tau^{*}\in[a,b]</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.4.m4.2d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ [ italic_a , italic_b ]</annotation></semantics></math> such that <math alttext="G(\tau^{*})=1/2" class="ltx_Math" display="inline" id="A1.SS1.1.p1.5.m5.1"><semantics id="A1.SS1.1.p1.5.m5.1a"><mrow id="A1.SS1.1.p1.5.m5.1.1" xref="A1.SS1.1.p1.5.m5.1.1.cmml"><mrow id="A1.SS1.1.p1.5.m5.1.1.1" xref="A1.SS1.1.p1.5.m5.1.1.1.cmml"><mi id="A1.SS1.1.p1.5.m5.1.1.1.3" xref="A1.SS1.1.p1.5.m5.1.1.1.3.cmml">G</mi><mo id="A1.SS1.1.p1.5.m5.1.1.1.2" xref="A1.SS1.1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.1.p1.5.m5.1.1.1.1.1" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="A1.SS1.1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.2" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.2.cmml">τ</mi><mo id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.3" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="A1.SS1.1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS1.1.p1.5.m5.1.1.2" xref="A1.SS1.1.p1.5.m5.1.1.2.cmml">=</mo><mrow id="A1.SS1.1.p1.5.m5.1.1.3" xref="A1.SS1.1.p1.5.m5.1.1.3.cmml"><mn id="A1.SS1.1.p1.5.m5.1.1.3.2" xref="A1.SS1.1.p1.5.m5.1.1.3.2.cmml">1</mn><mo id="A1.SS1.1.p1.5.m5.1.1.3.1" xref="A1.SS1.1.p1.5.m5.1.1.3.1.cmml">/</mo><mn id="A1.SS1.1.p1.5.m5.1.1.3.3" xref="A1.SS1.1.p1.5.m5.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.5.m5.1b"><apply id="A1.SS1.1.p1.5.m5.1.1.cmml" xref="A1.SS1.1.p1.5.m5.1.1"><eq id="A1.SS1.1.p1.5.m5.1.1.2.cmml" xref="A1.SS1.1.p1.5.m5.1.1.2"></eq><apply id="A1.SS1.1.p1.5.m5.1.1.1.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1"><times id="A1.SS1.1.p1.5.m5.1.1.1.2.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.2"></times><ci id="A1.SS1.1.p1.5.m5.1.1.1.3.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.3">𝐺</ci><apply id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.2">𝜏</ci><times id="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="A1.SS1.1.p1.5.m5.1.1.1.1.1.1.3"></times></apply></apply><apply id="A1.SS1.1.p1.5.m5.1.1.3.cmml" xref="A1.SS1.1.p1.5.m5.1.1.3"><divide id="A1.SS1.1.p1.5.m5.1.1.3.1.cmml" xref="A1.SS1.1.p1.5.m5.1.1.3.1"></divide><cn id="A1.SS1.1.p1.5.m5.1.1.3.2.cmml" type="integer" xref="A1.SS1.1.p1.5.m5.1.1.3.2">1</cn><cn id="A1.SS1.1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="A1.SS1.1.p1.5.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.5.m5.1c">G(\tau^{*})=1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.5.m5.1d">italic_G ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 1 / 2</annotation></semantics></math>. Then <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A1.SS1.1.p1.6.m6.1"><semantics id="A1.SS1.1.p1.6.m6.1a"><msup id="A1.SS1.1.p1.6.m6.1.1" xref="A1.SS1.1.p1.6.m6.1.1.cmml"><mi id="A1.SS1.1.p1.6.m6.1.1.2" xref="A1.SS1.1.p1.6.m6.1.1.2.cmml">τ</mi><mo id="A1.SS1.1.p1.6.m6.1.1.3" xref="A1.SS1.1.p1.6.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.1.p1.6.m6.1b"><apply id="A1.SS1.1.p1.6.m6.1.1.cmml" xref="A1.SS1.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS1.1.p1.6.m6.1.1.1.cmml" xref="A1.SS1.1.p1.6.m6.1.1">superscript</csymbol><ci id="A1.SS1.1.p1.6.m6.1.1.2.cmml" xref="A1.SS1.1.p1.6.m6.1.1.2">𝜏</ci><times id="A1.SS1.1.p1.6.m6.1.1.3.cmml" xref="A1.SS1.1.p1.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.1.p1.6.m6.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.1.p1.6.m6.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an equilibrium by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem1" title="Theorem 1. ‣ Results. ‣ 2.2 Equilibrium Characterization ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS1.p2"> <p class="ltx_p" id="A1.SS1.p2.3">Note in particular that if <math alttext="G(x)" class="ltx_Math" display="inline" id="A1.SS1.p2.1.m1.1"><semantics id="A1.SS1.p2.1.m1.1a"><mrow id="A1.SS1.p2.1.m1.1.2" xref="A1.SS1.p2.1.m1.1.2.cmml"><mi id="A1.SS1.p2.1.m1.1.2.2" xref="A1.SS1.p2.1.m1.1.2.2.cmml">G</mi><mo id="A1.SS1.p2.1.m1.1.2.1" xref="A1.SS1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="A1.SS1.p2.1.m1.1.2.3.2" xref="A1.SS1.p2.1.m1.1.2.cmml"><mo id="A1.SS1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="A1.SS1.p2.1.m1.1.2.cmml">(</mo><mi id="A1.SS1.p2.1.m1.1.1" xref="A1.SS1.p2.1.m1.1.1.cmml">x</mi><mo id="A1.SS1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="A1.SS1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p2.1.m1.1b"><apply id="A1.SS1.p2.1.m1.1.2.cmml" xref="A1.SS1.p2.1.m1.1.2"><times id="A1.SS1.p2.1.m1.1.2.1.cmml" xref="A1.SS1.p2.1.m1.1.2.1"></times><ci id="A1.SS1.p2.1.m1.1.2.2.cmml" xref="A1.SS1.p2.1.m1.1.2.2">𝐺</ci><ci id="A1.SS1.p2.1.m1.1.1.cmml" xref="A1.SS1.p2.1.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.1.m1.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.1.m1.1d">italic_G ( italic_x )</annotation></semantics></math> is monotone increasing and crosses 1/2, Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> is satisfied for the single point interval <math alttext="\tau" class="ltx_Math" display="inline" id="A1.SS1.p2.2.m2.1"><semantics id="A1.SS1.p2.2.m2.1a"><mi id="A1.SS1.p2.2.m2.1.1" xref="A1.SS1.p2.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.p2.2.m2.1b"><ci id="A1.SS1.p2.2.m2.1.1.cmml" xref="A1.SS1.p2.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.2.m2.1d">italic_τ</annotation></semantics></math> where <math alttext="G(\tau)=1/2" class="ltx_Math" display="inline" id="A1.SS1.p2.3.m3.1"><semantics id="A1.SS1.p2.3.m3.1a"><mrow id="A1.SS1.p2.3.m3.1.2" xref="A1.SS1.p2.3.m3.1.2.cmml"><mrow id="A1.SS1.p2.3.m3.1.2.2" xref="A1.SS1.p2.3.m3.1.2.2.cmml"><mi id="A1.SS1.p2.3.m3.1.2.2.2" xref="A1.SS1.p2.3.m3.1.2.2.2.cmml">G</mi><mo id="A1.SS1.p2.3.m3.1.2.2.1" xref="A1.SS1.p2.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS1.p2.3.m3.1.2.2.3.2" xref="A1.SS1.p2.3.m3.1.2.2.cmml"><mo id="A1.SS1.p2.3.m3.1.2.2.3.2.1" stretchy="false" xref="A1.SS1.p2.3.m3.1.2.2.cmml">(</mo><mi id="A1.SS1.p2.3.m3.1.1" xref="A1.SS1.p2.3.m3.1.1.cmml">τ</mi><mo id="A1.SS1.p2.3.m3.1.2.2.3.2.2" stretchy="false" xref="A1.SS1.p2.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.p2.3.m3.1.2.1" xref="A1.SS1.p2.3.m3.1.2.1.cmml">=</mo><mrow id="A1.SS1.p2.3.m3.1.2.3" xref="A1.SS1.p2.3.m3.1.2.3.cmml"><mn id="A1.SS1.p2.3.m3.1.2.3.2" xref="A1.SS1.p2.3.m3.1.2.3.2.cmml">1</mn><mo id="A1.SS1.p2.3.m3.1.2.3.1" xref="A1.SS1.p2.3.m3.1.2.3.1.cmml">/</mo><mn id="A1.SS1.p2.3.m3.1.2.3.3" xref="A1.SS1.p2.3.m3.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p2.3.m3.1b"><apply id="A1.SS1.p2.3.m3.1.2.cmml" xref="A1.SS1.p2.3.m3.1.2"><eq id="A1.SS1.p2.3.m3.1.2.1.cmml" xref="A1.SS1.p2.3.m3.1.2.1"></eq><apply id="A1.SS1.p2.3.m3.1.2.2.cmml" xref="A1.SS1.p2.3.m3.1.2.2"><times id="A1.SS1.p2.3.m3.1.2.2.1.cmml" xref="A1.SS1.p2.3.m3.1.2.2.1"></times><ci id="A1.SS1.p2.3.m3.1.2.2.2.cmml" xref="A1.SS1.p2.3.m3.1.2.2.2">𝐺</ci><ci id="A1.SS1.p2.3.m3.1.1.cmml" xref="A1.SS1.p2.3.m3.1.1">𝜏</ci></apply><apply id="A1.SS1.p2.3.m3.1.2.3.cmml" xref="A1.SS1.p2.3.m3.1.2.3"><divide id="A1.SS1.p2.3.m3.1.2.3.1.cmml" xref="A1.SS1.p2.3.m3.1.2.3.1"></divide><cn id="A1.SS1.p2.3.m3.1.2.3.2.cmml" type="integer" xref="A1.SS1.p2.3.m3.1.2.3.2">1</cn><cn id="A1.SS1.p2.3.m3.1.2.3.3.cmml" type="integer" xref="A1.SS1.p2.3.m3.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.3.m3.1c">G(\tau)=1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.3.m3.1d">italic_G ( italic_τ ) = 1 / 2</annotation></semantics></math> and we have a unique equilibrium.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition7.1.1.1">Proposition 7</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition7.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition7.p1"> <p class="ltx_p" id="Thmproposition7.p1.11">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>, with <math alttext="\Pr[X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="Thmproposition7.p1.1.m1.2"><semantics id="Thmproposition7.p1.1.m1.2a"><mrow id="Thmproposition7.p1.1.m1.2.2.1" xref="Thmproposition7.p1.1.m1.2.2.2.cmml"><mi id="Thmproposition7.p1.1.m1.1.1" xref="Thmproposition7.p1.1.m1.1.1.cmml">Pr</mi><mo id="Thmproposition7.p1.1.m1.2.2.1a" xref="Thmproposition7.p1.1.m1.2.2.2.cmml">⁡</mo><mrow id="Thmproposition7.p1.1.m1.2.2.1.1" xref="Thmproposition7.p1.1.m1.2.2.2.cmml"><mo id="Thmproposition7.p1.1.m1.2.2.1.1.2" stretchy="false" xref="Thmproposition7.p1.1.m1.2.2.2.cmml">[</mo><mrow id="Thmproposition7.p1.1.m1.2.2.1.1.1" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.cmml"><msup id="Thmproposition7.p1.1.m1.2.2.1.1.1.2" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2.cmml"><mi id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.2" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.3" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition7.p1.1.m1.2.2.1.1.1.3" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.3.cmml">≤</mo><mrow id="Thmproposition7.p1.1.m1.2.2.1.1.1.4" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.cmml"><mi id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.2" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.2.cmml">τ</mi><mo id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.1" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.1.cmml">∣</mo><mi id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.3" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.3.cmml">X</mi></mrow><mo id="Thmproposition7.p1.1.m1.2.2.1.1.1.5" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.5.cmml">=</mo><mi id="Thmproposition7.p1.1.m1.2.2.1.1.1.6" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.6.cmml">x</mi></mrow><mo id="Thmproposition7.p1.1.m1.2.2.1.1.3" stretchy="false" xref="Thmproposition7.p1.1.m1.2.2.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.1.m1.2b"><apply id="Thmproposition7.p1.1.m1.2.2.2.cmml" xref="Thmproposition7.p1.1.m1.2.2.1"><ci id="Thmproposition7.p1.1.m1.1.1.cmml" xref="Thmproposition7.p1.1.m1.1.1">Pr</ci><apply id="Thmproposition7.p1.1.m1.2.2.1.1.1.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1"><and id="Thmproposition7.p1.1.m1.2.2.1.1.1a.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1"></and><apply id="Thmproposition7.p1.1.m1.2.2.1.1.1b.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1"><leq id="Thmproposition7.p1.1.m1.2.2.1.1.1.3.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.3"></leq><apply id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.1.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2.2">𝑋</ci><ci id="Thmproposition7.p1.1.m1.2.2.1.1.1.2.3.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.2.3">′</ci></apply><apply id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4"><csymbol cd="latexml" id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.1.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.1">conditional</csymbol><ci id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.2.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.2">𝜏</ci><ci id="Thmproposition7.p1.1.m1.2.2.1.1.1.4.3.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="Thmproposition7.p1.1.m1.2.2.1.1.1c.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1"><eq id="Thmproposition7.p1.1.m1.2.2.1.1.1.5.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#Thmproposition7.p1.1.m1.2.2.1.1.1.4.cmml" id="Thmproposition7.p1.1.m1.2.2.1.1.1d.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1"></share><ci id="Thmproposition7.p1.1.m1.2.2.1.1.1.6.cmml" xref="Thmproposition7.p1.1.m1.2.2.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.1.m1.2c">\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math> monotone decreasing and continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmproposition7.p1.2.m2.1"><semantics id="Thmproposition7.p1.2.m2.1a"><mi id="Thmproposition7.p1.2.m2.1.1" xref="Thmproposition7.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.2.m2.1b"><ci id="Thmproposition7.p1.2.m2.1.1.cmml" xref="Thmproposition7.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.2.m2.1d">italic_x</annotation></semantics></math>, and <math alttext="G" class="ltx_Math" display="inline" id="Thmproposition7.p1.3.m3.1"><semantics id="Thmproposition7.p1.3.m3.1a"><mi id="Thmproposition7.p1.3.m3.1.1" xref="Thmproposition7.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.3.m3.1b"><ci id="Thmproposition7.p1.3.m3.1.1.cmml" xref="Thmproposition7.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.3.m3.1d">italic_G</annotation></semantics></math> continuous. Let <math alttext="m" class="ltx_Math" display="inline" id="Thmproposition7.p1.4.m4.1"><semantics id="Thmproposition7.p1.4.m4.1a"><mi id="Thmproposition7.p1.4.m4.1.1" xref="Thmproposition7.p1.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.4.m4.1b"><ci id="Thmproposition7.p1.4.m4.1.1.cmml" xref="Thmproposition7.p1.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.4.m4.1c">m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.4.m4.1d">italic_m</annotation></semantics></math> be the median of <math alttext="F" class="ltx_Math" display="inline" id="Thmproposition7.p1.5.m5.1"><semantics id="Thmproposition7.p1.5.m5.1a"><mi id="Thmproposition7.p1.5.m5.1.1" xref="Thmproposition7.p1.5.m5.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.5.m5.1b"><ci id="Thmproposition7.p1.5.m5.1.1.cmml" xref="Thmproposition7.p1.5.m5.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.5.m5.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.5.m5.1d">italic_F</annotation></semantics></math>, and consider running the OA mechanisim. Then if <math alttext="\Pr[X^{\prime}\leq m\mid m]>1/2" class="ltx_Math" display="inline" id="Thmproposition7.p1.6.m6.2"><semantics id="Thmproposition7.p1.6.m6.2a"><mrow id="Thmproposition7.p1.6.m6.2.2" xref="Thmproposition7.p1.6.m6.2.2.cmml"><mrow id="Thmproposition7.p1.6.m6.2.2.1.1" xref="Thmproposition7.p1.6.m6.2.2.1.2.cmml"><mi id="Thmproposition7.p1.6.m6.1.1" xref="Thmproposition7.p1.6.m6.1.1.cmml">Pr</mi><mo id="Thmproposition7.p1.6.m6.2.2.1.1a" xref="Thmproposition7.p1.6.m6.2.2.1.2.cmml">⁡</mo><mrow id="Thmproposition7.p1.6.m6.2.2.1.1.1" xref="Thmproposition7.p1.6.m6.2.2.1.2.cmml"><mo id="Thmproposition7.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="Thmproposition7.p1.6.m6.2.2.1.2.cmml">[</mo><mrow id="Thmproposition7.p1.6.m6.2.2.1.1.1.1" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.cmml"><msup id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.cmml"><mi id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.2" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.3" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.1" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.cmml"><mi id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.2" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.1" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.3" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="Thmproposition7.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="Thmproposition7.p1.6.m6.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="Thmproposition7.p1.6.m6.2.2.2" xref="Thmproposition7.p1.6.m6.2.2.2.cmml">></mo><mrow id="Thmproposition7.p1.6.m6.2.2.3" xref="Thmproposition7.p1.6.m6.2.2.3.cmml"><mn id="Thmproposition7.p1.6.m6.2.2.3.2" xref="Thmproposition7.p1.6.m6.2.2.3.2.cmml">1</mn><mo id="Thmproposition7.p1.6.m6.2.2.3.1" xref="Thmproposition7.p1.6.m6.2.2.3.1.cmml">/</mo><mn id="Thmproposition7.p1.6.m6.2.2.3.3" xref="Thmproposition7.p1.6.m6.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.6.m6.2b"><apply id="Thmproposition7.p1.6.m6.2.2.cmml" xref="Thmproposition7.p1.6.m6.2.2"><gt id="Thmproposition7.p1.6.m6.2.2.2.cmml" xref="Thmproposition7.p1.6.m6.2.2.2"></gt><apply id="Thmproposition7.p1.6.m6.2.2.1.2.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1"><ci id="Thmproposition7.p1.6.m6.1.1.cmml" xref="Thmproposition7.p1.6.m6.1.1">Pr</ci><apply id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1"><leq id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.1"></leq><apply id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.1.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2">superscript</csymbol><ci id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.2.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.2">𝑋</ci><ci id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.3.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.2.3">′</ci></apply><apply id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3"><csymbol cd="latexml" id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.1.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.2.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.3.cmml" xref="Thmproposition7.p1.6.m6.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="Thmproposition7.p1.6.m6.2.2.3.cmml" xref="Thmproposition7.p1.6.m6.2.2.3"><divide id="Thmproposition7.p1.6.m6.2.2.3.1.cmml" xref="Thmproposition7.p1.6.m6.2.2.3.1"></divide><cn id="Thmproposition7.p1.6.m6.2.2.3.2.cmml" type="integer" xref="Thmproposition7.p1.6.m6.2.2.3.2">1</cn><cn id="Thmproposition7.p1.6.m6.2.2.3.3.cmml" type="integer" xref="Thmproposition7.p1.6.m6.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.6.m6.2c">\Pr[X^{\prime}\leq m\mid m]>1/2</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.6.m6.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] > 1 / 2</annotation></semantics></math>, there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmproposition7.p1.7.m7.1"><semantics id="Thmproposition7.p1.7.m7.1a"><msup id="Thmproposition7.p1.7.m7.1.1" xref="Thmproposition7.p1.7.m7.1.1.cmml"><mi id="Thmproposition7.p1.7.m7.1.1.2" xref="Thmproposition7.p1.7.m7.1.1.2.cmml">τ</mi><mo id="Thmproposition7.p1.7.m7.1.1.3" xref="Thmproposition7.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.7.m7.1b"><apply id="Thmproposition7.p1.7.m7.1.1.cmml" xref="Thmproposition7.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmproposition7.p1.7.m7.1.1.1.cmml" xref="Thmproposition7.p1.7.m7.1.1">superscript</csymbol><ci id="Thmproposition7.p1.7.m7.1.1.2.cmml" xref="Thmproposition7.p1.7.m7.1.1.2">𝜏</ci><times id="Thmproposition7.p1.7.m7.1.1.3.cmml" xref="Thmproposition7.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.7.m7.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.7.m7.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\tau^{*}<m" class="ltx_Math" display="inline" id="Thmproposition7.p1.8.m8.1"><semantics id="Thmproposition7.p1.8.m8.1a"><mrow id="Thmproposition7.p1.8.m8.1.1" xref="Thmproposition7.p1.8.m8.1.1.cmml"><msup id="Thmproposition7.p1.8.m8.1.1.2" xref="Thmproposition7.p1.8.m8.1.1.2.cmml"><mi id="Thmproposition7.p1.8.m8.1.1.2.2" xref="Thmproposition7.p1.8.m8.1.1.2.2.cmml">τ</mi><mo id="Thmproposition7.p1.8.m8.1.1.2.3" xref="Thmproposition7.p1.8.m8.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition7.p1.8.m8.1.1.1" xref="Thmproposition7.p1.8.m8.1.1.1.cmml"><</mo><mi id="Thmproposition7.p1.8.m8.1.1.3" xref="Thmproposition7.p1.8.m8.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.8.m8.1b"><apply id="Thmproposition7.p1.8.m8.1.1.cmml" xref="Thmproposition7.p1.8.m8.1.1"><lt id="Thmproposition7.p1.8.m8.1.1.1.cmml" xref="Thmproposition7.p1.8.m8.1.1.1"></lt><apply id="Thmproposition7.p1.8.m8.1.1.2.cmml" xref="Thmproposition7.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="Thmproposition7.p1.8.m8.1.1.2.1.cmml" xref="Thmproposition7.p1.8.m8.1.1.2">superscript</csymbol><ci id="Thmproposition7.p1.8.m8.1.1.2.2.cmml" xref="Thmproposition7.p1.8.m8.1.1.2.2">𝜏</ci><times id="Thmproposition7.p1.8.m8.1.1.2.3.cmml" xref="Thmproposition7.p1.8.m8.1.1.2.3"></times></apply><ci id="Thmproposition7.p1.8.m8.1.1.3.cmml" xref="Thmproposition7.p1.8.m8.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.8.m8.1c">\tau^{*}<m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.8.m8.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT < italic_m</annotation></semantics></math>. Similarly, if <math alttext="\Pr[X^{\prime}\leq m\mid m]<1/2" class="ltx_Math" display="inline" id="Thmproposition7.p1.9.m9.2"><semantics id="Thmproposition7.p1.9.m9.2a"><mrow id="Thmproposition7.p1.9.m9.2.2" xref="Thmproposition7.p1.9.m9.2.2.cmml"><mrow id="Thmproposition7.p1.9.m9.2.2.1.1" xref="Thmproposition7.p1.9.m9.2.2.1.2.cmml"><mi id="Thmproposition7.p1.9.m9.1.1" xref="Thmproposition7.p1.9.m9.1.1.cmml">Pr</mi><mo id="Thmproposition7.p1.9.m9.2.2.1.1a" xref="Thmproposition7.p1.9.m9.2.2.1.2.cmml">⁡</mo><mrow id="Thmproposition7.p1.9.m9.2.2.1.1.1" xref="Thmproposition7.p1.9.m9.2.2.1.2.cmml"><mo id="Thmproposition7.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="Thmproposition7.p1.9.m9.2.2.1.2.cmml">[</mo><mrow id="Thmproposition7.p1.9.m9.2.2.1.1.1.1" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.cmml"><msup id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.cmml"><mi id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.2" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.3" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.1" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.cmml"><mi id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.2" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.1" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.3" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="Thmproposition7.p1.9.m9.2.2.1.1.1.3" stretchy="false" xref="Thmproposition7.p1.9.m9.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="Thmproposition7.p1.9.m9.2.2.2" xref="Thmproposition7.p1.9.m9.2.2.2.cmml"><</mo><mrow id="Thmproposition7.p1.9.m9.2.2.3" xref="Thmproposition7.p1.9.m9.2.2.3.cmml"><mn id="Thmproposition7.p1.9.m9.2.2.3.2" xref="Thmproposition7.p1.9.m9.2.2.3.2.cmml">1</mn><mo id="Thmproposition7.p1.9.m9.2.2.3.1" xref="Thmproposition7.p1.9.m9.2.2.3.1.cmml">/</mo><mn id="Thmproposition7.p1.9.m9.2.2.3.3" xref="Thmproposition7.p1.9.m9.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.9.m9.2b"><apply id="Thmproposition7.p1.9.m9.2.2.cmml" xref="Thmproposition7.p1.9.m9.2.2"><lt id="Thmproposition7.p1.9.m9.2.2.2.cmml" xref="Thmproposition7.p1.9.m9.2.2.2"></lt><apply id="Thmproposition7.p1.9.m9.2.2.1.2.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1"><ci id="Thmproposition7.p1.9.m9.1.1.cmml" xref="Thmproposition7.p1.9.m9.1.1">Pr</ci><apply id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1"><leq id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.1"></leq><apply id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.1.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2">superscript</csymbol><ci id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.2.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.2">𝑋</ci><ci id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.3.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.2.3">′</ci></apply><apply id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3"><csymbol cd="latexml" id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.1.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.2.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.3.cmml" xref="Thmproposition7.p1.9.m9.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="Thmproposition7.p1.9.m9.2.2.3.cmml" xref="Thmproposition7.p1.9.m9.2.2.3"><divide id="Thmproposition7.p1.9.m9.2.2.3.1.cmml" xref="Thmproposition7.p1.9.m9.2.2.3.1"></divide><cn id="Thmproposition7.p1.9.m9.2.2.3.2.cmml" type="integer" xref="Thmproposition7.p1.9.m9.2.2.3.2">1</cn><cn id="Thmproposition7.p1.9.m9.2.2.3.3.cmml" type="integer" xref="Thmproposition7.p1.9.m9.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.9.m9.2c">\Pr[X^{\prime}\leq m\mid m]<1/2</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.9.m9.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] < 1 / 2</annotation></semantics></math>, there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmproposition7.p1.10.m10.1"><semantics id="Thmproposition7.p1.10.m10.1a"><msup id="Thmproposition7.p1.10.m10.1.1" xref="Thmproposition7.p1.10.m10.1.1.cmml"><mi id="Thmproposition7.p1.10.m10.1.1.2" xref="Thmproposition7.p1.10.m10.1.1.2.cmml">τ</mi><mo id="Thmproposition7.p1.10.m10.1.1.3" xref="Thmproposition7.p1.10.m10.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.10.m10.1b"><apply id="Thmproposition7.p1.10.m10.1.1.cmml" xref="Thmproposition7.p1.10.m10.1.1"><csymbol cd="ambiguous" id="Thmproposition7.p1.10.m10.1.1.1.cmml" xref="Thmproposition7.p1.10.m10.1.1">superscript</csymbol><ci id="Thmproposition7.p1.10.m10.1.1.2.cmml" xref="Thmproposition7.p1.10.m10.1.1.2">𝜏</ci><times id="Thmproposition7.p1.10.m10.1.1.3.cmml" xref="Thmproposition7.p1.10.m10.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.10.m10.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.10.m10.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\tau^{*}>m" class="ltx_Math" display="inline" id="Thmproposition7.p1.11.m11.1"><semantics id="Thmproposition7.p1.11.m11.1a"><mrow id="Thmproposition7.p1.11.m11.1.1" xref="Thmproposition7.p1.11.m11.1.1.cmml"><msup id="Thmproposition7.p1.11.m11.1.1.2" xref="Thmproposition7.p1.11.m11.1.1.2.cmml"><mi id="Thmproposition7.p1.11.m11.1.1.2.2" xref="Thmproposition7.p1.11.m11.1.1.2.2.cmml">τ</mi><mo id="Thmproposition7.p1.11.m11.1.1.2.3" xref="Thmproposition7.p1.11.m11.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition7.p1.11.m11.1.1.1" xref="Thmproposition7.p1.11.m11.1.1.1.cmml">></mo><mi id="Thmproposition7.p1.11.m11.1.1.3" xref="Thmproposition7.p1.11.m11.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition7.p1.11.m11.1b"><apply id="Thmproposition7.p1.11.m11.1.1.cmml" xref="Thmproposition7.p1.11.m11.1.1"><gt id="Thmproposition7.p1.11.m11.1.1.1.cmml" xref="Thmproposition7.p1.11.m11.1.1.1"></gt><apply id="Thmproposition7.p1.11.m11.1.1.2.cmml" xref="Thmproposition7.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="Thmproposition7.p1.11.m11.1.1.2.1.cmml" xref="Thmproposition7.p1.11.m11.1.1.2">superscript</csymbol><ci id="Thmproposition7.p1.11.m11.1.1.2.2.cmml" xref="Thmproposition7.p1.11.m11.1.1.2.2">𝜏</ci><times id="Thmproposition7.p1.11.m11.1.1.2.3.cmml" xref="Thmproposition7.p1.11.m11.1.1.2.3"></times></apply><ci id="Thmproposition7.p1.11.m11.1.1.3.cmml" xref="Thmproposition7.p1.11.m11.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition7.p1.11.m11.1c">\tau^{*}>m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition7.p1.11.m11.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT > italic_m</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="A1.SS1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS1.2.p1"> <p class="ltx_p" id="A1.SS1.2.p1.17">First, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem7" title="Theorem 7. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a>, we know at least one equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.1.m1.1"><semantics id="A1.SS1.2.p1.1.m1.1a"><msup id="A1.SS1.2.p1.1.m1.1.1" xref="A1.SS1.2.p1.1.m1.1.1.cmml"><mi id="A1.SS1.2.p1.1.m1.1.1.2" xref="A1.SS1.2.p1.1.m1.1.1.2.cmml">τ</mi><mo id="A1.SS1.2.p1.1.m1.1.1.3" xref="A1.SS1.2.p1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.1.m1.1b"><apply id="A1.SS1.2.p1.1.m1.1.1.cmml" xref="A1.SS1.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.1.m1.1.1.1.cmml" xref="A1.SS1.2.p1.1.m1.1.1">superscript</csymbol><ci id="A1.SS1.2.p1.1.m1.1.1.2.cmml" xref="A1.SS1.2.p1.1.m1.1.1.2">𝜏</ci><times id="A1.SS1.2.p1.1.m1.1.1.3.cmml" xref="A1.SS1.2.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.1.m1.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> exists in the interval <math alttext="I" class="ltx_Math" display="inline" id="A1.SS1.2.p1.2.m2.1"><semantics id="A1.SS1.2.p1.2.m2.1a"><mi id="A1.SS1.2.p1.2.m2.1.1" xref="A1.SS1.2.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.2.m2.1b"><ci id="A1.SS1.2.p1.2.m2.1.1.cmml" xref="A1.SS1.2.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.2.m2.1d">italic_I</annotation></semantics></math>. Now assume that <math alttext="\Pr[X^{\prime}\leq m\mid m]=G(m)>1/2" class="ltx_Math" display="inline" id="A1.SS1.2.p1.3.m3.3"><semantics id="A1.SS1.2.p1.3.m3.3a"><mrow id="A1.SS1.2.p1.3.m3.3.3" xref="A1.SS1.2.p1.3.m3.3.3.cmml"><mrow id="A1.SS1.2.p1.3.m3.3.3.1.1" xref="A1.SS1.2.p1.3.m3.3.3.1.2.cmml"><mi id="A1.SS1.2.p1.3.m3.1.1" xref="A1.SS1.2.p1.3.m3.1.1.cmml">Pr</mi><mo id="A1.SS1.2.p1.3.m3.3.3.1.1a" xref="A1.SS1.2.p1.3.m3.3.3.1.2.cmml">⁡</mo><mrow id="A1.SS1.2.p1.3.m3.3.3.1.1.1" xref="A1.SS1.2.p1.3.m3.3.3.1.2.cmml"><mo id="A1.SS1.2.p1.3.m3.3.3.1.1.1.2" stretchy="false" xref="A1.SS1.2.p1.3.m3.3.3.1.2.cmml">[</mo><mrow id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.cmml"><msup id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.cmml"><mi id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.2" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.2.cmml">X</mi><mo id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.3" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.1" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.1.cmml">≤</mo><mrow id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.cmml"><mi id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.2" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.2.cmml">m</mi><mo id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.1" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.1.cmml">∣</mo><mi id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.3" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="A1.SS1.2.p1.3.m3.3.3.1.1.1.3" stretchy="false" xref="A1.SS1.2.p1.3.m3.3.3.1.2.cmml">]</mo></mrow></mrow><mo id="A1.SS1.2.p1.3.m3.3.3.3" xref="A1.SS1.2.p1.3.m3.3.3.3.cmml">=</mo><mrow id="A1.SS1.2.p1.3.m3.3.3.4" xref="A1.SS1.2.p1.3.m3.3.3.4.cmml"><mi id="A1.SS1.2.p1.3.m3.3.3.4.2" xref="A1.SS1.2.p1.3.m3.3.3.4.2.cmml">G</mi><mo id="A1.SS1.2.p1.3.m3.3.3.4.1" xref="A1.SS1.2.p1.3.m3.3.3.4.1.cmml">⁢</mo><mrow id="A1.SS1.2.p1.3.m3.3.3.4.3.2" xref="A1.SS1.2.p1.3.m3.3.3.4.cmml"><mo id="A1.SS1.2.p1.3.m3.3.3.4.3.2.1" stretchy="false" xref="A1.SS1.2.p1.3.m3.3.3.4.cmml">(</mo><mi id="A1.SS1.2.p1.3.m3.2.2" xref="A1.SS1.2.p1.3.m3.2.2.cmml">m</mi><mo id="A1.SS1.2.p1.3.m3.3.3.4.3.2.2" stretchy="false" xref="A1.SS1.2.p1.3.m3.3.3.4.cmml">)</mo></mrow></mrow><mo id="A1.SS1.2.p1.3.m3.3.3.5" xref="A1.SS1.2.p1.3.m3.3.3.5.cmml">></mo><mrow id="A1.SS1.2.p1.3.m3.3.3.6" xref="A1.SS1.2.p1.3.m3.3.3.6.cmml"><mn id="A1.SS1.2.p1.3.m3.3.3.6.2" xref="A1.SS1.2.p1.3.m3.3.3.6.2.cmml">1</mn><mo id="A1.SS1.2.p1.3.m3.3.3.6.1" xref="A1.SS1.2.p1.3.m3.3.3.6.1.cmml">/</mo><mn id="A1.SS1.2.p1.3.m3.3.3.6.3" xref="A1.SS1.2.p1.3.m3.3.3.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.3.m3.3b"><apply id="A1.SS1.2.p1.3.m3.3.3.cmml" xref="A1.SS1.2.p1.3.m3.3.3"><and id="A1.SS1.2.p1.3.m3.3.3a.cmml" xref="A1.SS1.2.p1.3.m3.3.3"></and><apply id="A1.SS1.2.p1.3.m3.3.3b.cmml" xref="A1.SS1.2.p1.3.m3.3.3"><eq id="A1.SS1.2.p1.3.m3.3.3.3.cmml" xref="A1.SS1.2.p1.3.m3.3.3.3"></eq><apply id="A1.SS1.2.p1.3.m3.3.3.1.2.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1"><ci id="A1.SS1.2.p1.3.m3.1.1.cmml" xref="A1.SS1.2.p1.3.m3.1.1">Pr</ci><apply id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1"><leq id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.1"></leq><apply id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2">superscript</csymbol><ci id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.2.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.2">𝑋</ci><ci id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.3.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.2.3">′</ci></apply><apply id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3"><csymbol cd="latexml" id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.1">conditional</csymbol><ci id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.2.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.2">𝑚</ci><ci id="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.3.cmml" xref="A1.SS1.2.p1.3.m3.3.3.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="A1.SS1.2.p1.3.m3.3.3.4.cmml" xref="A1.SS1.2.p1.3.m3.3.3.4"><times id="A1.SS1.2.p1.3.m3.3.3.4.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.4.1"></times><ci id="A1.SS1.2.p1.3.m3.3.3.4.2.cmml" xref="A1.SS1.2.p1.3.m3.3.3.4.2">𝐺</ci><ci id="A1.SS1.2.p1.3.m3.2.2.cmml" xref="A1.SS1.2.p1.3.m3.2.2">𝑚</ci></apply></apply><apply id="A1.SS1.2.p1.3.m3.3.3c.cmml" xref="A1.SS1.2.p1.3.m3.3.3"><gt id="A1.SS1.2.p1.3.m3.3.3.5.cmml" xref="A1.SS1.2.p1.3.m3.3.3.5"></gt><share href="https://arxiv.org/html/2503.16280v1#A1.SS1.2.p1.3.m3.3.3.4.cmml" id="A1.SS1.2.p1.3.m3.3.3d.cmml" xref="A1.SS1.2.p1.3.m3.3.3"></share><apply id="A1.SS1.2.p1.3.m3.3.3.6.cmml" xref="A1.SS1.2.p1.3.m3.3.3.6"><divide id="A1.SS1.2.p1.3.m3.3.3.6.1.cmml" xref="A1.SS1.2.p1.3.m3.3.3.6.1"></divide><cn id="A1.SS1.2.p1.3.m3.3.3.6.2.cmml" type="integer" xref="A1.SS1.2.p1.3.m3.3.3.6.2">1</cn><cn id="A1.SS1.2.p1.3.m3.3.3.6.3.cmml" type="integer" xref="A1.SS1.2.p1.3.m3.3.3.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.3.m3.3c">\Pr[X^{\prime}\leq m\mid m]=G(m)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.3.m3.3d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] = italic_G ( italic_m ) > 1 / 2</annotation></semantics></math>. It follows under Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> that either <math alttext="m>b" class="ltx_Math" display="inline" id="A1.SS1.2.p1.4.m4.1"><semantics id="A1.SS1.2.p1.4.m4.1a"><mrow id="A1.SS1.2.p1.4.m4.1.1" xref="A1.SS1.2.p1.4.m4.1.1.cmml"><mi id="A1.SS1.2.p1.4.m4.1.1.2" xref="A1.SS1.2.p1.4.m4.1.1.2.cmml">m</mi><mo id="A1.SS1.2.p1.4.m4.1.1.1" xref="A1.SS1.2.p1.4.m4.1.1.1.cmml">></mo><mi id="A1.SS1.2.p1.4.m4.1.1.3" xref="A1.SS1.2.p1.4.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.4.m4.1b"><apply id="A1.SS1.2.p1.4.m4.1.1.cmml" xref="A1.SS1.2.p1.4.m4.1.1"><gt id="A1.SS1.2.p1.4.m4.1.1.1.cmml" xref="A1.SS1.2.p1.4.m4.1.1.1"></gt><ci id="A1.SS1.2.p1.4.m4.1.1.2.cmml" xref="A1.SS1.2.p1.4.m4.1.1.2">𝑚</ci><ci id="A1.SS1.2.p1.4.m4.1.1.3.cmml" xref="A1.SS1.2.p1.4.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.4.m4.1c">m>b</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.4.m4.1d">italic_m > italic_b</annotation></semantics></math> or <math alttext="m\in I" class="ltx_Math" display="inline" id="A1.SS1.2.p1.5.m5.1"><semantics id="A1.SS1.2.p1.5.m5.1a"><mrow id="A1.SS1.2.p1.5.m5.1.1" xref="A1.SS1.2.p1.5.m5.1.1.cmml"><mi id="A1.SS1.2.p1.5.m5.1.1.2" xref="A1.SS1.2.p1.5.m5.1.1.2.cmml">m</mi><mo id="A1.SS1.2.p1.5.m5.1.1.1" xref="A1.SS1.2.p1.5.m5.1.1.1.cmml">∈</mo><mi id="A1.SS1.2.p1.5.m5.1.1.3" xref="A1.SS1.2.p1.5.m5.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.5.m5.1b"><apply id="A1.SS1.2.p1.5.m5.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1"><in id="A1.SS1.2.p1.5.m5.1.1.1.cmml" xref="A1.SS1.2.p1.5.m5.1.1.1"></in><ci id="A1.SS1.2.p1.5.m5.1.1.2.cmml" xref="A1.SS1.2.p1.5.m5.1.1.2">𝑚</ci><ci id="A1.SS1.2.p1.5.m5.1.1.3.cmml" xref="A1.SS1.2.p1.5.m5.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.5.m5.1c">m\in I</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.5.m5.1d">italic_m ∈ italic_I</annotation></semantics></math>. If <math alttext="m>b" class="ltx_Math" display="inline" id="A1.SS1.2.p1.6.m6.1"><semantics id="A1.SS1.2.p1.6.m6.1a"><mrow id="A1.SS1.2.p1.6.m6.1.1" xref="A1.SS1.2.p1.6.m6.1.1.cmml"><mi id="A1.SS1.2.p1.6.m6.1.1.2" xref="A1.SS1.2.p1.6.m6.1.1.2.cmml">m</mi><mo id="A1.SS1.2.p1.6.m6.1.1.1" xref="A1.SS1.2.p1.6.m6.1.1.1.cmml">></mo><mi id="A1.SS1.2.p1.6.m6.1.1.3" xref="A1.SS1.2.p1.6.m6.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.6.m6.1b"><apply id="A1.SS1.2.p1.6.m6.1.1.cmml" xref="A1.SS1.2.p1.6.m6.1.1"><gt id="A1.SS1.2.p1.6.m6.1.1.1.cmml" xref="A1.SS1.2.p1.6.m6.1.1.1"></gt><ci id="A1.SS1.2.p1.6.m6.1.1.2.cmml" xref="A1.SS1.2.p1.6.m6.1.1.2">𝑚</ci><ci id="A1.SS1.2.p1.6.m6.1.1.3.cmml" xref="A1.SS1.2.p1.6.m6.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.6.m6.1c">m>b</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.6.m6.1d">italic_m > italic_b</annotation></semantics></math>, then by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem7" title="Theorem 7. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a> there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.7.m7.1"><semantics id="A1.SS1.2.p1.7.m7.1a"><msup id="A1.SS1.2.p1.7.m7.1.1" xref="A1.SS1.2.p1.7.m7.1.1.cmml"><mi id="A1.SS1.2.p1.7.m7.1.1.2" xref="A1.SS1.2.p1.7.m7.1.1.2.cmml">τ</mi><mo id="A1.SS1.2.p1.7.m7.1.1.3" xref="A1.SS1.2.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.7.m7.1b"><apply id="A1.SS1.2.p1.7.m7.1.1.cmml" xref="A1.SS1.2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.7.m7.1.1.1.cmml" xref="A1.SS1.2.p1.7.m7.1.1">superscript</csymbol><ci id="A1.SS1.2.p1.7.m7.1.1.2.cmml" xref="A1.SS1.2.p1.7.m7.1.1.2">𝜏</ci><times id="A1.SS1.2.p1.7.m7.1.1.3.cmml" xref="A1.SS1.2.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.7.m7.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.7.m7.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> in the interval <math alttext="I" class="ltx_Math" display="inline" id="A1.SS1.2.p1.8.m8.1"><semantics id="A1.SS1.2.p1.8.m8.1a"><mi id="A1.SS1.2.p1.8.m8.1.1" xref="A1.SS1.2.p1.8.m8.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.8.m8.1b"><ci id="A1.SS1.2.p1.8.m8.1.1.cmml" xref="A1.SS1.2.p1.8.m8.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.8.m8.1c">I</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.8.m8.1d">italic_I</annotation></semantics></math>; <math alttext="\tau^{*}\leq b<m" class="ltx_Math" display="inline" id="A1.SS1.2.p1.9.m9.1"><semantics id="A1.SS1.2.p1.9.m9.1a"><mrow id="A1.SS1.2.p1.9.m9.1.1" xref="A1.SS1.2.p1.9.m9.1.1.cmml"><msup id="A1.SS1.2.p1.9.m9.1.1.2" xref="A1.SS1.2.p1.9.m9.1.1.2.cmml"><mi id="A1.SS1.2.p1.9.m9.1.1.2.2" xref="A1.SS1.2.p1.9.m9.1.1.2.2.cmml">τ</mi><mo id="A1.SS1.2.p1.9.m9.1.1.2.3" xref="A1.SS1.2.p1.9.m9.1.1.2.3.cmml">∗</mo></msup><mo id="A1.SS1.2.p1.9.m9.1.1.3" xref="A1.SS1.2.p1.9.m9.1.1.3.cmml">≤</mo><mi id="A1.SS1.2.p1.9.m9.1.1.4" xref="A1.SS1.2.p1.9.m9.1.1.4.cmml">b</mi><mo id="A1.SS1.2.p1.9.m9.1.1.5" xref="A1.SS1.2.p1.9.m9.1.1.5.cmml"><</mo><mi id="A1.SS1.2.p1.9.m9.1.1.6" xref="A1.SS1.2.p1.9.m9.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.9.m9.1b"><apply id="A1.SS1.2.p1.9.m9.1.1.cmml" xref="A1.SS1.2.p1.9.m9.1.1"><and id="A1.SS1.2.p1.9.m9.1.1a.cmml" xref="A1.SS1.2.p1.9.m9.1.1"></and><apply id="A1.SS1.2.p1.9.m9.1.1b.cmml" xref="A1.SS1.2.p1.9.m9.1.1"><leq id="A1.SS1.2.p1.9.m9.1.1.3.cmml" xref="A1.SS1.2.p1.9.m9.1.1.3"></leq><apply id="A1.SS1.2.p1.9.m9.1.1.2.cmml" xref="A1.SS1.2.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.9.m9.1.1.2.1.cmml" xref="A1.SS1.2.p1.9.m9.1.1.2">superscript</csymbol><ci id="A1.SS1.2.p1.9.m9.1.1.2.2.cmml" xref="A1.SS1.2.p1.9.m9.1.1.2.2">𝜏</ci><times id="A1.SS1.2.p1.9.m9.1.1.2.3.cmml" xref="A1.SS1.2.p1.9.m9.1.1.2.3"></times></apply><ci id="A1.SS1.2.p1.9.m9.1.1.4.cmml" xref="A1.SS1.2.p1.9.m9.1.1.4">𝑏</ci></apply><apply id="A1.SS1.2.p1.9.m9.1.1c.cmml" xref="A1.SS1.2.p1.9.m9.1.1"><lt id="A1.SS1.2.p1.9.m9.1.1.5.cmml" xref="A1.SS1.2.p1.9.m9.1.1.5"></lt><share href="https://arxiv.org/html/2503.16280v1#A1.SS1.2.p1.9.m9.1.1.4.cmml" id="A1.SS1.2.p1.9.m9.1.1d.cmml" xref="A1.SS1.2.p1.9.m9.1.1"></share><ci id="A1.SS1.2.p1.9.m9.1.1.6.cmml" xref="A1.SS1.2.p1.9.m9.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.9.m9.1c">\tau^{*}\leq b<m</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_b < italic_m</annotation></semantics></math>, so the result follows. If <math alttext="m\in I" class="ltx_Math" display="inline" id="A1.SS1.2.p1.10.m10.1"><semantics id="A1.SS1.2.p1.10.m10.1a"><mrow id="A1.SS1.2.p1.10.m10.1.1" xref="A1.SS1.2.p1.10.m10.1.1.cmml"><mi id="A1.SS1.2.p1.10.m10.1.1.2" xref="A1.SS1.2.p1.10.m10.1.1.2.cmml">m</mi><mo id="A1.SS1.2.p1.10.m10.1.1.1" xref="A1.SS1.2.p1.10.m10.1.1.1.cmml">∈</mo><mi id="A1.SS1.2.p1.10.m10.1.1.3" xref="A1.SS1.2.p1.10.m10.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.10.m10.1b"><apply id="A1.SS1.2.p1.10.m10.1.1.cmml" xref="A1.SS1.2.p1.10.m10.1.1"><in id="A1.SS1.2.p1.10.m10.1.1.1.cmml" xref="A1.SS1.2.p1.10.m10.1.1.1"></in><ci id="A1.SS1.2.p1.10.m10.1.1.2.cmml" xref="A1.SS1.2.p1.10.m10.1.1.2">𝑚</ci><ci id="A1.SS1.2.p1.10.m10.1.1.3.cmml" xref="A1.SS1.2.p1.10.m10.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.10.m10.1c">m\in I</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.10.m10.1d">italic_m ∈ italic_I</annotation></semantics></math>, then <math alttext="m\geq a" class="ltx_Math" display="inline" id="A1.SS1.2.p1.11.m11.1"><semantics id="A1.SS1.2.p1.11.m11.1a"><mrow id="A1.SS1.2.p1.11.m11.1.1" xref="A1.SS1.2.p1.11.m11.1.1.cmml"><mi id="A1.SS1.2.p1.11.m11.1.1.2" xref="A1.SS1.2.p1.11.m11.1.1.2.cmml">m</mi><mo id="A1.SS1.2.p1.11.m11.1.1.1" xref="A1.SS1.2.p1.11.m11.1.1.1.cmml">≥</mo><mi id="A1.SS1.2.p1.11.m11.1.1.3" xref="A1.SS1.2.p1.11.m11.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.11.m11.1b"><apply id="A1.SS1.2.p1.11.m11.1.1.cmml" xref="A1.SS1.2.p1.11.m11.1.1"><geq id="A1.SS1.2.p1.11.m11.1.1.1.cmml" xref="A1.SS1.2.p1.11.m11.1.1.1"></geq><ci id="A1.SS1.2.p1.11.m11.1.1.2.cmml" xref="A1.SS1.2.p1.11.m11.1.1.2">𝑚</ci><ci id="A1.SS1.2.p1.11.m11.1.1.3.cmml" xref="A1.SS1.2.p1.11.m11.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.11.m11.1c">m\geq a</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.11.m11.1d">italic_m ≥ italic_a</annotation></semantics></math>. Since <math alttext="G" class="ltx_Math" display="inline" id="A1.SS1.2.p1.12.m12.1"><semantics id="A1.SS1.2.p1.12.m12.1a"><mi id="A1.SS1.2.p1.12.m12.1.1" xref="A1.SS1.2.p1.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.12.m12.1b"><ci id="A1.SS1.2.p1.12.m12.1.1.cmml" xref="A1.SS1.2.p1.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.12.m12.1d">italic_G</annotation></semantics></math> is continuous, <math alttext="G(a)<1/2" class="ltx_Math" display="inline" id="A1.SS1.2.p1.13.m13.1"><semantics id="A1.SS1.2.p1.13.m13.1a"><mrow id="A1.SS1.2.p1.13.m13.1.2" xref="A1.SS1.2.p1.13.m13.1.2.cmml"><mrow id="A1.SS1.2.p1.13.m13.1.2.2" xref="A1.SS1.2.p1.13.m13.1.2.2.cmml"><mi id="A1.SS1.2.p1.13.m13.1.2.2.2" xref="A1.SS1.2.p1.13.m13.1.2.2.2.cmml">G</mi><mo id="A1.SS1.2.p1.13.m13.1.2.2.1" xref="A1.SS1.2.p1.13.m13.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS1.2.p1.13.m13.1.2.2.3.2" xref="A1.SS1.2.p1.13.m13.1.2.2.cmml"><mo id="A1.SS1.2.p1.13.m13.1.2.2.3.2.1" stretchy="false" xref="A1.SS1.2.p1.13.m13.1.2.2.cmml">(</mo><mi id="A1.SS1.2.p1.13.m13.1.1" xref="A1.SS1.2.p1.13.m13.1.1.cmml">a</mi><mo id="A1.SS1.2.p1.13.m13.1.2.2.3.2.2" stretchy="false" xref="A1.SS1.2.p1.13.m13.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS1.2.p1.13.m13.1.2.1" xref="A1.SS1.2.p1.13.m13.1.2.1.cmml"><</mo><mrow id="A1.SS1.2.p1.13.m13.1.2.3" xref="A1.SS1.2.p1.13.m13.1.2.3.cmml"><mn id="A1.SS1.2.p1.13.m13.1.2.3.2" xref="A1.SS1.2.p1.13.m13.1.2.3.2.cmml">1</mn><mo id="A1.SS1.2.p1.13.m13.1.2.3.1" xref="A1.SS1.2.p1.13.m13.1.2.3.1.cmml">/</mo><mn id="A1.SS1.2.p1.13.m13.1.2.3.3" xref="A1.SS1.2.p1.13.m13.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.13.m13.1b"><apply id="A1.SS1.2.p1.13.m13.1.2.cmml" xref="A1.SS1.2.p1.13.m13.1.2"><lt id="A1.SS1.2.p1.13.m13.1.2.1.cmml" xref="A1.SS1.2.p1.13.m13.1.2.1"></lt><apply id="A1.SS1.2.p1.13.m13.1.2.2.cmml" xref="A1.SS1.2.p1.13.m13.1.2.2"><times id="A1.SS1.2.p1.13.m13.1.2.2.1.cmml" xref="A1.SS1.2.p1.13.m13.1.2.2.1"></times><ci id="A1.SS1.2.p1.13.m13.1.2.2.2.cmml" xref="A1.SS1.2.p1.13.m13.1.2.2.2">𝐺</ci><ci id="A1.SS1.2.p1.13.m13.1.1.cmml" xref="A1.SS1.2.p1.13.m13.1.1">𝑎</ci></apply><apply id="A1.SS1.2.p1.13.m13.1.2.3.cmml" xref="A1.SS1.2.p1.13.m13.1.2.3"><divide id="A1.SS1.2.p1.13.m13.1.2.3.1.cmml" xref="A1.SS1.2.p1.13.m13.1.2.3.1"></divide><cn id="A1.SS1.2.p1.13.m13.1.2.3.2.cmml" type="integer" xref="A1.SS1.2.p1.13.m13.1.2.3.2">1</cn><cn id="A1.SS1.2.p1.13.m13.1.2.3.3.cmml" type="integer" xref="A1.SS1.2.p1.13.m13.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.13.m13.1c">G(a)<1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.13.m13.1d">italic_G ( italic_a ) < 1 / 2</annotation></semantics></math>, and <math alttext="G(m)>1/2" class="ltx_Math" display="inline" 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xref="A1.SS1.2.p1.14.m14.1.2.3.1.cmml">/</mo><mn id="A1.SS1.2.p1.14.m14.1.2.3.3" xref="A1.SS1.2.p1.14.m14.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.14.m14.1b"><apply id="A1.SS1.2.p1.14.m14.1.2.cmml" xref="A1.SS1.2.p1.14.m14.1.2"><gt id="A1.SS1.2.p1.14.m14.1.2.1.cmml" xref="A1.SS1.2.p1.14.m14.1.2.1"></gt><apply id="A1.SS1.2.p1.14.m14.1.2.2.cmml" xref="A1.SS1.2.p1.14.m14.1.2.2"><times id="A1.SS1.2.p1.14.m14.1.2.2.1.cmml" xref="A1.SS1.2.p1.14.m14.1.2.2.1"></times><ci id="A1.SS1.2.p1.14.m14.1.2.2.2.cmml" xref="A1.SS1.2.p1.14.m14.1.2.2.2">𝐺</ci><ci id="A1.SS1.2.p1.14.m14.1.1.cmml" xref="A1.SS1.2.p1.14.m14.1.1">𝑚</ci></apply><apply id="A1.SS1.2.p1.14.m14.1.2.3.cmml" xref="A1.SS1.2.p1.14.m14.1.2.3"><divide id="A1.SS1.2.p1.14.m14.1.2.3.1.cmml" xref="A1.SS1.2.p1.14.m14.1.2.3.1"></divide><cn id="A1.SS1.2.p1.14.m14.1.2.3.2.cmml" type="integer" xref="A1.SS1.2.p1.14.m14.1.2.3.2">1</cn><cn id="A1.SS1.2.p1.14.m14.1.2.3.3.cmml" type="integer" xref="A1.SS1.2.p1.14.m14.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.14.m14.1c">G(m)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.14.m14.1d">italic_G ( italic_m ) > 1 / 2</annotation></semantics></math>, by the Intermediate Value Theorem there exists a point <math alttext="\tau^{*}\in[a,m]" class="ltx_Math" display="inline" id="A1.SS1.2.p1.15.m15.2"><semantics id="A1.SS1.2.p1.15.m15.2a"><mrow id="A1.SS1.2.p1.15.m15.2.3" xref="A1.SS1.2.p1.15.m15.2.3.cmml"><msup id="A1.SS1.2.p1.15.m15.2.3.2" xref="A1.SS1.2.p1.15.m15.2.3.2.cmml"><mi id="A1.SS1.2.p1.15.m15.2.3.2.2" xref="A1.SS1.2.p1.15.m15.2.3.2.2.cmml">τ</mi><mo id="A1.SS1.2.p1.15.m15.2.3.2.3" xref="A1.SS1.2.p1.15.m15.2.3.2.3.cmml">∗</mo></msup><mo id="A1.SS1.2.p1.15.m15.2.3.1" xref="A1.SS1.2.p1.15.m15.2.3.1.cmml">∈</mo><mrow id="A1.SS1.2.p1.15.m15.2.3.3.2" xref="A1.SS1.2.p1.15.m15.2.3.3.1.cmml"><mo id="A1.SS1.2.p1.15.m15.2.3.3.2.1" stretchy="false" xref="A1.SS1.2.p1.15.m15.2.3.3.1.cmml">[</mo><mi id="A1.SS1.2.p1.15.m15.1.1" xref="A1.SS1.2.p1.15.m15.1.1.cmml">a</mi><mo id="A1.SS1.2.p1.15.m15.2.3.3.2.2" xref="A1.SS1.2.p1.15.m15.2.3.3.1.cmml">,</mo><mi id="A1.SS1.2.p1.15.m15.2.2" xref="A1.SS1.2.p1.15.m15.2.2.cmml">m</mi><mo id="A1.SS1.2.p1.15.m15.2.3.3.2.3" stretchy="false" xref="A1.SS1.2.p1.15.m15.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.15.m15.2b"><apply id="A1.SS1.2.p1.15.m15.2.3.cmml" xref="A1.SS1.2.p1.15.m15.2.3"><in id="A1.SS1.2.p1.15.m15.2.3.1.cmml" xref="A1.SS1.2.p1.15.m15.2.3.1"></in><apply id="A1.SS1.2.p1.15.m15.2.3.2.cmml" xref="A1.SS1.2.p1.15.m15.2.3.2"><csymbol cd="ambiguous" id="A1.SS1.2.p1.15.m15.2.3.2.1.cmml" xref="A1.SS1.2.p1.15.m15.2.3.2">superscript</csymbol><ci id="A1.SS1.2.p1.15.m15.2.3.2.2.cmml" xref="A1.SS1.2.p1.15.m15.2.3.2.2">𝜏</ci><times id="A1.SS1.2.p1.15.m15.2.3.2.3.cmml" xref="A1.SS1.2.p1.15.m15.2.3.2.3"></times></apply><interval closure="closed" id="A1.SS1.2.p1.15.m15.2.3.3.1.cmml" xref="A1.SS1.2.p1.15.m15.2.3.3.2"><ci id="A1.SS1.2.p1.15.m15.1.1.cmml" xref="A1.SS1.2.p1.15.m15.1.1">𝑎</ci><ci id="A1.SS1.2.p1.15.m15.2.2.cmml" xref="A1.SS1.2.p1.15.m15.2.2">𝑚</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.15.m15.2c">\tau^{*}\in[a,m]</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.15.m15.2d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ [ italic_a , italic_m ]</annotation></semantics></math> such that <math alttext="G(\tau^{*})=1/2" class="ltx_Math" display="inline" id="A1.SS1.2.p1.16.m16.1"><semantics id="A1.SS1.2.p1.16.m16.1a"><mrow id="A1.SS1.2.p1.16.m16.1.1" xref="A1.SS1.2.p1.16.m16.1.1.cmml"><mrow id="A1.SS1.2.p1.16.m16.1.1.1" xref="A1.SS1.2.p1.16.m16.1.1.1.cmml"><mi id="A1.SS1.2.p1.16.m16.1.1.1.3" xref="A1.SS1.2.p1.16.m16.1.1.1.3.cmml">G</mi><mo id="A1.SS1.2.p1.16.m16.1.1.1.2" xref="A1.SS1.2.p1.16.m16.1.1.1.2.cmml">⁢</mo><mrow 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xref="A1.SS1.2.p1.16.m16.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.16.m16.1b"><apply id="A1.SS1.2.p1.16.m16.1.1.cmml" xref="A1.SS1.2.p1.16.m16.1.1"><eq id="A1.SS1.2.p1.16.m16.1.1.2.cmml" xref="A1.SS1.2.p1.16.m16.1.1.2"></eq><apply id="A1.SS1.2.p1.16.m16.1.1.1.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1"><times id="A1.SS1.2.p1.16.m16.1.1.1.2.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.2"></times><ci id="A1.SS1.2.p1.16.m16.1.1.1.3.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.3">𝐺</ci><apply id="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.1.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.1.1">superscript</csymbol><ci id="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.2.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.2">𝜏</ci><times id="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.3.cmml" xref="A1.SS1.2.p1.16.m16.1.1.1.1.1.1.3"></times></apply></apply><apply id="A1.SS1.2.p1.16.m16.1.1.3.cmml" xref="A1.SS1.2.p1.16.m16.1.1.3"><divide id="A1.SS1.2.p1.16.m16.1.1.3.1.cmml" xref="A1.SS1.2.p1.16.m16.1.1.3.1"></divide><cn id="A1.SS1.2.p1.16.m16.1.1.3.2.cmml" type="integer" xref="A1.SS1.2.p1.16.m16.1.1.3.2">1</cn><cn id="A1.SS1.2.p1.16.m16.1.1.3.3.cmml" type="integer" xref="A1.SS1.2.p1.16.m16.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.16.m16.1c">G(\tau^{*})=1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.16.m16.1d">italic_G ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 1 / 2</annotation></semantics></math>. By our characterization in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem3" title="Theorem 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>, <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A1.SS1.2.p1.17.m17.1"><semantics id="A1.SS1.2.p1.17.m17.1a"><msup id="A1.SS1.2.p1.17.m17.1.1" xref="A1.SS1.2.p1.17.m17.1.1.cmml"><mi id="A1.SS1.2.p1.17.m17.1.1.2" xref="A1.SS1.2.p1.17.m17.1.1.2.cmml">τ</mi><mo id="A1.SS1.2.p1.17.m17.1.1.3" xref="A1.SS1.2.p1.17.m17.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.2.p1.17.m17.1b"><apply id="A1.SS1.2.p1.17.m17.1.1.cmml" xref="A1.SS1.2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="A1.SS1.2.p1.17.m17.1.1.1.cmml" xref="A1.SS1.2.p1.17.m17.1.1">superscript</csymbol><ci id="A1.SS1.2.p1.17.m17.1.1.2.cmml" xref="A1.SS1.2.p1.17.m17.1.1.2">𝜏</ci><times id="A1.SS1.2.p1.17.m17.1.1.3.cmml" xref="A1.SS1.2.p1.17.m17.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.2.p1.17.m17.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.2.p1.17.m17.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an equilibrium.</p> </div> <div class="ltx_para" id="A1.SS1.3.p2"> <p class="ltx_p" id="A1.SS1.3.p2.1">A symmetric argument holds for when <math alttext="\Pr[X^{\prime}\leq m\mid m]<1/2" class="ltx_Math" display="inline" id="A1.SS1.3.p2.1.m1.2"><semantics id="A1.SS1.3.p2.1.m1.2a"><mrow id="A1.SS1.3.p2.1.m1.2.2" xref="A1.SS1.3.p2.1.m1.2.2.cmml"><mrow id="A1.SS1.3.p2.1.m1.2.2.1.1" xref="A1.SS1.3.p2.1.m1.2.2.1.2.cmml"><mi id="A1.SS1.3.p2.1.m1.1.1" xref="A1.SS1.3.p2.1.m1.1.1.cmml">Pr</mi><mo id="A1.SS1.3.p2.1.m1.2.2.1.1a" xref="A1.SS1.3.p2.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="A1.SS1.3.p2.1.m1.2.2.1.1.1" 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stretchy="false" xref="A1.SS1.3.p2.1.m1.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="A1.SS1.3.p2.1.m1.2.2.2" xref="A1.SS1.3.p2.1.m1.2.2.2.cmml"><</mo><mrow id="A1.SS1.3.p2.1.m1.2.2.3" xref="A1.SS1.3.p2.1.m1.2.2.3.cmml"><mn id="A1.SS1.3.p2.1.m1.2.2.3.2" xref="A1.SS1.3.p2.1.m1.2.2.3.2.cmml">1</mn><mo id="A1.SS1.3.p2.1.m1.2.2.3.1" xref="A1.SS1.3.p2.1.m1.2.2.3.1.cmml">/</mo><mn id="A1.SS1.3.p2.1.m1.2.2.3.3" xref="A1.SS1.3.p2.1.m1.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.3.p2.1.m1.2b"><apply id="A1.SS1.3.p2.1.m1.2.2.cmml" xref="A1.SS1.3.p2.1.m1.2.2"><lt id="A1.SS1.3.p2.1.m1.2.2.2.cmml" xref="A1.SS1.3.p2.1.m1.2.2.2"></lt><apply id="A1.SS1.3.p2.1.m1.2.2.1.2.cmml" xref="A1.SS1.3.p2.1.m1.2.2.1.1"><ci id="A1.SS1.3.p2.1.m1.1.1.cmml" xref="A1.SS1.3.p2.1.m1.1.1">Pr</ci><apply id="A1.SS1.3.p2.1.m1.2.2.1.1.1.1.cmml" xref="A1.SS1.3.p2.1.m1.2.2.1.1.1.1"><leq id="A1.SS1.3.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="A1.SS1.3.p2.1.m1.2.2.1.1.1.1.1"></leq><apply 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id="A1.SS1.3.p2.1.m1.2.2.3.2.cmml" type="integer" xref="A1.SS1.3.p2.1.m1.2.2.3.2">1</cn><cn id="A1.SS1.3.p2.1.m1.2.2.3.3.cmml" type="integer" xref="A1.SS1.3.p2.1.m1.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.3.p2.1.m1.2c">\Pr[X^{\prime}\leq m\mid m]<1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.3.p2.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] < 1 / 2</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="A1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.2 </span>Dynamics Generalization</h3> <div class="ltx_para" id="A1.SS2.p1"> <p class="ltx_p" id="A1.SS2.p1.1">We can observe instability of equilibria under Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition8.1.1.1">Proposition 8</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition8.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition8.p1"> <p class="ltx_p" id="Thmproposition8.p1.4">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> for interval <math alttext="I" class="ltx_Math" display="inline" id="Thmproposition8.p1.1.m1.1"><semantics id="Thmproposition8.p1.1.m1.1a"><mi id="Thmproposition8.p1.1.m1.1.1" xref="Thmproposition8.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="Thmproposition8.p1.1.m1.1b"><ci id="Thmproposition8.p1.1.m1.1.1.cmml" xref="Thmproposition8.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition8.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="Thmproposition8.p1.1.m1.1d">italic_I</annotation></semantics></math>, and such that <math alttext="G" class="ltx_Math" display="inline" id="Thmproposition8.p1.2.m2.1"><semantics id="Thmproposition8.p1.2.m2.1a"><mi id="Thmproposition8.p1.2.m2.1.1" xref="Thmproposition8.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmproposition8.p1.2.m2.1b"><ci id="Thmproposition8.p1.2.m2.1.1.cmml" xref="Thmproposition8.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition8.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmproposition8.p1.2.m2.1d">italic_G</annotation></semantics></math> is continuous and differentiable. Then there exists an equilibrium <math alttext="\tau^{*}\in I" class="ltx_Math" display="inline" id="Thmproposition8.p1.3.m3.1"><semantics id="Thmproposition8.p1.3.m3.1a"><mrow id="Thmproposition8.p1.3.m3.1.1" xref="Thmproposition8.p1.3.m3.1.1.cmml"><msup id="Thmproposition8.p1.3.m3.1.1.2" xref="Thmproposition8.p1.3.m3.1.1.2.cmml"><mi id="Thmproposition8.p1.3.m3.1.1.2.2" xref="Thmproposition8.p1.3.m3.1.1.2.2.cmml">τ</mi><mo id="Thmproposition8.p1.3.m3.1.1.2.3" xref="Thmproposition8.p1.3.m3.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition8.p1.3.m3.1.1.1" xref="Thmproposition8.p1.3.m3.1.1.1.cmml">∈</mo><mi id="Thmproposition8.p1.3.m3.1.1.3" xref="Thmproposition8.p1.3.m3.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition8.p1.3.m3.1b"><apply id="Thmproposition8.p1.3.m3.1.1.cmml" xref="Thmproposition8.p1.3.m3.1.1"><in id="Thmproposition8.p1.3.m3.1.1.1.cmml" xref="Thmproposition8.p1.3.m3.1.1.1"></in><apply id="Thmproposition8.p1.3.m3.1.1.2.cmml" xref="Thmproposition8.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="Thmproposition8.p1.3.m3.1.1.2.1.cmml" xref="Thmproposition8.p1.3.m3.1.1.2">superscript</csymbol><ci id="Thmproposition8.p1.3.m3.1.1.2.2.cmml" xref="Thmproposition8.p1.3.m3.1.1.2.2">𝜏</ci><times id="Thmproposition8.p1.3.m3.1.1.2.3.cmml" xref="Thmproposition8.p1.3.m3.1.1.2.3"></times></apply><ci id="Thmproposition8.p1.3.m3.1.1.3.cmml" xref="Thmproposition8.p1.3.m3.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition8.p1.3.m3.1c">\tau^{*}\in I</annotation><annotation encoding="application/x-llamapun" id="Thmproposition8.p1.3.m3.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math> which is unstable, while the equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmproposition8.p1.4.m4.1"><semantics id="Thmproposition8.p1.4.m4.1a"><mrow id="Thmproposition8.p1.4.m4.1.1" xref="Thmproposition8.p1.4.m4.1.1.cmml"><mo id="Thmproposition8.p1.4.m4.1.1a" xref="Thmproposition8.p1.4.m4.1.1.cmml">±</mo><mi id="Thmproposition8.p1.4.m4.1.1.2" mathvariant="normal" xref="Thmproposition8.p1.4.m4.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition8.p1.4.m4.1b"><apply id="Thmproposition8.p1.4.m4.1.1.cmml" xref="Thmproposition8.p1.4.m4.1.1"><csymbol cd="latexml" id="Thmproposition8.p1.4.m4.1.1.1.cmml" xref="Thmproposition8.p1.4.m4.1.1">plus-or-minus</csymbol><infinity id="Thmproposition8.p1.4.m4.1.1.2.cmml" xref="Thmproposition8.p1.4.m4.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition8.p1.4.m4.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition8.p1.4.m4.1d">± ∞</annotation></semantics></math> are stable.</p> </div> </div> <div class="ltx_proof" id="A1.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.SS2.1.p1"> <p class="ltx_p" id="A1.SS2.1.p1.11">We know already from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem7" title="Theorem 7. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">7</span></a> that at least one equilibrium exists in the interval <math alttext="I" class="ltx_Math" display="inline" id="A1.SS2.1.p1.1.m1.1"><semantics id="A1.SS2.1.p1.1.m1.1a"><mi id="A1.SS2.1.p1.1.m1.1.1" xref="A1.SS2.1.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.1.m1.1b"><ci id="A1.SS2.1.p1.1.m1.1.1.cmml" xref="A1.SS2.1.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.1.m1.1d">italic_I</annotation></semantics></math>. Note first that since <math alttext="G(a)<1/2" class="ltx_Math" display="inline" id="A1.SS2.1.p1.2.m2.1"><semantics id="A1.SS2.1.p1.2.m2.1a"><mrow id="A1.SS2.1.p1.2.m2.1.2" xref="A1.SS2.1.p1.2.m2.1.2.cmml"><mrow id="A1.SS2.1.p1.2.m2.1.2.2" xref="A1.SS2.1.p1.2.m2.1.2.2.cmml"><mi id="A1.SS2.1.p1.2.m2.1.2.2.2" xref="A1.SS2.1.p1.2.m2.1.2.2.2.cmml">G</mi><mo id="A1.SS2.1.p1.2.m2.1.2.2.1" xref="A1.SS2.1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.1.p1.2.m2.1.2.2.3.2" xref="A1.SS2.1.p1.2.m2.1.2.2.cmml"><mo id="A1.SS2.1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.1.p1.2.m2.1.2.2.cmml">(</mo><mi id="A1.SS2.1.p1.2.m2.1.1" xref="A1.SS2.1.p1.2.m2.1.1.cmml">a</mi><mo id="A1.SS2.1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.1.p1.2.m2.1.2.1" xref="A1.SS2.1.p1.2.m2.1.2.1.cmml"><</mo><mrow id="A1.SS2.1.p1.2.m2.1.2.3" xref="A1.SS2.1.p1.2.m2.1.2.3.cmml"><mn id="A1.SS2.1.p1.2.m2.1.2.3.2" xref="A1.SS2.1.p1.2.m2.1.2.3.2.cmml">1</mn><mo id="A1.SS2.1.p1.2.m2.1.2.3.1" xref="A1.SS2.1.p1.2.m2.1.2.3.1.cmml">/</mo><mn id="A1.SS2.1.p1.2.m2.1.2.3.3" xref="A1.SS2.1.p1.2.m2.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.2.m2.1b"><apply id="A1.SS2.1.p1.2.m2.1.2.cmml" xref="A1.SS2.1.p1.2.m2.1.2"><lt id="A1.SS2.1.p1.2.m2.1.2.1.cmml" xref="A1.SS2.1.p1.2.m2.1.2.1"></lt><apply id="A1.SS2.1.p1.2.m2.1.2.2.cmml" xref="A1.SS2.1.p1.2.m2.1.2.2"><times id="A1.SS2.1.p1.2.m2.1.2.2.1.cmml" xref="A1.SS2.1.p1.2.m2.1.2.2.1"></times><ci id="A1.SS2.1.p1.2.m2.1.2.2.2.cmml" xref="A1.SS2.1.p1.2.m2.1.2.2.2">𝐺</ci><ci id="A1.SS2.1.p1.2.m2.1.1.cmml" xref="A1.SS2.1.p1.2.m2.1.1">𝑎</ci></apply><apply id="A1.SS2.1.p1.2.m2.1.2.3.cmml" xref="A1.SS2.1.p1.2.m2.1.2.3"><divide id="A1.SS2.1.p1.2.m2.1.2.3.1.cmml" xref="A1.SS2.1.p1.2.m2.1.2.3.1"></divide><cn id="A1.SS2.1.p1.2.m2.1.2.3.2.cmml" type="integer" xref="A1.SS2.1.p1.2.m2.1.2.3.2">1</cn><cn id="A1.SS2.1.p1.2.m2.1.2.3.3.cmml" type="integer" xref="A1.SS2.1.p1.2.m2.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.2.m2.1c">G(a)<1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.2.m2.1d">italic_G ( italic_a ) < 1 / 2</annotation></semantics></math> and <math alttext="G(b)>1/2" class="ltx_Math" display="inline" id="A1.SS2.1.p1.3.m3.1"><semantics id="A1.SS2.1.p1.3.m3.1a"><mrow id="A1.SS2.1.p1.3.m3.1.2" xref="A1.SS2.1.p1.3.m3.1.2.cmml"><mrow id="A1.SS2.1.p1.3.m3.1.2.2" xref="A1.SS2.1.p1.3.m3.1.2.2.cmml"><mi id="A1.SS2.1.p1.3.m3.1.2.2.2" xref="A1.SS2.1.p1.3.m3.1.2.2.2.cmml">G</mi><mo id="A1.SS2.1.p1.3.m3.1.2.2.1" xref="A1.SS2.1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.1.p1.3.m3.1.2.2.3.2" xref="A1.SS2.1.p1.3.m3.1.2.2.cmml"><mo id="A1.SS2.1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.1.p1.3.m3.1.2.2.cmml">(</mo><mi id="A1.SS2.1.p1.3.m3.1.1" xref="A1.SS2.1.p1.3.m3.1.1.cmml">b</mi><mo id="A1.SS2.1.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.1.p1.3.m3.1.2.1" xref="A1.SS2.1.p1.3.m3.1.2.1.cmml">></mo><mrow id="A1.SS2.1.p1.3.m3.1.2.3" xref="A1.SS2.1.p1.3.m3.1.2.3.cmml"><mn id="A1.SS2.1.p1.3.m3.1.2.3.2" xref="A1.SS2.1.p1.3.m3.1.2.3.2.cmml">1</mn><mo id="A1.SS2.1.p1.3.m3.1.2.3.1" xref="A1.SS2.1.p1.3.m3.1.2.3.1.cmml">/</mo><mn id="A1.SS2.1.p1.3.m3.1.2.3.3" xref="A1.SS2.1.p1.3.m3.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.3.m3.1b"><apply id="A1.SS2.1.p1.3.m3.1.2.cmml" xref="A1.SS2.1.p1.3.m3.1.2"><gt id="A1.SS2.1.p1.3.m3.1.2.1.cmml" xref="A1.SS2.1.p1.3.m3.1.2.1"></gt><apply id="A1.SS2.1.p1.3.m3.1.2.2.cmml" xref="A1.SS2.1.p1.3.m3.1.2.2"><times id="A1.SS2.1.p1.3.m3.1.2.2.1.cmml" xref="A1.SS2.1.p1.3.m3.1.2.2.1"></times><ci id="A1.SS2.1.p1.3.m3.1.2.2.2.cmml" xref="A1.SS2.1.p1.3.m3.1.2.2.2">𝐺</ci><ci id="A1.SS2.1.p1.3.m3.1.1.cmml" xref="A1.SS2.1.p1.3.m3.1.1">𝑏</ci></apply><apply id="A1.SS2.1.p1.3.m3.1.2.3.cmml" xref="A1.SS2.1.p1.3.m3.1.2.3"><divide id="A1.SS2.1.p1.3.m3.1.2.3.1.cmml" xref="A1.SS2.1.p1.3.m3.1.2.3.1"></divide><cn id="A1.SS2.1.p1.3.m3.1.2.3.2.cmml" type="integer" xref="A1.SS2.1.p1.3.m3.1.2.3.2">1</cn><cn id="A1.SS2.1.p1.3.m3.1.2.3.3.cmml" type="integer" xref="A1.SS2.1.p1.3.m3.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.3.m3.1c">G(b)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.3.m3.1d">italic_G ( italic_b ) > 1 / 2</annotation></semantics></math>, by continuity and differentiability <math alttext="G" class="ltx_Math" display="inline" id="A1.SS2.1.p1.4.m4.1"><semantics id="A1.SS2.1.p1.4.m4.1a"><mi id="A1.SS2.1.p1.4.m4.1.1" xref="A1.SS2.1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.4.m4.1b"><ci id="A1.SS2.1.p1.4.m4.1.1.cmml" xref="A1.SS2.1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.4.m4.1d">italic_G</annotation></semantics></math> must cross 1/2 at some point <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="A1.SS2.1.p1.5.m5.1"><semantics id="A1.SS2.1.p1.5.m5.1a"><msub id="A1.SS2.1.p1.5.m5.1.1" xref="A1.SS2.1.p1.5.m5.1.1.cmml"><mi id="A1.SS2.1.p1.5.m5.1.1.2" xref="A1.SS2.1.p1.5.m5.1.1.2.cmml">τ</mi><mn id="A1.SS2.1.p1.5.m5.1.1.3" xref="A1.SS2.1.p1.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.5.m5.1b"><apply id="A1.SS2.1.p1.5.m5.1.1.cmml" xref="A1.SS2.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.p1.5.m5.1.1.1.cmml" xref="A1.SS2.1.p1.5.m5.1.1">subscript</csymbol><ci id="A1.SS2.1.p1.5.m5.1.1.2.cmml" xref="A1.SS2.1.p1.5.m5.1.1.2">𝜏</ci><cn id="A1.SS2.1.p1.5.m5.1.1.3.cmml" type="integer" xref="A1.SS2.1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.5.m5.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.5.m5.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="G^{\prime}(\tau_{0})>0" class="ltx_Math" display="inline" id="A1.SS2.1.p1.6.m6.1"><semantics id="A1.SS2.1.p1.6.m6.1a"><mrow id="A1.SS2.1.p1.6.m6.1.1" xref="A1.SS2.1.p1.6.m6.1.1.cmml"><mrow id="A1.SS2.1.p1.6.m6.1.1.1" xref="A1.SS2.1.p1.6.m6.1.1.1.cmml"><msup id="A1.SS2.1.p1.6.m6.1.1.1.3" xref="A1.SS2.1.p1.6.m6.1.1.1.3.cmml"><mi id="A1.SS2.1.p1.6.m6.1.1.1.3.2" xref="A1.SS2.1.p1.6.m6.1.1.1.3.2.cmml">G</mi><mo id="A1.SS2.1.p1.6.m6.1.1.1.3.3" xref="A1.SS2.1.p1.6.m6.1.1.1.3.3.cmml">′</mo></msup><mo id="A1.SS2.1.p1.6.m6.1.1.1.2" xref="A1.SS2.1.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS2.1.p1.6.m6.1.1.1.1.1" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="A1.SS2.1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.2" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.3" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="A1.SS2.1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.SS2.1.p1.6.m6.1.1.2" xref="A1.SS2.1.p1.6.m6.1.1.2.cmml">></mo><mn id="A1.SS2.1.p1.6.m6.1.1.3" xref="A1.SS2.1.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.6.m6.1b"><apply id="A1.SS2.1.p1.6.m6.1.1.cmml" xref="A1.SS2.1.p1.6.m6.1.1"><gt id="A1.SS2.1.p1.6.m6.1.1.2.cmml" xref="A1.SS2.1.p1.6.m6.1.1.2"></gt><apply id="A1.SS2.1.p1.6.m6.1.1.1.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1"><times id="A1.SS2.1.p1.6.m6.1.1.1.2.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.2"></times><apply id="A1.SS2.1.p1.6.m6.1.1.1.3.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.1.p1.6.m6.1.1.1.3.1.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.3">superscript</csymbol><ci id="A1.SS2.1.p1.6.m6.1.1.1.3.2.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.3.2">𝐺</ci><ci id="A1.SS2.1.p1.6.m6.1.1.1.3.3.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.3.3">′</ci></apply><apply id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.2">𝜏</ci><cn id="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.SS2.1.p1.6.m6.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="A1.SS2.1.p1.6.m6.1.1.3.cmml" type="integer" xref="A1.SS2.1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.6.m6.1c">G^{\prime}(\tau_{0})>0</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.6.m6.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) > 0</annotation></semantics></math>. Assume not: then <math alttext="G(x)\leq 1/2" class="ltx_Math" display="inline" id="A1.SS2.1.p1.7.m7.1"><semantics id="A1.SS2.1.p1.7.m7.1a"><mrow id="A1.SS2.1.p1.7.m7.1.2" xref="A1.SS2.1.p1.7.m7.1.2.cmml"><mrow id="A1.SS2.1.p1.7.m7.1.2.2" xref="A1.SS2.1.p1.7.m7.1.2.2.cmml"><mi id="A1.SS2.1.p1.7.m7.1.2.2.2" xref="A1.SS2.1.p1.7.m7.1.2.2.2.cmml">G</mi><mo id="A1.SS2.1.p1.7.m7.1.2.2.1" xref="A1.SS2.1.p1.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.1.p1.7.m7.1.2.2.3.2" xref="A1.SS2.1.p1.7.m7.1.2.2.cmml"><mo id="A1.SS2.1.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.1.p1.7.m7.1.2.2.cmml">(</mo><mi id="A1.SS2.1.p1.7.m7.1.1" xref="A1.SS2.1.p1.7.m7.1.1.cmml">x</mi><mo id="A1.SS2.1.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.1.p1.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.1.p1.7.m7.1.2.1" xref="A1.SS2.1.p1.7.m7.1.2.1.cmml">≤</mo><mrow id="A1.SS2.1.p1.7.m7.1.2.3" xref="A1.SS2.1.p1.7.m7.1.2.3.cmml"><mn id="A1.SS2.1.p1.7.m7.1.2.3.2" xref="A1.SS2.1.p1.7.m7.1.2.3.2.cmml">1</mn><mo id="A1.SS2.1.p1.7.m7.1.2.3.1" xref="A1.SS2.1.p1.7.m7.1.2.3.1.cmml">/</mo><mn id="A1.SS2.1.p1.7.m7.1.2.3.3" xref="A1.SS2.1.p1.7.m7.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.7.m7.1b"><apply id="A1.SS2.1.p1.7.m7.1.2.cmml" xref="A1.SS2.1.p1.7.m7.1.2"><leq id="A1.SS2.1.p1.7.m7.1.2.1.cmml" xref="A1.SS2.1.p1.7.m7.1.2.1"></leq><apply id="A1.SS2.1.p1.7.m7.1.2.2.cmml" xref="A1.SS2.1.p1.7.m7.1.2.2"><times id="A1.SS2.1.p1.7.m7.1.2.2.1.cmml" xref="A1.SS2.1.p1.7.m7.1.2.2.1"></times><ci id="A1.SS2.1.p1.7.m7.1.2.2.2.cmml" xref="A1.SS2.1.p1.7.m7.1.2.2.2">𝐺</ci><ci id="A1.SS2.1.p1.7.m7.1.1.cmml" xref="A1.SS2.1.p1.7.m7.1.1">𝑥</ci></apply><apply id="A1.SS2.1.p1.7.m7.1.2.3.cmml" xref="A1.SS2.1.p1.7.m7.1.2.3"><divide id="A1.SS2.1.p1.7.m7.1.2.3.1.cmml" xref="A1.SS2.1.p1.7.m7.1.2.3.1"></divide><cn id="A1.SS2.1.p1.7.m7.1.2.3.2.cmml" type="integer" xref="A1.SS2.1.p1.7.m7.1.2.3.2">1</cn><cn id="A1.SS2.1.p1.7.m7.1.2.3.3.cmml" type="integer" xref="A1.SS2.1.p1.7.m7.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.7.m7.1c">G(x)\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.7.m7.1d">italic_G ( italic_x ) ≤ 1 / 2</annotation></semantics></math> for all <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.1.p1.8.m8.1"><semantics id="A1.SS2.1.p1.8.m8.1a"><mi id="A1.SS2.1.p1.8.m8.1.1" xref="A1.SS2.1.p1.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.8.m8.1b"><ci id="A1.SS2.1.p1.8.m8.1.1.cmml" xref="A1.SS2.1.p1.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.8.m8.1d">italic_x</annotation></semantics></math>, which contradicts the fact that <math alttext="G(x)>1/2" class="ltx_Math" display="inline" id="A1.SS2.1.p1.9.m9.1"><semantics id="A1.SS2.1.p1.9.m9.1a"><mrow id="A1.SS2.1.p1.9.m9.1.2" xref="A1.SS2.1.p1.9.m9.1.2.cmml"><mrow id="A1.SS2.1.p1.9.m9.1.2.2" xref="A1.SS2.1.p1.9.m9.1.2.2.cmml"><mi id="A1.SS2.1.p1.9.m9.1.2.2.2" xref="A1.SS2.1.p1.9.m9.1.2.2.2.cmml">G</mi><mo id="A1.SS2.1.p1.9.m9.1.2.2.1" xref="A1.SS2.1.p1.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.1.p1.9.m9.1.2.2.3.2" xref="A1.SS2.1.p1.9.m9.1.2.2.cmml"><mo id="A1.SS2.1.p1.9.m9.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.1.p1.9.m9.1.2.2.cmml">(</mo><mi id="A1.SS2.1.p1.9.m9.1.1" xref="A1.SS2.1.p1.9.m9.1.1.cmml">x</mi><mo id="A1.SS2.1.p1.9.m9.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.1.p1.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.1.p1.9.m9.1.2.1" xref="A1.SS2.1.p1.9.m9.1.2.1.cmml">></mo><mrow id="A1.SS2.1.p1.9.m9.1.2.3" xref="A1.SS2.1.p1.9.m9.1.2.3.cmml"><mn id="A1.SS2.1.p1.9.m9.1.2.3.2" xref="A1.SS2.1.p1.9.m9.1.2.3.2.cmml">1</mn><mo id="A1.SS2.1.p1.9.m9.1.2.3.1" xref="A1.SS2.1.p1.9.m9.1.2.3.1.cmml">/</mo><mn id="A1.SS2.1.p1.9.m9.1.2.3.3" xref="A1.SS2.1.p1.9.m9.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.9.m9.1b"><apply id="A1.SS2.1.p1.9.m9.1.2.cmml" xref="A1.SS2.1.p1.9.m9.1.2"><gt id="A1.SS2.1.p1.9.m9.1.2.1.cmml" xref="A1.SS2.1.p1.9.m9.1.2.1"></gt><apply id="A1.SS2.1.p1.9.m9.1.2.2.cmml" xref="A1.SS2.1.p1.9.m9.1.2.2"><times id="A1.SS2.1.p1.9.m9.1.2.2.1.cmml" xref="A1.SS2.1.p1.9.m9.1.2.2.1"></times><ci id="A1.SS2.1.p1.9.m9.1.2.2.2.cmml" xref="A1.SS2.1.p1.9.m9.1.2.2.2">𝐺</ci><ci id="A1.SS2.1.p1.9.m9.1.1.cmml" xref="A1.SS2.1.p1.9.m9.1.1">𝑥</ci></apply><apply id="A1.SS2.1.p1.9.m9.1.2.3.cmml" xref="A1.SS2.1.p1.9.m9.1.2.3"><divide id="A1.SS2.1.p1.9.m9.1.2.3.1.cmml" xref="A1.SS2.1.p1.9.m9.1.2.3.1"></divide><cn id="A1.SS2.1.p1.9.m9.1.2.3.2.cmml" type="integer" xref="A1.SS2.1.p1.9.m9.1.2.3.2">1</cn><cn id="A1.SS2.1.p1.9.m9.1.2.3.3.cmml" type="integer" xref="A1.SS2.1.p1.9.m9.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.9.m9.1c">G(x)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.9.m9.1d">italic_G ( italic_x ) > 1 / 2</annotation></semantics></math> for <math alttext="x\geq b" class="ltx_Math" display="inline" id="A1.SS2.1.p1.10.m10.1"><semantics id="A1.SS2.1.p1.10.m10.1a"><mrow id="A1.SS2.1.p1.10.m10.1.1" xref="A1.SS2.1.p1.10.m10.1.1.cmml"><mi id="A1.SS2.1.p1.10.m10.1.1.2" xref="A1.SS2.1.p1.10.m10.1.1.2.cmml">x</mi><mo id="A1.SS2.1.p1.10.m10.1.1.1" xref="A1.SS2.1.p1.10.m10.1.1.1.cmml">≥</mo><mi id="A1.SS2.1.p1.10.m10.1.1.3" xref="A1.SS2.1.p1.10.m10.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.10.m10.1b"><apply id="A1.SS2.1.p1.10.m10.1.1.cmml" xref="A1.SS2.1.p1.10.m10.1.1"><geq id="A1.SS2.1.p1.10.m10.1.1.1.cmml" xref="A1.SS2.1.p1.10.m10.1.1.1"></geq><ci id="A1.SS2.1.p1.10.m10.1.1.2.cmml" xref="A1.SS2.1.p1.10.m10.1.1.2">𝑥</ci><ci id="A1.SS2.1.p1.10.m10.1.1.3.cmml" xref="A1.SS2.1.p1.10.m10.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.10.m10.1c">x\geq b</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.10.m10.1d">italic_x ≥ italic_b</annotation></semantics></math>. It follows by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem2" title="Theorem 2. ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> that <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="A1.SS2.1.p1.11.m11.1"><semantics id="A1.SS2.1.p1.11.m11.1a"><msub id="A1.SS2.1.p1.11.m11.1.1" xref="A1.SS2.1.p1.11.m11.1.1.cmml"><mi id="A1.SS2.1.p1.11.m11.1.1.2" xref="A1.SS2.1.p1.11.m11.1.1.2.cmml">τ</mi><mn id="A1.SS2.1.p1.11.m11.1.1.3" xref="A1.SS2.1.p1.11.m11.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.11.m11.1b"><apply id="A1.SS2.1.p1.11.m11.1.1.cmml" xref="A1.SS2.1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.p1.11.m11.1.1.1.cmml" xref="A1.SS2.1.p1.11.m11.1.1">subscript</csymbol><ci id="A1.SS2.1.p1.11.m11.1.1.2.cmml" xref="A1.SS2.1.p1.11.m11.1.1.2">𝜏</ci><cn id="A1.SS2.1.p1.11.m11.1.1.3.cmml" type="integer" xref="A1.SS2.1.p1.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.11.m11.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.1.p1.11.m11.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is an unstable equilibrium.</p> </div> <div class="ltx_para" id="A1.SS2.2.p2"> <ol class="ltx_enumerate" id="A1.I1"> <li class="ltx_item" id="A1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="A1.I1.i1.p1"> <p class="ltx_p" id="A1.I1.i1.p1.9">Now, WLOG we pick the equilibrium <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.I1.i1.p1.1.m1.1"><semantics id="A1.I1.i1.p1.1.m1.1a"><msub id="A1.I1.i1.p1.1.m1.1.1" xref="A1.I1.i1.p1.1.m1.1.1.cmml"><mi id="A1.I1.i1.p1.1.m1.1.1.2" xref="A1.I1.i1.p1.1.m1.1.1.2.cmml">τ</mi><mn id="A1.I1.i1.p1.1.m1.1.1.3" xref="A1.I1.i1.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.1.m1.1b"><apply id="A1.I1.i1.p1.1.m1.1.1.cmml" xref="A1.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.I1.i1.p1.1.m1.1.1.1.cmml" xref="A1.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.I1.i1.p1.1.m1.1.1.2.cmml" xref="A1.I1.i1.p1.1.m1.1.1.2">𝜏</ci><cn id="A1.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.1.m1.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.1.m1.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> which is closest to <math alttext="a" class="ltx_Math" display="inline" id="A1.I1.i1.p1.2.m2.1"><semantics id="A1.I1.i1.p1.2.m2.1a"><mi id="A1.I1.i1.p1.2.m2.1.1" xref="A1.I1.i1.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.2.m2.1b"><ci id="A1.I1.i1.p1.2.m2.1.1.cmml" xref="A1.I1.i1.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.2.m2.1d">italic_a</annotation></semantics></math>. By Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> and since <math alttext="G(x)" class="ltx_Math" display="inline" id="A1.I1.i1.p1.3.m3.1"><semantics id="A1.I1.i1.p1.3.m3.1a"><mrow id="A1.I1.i1.p1.3.m3.1.2" xref="A1.I1.i1.p1.3.m3.1.2.cmml"><mi id="A1.I1.i1.p1.3.m3.1.2.2" xref="A1.I1.i1.p1.3.m3.1.2.2.cmml">G</mi><mo id="A1.I1.i1.p1.3.m3.1.2.1" xref="A1.I1.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="A1.I1.i1.p1.3.m3.1.2.3.2" xref="A1.I1.i1.p1.3.m3.1.2.cmml"><mo id="A1.I1.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="A1.I1.i1.p1.3.m3.1.2.cmml">(</mo><mi id="A1.I1.i1.p1.3.m3.1.1" xref="A1.I1.i1.p1.3.m3.1.1.cmml">x</mi><mo id="A1.I1.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="A1.I1.i1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.3.m3.1b"><apply id="A1.I1.i1.p1.3.m3.1.2.cmml" xref="A1.I1.i1.p1.3.m3.1.2"><times id="A1.I1.i1.p1.3.m3.1.2.1.cmml" xref="A1.I1.i1.p1.3.m3.1.2.1"></times><ci id="A1.I1.i1.p1.3.m3.1.2.2.cmml" xref="A1.I1.i1.p1.3.m3.1.2.2">𝐺</ci><ci id="A1.I1.i1.p1.3.m3.1.1.cmml" xref="A1.I1.i1.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.3.m3.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.3.m3.1d">italic_G ( italic_x )</annotation></semantics></math> is continuous, <math alttext="G(x)<1/2" class="ltx_Math" display="inline" id="A1.I1.i1.p1.4.m4.1"><semantics id="A1.I1.i1.p1.4.m4.1a"><mrow id="A1.I1.i1.p1.4.m4.1.2" xref="A1.I1.i1.p1.4.m4.1.2.cmml"><mrow id="A1.I1.i1.p1.4.m4.1.2.2" xref="A1.I1.i1.p1.4.m4.1.2.2.cmml"><mi id="A1.I1.i1.p1.4.m4.1.2.2.2" xref="A1.I1.i1.p1.4.m4.1.2.2.2.cmml">G</mi><mo id="A1.I1.i1.p1.4.m4.1.2.2.1" xref="A1.I1.i1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="A1.I1.i1.p1.4.m4.1.2.2.3.2" xref="A1.I1.i1.p1.4.m4.1.2.2.cmml"><mo id="A1.I1.i1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="A1.I1.i1.p1.4.m4.1.2.2.cmml">(</mo><mi id="A1.I1.i1.p1.4.m4.1.1" xref="A1.I1.i1.p1.4.m4.1.1.cmml">x</mi><mo id="A1.I1.i1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="A1.I1.i1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.I1.i1.p1.4.m4.1.2.1" xref="A1.I1.i1.p1.4.m4.1.2.1.cmml"><</mo><mrow id="A1.I1.i1.p1.4.m4.1.2.3" xref="A1.I1.i1.p1.4.m4.1.2.3.cmml"><mn id="A1.I1.i1.p1.4.m4.1.2.3.2" xref="A1.I1.i1.p1.4.m4.1.2.3.2.cmml">1</mn><mo id="A1.I1.i1.p1.4.m4.1.2.3.1" xref="A1.I1.i1.p1.4.m4.1.2.3.1.cmml">/</mo><mn id="A1.I1.i1.p1.4.m4.1.2.3.3" xref="A1.I1.i1.p1.4.m4.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.4.m4.1b"><apply id="A1.I1.i1.p1.4.m4.1.2.cmml" xref="A1.I1.i1.p1.4.m4.1.2"><lt id="A1.I1.i1.p1.4.m4.1.2.1.cmml" xref="A1.I1.i1.p1.4.m4.1.2.1"></lt><apply id="A1.I1.i1.p1.4.m4.1.2.2.cmml" xref="A1.I1.i1.p1.4.m4.1.2.2"><times id="A1.I1.i1.p1.4.m4.1.2.2.1.cmml" xref="A1.I1.i1.p1.4.m4.1.2.2.1"></times><ci id="A1.I1.i1.p1.4.m4.1.2.2.2.cmml" xref="A1.I1.i1.p1.4.m4.1.2.2.2">𝐺</ci><ci id="A1.I1.i1.p1.4.m4.1.1.cmml" xref="A1.I1.i1.p1.4.m4.1.1">𝑥</ci></apply><apply id="A1.I1.i1.p1.4.m4.1.2.3.cmml" xref="A1.I1.i1.p1.4.m4.1.2.3"><divide id="A1.I1.i1.p1.4.m4.1.2.3.1.cmml" xref="A1.I1.i1.p1.4.m4.1.2.3.1"></divide><cn id="A1.I1.i1.p1.4.m4.1.2.3.2.cmml" type="integer" xref="A1.I1.i1.p1.4.m4.1.2.3.2">1</cn><cn id="A1.I1.i1.p1.4.m4.1.2.3.3.cmml" type="integer" xref="A1.I1.i1.p1.4.m4.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.4.m4.1c">G(x)<1/2</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.4.m4.1d">italic_G ( italic_x ) < 1 / 2</annotation></semantics></math> for <math alttext="x<\tau_{1}" class="ltx_Math" display="inline" id="A1.I1.i1.p1.5.m5.1"><semantics id="A1.I1.i1.p1.5.m5.1a"><mrow id="A1.I1.i1.p1.5.m5.1.1" xref="A1.I1.i1.p1.5.m5.1.1.cmml"><mi id="A1.I1.i1.p1.5.m5.1.1.2" xref="A1.I1.i1.p1.5.m5.1.1.2.cmml">x</mi><mo id="A1.I1.i1.p1.5.m5.1.1.1" xref="A1.I1.i1.p1.5.m5.1.1.1.cmml"><</mo><msub id="A1.I1.i1.p1.5.m5.1.1.3" xref="A1.I1.i1.p1.5.m5.1.1.3.cmml"><mi id="A1.I1.i1.p1.5.m5.1.1.3.2" xref="A1.I1.i1.p1.5.m5.1.1.3.2.cmml">τ</mi><mn id="A1.I1.i1.p1.5.m5.1.1.3.3" xref="A1.I1.i1.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.5.m5.1b"><apply id="A1.I1.i1.p1.5.m5.1.1.cmml" xref="A1.I1.i1.p1.5.m5.1.1"><lt id="A1.I1.i1.p1.5.m5.1.1.1.cmml" xref="A1.I1.i1.p1.5.m5.1.1.1"></lt><ci id="A1.I1.i1.p1.5.m5.1.1.2.cmml" xref="A1.I1.i1.p1.5.m5.1.1.2">𝑥</ci><apply id="A1.I1.i1.p1.5.m5.1.1.3.cmml" xref="A1.I1.i1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i1.p1.5.m5.1.1.3.1.cmml" xref="A1.I1.i1.p1.5.m5.1.1.3">subscript</csymbol><ci id="A1.I1.i1.p1.5.m5.1.1.3.2.cmml" xref="A1.I1.i1.p1.5.m5.1.1.3.2">𝜏</ci><cn id="A1.I1.i1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="A1.I1.i1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.5.m5.1c">x<\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.5.m5.1d">italic_x < italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then it must follow that <math alttext="G^{\prime}(\tau_{1})\geq 0" class="ltx_Math" display="inline" id="A1.I1.i1.p1.6.m6.1"><semantics id="A1.I1.i1.p1.6.m6.1a"><mrow id="A1.I1.i1.p1.6.m6.1.1" xref="A1.I1.i1.p1.6.m6.1.1.cmml"><mrow id="A1.I1.i1.p1.6.m6.1.1.1" xref="A1.I1.i1.p1.6.m6.1.1.1.cmml"><msup id="A1.I1.i1.p1.6.m6.1.1.1.3" xref="A1.I1.i1.p1.6.m6.1.1.1.3.cmml"><mi id="A1.I1.i1.p1.6.m6.1.1.1.3.2" xref="A1.I1.i1.p1.6.m6.1.1.1.3.2.cmml">G</mi><mo id="A1.I1.i1.p1.6.m6.1.1.1.3.3" xref="A1.I1.i1.p1.6.m6.1.1.1.3.3.cmml">′</mo></msup><mo id="A1.I1.i1.p1.6.m6.1.1.1.2" xref="A1.I1.i1.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="A1.I1.i1.p1.6.m6.1.1.1.1.1" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="A1.I1.i1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.2" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.3" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A1.I1.i1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.I1.i1.p1.6.m6.1.1.2" xref="A1.I1.i1.p1.6.m6.1.1.2.cmml">≥</mo><mn id="A1.I1.i1.p1.6.m6.1.1.3" xref="A1.I1.i1.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.6.m6.1b"><apply id="A1.I1.i1.p1.6.m6.1.1.cmml" xref="A1.I1.i1.p1.6.m6.1.1"><geq id="A1.I1.i1.p1.6.m6.1.1.2.cmml" xref="A1.I1.i1.p1.6.m6.1.1.2"></geq><apply id="A1.I1.i1.p1.6.m6.1.1.1.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1"><times id="A1.I1.i1.p1.6.m6.1.1.1.2.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.2"></times><apply id="A1.I1.i1.p1.6.m6.1.1.1.3.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i1.p1.6.m6.1.1.1.3.1.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.3">superscript</csymbol><ci id="A1.I1.i1.p1.6.m6.1.1.1.3.2.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.3.2">𝐺</ci><ci id="A1.I1.i1.p1.6.m6.1.1.1.3.3.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.3.3">′</ci></apply><apply id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.2">𝜏</ci><cn id="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.6.m6.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="A1.I1.i1.p1.6.m6.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.6.m6.1c">G^{\prime}(\tau_{1})\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.6.m6.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≥ 0</annotation></semantics></math>. Thus <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.I1.i1.p1.7.m7.1"><semantics id="A1.I1.i1.p1.7.m7.1a"><msub id="A1.I1.i1.p1.7.m7.1.1" xref="A1.I1.i1.p1.7.m7.1.1.cmml"><mi id="A1.I1.i1.p1.7.m7.1.1.2" xref="A1.I1.i1.p1.7.m7.1.1.2.cmml">τ</mi><mn id="A1.I1.i1.p1.7.m7.1.1.3" xref="A1.I1.i1.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.7.m7.1b"><apply id="A1.I1.i1.p1.7.m7.1.1.cmml" xref="A1.I1.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A1.I1.i1.p1.7.m7.1.1.1.cmml" xref="A1.I1.i1.p1.7.m7.1.1">subscript</csymbol><ci id="A1.I1.i1.p1.7.m7.1.1.2.cmml" xref="A1.I1.i1.p1.7.m7.1.1.2">𝜏</ci><cn id="A1.I1.i1.p1.7.m7.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.7.m7.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.7.m7.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is unstable, at least on the left (if <math alttext="G^{\prime}(\tau_{1})=0" class="ltx_Math" display="inline" id="A1.I1.i1.p1.8.m8.1"><semantics id="A1.I1.i1.p1.8.m8.1a"><mrow id="A1.I1.i1.p1.8.m8.1.1" xref="A1.I1.i1.p1.8.m8.1.1.cmml"><mrow id="A1.I1.i1.p1.8.m8.1.1.1" xref="A1.I1.i1.p1.8.m8.1.1.1.cmml"><msup id="A1.I1.i1.p1.8.m8.1.1.1.3" xref="A1.I1.i1.p1.8.m8.1.1.1.3.cmml"><mi id="A1.I1.i1.p1.8.m8.1.1.1.3.2" xref="A1.I1.i1.p1.8.m8.1.1.1.3.2.cmml">G</mi><mo id="A1.I1.i1.p1.8.m8.1.1.1.3.3" xref="A1.I1.i1.p1.8.m8.1.1.1.3.3.cmml">′</mo></msup><mo id="A1.I1.i1.p1.8.m8.1.1.1.2" xref="A1.I1.i1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="A1.I1.i1.p1.8.m8.1.1.1.1.1" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="A1.I1.i1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.2" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.3" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A1.I1.i1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.I1.i1.p1.8.m8.1.1.2" xref="A1.I1.i1.p1.8.m8.1.1.2.cmml">=</mo><mn id="A1.I1.i1.p1.8.m8.1.1.3" xref="A1.I1.i1.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.8.m8.1b"><apply id="A1.I1.i1.p1.8.m8.1.1.cmml" xref="A1.I1.i1.p1.8.m8.1.1"><eq id="A1.I1.i1.p1.8.m8.1.1.2.cmml" xref="A1.I1.i1.p1.8.m8.1.1.2"></eq><apply id="A1.I1.i1.p1.8.m8.1.1.1.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1"><times id="A1.I1.i1.p1.8.m8.1.1.1.2.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.2"></times><apply id="A1.I1.i1.p1.8.m8.1.1.1.3.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i1.p1.8.m8.1.1.1.3.1.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.3">superscript</csymbol><ci id="A1.I1.i1.p1.8.m8.1.1.1.3.2.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.3.2">𝐺</ci><ci id="A1.I1.i1.p1.8.m8.1.1.1.3.3.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.3.3">′</ci></apply><apply id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.2">𝜏</ci><cn id="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.8.m8.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="A1.I1.i1.p1.8.m8.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.8.m8.1c">G^{\prime}(\tau_{1})=0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.8.m8.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, then <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.I1.i1.p1.9.m9.1"><semantics id="A1.I1.i1.p1.9.m9.1a"><msub id="A1.I1.i1.p1.9.m9.1.1" xref="A1.I1.i1.p1.9.m9.1.1.cmml"><mi id="A1.I1.i1.p1.9.m9.1.1.2" xref="A1.I1.i1.p1.9.m9.1.1.2.cmml">τ</mi><mn id="A1.I1.i1.p1.9.m9.1.1.3" xref="A1.I1.i1.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i1.p1.9.m9.1b"><apply id="A1.I1.i1.p1.9.m9.1.1.cmml" xref="A1.I1.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A1.I1.i1.p1.9.m9.1.1.1.cmml" xref="A1.I1.i1.p1.9.m9.1.1">subscript</csymbol><ci id="A1.I1.i1.p1.9.m9.1.1.2.cmml" xref="A1.I1.i1.p1.9.m9.1.1.2">𝜏</ci><cn id="A1.I1.i1.p1.9.m9.1.1.3.cmml" type="integer" xref="A1.I1.i1.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.9.m9.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.9.m9.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is unstable on the left and stable on the right).</p> </div> </li> <li class="ltx_item" id="A1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="A1.I1.i2.p1"> <p class="ltx_p" id="A1.I1.i2.p1.10">Pick the equilibrium <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.1.m1.1"><semantics id="A1.I1.i2.p1.1.m1.1a"><msub id="A1.I1.i2.p1.1.m1.1.1" xref="A1.I1.i2.p1.1.m1.1.1.cmml"><mi id="A1.I1.i2.p1.1.m1.1.1.2" xref="A1.I1.i2.p1.1.m1.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.1.m1.1.1.3" xref="A1.I1.i2.p1.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.1.m1.1b"><apply id="A1.I1.i2.p1.1.m1.1.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.1.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.1.m1.1.1.2.cmml" xref="A1.I1.i2.p1.1.m1.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.1.m1.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.1.m1.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> which is closest to <math alttext="b" class="ltx_Math" display="inline" id="A1.I1.i2.p1.2.m2.1"><semantics id="A1.I1.i2.p1.2.m2.1a"><mi id="A1.I1.i2.p1.2.m2.1.1" xref="A1.I1.i2.p1.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.2.m2.1b"><ci id="A1.I1.i2.p1.2.m2.1.1.cmml" xref="A1.I1.i2.p1.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.2.m2.1d">italic_b</annotation></semantics></math> (this could be the same as the <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.3.m3.1"><semantics id="A1.I1.i2.p1.3.m3.1a"><msub id="A1.I1.i2.p1.3.m3.1.1" xref="A1.I1.i2.p1.3.m3.1.1.cmml"><mi id="A1.I1.i2.p1.3.m3.1.1.2" xref="A1.I1.i2.p1.3.m3.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.3.m3.1.1.3" xref="A1.I1.i2.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.3.m3.1b"><apply id="A1.I1.i2.p1.3.m3.1.1.cmml" xref="A1.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.3.m3.1.1.1.cmml" xref="A1.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.3.m3.1.1.2.cmml" xref="A1.I1.i2.p1.3.m3.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.3.m3.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.3.m3.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we chose in the previous step). By Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> and since <math alttext="G(x)" class="ltx_Math" display="inline" id="A1.I1.i2.p1.4.m4.1"><semantics id="A1.I1.i2.p1.4.m4.1a"><mrow id="A1.I1.i2.p1.4.m4.1.2" xref="A1.I1.i2.p1.4.m4.1.2.cmml"><mi id="A1.I1.i2.p1.4.m4.1.2.2" xref="A1.I1.i2.p1.4.m4.1.2.2.cmml">G</mi><mo id="A1.I1.i2.p1.4.m4.1.2.1" xref="A1.I1.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.I1.i2.p1.4.m4.1.2.3.2" xref="A1.I1.i2.p1.4.m4.1.2.cmml"><mo id="A1.I1.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="A1.I1.i2.p1.4.m4.1.2.cmml">(</mo><mi id="A1.I1.i2.p1.4.m4.1.1" xref="A1.I1.i2.p1.4.m4.1.1.cmml">x</mi><mo id="A1.I1.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="A1.I1.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.4.m4.1b"><apply id="A1.I1.i2.p1.4.m4.1.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2"><times id="A1.I1.i2.p1.4.m4.1.2.1.cmml" xref="A1.I1.i2.p1.4.m4.1.2.1"></times><ci id="A1.I1.i2.p1.4.m4.1.2.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2.2">𝐺</ci><ci id="A1.I1.i2.p1.4.m4.1.1.cmml" xref="A1.I1.i2.p1.4.m4.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.4.m4.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.4.m4.1d">italic_G ( italic_x )</annotation></semantics></math> is continuous, <math alttext="G(x)>1/2" class="ltx_Math" display="inline" id="A1.I1.i2.p1.5.m5.1"><semantics id="A1.I1.i2.p1.5.m5.1a"><mrow id="A1.I1.i2.p1.5.m5.1.2" xref="A1.I1.i2.p1.5.m5.1.2.cmml"><mrow id="A1.I1.i2.p1.5.m5.1.2.2" xref="A1.I1.i2.p1.5.m5.1.2.2.cmml"><mi id="A1.I1.i2.p1.5.m5.1.2.2.2" xref="A1.I1.i2.p1.5.m5.1.2.2.2.cmml">G</mi><mo id="A1.I1.i2.p1.5.m5.1.2.2.1" xref="A1.I1.i2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="A1.I1.i2.p1.5.m5.1.2.2.3.2" xref="A1.I1.i2.p1.5.m5.1.2.2.cmml"><mo id="A1.I1.i2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="A1.I1.i2.p1.5.m5.1.2.2.cmml">(</mo><mi id="A1.I1.i2.p1.5.m5.1.1" xref="A1.I1.i2.p1.5.m5.1.1.cmml">x</mi><mo id="A1.I1.i2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="A1.I1.i2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.I1.i2.p1.5.m5.1.2.1" xref="A1.I1.i2.p1.5.m5.1.2.1.cmml">></mo><mrow id="A1.I1.i2.p1.5.m5.1.2.3" xref="A1.I1.i2.p1.5.m5.1.2.3.cmml"><mn id="A1.I1.i2.p1.5.m5.1.2.3.2" xref="A1.I1.i2.p1.5.m5.1.2.3.2.cmml">1</mn><mo id="A1.I1.i2.p1.5.m5.1.2.3.1" xref="A1.I1.i2.p1.5.m5.1.2.3.1.cmml">/</mo><mn id="A1.I1.i2.p1.5.m5.1.2.3.3" xref="A1.I1.i2.p1.5.m5.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.5.m5.1b"><apply id="A1.I1.i2.p1.5.m5.1.2.cmml" xref="A1.I1.i2.p1.5.m5.1.2"><gt id="A1.I1.i2.p1.5.m5.1.2.1.cmml" xref="A1.I1.i2.p1.5.m5.1.2.1"></gt><apply id="A1.I1.i2.p1.5.m5.1.2.2.cmml" xref="A1.I1.i2.p1.5.m5.1.2.2"><times id="A1.I1.i2.p1.5.m5.1.2.2.1.cmml" xref="A1.I1.i2.p1.5.m5.1.2.2.1"></times><ci id="A1.I1.i2.p1.5.m5.1.2.2.2.cmml" xref="A1.I1.i2.p1.5.m5.1.2.2.2">𝐺</ci><ci id="A1.I1.i2.p1.5.m5.1.1.cmml" xref="A1.I1.i2.p1.5.m5.1.1">𝑥</ci></apply><apply id="A1.I1.i2.p1.5.m5.1.2.3.cmml" xref="A1.I1.i2.p1.5.m5.1.2.3"><divide id="A1.I1.i2.p1.5.m5.1.2.3.1.cmml" xref="A1.I1.i2.p1.5.m5.1.2.3.1"></divide><cn id="A1.I1.i2.p1.5.m5.1.2.3.2.cmml" type="integer" xref="A1.I1.i2.p1.5.m5.1.2.3.2">1</cn><cn id="A1.I1.i2.p1.5.m5.1.2.3.3.cmml" type="integer" xref="A1.I1.i2.p1.5.m5.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.5.m5.1c">G(x)>1/2</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.5.m5.1d">italic_G ( italic_x ) > 1 / 2</annotation></semantics></math> for <math alttext="x>\tau_{2}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.6.m6.1"><semantics id="A1.I1.i2.p1.6.m6.1a"><mrow id="A1.I1.i2.p1.6.m6.1.1" xref="A1.I1.i2.p1.6.m6.1.1.cmml"><mi id="A1.I1.i2.p1.6.m6.1.1.2" xref="A1.I1.i2.p1.6.m6.1.1.2.cmml">x</mi><mo id="A1.I1.i2.p1.6.m6.1.1.1" xref="A1.I1.i2.p1.6.m6.1.1.1.cmml">></mo><msub id="A1.I1.i2.p1.6.m6.1.1.3" xref="A1.I1.i2.p1.6.m6.1.1.3.cmml"><mi id="A1.I1.i2.p1.6.m6.1.1.3.2" xref="A1.I1.i2.p1.6.m6.1.1.3.2.cmml">τ</mi><mn id="A1.I1.i2.p1.6.m6.1.1.3.3" xref="A1.I1.i2.p1.6.m6.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.6.m6.1b"><apply id="A1.I1.i2.p1.6.m6.1.1.cmml" xref="A1.I1.i2.p1.6.m6.1.1"><gt id="A1.I1.i2.p1.6.m6.1.1.1.cmml" xref="A1.I1.i2.p1.6.m6.1.1.1"></gt><ci id="A1.I1.i2.p1.6.m6.1.1.2.cmml" xref="A1.I1.i2.p1.6.m6.1.1.2">𝑥</ci><apply id="A1.I1.i2.p1.6.m6.1.1.3.cmml" xref="A1.I1.i2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i2.p1.6.m6.1.1.3.1.cmml" xref="A1.I1.i2.p1.6.m6.1.1.3">subscript</csymbol><ci id="A1.I1.i2.p1.6.m6.1.1.3.2.cmml" xref="A1.I1.i2.p1.6.m6.1.1.3.2">𝜏</ci><cn id="A1.I1.i2.p1.6.m6.1.1.3.3.cmml" type="integer" xref="A1.I1.i2.p1.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.6.m6.1c">x>\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.6.m6.1d">italic_x > italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Then it must follow that <math alttext="G^{\prime}(\tau_{2})\geq 0" class="ltx_Math" display="inline" id="A1.I1.i2.p1.7.m7.1"><semantics id="A1.I1.i2.p1.7.m7.1a"><mrow id="A1.I1.i2.p1.7.m7.1.1" xref="A1.I1.i2.p1.7.m7.1.1.cmml"><mrow id="A1.I1.i2.p1.7.m7.1.1.1" xref="A1.I1.i2.p1.7.m7.1.1.1.cmml"><msup id="A1.I1.i2.p1.7.m7.1.1.1.3" xref="A1.I1.i2.p1.7.m7.1.1.1.3.cmml"><mi id="A1.I1.i2.p1.7.m7.1.1.1.3.2" xref="A1.I1.i2.p1.7.m7.1.1.1.3.2.cmml">G</mi><mo id="A1.I1.i2.p1.7.m7.1.1.1.3.3" xref="A1.I1.i2.p1.7.m7.1.1.1.3.3.cmml">′</mo></msup><mo id="A1.I1.i2.p1.7.m7.1.1.1.2" xref="A1.I1.i2.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="A1.I1.i2.p1.7.m7.1.1.1.1.1" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="A1.I1.i2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.2" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.3" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="A1.I1.i2.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.I1.i2.p1.7.m7.1.1.2" xref="A1.I1.i2.p1.7.m7.1.1.2.cmml">≥</mo><mn id="A1.I1.i2.p1.7.m7.1.1.3" xref="A1.I1.i2.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.7.m7.1b"><apply id="A1.I1.i2.p1.7.m7.1.1.cmml" xref="A1.I1.i2.p1.7.m7.1.1"><geq id="A1.I1.i2.p1.7.m7.1.1.2.cmml" xref="A1.I1.i2.p1.7.m7.1.1.2"></geq><apply id="A1.I1.i2.p1.7.m7.1.1.1.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1"><times id="A1.I1.i2.p1.7.m7.1.1.1.2.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.2"></times><apply id="A1.I1.i2.p1.7.m7.1.1.1.3.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i2.p1.7.m7.1.1.1.3.1.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.3">superscript</csymbol><ci id="A1.I1.i2.p1.7.m7.1.1.1.3.2.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.3.2">𝐺</ci><ci id="A1.I1.i2.p1.7.m7.1.1.1.3.3.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.3.3">′</ci></apply><apply id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.7.m7.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="A1.I1.i2.p1.7.m7.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.7.m7.1c">G^{\prime}(\tau_{2})\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.7.m7.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 0</annotation></semantics></math>. Thus <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.8.m8.1"><semantics id="A1.I1.i2.p1.8.m8.1a"><msub id="A1.I1.i2.p1.8.m8.1.1" xref="A1.I1.i2.p1.8.m8.1.1.cmml"><mi id="A1.I1.i2.p1.8.m8.1.1.2" xref="A1.I1.i2.p1.8.m8.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.8.m8.1.1.3" xref="A1.I1.i2.p1.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.8.m8.1b"><apply id="A1.I1.i2.p1.8.m8.1.1.cmml" xref="A1.I1.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.8.m8.1.1.1.cmml" xref="A1.I1.i2.p1.8.m8.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.8.m8.1.1.2.cmml" xref="A1.I1.i2.p1.8.m8.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.8.m8.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.8.m8.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.8.m8.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is unstable, at least on the right (if <math alttext="G^{\prime}(\tau_{2})=0" class="ltx_Math" display="inline" id="A1.I1.i2.p1.9.m9.1"><semantics id="A1.I1.i2.p1.9.m9.1a"><mrow id="A1.I1.i2.p1.9.m9.1.1" xref="A1.I1.i2.p1.9.m9.1.1.cmml"><mrow id="A1.I1.i2.p1.9.m9.1.1.1" xref="A1.I1.i2.p1.9.m9.1.1.1.cmml"><msup id="A1.I1.i2.p1.9.m9.1.1.1.3" xref="A1.I1.i2.p1.9.m9.1.1.1.3.cmml"><mi id="A1.I1.i2.p1.9.m9.1.1.1.3.2" xref="A1.I1.i2.p1.9.m9.1.1.1.3.2.cmml">G</mi><mo id="A1.I1.i2.p1.9.m9.1.1.1.3.3" xref="A1.I1.i2.p1.9.m9.1.1.1.3.3.cmml">′</mo></msup><mo id="A1.I1.i2.p1.9.m9.1.1.1.2" xref="A1.I1.i2.p1.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="A1.I1.i2.p1.9.m9.1.1.1.1.1" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml"><mo id="A1.I1.i2.p1.9.m9.1.1.1.1.1.2" stretchy="false" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml"><mi id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.2" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.3" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="A1.I1.i2.p1.9.m9.1.1.1.1.1.3" stretchy="false" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.I1.i2.p1.9.m9.1.1.2" xref="A1.I1.i2.p1.9.m9.1.1.2.cmml">=</mo><mn id="A1.I1.i2.p1.9.m9.1.1.3" xref="A1.I1.i2.p1.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.9.m9.1b"><apply id="A1.I1.i2.p1.9.m9.1.1.cmml" xref="A1.I1.i2.p1.9.m9.1.1"><eq id="A1.I1.i2.p1.9.m9.1.1.2.cmml" xref="A1.I1.i2.p1.9.m9.1.1.2"></eq><apply id="A1.I1.i2.p1.9.m9.1.1.1.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1"><times id="A1.I1.i2.p1.9.m9.1.1.1.2.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.2"></times><apply id="A1.I1.i2.p1.9.m9.1.1.1.3.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="A1.I1.i2.p1.9.m9.1.1.1.3.1.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.3">superscript</csymbol><ci id="A1.I1.i2.p1.9.m9.1.1.1.3.2.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.3.2">𝐺</ci><ci id="A1.I1.i2.p1.9.m9.1.1.1.3.3.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.3.3">′</ci></apply><apply id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.1.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.2.cmml" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.9.m9.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="A1.I1.i2.p1.9.m9.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.9.m9.1c">G^{\prime}(\tau_{2})=0</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.9.m9.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, then <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.10.m10.1"><semantics id="A1.I1.i2.p1.10.m10.1a"><msub id="A1.I1.i2.p1.10.m10.1.1" xref="A1.I1.i2.p1.10.m10.1.1.cmml"><mi id="A1.I1.i2.p1.10.m10.1.1.2" xref="A1.I1.i2.p1.10.m10.1.1.2.cmml">τ</mi><mn id="A1.I1.i2.p1.10.m10.1.1.3" xref="A1.I1.i2.p1.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.10.m10.1b"><apply id="A1.I1.i2.p1.10.m10.1.1.cmml" xref="A1.I1.i2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.10.m10.1.1.1.cmml" xref="A1.I1.i2.p1.10.m10.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.10.m10.1.1.2.cmml" xref="A1.I1.i2.p1.10.m10.1.1.2">𝜏</ci><cn id="A1.I1.i2.p1.10.m10.1.1.3.cmml" type="integer" xref="A1.I1.i2.p1.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.10.m10.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.10.m10.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is unstable on the right and stable on the left).</p> </div> </li> </ol> <p class="ltx_p" id="A1.SS2.2.p2.10">We know by Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> that <math alttext="G(x)\neq 1/2" class="ltx_Math" display="inline" id="A1.SS2.2.p2.1.m1.1"><semantics id="A1.SS2.2.p2.1.m1.1a"><mrow id="A1.SS2.2.p2.1.m1.1.2" xref="A1.SS2.2.p2.1.m1.1.2.cmml"><mrow id="A1.SS2.2.p2.1.m1.1.2.2" xref="A1.SS2.2.p2.1.m1.1.2.2.cmml"><mi id="A1.SS2.2.p2.1.m1.1.2.2.2" xref="A1.SS2.2.p2.1.m1.1.2.2.2.cmml">G</mi><mo id="A1.SS2.2.p2.1.m1.1.2.2.1" xref="A1.SS2.2.p2.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="A1.SS2.2.p2.1.m1.1.2.2.3.2" xref="A1.SS2.2.p2.1.m1.1.2.2.cmml"><mo id="A1.SS2.2.p2.1.m1.1.2.2.3.2.1" stretchy="false" xref="A1.SS2.2.p2.1.m1.1.2.2.cmml">(</mo><mi id="A1.SS2.2.p2.1.m1.1.1" xref="A1.SS2.2.p2.1.m1.1.1.cmml">x</mi><mo id="A1.SS2.2.p2.1.m1.1.2.2.3.2.2" stretchy="false" xref="A1.SS2.2.p2.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.SS2.2.p2.1.m1.1.2.1" xref="A1.SS2.2.p2.1.m1.1.2.1.cmml">≠</mo><mrow id="A1.SS2.2.p2.1.m1.1.2.3" xref="A1.SS2.2.p2.1.m1.1.2.3.cmml"><mn id="A1.SS2.2.p2.1.m1.1.2.3.2" xref="A1.SS2.2.p2.1.m1.1.2.3.2.cmml">1</mn><mo id="A1.SS2.2.p2.1.m1.1.2.3.1" xref="A1.SS2.2.p2.1.m1.1.2.3.1.cmml">/</mo><mn id="A1.SS2.2.p2.1.m1.1.2.3.3" xref="A1.SS2.2.p2.1.m1.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.1.m1.1b"><apply id="A1.SS2.2.p2.1.m1.1.2.cmml" xref="A1.SS2.2.p2.1.m1.1.2"><neq id="A1.SS2.2.p2.1.m1.1.2.1.cmml" xref="A1.SS2.2.p2.1.m1.1.2.1"></neq><apply id="A1.SS2.2.p2.1.m1.1.2.2.cmml" xref="A1.SS2.2.p2.1.m1.1.2.2"><times id="A1.SS2.2.p2.1.m1.1.2.2.1.cmml" xref="A1.SS2.2.p2.1.m1.1.2.2.1"></times><ci id="A1.SS2.2.p2.1.m1.1.2.2.2.cmml" xref="A1.SS2.2.p2.1.m1.1.2.2.2">𝐺</ci><ci id="A1.SS2.2.p2.1.m1.1.1.cmml" xref="A1.SS2.2.p2.1.m1.1.1">𝑥</ci></apply><apply id="A1.SS2.2.p2.1.m1.1.2.3.cmml" xref="A1.SS2.2.p2.1.m1.1.2.3"><divide id="A1.SS2.2.p2.1.m1.1.2.3.1.cmml" xref="A1.SS2.2.p2.1.m1.1.2.3.1"></divide><cn id="A1.SS2.2.p2.1.m1.1.2.3.2.cmml" type="integer" xref="A1.SS2.2.p2.1.m1.1.2.3.2">1</cn><cn id="A1.SS2.2.p2.1.m1.1.2.3.3.cmml" type="integer" xref="A1.SS2.2.p2.1.m1.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.1.m1.1c">G(x)\neq 1/2</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.1.m1.1d">italic_G ( italic_x ) ≠ 1 / 2</annotation></semantics></math> for <math alttext="x\notin[a,b]" class="ltx_Math" display="inline" id="A1.SS2.2.p2.2.m2.2"><semantics id="A1.SS2.2.p2.2.m2.2a"><mrow id="A1.SS2.2.p2.2.m2.2.3" xref="A1.SS2.2.p2.2.m2.2.3.cmml"><mi id="A1.SS2.2.p2.2.m2.2.3.2" xref="A1.SS2.2.p2.2.m2.2.3.2.cmml">x</mi><mo id="A1.SS2.2.p2.2.m2.2.3.1" xref="A1.SS2.2.p2.2.m2.2.3.1.cmml">∉</mo><mrow id="A1.SS2.2.p2.2.m2.2.3.3.2" xref="A1.SS2.2.p2.2.m2.2.3.3.1.cmml"><mo id="A1.SS2.2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="A1.SS2.2.p2.2.m2.2.3.3.1.cmml">[</mo><mi id="A1.SS2.2.p2.2.m2.1.1" xref="A1.SS2.2.p2.2.m2.1.1.cmml">a</mi><mo id="A1.SS2.2.p2.2.m2.2.3.3.2.2" xref="A1.SS2.2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="A1.SS2.2.p2.2.m2.2.2" xref="A1.SS2.2.p2.2.m2.2.2.cmml">b</mi><mo id="A1.SS2.2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="A1.SS2.2.p2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.2.m2.2b"><apply id="A1.SS2.2.p2.2.m2.2.3.cmml" xref="A1.SS2.2.p2.2.m2.2.3"><notin id="A1.SS2.2.p2.2.m2.2.3.1.cmml" xref="A1.SS2.2.p2.2.m2.2.3.1"></notin><ci id="A1.SS2.2.p2.2.m2.2.3.2.cmml" xref="A1.SS2.2.p2.2.m2.2.3.2">𝑥</ci><interval closure="closed" id="A1.SS2.2.p2.2.m2.2.3.3.1.cmml" xref="A1.SS2.2.p2.2.m2.2.3.3.2"><ci id="A1.SS2.2.p2.2.m2.1.1.cmml" xref="A1.SS2.2.p2.2.m2.1.1">𝑎</ci><ci id="A1.SS2.2.p2.2.m2.2.2.cmml" xref="A1.SS2.2.p2.2.m2.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.2.m2.2c">x\notin[a,b]</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.2.m2.2d">italic_x ∉ [ italic_a , italic_b ]</annotation></semantics></math>, so that no equilibria occur outside the interval <math alttext="I" class="ltx_Math" display="inline" id="A1.SS2.2.p2.3.m3.1"><semantics id="A1.SS2.2.p2.3.m3.1a"><mi id="A1.SS2.2.p2.3.m3.1.1" xref="A1.SS2.2.p2.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.3.m3.1b"><ci id="A1.SS2.2.p2.3.m3.1.1.cmml" xref="A1.SS2.2.p2.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.3.m3.1d">italic_I</annotation></semantics></math> other than <math alttext="\pm\infty" class="ltx_Math" display="inline" id="A1.SS2.2.p2.4.m4.1"><semantics id="A1.SS2.2.p2.4.m4.1a"><mrow id="A1.SS2.2.p2.4.m4.1.1" xref="A1.SS2.2.p2.4.m4.1.1.cmml"><mo id="A1.SS2.2.p2.4.m4.1.1a" xref="A1.SS2.2.p2.4.m4.1.1.cmml">±</mo><mi id="A1.SS2.2.p2.4.m4.1.1.2" mathvariant="normal" xref="A1.SS2.2.p2.4.m4.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.4.m4.1b"><apply id="A1.SS2.2.p2.4.m4.1.1.cmml" xref="A1.SS2.2.p2.4.m4.1.1"><csymbol cd="latexml" id="A1.SS2.2.p2.4.m4.1.1.1.cmml" xref="A1.SS2.2.p2.4.m4.1.1">plus-or-minus</csymbol><infinity id="A1.SS2.2.p2.4.m4.1.1.2.cmml" xref="A1.SS2.2.p2.4.m4.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.4.m4.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.4.m4.1d">± ∞</annotation></semantics></math>. Since both <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.SS2.2.p2.5.m5.1"><semantics id="A1.SS2.2.p2.5.m5.1a"><msub id="A1.SS2.2.p2.5.m5.1.1" xref="A1.SS2.2.p2.5.m5.1.1.cmml"><mi id="A1.SS2.2.p2.5.m5.1.1.2" xref="A1.SS2.2.p2.5.m5.1.1.2.cmml">τ</mi><mn id="A1.SS2.2.p2.5.m5.1.1.3" xref="A1.SS2.2.p2.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.5.m5.1b"><apply id="A1.SS2.2.p2.5.m5.1.1.cmml" xref="A1.SS2.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p2.5.m5.1.1.1.cmml" xref="A1.SS2.2.p2.5.m5.1.1">subscript</csymbol><ci id="A1.SS2.2.p2.5.m5.1.1.2.cmml" xref="A1.SS2.2.p2.5.m5.1.1.2">𝜏</ci><cn id="A1.SS2.2.p2.5.m5.1.1.3.cmml" type="integer" xref="A1.SS2.2.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.5.m5.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.5.m5.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A1.SS2.2.p2.6.m6.1"><semantics id="A1.SS2.2.p2.6.m6.1a"><msub id="A1.SS2.2.p2.6.m6.1.1" xref="A1.SS2.2.p2.6.m6.1.1.cmml"><mi id="A1.SS2.2.p2.6.m6.1.1.2" xref="A1.SS2.2.p2.6.m6.1.1.2.cmml">τ</mi><mn id="A1.SS2.2.p2.6.m6.1.1.3" xref="A1.SS2.2.p2.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.6.m6.1b"><apply id="A1.SS2.2.p2.6.m6.1.1.cmml" xref="A1.SS2.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p2.6.m6.1.1.1.cmml" xref="A1.SS2.2.p2.6.m6.1.1">subscript</csymbol><ci id="A1.SS2.2.p2.6.m6.1.1.2.cmml" xref="A1.SS2.2.p2.6.m6.1.1.2">𝜏</ci><cn id="A1.SS2.2.p2.6.m6.1.1.3.cmml" type="integer" xref="A1.SS2.2.p2.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.6.m6.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.6.m6.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> (or just <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.SS2.2.p2.7.m7.1"><semantics id="A1.SS2.2.p2.7.m7.1a"><msub id="A1.SS2.2.p2.7.m7.1.1" xref="A1.SS2.2.p2.7.m7.1.1.cmml"><mi id="A1.SS2.2.p2.7.m7.1.1.2" xref="A1.SS2.2.p2.7.m7.1.1.2.cmml">τ</mi><mn id="A1.SS2.2.p2.7.m7.1.1.3" xref="A1.SS2.2.p2.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.7.m7.1b"><apply id="A1.SS2.2.p2.7.m7.1.1.cmml" xref="A1.SS2.2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p2.7.m7.1.1.1.cmml" xref="A1.SS2.2.p2.7.m7.1.1">subscript</csymbol><ci id="A1.SS2.2.p2.7.m7.1.1.2.cmml" xref="A1.SS2.2.p2.7.m7.1.1.2">𝜏</ci><cn id="A1.SS2.2.p2.7.m7.1.1.3.cmml" type="integer" xref="A1.SS2.2.p2.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.7.m7.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.7.m7.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, if they are the same equilibrium) are unstable (at least to the left for <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A1.SS2.2.p2.8.m8.1"><semantics id="A1.SS2.2.p2.8.m8.1a"><msub id="A1.SS2.2.p2.8.m8.1.1" xref="A1.SS2.2.p2.8.m8.1.1.cmml"><mi id="A1.SS2.2.p2.8.m8.1.1.2" xref="A1.SS2.2.p2.8.m8.1.1.2.cmml">τ</mi><mn id="A1.SS2.2.p2.8.m8.1.1.3" xref="A1.SS2.2.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.8.m8.1b"><apply id="A1.SS2.2.p2.8.m8.1.1.cmml" xref="A1.SS2.2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p2.8.m8.1.1.1.cmml" xref="A1.SS2.2.p2.8.m8.1.1">subscript</csymbol><ci id="A1.SS2.2.p2.8.m8.1.1.2.cmml" xref="A1.SS2.2.p2.8.m8.1.1.2">𝜏</ci><cn id="A1.SS2.2.p2.8.m8.1.1.3.cmml" type="integer" xref="A1.SS2.2.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.8.m8.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.8.m8.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and right for <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A1.SS2.2.p2.9.m9.1"><semantics id="A1.SS2.2.p2.9.m9.1a"><msub id="A1.SS2.2.p2.9.m9.1.1" xref="A1.SS2.2.p2.9.m9.1.1.cmml"><mi id="A1.SS2.2.p2.9.m9.1.1.2" xref="A1.SS2.2.p2.9.m9.1.1.2.cmml">τ</mi><mn id="A1.SS2.2.p2.9.m9.1.1.3" xref="A1.SS2.2.p2.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.9.m9.1b"><apply id="A1.SS2.2.p2.9.m9.1.1.cmml" xref="A1.SS2.2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p2.9.m9.1.1.1.cmml" xref="A1.SS2.2.p2.9.m9.1.1">subscript</csymbol><ci id="A1.SS2.2.p2.9.m9.1.1.2.cmml" xref="A1.SS2.2.p2.9.m9.1.1.2">𝜏</ci><cn id="A1.SS2.2.p2.9.m9.1.1.3.cmml" type="integer" xref="A1.SS2.2.p2.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.9.m9.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.9.m9.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>), it is topologically necessary that the uninformative equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="A1.SS2.2.p2.10.m10.1"><semantics id="A1.SS2.2.p2.10.m10.1a"><mrow id="A1.SS2.2.p2.10.m10.1.1" xref="A1.SS2.2.p2.10.m10.1.1.cmml"><mo id="A1.SS2.2.p2.10.m10.1.1a" xref="A1.SS2.2.p2.10.m10.1.1.cmml">±</mo><mi id="A1.SS2.2.p2.10.m10.1.1.2" mathvariant="normal" xref="A1.SS2.2.p2.10.m10.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p2.10.m10.1b"><apply id="A1.SS2.2.p2.10.m10.1.1.cmml" xref="A1.SS2.2.p2.10.m10.1.1"><csymbol cd="latexml" id="A1.SS2.2.p2.10.m10.1.1.1.cmml" xref="A1.SS2.2.p2.10.m10.1.1">plus-or-minus</csymbol><infinity id="A1.SS2.2.p2.10.m10.1.1.2.cmml" xref="A1.SS2.2.p2.10.m10.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p2.10.m10.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.2.p2.10.m10.1d">± ∞</annotation></semantics></math> are stable. ∎</p> </div> </div> <div class="ltx_para" id="A1.SS2.p2"> <p class="ltx_p" id="A1.SS2.p2.5">In most reasonable settings, one would expect <math alttext="G(\tau)" class="ltx_Math" display="inline" id="A1.SS2.p2.1.m1.1"><semantics id="A1.SS2.p2.1.m1.1a"><mrow id="A1.SS2.p2.1.m1.1.2" xref="A1.SS2.p2.1.m1.1.2.cmml"><mi id="A1.SS2.p2.1.m1.1.2.2" xref="A1.SS2.p2.1.m1.1.2.2.cmml">G</mi><mo id="A1.SS2.p2.1.m1.1.2.1" xref="A1.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="A1.SS2.p2.1.m1.1.2.3.2" xref="A1.SS2.p2.1.m1.1.2.cmml"><mo id="A1.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="A1.SS2.p2.1.m1.1.2.cmml">(</mo><mi id="A1.SS2.p2.1.m1.1.1" xref="A1.SS2.p2.1.m1.1.1.cmml">τ</mi><mo id="A1.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="A1.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.1.m1.1b"><apply id="A1.SS2.p2.1.m1.1.2.cmml" xref="A1.SS2.p2.1.m1.1.2"><times id="A1.SS2.p2.1.m1.1.2.1.cmml" xref="A1.SS2.p2.1.m1.1.2.1"></times><ci id="A1.SS2.p2.1.m1.1.2.2.cmml" xref="A1.SS2.p2.1.m1.1.2.2">𝐺</ci><ci id="A1.SS2.p2.1.m1.1.1.cmml" xref="A1.SS2.p2.1.m1.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.1.m1.1c">G(\tau)</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.1.m1.1d">italic_G ( italic_τ )</annotation></semantics></math> to behave according to Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition1" title="Condition 1. ‣ A.1 Equilibrium Characterization Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a>. Specifically, as a signal <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.p2.2.m2.1"><semantics id="A1.SS2.p2.2.m2.1a"><mi id="A1.SS2.p2.2.m2.1.1" xref="A1.SS2.p2.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.2.m2.1b"><ci id="A1.SS2.p2.2.m2.1.1.cmml" xref="A1.SS2.p2.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.2.m2.1d">italic_x</annotation></semantics></math> becomes increasingly small, the probability another agent receives a signal smaller than <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.p2.3.m3.1"><semantics id="A1.SS2.p2.3.m3.1a"><mi id="A1.SS2.p2.3.m3.1.1" xref="A1.SS2.p2.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.3.m3.1b"><ci id="A1.SS2.p2.3.m3.1.1.cmml" xref="A1.SS2.p2.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.3.m3.1d">italic_x</annotation></semantics></math> should approach 0. Meanwhile, as a signal <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.p2.4.m4.1"><semantics id="A1.SS2.p2.4.m4.1a"><mi id="A1.SS2.p2.4.m4.1.1" xref="A1.SS2.p2.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.4.m4.1b"><ci id="A1.SS2.p2.4.m4.1.1.cmml" xref="A1.SS2.p2.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.4.m4.1d">italic_x</annotation></semantics></math> grows large, the probability another agent receives a signal smaller than <math alttext="x" class="ltx_Math" display="inline" id="A1.SS2.p2.5.m5.1"><semantics id="A1.SS2.p2.5.m5.1a"><mi id="A1.SS2.p2.5.m5.1.1" xref="A1.SS2.p2.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A1.SS2.p2.5.m5.1b"><ci id="A1.SS2.p2.5.m5.1.1.cmml" xref="A1.SS2.p2.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p2.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.p2.5.m5.1d">italic_x</annotation></semantics></math> should approach 1. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition8" title="Proposition 8. ‣ A.2 Dynamics Generalization ‣ Appendix A Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a> thus suggests that when information corresponds predictably and monotonically to quality of an essay or task, over time agents will naturally move toward uninformative equilibria. Intuitively, these dynamics make sense: an agent who thinks others are submitting more “high” reports will shift their threshold down to match, reinforcing larger and larger thresholds over time.</p> </div> </section> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Dasgupta-Ghosh</h2> <section class="ltx_subsection" id="A2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.1 </span>Omitted Proofs</h3> <div class="ltx_proof" id="A2.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition3" title="Proposition 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>.</h6> <div class="ltx_para" id="A2.SS1.1.p1"> <p class="ltx_p" id="A2.SS1.1.p1.13">Let <math alttext="\tau^{*}=\infty" class="ltx_Math" display="inline" id="A2.SS1.1.p1.1.m1.1"><semantics id="A2.SS1.1.p1.1.m1.1a"><mrow id="A2.SS1.1.p1.1.m1.1.1" xref="A2.SS1.1.p1.1.m1.1.1.cmml"><msup id="A2.SS1.1.p1.1.m1.1.1.2" xref="A2.SS1.1.p1.1.m1.1.1.2.cmml"><mi id="A2.SS1.1.p1.1.m1.1.1.2.2" xref="A2.SS1.1.p1.1.m1.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.1.p1.1.m1.1.1.2.3" xref="A2.SS1.1.p1.1.m1.1.1.2.3.cmml">∗</mo></msup><mo id="A2.SS1.1.p1.1.m1.1.1.1" xref="A2.SS1.1.p1.1.m1.1.1.1.cmml">=</mo><mi id="A2.SS1.1.p1.1.m1.1.1.3" mathvariant="normal" xref="A2.SS1.1.p1.1.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.1.m1.1b"><apply id="A2.SS1.1.p1.1.m1.1.1.cmml" xref="A2.SS1.1.p1.1.m1.1.1"><eq id="A2.SS1.1.p1.1.m1.1.1.1.cmml" xref="A2.SS1.1.p1.1.m1.1.1.1"></eq><apply id="A2.SS1.1.p1.1.m1.1.1.2.cmml" xref="A2.SS1.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.1.m1.1.1.2.1.cmml" xref="A2.SS1.1.p1.1.m1.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.1.m1.1.1.2.2.cmml" xref="A2.SS1.1.p1.1.m1.1.1.2.2">𝜏</ci><times id="A2.SS1.1.p1.1.m1.1.1.2.3.cmml" xref="A2.SS1.1.p1.1.m1.1.1.2.3"></times></apply><infinity id="A2.SS1.1.p1.1.m1.1.1.3.cmml" xref="A2.SS1.1.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.1.m1.1c">\tau^{*}=\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ∞</annotation></semantics></math>. Then for all signals <math alttext="x" class="ltx_Math" display="inline" id="A2.SS1.1.p1.2.m2.1"><semantics id="A2.SS1.1.p1.2.m2.1a"><mi id="A2.SS1.1.p1.2.m2.1.1" xref="A2.SS1.1.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.2.m2.1b"><ci id="A2.SS1.1.p1.2.m2.1.1.cmml" xref="A2.SS1.1.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.2.m2.1d">italic_x</annotation></semantics></math> such that <math alttext="x\leq\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.3.m3.1"><semantics id="A2.SS1.1.p1.3.m3.1a"><mrow id="A2.SS1.1.p1.3.m3.1.1" xref="A2.SS1.1.p1.3.m3.1.1.cmml"><mi id="A2.SS1.1.p1.3.m3.1.1.2" xref="A2.SS1.1.p1.3.m3.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.3.m3.1.1.1" xref="A2.SS1.1.p1.3.m3.1.1.1.cmml">≤</mo><msup id="A2.SS1.1.p1.3.m3.1.1.3" xref="A2.SS1.1.p1.3.m3.1.1.3.cmml"><mi id="A2.SS1.1.p1.3.m3.1.1.3.2" xref="A2.SS1.1.p1.3.m3.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.1.p1.3.m3.1.1.3.3" xref="A2.SS1.1.p1.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.3.m3.1b"><apply id="A2.SS1.1.p1.3.m3.1.1.cmml" xref="A2.SS1.1.p1.3.m3.1.1"><leq id="A2.SS1.1.p1.3.m3.1.1.1.cmml" xref="A2.SS1.1.p1.3.m3.1.1.1"></leq><ci id="A2.SS1.1.p1.3.m3.1.1.2.cmml" xref="A2.SS1.1.p1.3.m3.1.1.2">𝑥</ci><apply id="A2.SS1.1.p1.3.m3.1.1.3.cmml" xref="A2.SS1.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.1.p1.3.m3.1.1.3.1.cmml" xref="A2.SS1.1.p1.3.m3.1.1.3">superscript</csymbol><ci id="A2.SS1.1.p1.3.m3.1.1.3.2.cmml" xref="A2.SS1.1.p1.3.m3.1.1.3.2">𝜏</ci><times id="A2.SS1.1.p1.3.m3.1.1.3.3.cmml" xref="A2.SS1.1.p1.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.3.m3.1c">x\leq\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.3.m3.1d">italic_x ≤ italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P(\tau^{*};x)=1\geq\Pr[X^{\prime}\leq\infty]" class="ltx_Math" display="inline" id="A2.SS1.1.p1.4.m4.4"><semantics id="A2.SS1.1.p1.4.m4.4a"><mrow id="A2.SS1.1.p1.4.m4.4.4" xref="A2.SS1.1.p1.4.m4.4.4.cmml"><mrow id="A2.SS1.1.p1.4.m4.3.3.1" xref="A2.SS1.1.p1.4.m4.3.3.1.cmml"><mi id="A2.SS1.1.p1.4.m4.3.3.1.3" xref="A2.SS1.1.p1.4.m4.3.3.1.3.cmml">P</mi><mo id="A2.SS1.1.p1.4.m4.3.3.1.2" xref="A2.SS1.1.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="A2.SS1.1.p1.4.m4.3.3.1.1.1" xref="A2.SS1.1.p1.4.m4.3.3.1.1.2.cmml"><mo id="A2.SS1.1.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="A2.SS1.1.p1.4.m4.3.3.1.1.2.cmml">(</mo><msup id="A2.SS1.1.p1.4.m4.3.3.1.1.1.1" xref="A2.SS1.1.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="A2.SS1.1.p1.4.m4.3.3.1.1.1.1.2" xref="A2.SS1.1.p1.4.m4.3.3.1.1.1.1.2.cmml">τ</mi><mo id="A2.SS1.1.p1.4.m4.3.3.1.1.1.1.3" xref="A2.SS1.1.p1.4.m4.3.3.1.1.1.1.3.cmml">∗</mo></msup><mo id="A2.SS1.1.p1.4.m4.3.3.1.1.1.3" xref="A2.SS1.1.p1.4.m4.3.3.1.1.2.cmml">;</mo><mi id="A2.SS1.1.p1.4.m4.1.1" xref="A2.SS1.1.p1.4.m4.1.1.cmml">x</mi><mo id="A2.SS1.1.p1.4.m4.3.3.1.1.1.4" stretchy="false" xref="A2.SS1.1.p1.4.m4.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.1.p1.4.m4.4.4.4" xref="A2.SS1.1.p1.4.m4.4.4.4.cmml">=</mo><mn id="A2.SS1.1.p1.4.m4.4.4.5" xref="A2.SS1.1.p1.4.m4.4.4.5.cmml">1</mn><mo id="A2.SS1.1.p1.4.m4.4.4.6" xref="A2.SS1.1.p1.4.m4.4.4.6.cmml">≥</mo><mrow id="A2.SS1.1.p1.4.m4.4.4.2.1" xref="A2.SS1.1.p1.4.m4.4.4.2.2.cmml"><mi id="A2.SS1.1.p1.4.m4.2.2" xref="A2.SS1.1.p1.4.m4.2.2.cmml">Pr</mi><mo id="A2.SS1.1.p1.4.m4.4.4.2.1a" xref="A2.SS1.1.p1.4.m4.4.4.2.2.cmml">⁡</mo><mrow id="A2.SS1.1.p1.4.m4.4.4.2.1.1" xref="A2.SS1.1.p1.4.m4.4.4.2.2.cmml"><mo id="A2.SS1.1.p1.4.m4.4.4.2.1.1.2" stretchy="false" xref="A2.SS1.1.p1.4.m4.4.4.2.2.cmml">[</mo><mrow id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.cmml"><msup id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.cmml"><mi id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.2" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.2.cmml">X</mi><mo id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.3" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.1" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.1.cmml">≤</mo><mi id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.3" mathvariant="normal" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.3.cmml">∞</mi></mrow><mo id="A2.SS1.1.p1.4.m4.4.4.2.1.1.3" stretchy="false" xref="A2.SS1.1.p1.4.m4.4.4.2.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.4.m4.4b"><apply id="A2.SS1.1.p1.4.m4.4.4.cmml" xref="A2.SS1.1.p1.4.m4.4.4"><and id="A2.SS1.1.p1.4.m4.4.4a.cmml" xref="A2.SS1.1.p1.4.m4.4.4"></and><apply id="A2.SS1.1.p1.4.m4.4.4b.cmml" xref="A2.SS1.1.p1.4.m4.4.4"><eq id="A2.SS1.1.p1.4.m4.4.4.4.cmml" xref="A2.SS1.1.p1.4.m4.4.4.4"></eq><apply id="A2.SS1.1.p1.4.m4.3.3.1.cmml" xref="A2.SS1.1.p1.4.m4.3.3.1"><times 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href="https://arxiv.org/html/2503.16280v1#A2.SS1.1.p1.4.m4.4.4.5.cmml" id="A2.SS1.1.p1.4.m4.4.4d.cmml" xref="A2.SS1.1.p1.4.m4.4.4"></share><apply id="A2.SS1.1.p1.4.m4.4.4.2.2.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1"><ci id="A2.SS1.1.p1.4.m4.2.2.cmml" xref="A2.SS1.1.p1.4.m4.2.2">Pr</ci><apply id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1"><leq id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.1.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.1"></leq><apply id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.1.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.2.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.2">𝑋</ci><ci id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.3.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.2.3">′</ci></apply><infinity id="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.3.cmml" xref="A2.SS1.1.p1.4.m4.4.4.2.1.1.1.3"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.4.m4.4c">P(\tau^{*};x)=1\geq\Pr[X^{\prime}\leq\infty]</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.4.m4.4d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x ) = 1 ≥ roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ ∞ ]</annotation></semantics></math> (since any <math alttext="X^{\prime}\in\mathbb{R}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.5.m5.1"><semantics id="A2.SS1.1.p1.5.m5.1a"><mrow id="A2.SS1.1.p1.5.m5.1.1" xref="A2.SS1.1.p1.5.m5.1.1.cmml"><msup id="A2.SS1.1.p1.5.m5.1.1.2" xref="A2.SS1.1.p1.5.m5.1.1.2.cmml"><mi id="A2.SS1.1.p1.5.m5.1.1.2.2" xref="A2.SS1.1.p1.5.m5.1.1.2.2.cmml">X</mi><mo id="A2.SS1.1.p1.5.m5.1.1.2.3" xref="A2.SS1.1.p1.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.1.p1.5.m5.1.1.1" xref="A2.SS1.1.p1.5.m5.1.1.1.cmml">∈</mo><mi id="A2.SS1.1.p1.5.m5.1.1.3" xref="A2.SS1.1.p1.5.m5.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.5.m5.1b"><apply id="A2.SS1.1.p1.5.m5.1.1.cmml" xref="A2.SS1.1.p1.5.m5.1.1"><in id="A2.SS1.1.p1.5.m5.1.1.1.cmml" xref="A2.SS1.1.p1.5.m5.1.1.1"></in><apply id="A2.SS1.1.p1.5.m5.1.1.2.cmml" xref="A2.SS1.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.5.m5.1.1.2.1.cmml" xref="A2.SS1.1.p1.5.m5.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.5.m5.1.1.2.2.cmml" xref="A2.SS1.1.p1.5.m5.1.1.2.2">𝑋</ci><ci id="A2.SS1.1.p1.5.m5.1.1.2.3.cmml" xref="A2.SS1.1.p1.5.m5.1.1.2.3">′</ci></apply><ci id="A2.SS1.1.p1.5.m5.1.1.3.cmml" xref="A2.SS1.1.p1.5.m5.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.5.m5.1c">X^{\prime}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.5.m5.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="X^{\prime}<\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.6.m6.1"><semantics id="A2.SS1.1.p1.6.m6.1a"><mrow id="A2.SS1.1.p1.6.m6.1.1" xref="A2.SS1.1.p1.6.m6.1.1.cmml"><msup id="A2.SS1.1.p1.6.m6.1.1.2" xref="A2.SS1.1.p1.6.m6.1.1.2.cmml"><mi id="A2.SS1.1.p1.6.m6.1.1.2.2" xref="A2.SS1.1.p1.6.m6.1.1.2.2.cmml">X</mi><mo id="A2.SS1.1.p1.6.m6.1.1.2.3" xref="A2.SS1.1.p1.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.1.p1.6.m6.1.1.1" xref="A2.SS1.1.p1.6.m6.1.1.1.cmml"><</mo><msup id="A2.SS1.1.p1.6.m6.1.1.3" xref="A2.SS1.1.p1.6.m6.1.1.3.cmml"><mi id="A2.SS1.1.p1.6.m6.1.1.3.2" xref="A2.SS1.1.p1.6.m6.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.1.p1.6.m6.1.1.3.3" xref="A2.SS1.1.p1.6.m6.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.6.m6.1b"><apply id="A2.SS1.1.p1.6.m6.1.1.cmml" xref="A2.SS1.1.p1.6.m6.1.1"><lt id="A2.SS1.1.p1.6.m6.1.1.1.cmml" xref="A2.SS1.1.p1.6.m6.1.1.1"></lt><apply id="A2.SS1.1.p1.6.m6.1.1.2.cmml" xref="A2.SS1.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.6.m6.1.1.2.1.cmml" xref="A2.SS1.1.p1.6.m6.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.6.m6.1.1.2.2.cmml" xref="A2.SS1.1.p1.6.m6.1.1.2.2">𝑋</ci><ci id="A2.SS1.1.p1.6.m6.1.1.2.3.cmml" xref="A2.SS1.1.p1.6.m6.1.1.2.3">′</ci></apply><apply id="A2.SS1.1.p1.6.m6.1.1.3.cmml" xref="A2.SS1.1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.1.p1.6.m6.1.1.3.1.cmml" xref="A2.SS1.1.p1.6.m6.1.1.3">superscript</csymbol><ci id="A2.SS1.1.p1.6.m6.1.1.3.2.cmml" xref="A2.SS1.1.p1.6.m6.1.1.3.2">𝜏</ci><times id="A2.SS1.1.p1.6.m6.1.1.3.3.cmml" xref="A2.SS1.1.p1.6.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.6.m6.1c">X^{\prime}<\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.6.m6.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with probability one.) Meanwhile, Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E12" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">12</span></a> is vacuous since no signal <math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.7.m7.1"><semantics id="A2.SS1.1.p1.7.m7.1a"><mrow id="A2.SS1.1.p1.7.m7.1.1" xref="A2.SS1.1.p1.7.m7.1.1.cmml"><mi id="A2.SS1.1.p1.7.m7.1.1.2" xref="A2.SS1.1.p1.7.m7.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.7.m7.1.1.1" xref="A2.SS1.1.p1.7.m7.1.1.1.cmml">∈</mo><mi id="A2.SS1.1.p1.7.m7.1.1.3" xref="A2.SS1.1.p1.7.m7.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.7.m7.1b"><apply id="A2.SS1.1.p1.7.m7.1.1.cmml" xref="A2.SS1.1.p1.7.m7.1.1"><in id="A2.SS1.1.p1.7.m7.1.1.1.cmml" xref="A2.SS1.1.p1.7.m7.1.1.1"></in><ci id="A2.SS1.1.p1.7.m7.1.1.2.cmml" xref="A2.SS1.1.p1.7.m7.1.1.2">𝑥</ci><ci id="A2.SS1.1.p1.7.m7.1.1.3.cmml" xref="A2.SS1.1.p1.7.m7.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.7.m7.1c">x\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.7.m7.1d">italic_x ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="x>\infty" class="ltx_Math" display="inline" id="A2.SS1.1.p1.8.m8.1"><semantics id="A2.SS1.1.p1.8.m8.1a"><mrow id="A2.SS1.1.p1.8.m8.1.1" xref="A2.SS1.1.p1.8.m8.1.1.cmml"><mi id="A2.SS1.1.p1.8.m8.1.1.2" xref="A2.SS1.1.p1.8.m8.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.8.m8.1.1.1" xref="A2.SS1.1.p1.8.m8.1.1.1.cmml">></mo><mi id="A2.SS1.1.p1.8.m8.1.1.3" mathvariant="normal" xref="A2.SS1.1.p1.8.m8.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.8.m8.1b"><apply id="A2.SS1.1.p1.8.m8.1.1.cmml" xref="A2.SS1.1.p1.8.m8.1.1"><gt id="A2.SS1.1.p1.8.m8.1.1.1.cmml" xref="A2.SS1.1.p1.8.m8.1.1.1"></gt><ci id="A2.SS1.1.p1.8.m8.1.1.2.cmml" xref="A2.SS1.1.p1.8.m8.1.1.2">𝑥</ci><infinity id="A2.SS1.1.p1.8.m8.1.1.3.cmml" xref="A2.SS1.1.p1.8.m8.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.8.m8.1c">x>\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.8.m8.1d">italic_x > ∞</annotation></semantics></math>. A similar argument follows for <math alttext="\tau^{*}=-\infty" class="ltx_Math" display="inline" id="A2.SS1.1.p1.9.m9.1"><semantics id="A2.SS1.1.p1.9.m9.1a"><mrow id="A2.SS1.1.p1.9.m9.1.1" xref="A2.SS1.1.p1.9.m9.1.1.cmml"><msup id="A2.SS1.1.p1.9.m9.1.1.2" xref="A2.SS1.1.p1.9.m9.1.1.2.cmml"><mi id="A2.SS1.1.p1.9.m9.1.1.2.2" xref="A2.SS1.1.p1.9.m9.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.1.p1.9.m9.1.1.2.3" xref="A2.SS1.1.p1.9.m9.1.1.2.3.cmml">∗</mo></msup><mo id="A2.SS1.1.p1.9.m9.1.1.1" xref="A2.SS1.1.p1.9.m9.1.1.1.cmml">=</mo><mrow id="A2.SS1.1.p1.9.m9.1.1.3" xref="A2.SS1.1.p1.9.m9.1.1.3.cmml"><mo id="A2.SS1.1.p1.9.m9.1.1.3a" xref="A2.SS1.1.p1.9.m9.1.1.3.cmml">−</mo><mi id="A2.SS1.1.p1.9.m9.1.1.3.2" mathvariant="normal" xref="A2.SS1.1.p1.9.m9.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.9.m9.1b"><apply id="A2.SS1.1.p1.9.m9.1.1.cmml" xref="A2.SS1.1.p1.9.m9.1.1"><eq id="A2.SS1.1.p1.9.m9.1.1.1.cmml" xref="A2.SS1.1.p1.9.m9.1.1.1"></eq><apply id="A2.SS1.1.p1.9.m9.1.1.2.cmml" xref="A2.SS1.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.9.m9.1.1.2.1.cmml" xref="A2.SS1.1.p1.9.m9.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.9.m9.1.1.2.2.cmml" xref="A2.SS1.1.p1.9.m9.1.1.2.2">𝜏</ci><times id="A2.SS1.1.p1.9.m9.1.1.2.3.cmml" xref="A2.SS1.1.p1.9.m9.1.1.2.3"></times></apply><apply id="A2.SS1.1.p1.9.m9.1.1.3.cmml" xref="A2.SS1.1.p1.9.m9.1.1.3"><minus id="A2.SS1.1.p1.9.m9.1.1.3.1.cmml" xref="A2.SS1.1.p1.9.m9.1.1.3"></minus><infinity id="A2.SS1.1.p1.9.m9.1.1.3.2.cmml" xref="A2.SS1.1.p1.9.m9.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.9.m9.1c">\tau^{*}=-\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = - ∞</annotation></semantics></math>: for all signals <math alttext="x>\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.10.m10.1"><semantics id="A2.SS1.1.p1.10.m10.1a"><mrow id="A2.SS1.1.p1.10.m10.1.1" xref="A2.SS1.1.p1.10.m10.1.1.cmml"><mi id="A2.SS1.1.p1.10.m10.1.1.2" xref="A2.SS1.1.p1.10.m10.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.10.m10.1.1.1" xref="A2.SS1.1.p1.10.m10.1.1.1.cmml">></mo><msup id="A2.SS1.1.p1.10.m10.1.1.3" xref="A2.SS1.1.p1.10.m10.1.1.3.cmml"><mi id="A2.SS1.1.p1.10.m10.1.1.3.2" xref="A2.SS1.1.p1.10.m10.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.1.p1.10.m10.1.1.3.3" xref="A2.SS1.1.p1.10.m10.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.10.m10.1b"><apply id="A2.SS1.1.p1.10.m10.1.1.cmml" xref="A2.SS1.1.p1.10.m10.1.1"><gt id="A2.SS1.1.p1.10.m10.1.1.1.cmml" xref="A2.SS1.1.p1.10.m10.1.1.1"></gt><ci id="A2.SS1.1.p1.10.m10.1.1.2.cmml" xref="A2.SS1.1.p1.10.m10.1.1.2">𝑥</ci><apply id="A2.SS1.1.p1.10.m10.1.1.3.cmml" xref="A2.SS1.1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.1.p1.10.m10.1.1.3.1.cmml" xref="A2.SS1.1.p1.10.m10.1.1.3">superscript</csymbol><ci id="A2.SS1.1.p1.10.m10.1.1.3.2.cmml" xref="A2.SS1.1.p1.10.m10.1.1.3.2">𝜏</ci><times id="A2.SS1.1.p1.10.m10.1.1.3.3.cmml" xref="A2.SS1.1.p1.10.m10.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.10.m10.1c">x>\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.10.m10.1d">italic_x > italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P(\tau^{*};x)=0\leq\Pr[X^{\prime}\leq-\infty]" class="ltx_Math" display="inline" id="A2.SS1.1.p1.11.m11.4"><semantics id="A2.SS1.1.p1.11.m11.4a"><mrow id="A2.SS1.1.p1.11.m11.4.4" xref="A2.SS1.1.p1.11.m11.4.4.cmml"><mrow id="A2.SS1.1.p1.11.m11.3.3.1" xref="A2.SS1.1.p1.11.m11.3.3.1.cmml"><mi id="A2.SS1.1.p1.11.m11.3.3.1.3" xref="A2.SS1.1.p1.11.m11.3.3.1.3.cmml">P</mi><mo id="A2.SS1.1.p1.11.m11.3.3.1.2" xref="A2.SS1.1.p1.11.m11.3.3.1.2.cmml">⁢</mo><mrow id="A2.SS1.1.p1.11.m11.3.3.1.1.1" xref="A2.SS1.1.p1.11.m11.3.3.1.1.2.cmml"><mo id="A2.SS1.1.p1.11.m11.3.3.1.1.1.2" stretchy="false" xref="A2.SS1.1.p1.11.m11.3.3.1.1.2.cmml">(</mo><msup id="A2.SS1.1.p1.11.m11.3.3.1.1.1.1" xref="A2.SS1.1.p1.11.m11.3.3.1.1.1.1.cmml"><mi id="A2.SS1.1.p1.11.m11.3.3.1.1.1.1.2" xref="A2.SS1.1.p1.11.m11.3.3.1.1.1.1.2.cmml">τ</mi><mo id="A2.SS1.1.p1.11.m11.3.3.1.1.1.1.3" xref="A2.SS1.1.p1.11.m11.3.3.1.1.1.1.3.cmml">∗</mo></msup><mo id="A2.SS1.1.p1.11.m11.3.3.1.1.1.3" xref="A2.SS1.1.p1.11.m11.3.3.1.1.2.cmml">;</mo><mi id="A2.SS1.1.p1.11.m11.1.1" xref="A2.SS1.1.p1.11.m11.1.1.cmml">x</mi><mo id="A2.SS1.1.p1.11.m11.3.3.1.1.1.4" stretchy="false" xref="A2.SS1.1.p1.11.m11.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.1.p1.11.m11.4.4.4" xref="A2.SS1.1.p1.11.m11.4.4.4.cmml">=</mo><mn id="A2.SS1.1.p1.11.m11.4.4.5" xref="A2.SS1.1.p1.11.m11.4.4.5.cmml">0</mn><mo id="A2.SS1.1.p1.11.m11.4.4.6" xref="A2.SS1.1.p1.11.m11.4.4.6.cmml">≤</mo><mrow id="A2.SS1.1.p1.11.m11.4.4.2.1" 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id="A2.SS1.1.p1.11.m11.2.2.cmml" xref="A2.SS1.1.p1.11.m11.2.2">Pr</ci><apply id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1"><leq id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.1.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.1"></leq><apply id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.1.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2">superscript</csymbol><ci id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.2.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.2">𝑋</ci><ci id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.3.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.2.3">′</ci></apply><apply id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3"><minus id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3.1.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3"></minus><infinity id="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3.2.cmml" xref="A2.SS1.1.p1.11.m11.4.4.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.11.m11.4c">P(\tau^{*};x)=0\leq\Pr[X^{\prime}\leq-\infty]</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.11.m11.4d">italic_P ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ; italic_x ) = 0 ≤ roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ - ∞ ]</annotation></semantics></math>, while Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E11" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">11</span></a> is vacuous since no signal <math alttext="x\in\mathbb{R}" class="ltx_Math" display="inline" id="A2.SS1.1.p1.12.m12.1"><semantics id="A2.SS1.1.p1.12.m12.1a"><mrow id="A2.SS1.1.p1.12.m12.1.1" xref="A2.SS1.1.p1.12.m12.1.1.cmml"><mi id="A2.SS1.1.p1.12.m12.1.1.2" xref="A2.SS1.1.p1.12.m12.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.12.m12.1.1.1" xref="A2.SS1.1.p1.12.m12.1.1.1.cmml">∈</mo><mi id="A2.SS1.1.p1.12.m12.1.1.3" xref="A2.SS1.1.p1.12.m12.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.12.m12.1b"><apply id="A2.SS1.1.p1.12.m12.1.1.cmml" xref="A2.SS1.1.p1.12.m12.1.1"><in id="A2.SS1.1.p1.12.m12.1.1.1.cmml" xref="A2.SS1.1.p1.12.m12.1.1.1"></in><ci id="A2.SS1.1.p1.12.m12.1.1.2.cmml" xref="A2.SS1.1.p1.12.m12.1.1.2">𝑥</ci><ci id="A2.SS1.1.p1.12.m12.1.1.3.cmml" xref="A2.SS1.1.p1.12.m12.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.12.m12.1c">x\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.12.m12.1d">italic_x ∈ blackboard_R</annotation></semantics></math> satisfies <math alttext="x<-\infty" class="ltx_Math" display="inline" id="A2.SS1.1.p1.13.m13.1"><semantics id="A2.SS1.1.p1.13.m13.1a"><mrow id="A2.SS1.1.p1.13.m13.1.1" xref="A2.SS1.1.p1.13.m13.1.1.cmml"><mi id="A2.SS1.1.p1.13.m13.1.1.2" xref="A2.SS1.1.p1.13.m13.1.1.2.cmml">x</mi><mo id="A2.SS1.1.p1.13.m13.1.1.1" xref="A2.SS1.1.p1.13.m13.1.1.1.cmml"><</mo><mrow id="A2.SS1.1.p1.13.m13.1.1.3" xref="A2.SS1.1.p1.13.m13.1.1.3.cmml"><mo id="A2.SS1.1.p1.13.m13.1.1.3a" xref="A2.SS1.1.p1.13.m13.1.1.3.cmml">−</mo><mi id="A2.SS1.1.p1.13.m13.1.1.3.2" mathvariant="normal" xref="A2.SS1.1.p1.13.m13.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.1.p1.13.m13.1b"><apply id="A2.SS1.1.p1.13.m13.1.1.cmml" xref="A2.SS1.1.p1.13.m13.1.1"><lt id="A2.SS1.1.p1.13.m13.1.1.1.cmml" xref="A2.SS1.1.p1.13.m13.1.1.1"></lt><ci id="A2.SS1.1.p1.13.m13.1.1.2.cmml" xref="A2.SS1.1.p1.13.m13.1.1.2">𝑥</ci><apply id="A2.SS1.1.p1.13.m13.1.1.3.cmml" xref="A2.SS1.1.p1.13.m13.1.1.3"><minus id="A2.SS1.1.p1.13.m13.1.1.3.1.cmml" xref="A2.SS1.1.p1.13.m13.1.1.3"></minus><infinity id="A2.SS1.1.p1.13.m13.1.1.3.2.cmml" xref="A2.SS1.1.p1.13.m13.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.1.p1.13.m13.1c">x<-\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.1.p1.13.m13.1d">italic_x < - ∞</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_proof" id="A2.SS1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem3" title="Theorem 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>.</h6> <div class="ltx_para" id="A2.SS1.2.p1"> <p class="ltx_p" id="A2.SS1.2.p1.6">For necessity, assume that <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.2.p1.1.m1.1"><semantics id="A2.SS1.2.p1.1.m1.1a"><mi id="A2.SS1.2.p1.1.m1.1.1" xref="A2.SS1.2.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.2.p1.1.m1.1b"><ci id="A2.SS1.2.p1.1.m1.1.1.cmml" xref="A2.SS1.2.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.1.m1.1d">italic_τ</annotation></semantics></math> is a threshold equilibrium, so Equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E11" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">11</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E12" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">12</span></a>) hold. Since <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="A2.SS1.2.p1.2.m2.2"><semantics id="A2.SS1.2.p1.2.m2.2a"><mrow id="A2.SS1.2.p1.2.m2.2.3" xref="A2.SS1.2.p1.2.m2.2.3.cmml"><mi id="A2.SS1.2.p1.2.m2.2.3.2" xref="A2.SS1.2.p1.2.m2.2.3.2.cmml">P</mi><mo id="A2.SS1.2.p1.2.m2.2.3.1" xref="A2.SS1.2.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.2.m2.2.3.3.2" xref="A2.SS1.2.p1.2.m2.2.3.3.1.cmml"><mo id="A2.SS1.2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="A2.SS1.2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="A2.SS1.2.p1.2.m2.1.1" xref="A2.SS1.2.p1.2.m2.1.1.cmml">τ</mi><mo id="A2.SS1.2.p1.2.m2.2.3.3.2.2" xref="A2.SS1.2.p1.2.m2.2.3.3.1.cmml">;</mo><mi id="A2.SS1.2.p1.2.m2.2.2" xref="A2.SS1.2.p1.2.m2.2.2.cmml">x</mi><mo id="A2.SS1.2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="A2.SS1.2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.2.p1.2.m2.2b"><apply id="A2.SS1.2.p1.2.m2.2.3.cmml" xref="A2.SS1.2.p1.2.m2.2.3"><times id="A2.SS1.2.p1.2.m2.2.3.1.cmml" xref="A2.SS1.2.p1.2.m2.2.3.1"></times><ci id="A2.SS1.2.p1.2.m2.2.3.2.cmml" xref="A2.SS1.2.p1.2.m2.2.3.2">𝑃</ci><list id="A2.SS1.2.p1.2.m2.2.3.3.1.cmml" xref="A2.SS1.2.p1.2.m2.2.3.3.2"><ci id="A2.SS1.2.p1.2.m2.1.1.cmml" xref="A2.SS1.2.p1.2.m2.1.1">𝜏</ci><ci id="A2.SS1.2.p1.2.m2.2.2.cmml" xref="A2.SS1.2.p1.2.m2.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.2.m2.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.2.m2.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is continuous, we must have <math alttext="\lim_{x\to\tau^{+}}P(\tau;x)=\lim_{x\to\tau^{-}}P(\tau;x)=P(\tau;\tau)." class="ltx_Math" display="inline" id="A2.SS1.2.p1.3.m3.7"><semantics id="A2.SS1.2.p1.3.m3.7a"><mrow id="A2.SS1.2.p1.3.m3.7.7.1" xref="A2.SS1.2.p1.3.m3.7.7.1.1.cmml"><mrow id="A2.SS1.2.p1.3.m3.7.7.1.1" xref="A2.SS1.2.p1.3.m3.7.7.1.1.cmml"><mrow 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id="A2.SS1.2.p1.3.m3.7.7.1.1.4.2.3.1.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.4.2.3.2"><ci id="A2.SS1.2.p1.3.m3.3.3.cmml" xref="A2.SS1.2.p1.3.m3.3.3">𝜏</ci><ci id="A2.SS1.2.p1.3.m3.4.4.cmml" xref="A2.SS1.2.p1.3.m3.4.4">𝑥</ci></list></apply></apply></apply><apply id="A2.SS1.2.p1.3.m3.7.7.1.1c.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1"><eq id="A2.SS1.2.p1.3.m3.7.7.1.1.5.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.2.p1.3.m3.7.7.1.1.4.cmml" id="A2.SS1.2.p1.3.m3.7.7.1.1d.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1"></share><apply id="A2.SS1.2.p1.3.m3.7.7.1.1.6.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.6"><times id="A2.SS1.2.p1.3.m3.7.7.1.1.6.1.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.6.1"></times><ci id="A2.SS1.2.p1.3.m3.7.7.1.1.6.2.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.6.2">𝑃</ci><list id="A2.SS1.2.p1.3.m3.7.7.1.1.6.3.1.cmml" xref="A2.SS1.2.p1.3.m3.7.7.1.1.6.3.2"><ci id="A2.SS1.2.p1.3.m3.5.5.cmml" xref="A2.SS1.2.p1.3.m3.5.5">𝜏</ci><ci id="A2.SS1.2.p1.3.m3.6.6.cmml" xref="A2.SS1.2.p1.3.m3.6.6">𝜏</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.3.m3.7c">\lim_{x\to\tau^{+}}P(\tau;x)=\lim_{x\to\tau^{-}}P(\tau;x)=P(\tau;\tau).</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.3.m3.7d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) = roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) = italic_P ( italic_τ ; italic_τ ) .</annotation></semantics></math> Further, <math alttext="\lim_{x\to\tau^{+}}P(\tau;x)\leq\pi_{L}" class="ltx_Math" display="inline" id="A2.SS1.2.p1.4.m4.2"><semantics id="A2.SS1.2.p1.4.m4.2a"><mrow id="A2.SS1.2.p1.4.m4.2.3" xref="A2.SS1.2.p1.4.m4.2.3.cmml"><mrow id="A2.SS1.2.p1.4.m4.2.3.2" xref="A2.SS1.2.p1.4.m4.2.3.2.cmml"><msub id="A2.SS1.2.p1.4.m4.2.3.2.1" xref="A2.SS1.2.p1.4.m4.2.3.2.1.cmml"><mo id="A2.SS1.2.p1.4.m4.2.3.2.1.2" xref="A2.SS1.2.p1.4.m4.2.3.2.1.2.cmml">lim</mo><mrow id="A2.SS1.2.p1.4.m4.2.3.2.1.3" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.cmml"><mi id="A2.SS1.2.p1.4.m4.2.3.2.1.3.2" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.2.cmml">x</mi><mo id="A2.SS1.2.p1.4.m4.2.3.2.1.3.1" stretchy="false" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.1.cmml">→</mo><msup id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.cmml"><mi id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.2" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.2.cmml">τ</mi><mo id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.3" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.3.cmml">+</mo></msup></mrow></msub><mrow id="A2.SS1.2.p1.4.m4.2.3.2.2" xref="A2.SS1.2.p1.4.m4.2.3.2.2.cmml"><mi id="A2.SS1.2.p1.4.m4.2.3.2.2.2" xref="A2.SS1.2.p1.4.m4.2.3.2.2.2.cmml">P</mi><mo id="A2.SS1.2.p1.4.m4.2.3.2.2.1" xref="A2.SS1.2.p1.4.m4.2.3.2.2.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.4.m4.2.3.2.2.3.2" xref="A2.SS1.2.p1.4.m4.2.3.2.2.3.1.cmml"><mo id="A2.SS1.2.p1.4.m4.2.3.2.2.3.2.1" stretchy="false" xref="A2.SS1.2.p1.4.m4.2.3.2.2.3.1.cmml">(</mo><mi id="A2.SS1.2.p1.4.m4.1.1" xref="A2.SS1.2.p1.4.m4.1.1.cmml">τ</mi><mo id="A2.SS1.2.p1.4.m4.2.3.2.2.3.2.2" xref="A2.SS1.2.p1.4.m4.2.3.2.2.3.1.cmml">;</mo><mi id="A2.SS1.2.p1.4.m4.2.2" xref="A2.SS1.2.p1.4.m4.2.2.cmml">x</mi><mo id="A2.SS1.2.p1.4.m4.2.3.2.2.3.2.3" stretchy="false" xref="A2.SS1.2.p1.4.m4.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A2.SS1.2.p1.4.m4.2.3.1" xref="A2.SS1.2.p1.4.m4.2.3.1.cmml">≤</mo><msub id="A2.SS1.2.p1.4.m4.2.3.3" xref="A2.SS1.2.p1.4.m4.2.3.3.cmml"><mi id="A2.SS1.2.p1.4.m4.2.3.3.2" xref="A2.SS1.2.p1.4.m4.2.3.3.2.cmml">π</mi><mi id="A2.SS1.2.p1.4.m4.2.3.3.3" xref="A2.SS1.2.p1.4.m4.2.3.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.2.p1.4.m4.2b"><apply id="A2.SS1.2.p1.4.m4.2.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3"><leq id="A2.SS1.2.p1.4.m4.2.3.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.1"></leq><apply id="A2.SS1.2.p1.4.m4.2.3.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2"><apply id="A2.SS1.2.p1.4.m4.2.3.2.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1"><csymbol cd="ambiguous" id="A2.SS1.2.p1.4.m4.2.3.2.1.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1">subscript</csymbol><limit id="A2.SS1.2.p1.4.m4.2.3.2.1.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.2"></limit><apply id="A2.SS1.2.p1.4.m4.2.3.2.1.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3"><ci id="A2.SS1.2.p1.4.m4.2.3.2.1.3.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.1">→</ci><ci id="A2.SS1.2.p1.4.m4.2.3.2.1.3.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.2">𝑥</ci><apply id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3"><csymbol cd="ambiguous" id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3">superscript</csymbol><ci id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.2">𝜏</ci><plus id="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.1.3.3.3"></plus></apply></apply></apply><apply id="A2.SS1.2.p1.4.m4.2.3.2.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.2"><times id="A2.SS1.2.p1.4.m4.2.3.2.2.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.2.1"></times><ci id="A2.SS1.2.p1.4.m4.2.3.2.2.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.2.2">𝑃</ci><list id="A2.SS1.2.p1.4.m4.2.3.2.2.3.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.2.2.3.2"><ci id="A2.SS1.2.p1.4.m4.1.1.cmml" xref="A2.SS1.2.p1.4.m4.1.1">𝜏</ci><ci id="A2.SS1.2.p1.4.m4.2.2.cmml" xref="A2.SS1.2.p1.4.m4.2.2">𝑥</ci></list></apply></apply><apply id="A2.SS1.2.p1.4.m4.2.3.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3.3"><csymbol cd="ambiguous" id="A2.SS1.2.p1.4.m4.2.3.3.1.cmml" xref="A2.SS1.2.p1.4.m4.2.3.3">subscript</csymbol><ci id="A2.SS1.2.p1.4.m4.2.3.3.2.cmml" xref="A2.SS1.2.p1.4.m4.2.3.3.2">𝜋</ci><ci id="A2.SS1.2.p1.4.m4.2.3.3.3.cmml" xref="A2.SS1.2.p1.4.m4.2.3.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.4.m4.2c">\lim_{x\to\tau^{+}}P(\tau;x)\leq\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.4.m4.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) ≤ italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lim_{x\to\tau^{-}}P(\tau;x)\geq\pi_{L}" class="ltx_Math" display="inline" id="A2.SS1.2.p1.5.m5.2"><semantics id="A2.SS1.2.p1.5.m5.2a"><mrow id="A2.SS1.2.p1.5.m5.2.3" xref="A2.SS1.2.p1.5.m5.2.3.cmml"><mrow id="A2.SS1.2.p1.5.m5.2.3.2" xref="A2.SS1.2.p1.5.m5.2.3.2.cmml"><msub id="A2.SS1.2.p1.5.m5.2.3.2.1" xref="A2.SS1.2.p1.5.m5.2.3.2.1.cmml"><mo id="A2.SS1.2.p1.5.m5.2.3.2.1.2" xref="A2.SS1.2.p1.5.m5.2.3.2.1.2.cmml">lim</mo><mrow id="A2.SS1.2.p1.5.m5.2.3.2.1.3" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.cmml"><mi id="A2.SS1.2.p1.5.m5.2.3.2.1.3.2" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.2.cmml">x</mi><mo id="A2.SS1.2.p1.5.m5.2.3.2.1.3.1" stretchy="false" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.1.cmml">→</mo><msup id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.cmml"><mi id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.2" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.2.cmml">τ</mi><mo id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.3" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.3.cmml">−</mo></msup></mrow></msub><mrow id="A2.SS1.2.p1.5.m5.2.3.2.2" xref="A2.SS1.2.p1.5.m5.2.3.2.2.cmml"><mi id="A2.SS1.2.p1.5.m5.2.3.2.2.2" xref="A2.SS1.2.p1.5.m5.2.3.2.2.2.cmml">P</mi><mo id="A2.SS1.2.p1.5.m5.2.3.2.2.1" xref="A2.SS1.2.p1.5.m5.2.3.2.2.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.5.m5.2.3.2.2.3.2" xref="A2.SS1.2.p1.5.m5.2.3.2.2.3.1.cmml"><mo id="A2.SS1.2.p1.5.m5.2.3.2.2.3.2.1" stretchy="false" xref="A2.SS1.2.p1.5.m5.2.3.2.2.3.1.cmml">(</mo><mi id="A2.SS1.2.p1.5.m5.1.1" xref="A2.SS1.2.p1.5.m5.1.1.cmml">τ</mi><mo id="A2.SS1.2.p1.5.m5.2.3.2.2.3.2.2" xref="A2.SS1.2.p1.5.m5.2.3.2.2.3.1.cmml">;</mo><mi id="A2.SS1.2.p1.5.m5.2.2" xref="A2.SS1.2.p1.5.m5.2.2.cmml">x</mi><mo id="A2.SS1.2.p1.5.m5.2.3.2.2.3.2.3" stretchy="false" xref="A2.SS1.2.p1.5.m5.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A2.SS1.2.p1.5.m5.2.3.1" xref="A2.SS1.2.p1.5.m5.2.3.1.cmml">≥</mo><msub id="A2.SS1.2.p1.5.m5.2.3.3" xref="A2.SS1.2.p1.5.m5.2.3.3.cmml"><mi id="A2.SS1.2.p1.5.m5.2.3.3.2" xref="A2.SS1.2.p1.5.m5.2.3.3.2.cmml">π</mi><mi id="A2.SS1.2.p1.5.m5.2.3.3.3" xref="A2.SS1.2.p1.5.m5.2.3.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.2.p1.5.m5.2b"><apply id="A2.SS1.2.p1.5.m5.2.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3"><geq id="A2.SS1.2.p1.5.m5.2.3.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.1"></geq><apply id="A2.SS1.2.p1.5.m5.2.3.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2"><apply id="A2.SS1.2.p1.5.m5.2.3.2.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1"><csymbol cd="ambiguous" id="A2.SS1.2.p1.5.m5.2.3.2.1.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1">subscript</csymbol><limit id="A2.SS1.2.p1.5.m5.2.3.2.1.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.2"></limit><apply id="A2.SS1.2.p1.5.m5.2.3.2.1.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3"><ci id="A2.SS1.2.p1.5.m5.2.3.2.1.3.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.1">→</ci><ci id="A2.SS1.2.p1.5.m5.2.3.2.1.3.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.2">𝑥</ci><apply id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3"><csymbol cd="ambiguous" id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3">superscript</csymbol><ci id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.2">𝜏</ci><minus id="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.1.3.3.3"></minus></apply></apply></apply><apply id="A2.SS1.2.p1.5.m5.2.3.2.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.2"><times id="A2.SS1.2.p1.5.m5.2.3.2.2.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.2.1"></times><ci id="A2.SS1.2.p1.5.m5.2.3.2.2.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.2.2">𝑃</ci><list id="A2.SS1.2.p1.5.m5.2.3.2.2.3.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.2.2.3.2"><ci id="A2.SS1.2.p1.5.m5.1.1.cmml" xref="A2.SS1.2.p1.5.m5.1.1">𝜏</ci><ci id="A2.SS1.2.p1.5.m5.2.2.cmml" xref="A2.SS1.2.p1.5.m5.2.2">𝑥</ci></list></apply></apply><apply id="A2.SS1.2.p1.5.m5.2.3.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3.3"><csymbol cd="ambiguous" id="A2.SS1.2.p1.5.m5.2.3.3.1.cmml" xref="A2.SS1.2.p1.5.m5.2.3.3">subscript</csymbol><ci id="A2.SS1.2.p1.5.m5.2.3.3.2.cmml" xref="A2.SS1.2.p1.5.m5.2.3.3.2">𝜋</ci><ci id="A2.SS1.2.p1.5.m5.2.3.3.3.cmml" xref="A2.SS1.2.p1.5.m5.2.3.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.5.m5.2c">\lim_{x\to\tau^{-}}P(\tau;x)\geq\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.5.m5.2d">roman_lim start_POSTSUBSCRIPT italic_x → italic_τ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_τ ; italic_x ) ≥ italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="G(\tau)=P(\tau;\tau)=\pi_{L}=F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.2.p1.6.m6.4"><semantics id="A2.SS1.2.p1.6.m6.4a"><mrow id="A2.SS1.2.p1.6.m6.4.5" xref="A2.SS1.2.p1.6.m6.4.5.cmml"><mrow id="A2.SS1.2.p1.6.m6.4.5.2" xref="A2.SS1.2.p1.6.m6.4.5.2.cmml"><mi id="A2.SS1.2.p1.6.m6.4.5.2.2" xref="A2.SS1.2.p1.6.m6.4.5.2.2.cmml">G</mi><mo id="A2.SS1.2.p1.6.m6.4.5.2.1" xref="A2.SS1.2.p1.6.m6.4.5.2.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.6.m6.4.5.2.3.2" xref="A2.SS1.2.p1.6.m6.4.5.2.cmml"><mo id="A2.SS1.2.p1.6.m6.4.5.2.3.2.1" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.2.cmml">(</mo><mi id="A2.SS1.2.p1.6.m6.1.1" xref="A2.SS1.2.p1.6.m6.1.1.cmml">τ</mi><mo id="A2.SS1.2.p1.6.m6.4.5.2.3.2.2" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.2.p1.6.m6.4.5.3" xref="A2.SS1.2.p1.6.m6.4.5.3.cmml">=</mo><mrow id="A2.SS1.2.p1.6.m6.4.5.4" xref="A2.SS1.2.p1.6.m6.4.5.4.cmml"><mi id="A2.SS1.2.p1.6.m6.4.5.4.2" xref="A2.SS1.2.p1.6.m6.4.5.4.2.cmml">P</mi><mo id="A2.SS1.2.p1.6.m6.4.5.4.1" xref="A2.SS1.2.p1.6.m6.4.5.4.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.6.m6.4.5.4.3.2" xref="A2.SS1.2.p1.6.m6.4.5.4.3.1.cmml"><mo id="A2.SS1.2.p1.6.m6.4.5.4.3.2.1" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.4.3.1.cmml">(</mo><mi id="A2.SS1.2.p1.6.m6.2.2" xref="A2.SS1.2.p1.6.m6.2.2.cmml">τ</mi><mo id="A2.SS1.2.p1.6.m6.4.5.4.3.2.2" xref="A2.SS1.2.p1.6.m6.4.5.4.3.1.cmml">;</mo><mi id="A2.SS1.2.p1.6.m6.3.3" xref="A2.SS1.2.p1.6.m6.3.3.cmml">τ</mi><mo id="A2.SS1.2.p1.6.m6.4.5.4.3.2.3" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.2.p1.6.m6.4.5.5" xref="A2.SS1.2.p1.6.m6.4.5.5.cmml">=</mo><msub id="A2.SS1.2.p1.6.m6.4.5.6" xref="A2.SS1.2.p1.6.m6.4.5.6.cmml"><mi id="A2.SS1.2.p1.6.m6.4.5.6.2" xref="A2.SS1.2.p1.6.m6.4.5.6.2.cmml">π</mi><mi id="A2.SS1.2.p1.6.m6.4.5.6.3" xref="A2.SS1.2.p1.6.m6.4.5.6.3.cmml">L</mi></msub><mo id="A2.SS1.2.p1.6.m6.4.5.7" xref="A2.SS1.2.p1.6.m6.4.5.7.cmml">=</mo><mrow id="A2.SS1.2.p1.6.m6.4.5.8" xref="A2.SS1.2.p1.6.m6.4.5.8.cmml"><mi id="A2.SS1.2.p1.6.m6.4.5.8.2" xref="A2.SS1.2.p1.6.m6.4.5.8.2.cmml">F</mi><mo id="A2.SS1.2.p1.6.m6.4.5.8.1" xref="A2.SS1.2.p1.6.m6.4.5.8.1.cmml">⁢</mo><mrow id="A2.SS1.2.p1.6.m6.4.5.8.3.2" xref="A2.SS1.2.p1.6.m6.4.5.8.cmml"><mo id="A2.SS1.2.p1.6.m6.4.5.8.3.2.1" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.8.cmml">(</mo><mi id="A2.SS1.2.p1.6.m6.4.4" xref="A2.SS1.2.p1.6.m6.4.4.cmml">τ</mi><mo id="A2.SS1.2.p1.6.m6.4.5.8.3.2.2" stretchy="false" xref="A2.SS1.2.p1.6.m6.4.5.8.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.2.p1.6.m6.4b"><apply id="A2.SS1.2.p1.6.m6.4.5.cmml" xref="A2.SS1.2.p1.6.m6.4.5"><and id="A2.SS1.2.p1.6.m6.4.5a.cmml" xref="A2.SS1.2.p1.6.m6.4.5"></and><apply id="A2.SS1.2.p1.6.m6.4.5b.cmml" xref="A2.SS1.2.p1.6.m6.4.5"><eq id="A2.SS1.2.p1.6.m6.4.5.3.cmml" xref="A2.SS1.2.p1.6.m6.4.5.3"></eq><apply id="A2.SS1.2.p1.6.m6.4.5.2.cmml" xref="A2.SS1.2.p1.6.m6.4.5.2"><times id="A2.SS1.2.p1.6.m6.4.5.2.1.cmml" xref="A2.SS1.2.p1.6.m6.4.5.2.1"></times><ci id="A2.SS1.2.p1.6.m6.4.5.2.2.cmml" xref="A2.SS1.2.p1.6.m6.4.5.2.2">𝐺</ci><ci id="A2.SS1.2.p1.6.m6.1.1.cmml" xref="A2.SS1.2.p1.6.m6.1.1">𝜏</ci></apply><apply id="A2.SS1.2.p1.6.m6.4.5.4.cmml" xref="A2.SS1.2.p1.6.m6.4.5.4"><times id="A2.SS1.2.p1.6.m6.4.5.4.1.cmml" xref="A2.SS1.2.p1.6.m6.4.5.4.1"></times><ci id="A2.SS1.2.p1.6.m6.4.5.4.2.cmml" xref="A2.SS1.2.p1.6.m6.4.5.4.2">𝑃</ci><list id="A2.SS1.2.p1.6.m6.4.5.4.3.1.cmml" xref="A2.SS1.2.p1.6.m6.4.5.4.3.2"><ci id="A2.SS1.2.p1.6.m6.2.2.cmml" xref="A2.SS1.2.p1.6.m6.2.2">𝜏</ci><ci id="A2.SS1.2.p1.6.m6.3.3.cmml" xref="A2.SS1.2.p1.6.m6.3.3">𝜏</ci></list></apply></apply><apply id="A2.SS1.2.p1.6.m6.4.5c.cmml" xref="A2.SS1.2.p1.6.m6.4.5"><eq id="A2.SS1.2.p1.6.m6.4.5.5.cmml" xref="A2.SS1.2.p1.6.m6.4.5.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.2.p1.6.m6.4.5.4.cmml" id="A2.SS1.2.p1.6.m6.4.5d.cmml" xref="A2.SS1.2.p1.6.m6.4.5"></share><apply id="A2.SS1.2.p1.6.m6.4.5.6.cmml" xref="A2.SS1.2.p1.6.m6.4.5.6"><csymbol cd="ambiguous" id="A2.SS1.2.p1.6.m6.4.5.6.1.cmml" xref="A2.SS1.2.p1.6.m6.4.5.6">subscript</csymbol><ci id="A2.SS1.2.p1.6.m6.4.5.6.2.cmml" xref="A2.SS1.2.p1.6.m6.4.5.6.2">𝜋</ci><ci id="A2.SS1.2.p1.6.m6.4.5.6.3.cmml" xref="A2.SS1.2.p1.6.m6.4.5.6.3">𝐿</ci></apply></apply><apply id="A2.SS1.2.p1.6.m6.4.5e.cmml" xref="A2.SS1.2.p1.6.m6.4.5"><eq id="A2.SS1.2.p1.6.m6.4.5.7.cmml" xref="A2.SS1.2.p1.6.m6.4.5.7"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.2.p1.6.m6.4.5.6.cmml" id="A2.SS1.2.p1.6.m6.4.5f.cmml" xref="A2.SS1.2.p1.6.m6.4.5"></share><apply id="A2.SS1.2.p1.6.m6.4.5.8.cmml" xref="A2.SS1.2.p1.6.m6.4.5.8"><times id="A2.SS1.2.p1.6.m6.4.5.8.1.cmml" xref="A2.SS1.2.p1.6.m6.4.5.8.1"></times><ci id="A2.SS1.2.p1.6.m6.4.5.8.2.cmml" xref="A2.SS1.2.p1.6.m6.4.5.8.2">𝐹</ci><ci id="A2.SS1.2.p1.6.m6.4.4.cmml" xref="A2.SS1.2.p1.6.m6.4.4">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.2.p1.6.m6.4c">G(\tau)=P(\tau;\tau)=\pi_{L}=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.2.p1.6.m6.4d">italic_G ( italic_τ ) = italic_P ( italic_τ ; italic_τ ) = italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_F ( italic_τ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS1.3.p2"> <p class="ltx_p" id="A2.SS1.3.p2.8">For sufficiency, assume <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.3.p2.1.m1.2"><semantics id="A2.SS1.3.p2.1.m1.2a"><mrow id="A2.SS1.3.p2.1.m1.2.3" xref="A2.SS1.3.p2.1.m1.2.3.cmml"><mrow id="A2.SS1.3.p2.1.m1.2.3.2" xref="A2.SS1.3.p2.1.m1.2.3.2.cmml"><mi id="A2.SS1.3.p2.1.m1.2.3.2.2" xref="A2.SS1.3.p2.1.m1.2.3.2.2.cmml">G</mi><mo id="A2.SS1.3.p2.1.m1.2.3.2.1" xref="A2.SS1.3.p2.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.1.m1.2.3.2.3.2" xref="A2.SS1.3.p2.1.m1.2.3.2.cmml"><mo id="A2.SS1.3.p2.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.3.p2.1.m1.2.3.2.cmml">(</mo><mi id="A2.SS1.3.p2.1.m1.1.1" xref="A2.SS1.3.p2.1.m1.1.1.cmml">τ</mi><mo id="A2.SS1.3.p2.1.m1.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.3.p2.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.1.m1.2.3.1" xref="A2.SS1.3.p2.1.m1.2.3.1.cmml">=</mo><mrow id="A2.SS1.3.p2.1.m1.2.3.3" xref="A2.SS1.3.p2.1.m1.2.3.3.cmml"><mi id="A2.SS1.3.p2.1.m1.2.3.3.2" xref="A2.SS1.3.p2.1.m1.2.3.3.2.cmml">F</mi><mo id="A2.SS1.3.p2.1.m1.2.3.3.1" xref="A2.SS1.3.p2.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.1.m1.2.3.3.3.2" xref="A2.SS1.3.p2.1.m1.2.3.3.cmml"><mo id="A2.SS1.3.p2.1.m1.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.3.p2.1.m1.2.3.3.cmml">(</mo><mi id="A2.SS1.3.p2.1.m1.2.2" xref="A2.SS1.3.p2.1.m1.2.2.cmml">τ</mi><mo id="A2.SS1.3.p2.1.m1.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.3.p2.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.1.m1.2b"><apply id="A2.SS1.3.p2.1.m1.2.3.cmml" xref="A2.SS1.3.p2.1.m1.2.3"><eq id="A2.SS1.3.p2.1.m1.2.3.1.cmml" xref="A2.SS1.3.p2.1.m1.2.3.1"></eq><apply id="A2.SS1.3.p2.1.m1.2.3.2.cmml" xref="A2.SS1.3.p2.1.m1.2.3.2"><times id="A2.SS1.3.p2.1.m1.2.3.2.1.cmml" xref="A2.SS1.3.p2.1.m1.2.3.2.1"></times><ci id="A2.SS1.3.p2.1.m1.2.3.2.2.cmml" xref="A2.SS1.3.p2.1.m1.2.3.2.2">𝐺</ci><ci id="A2.SS1.3.p2.1.m1.1.1.cmml" xref="A2.SS1.3.p2.1.m1.1.1">𝜏</ci></apply><apply id="A2.SS1.3.p2.1.m1.2.3.3.cmml" xref="A2.SS1.3.p2.1.m1.2.3.3"><times id="A2.SS1.3.p2.1.m1.2.3.3.1.cmml" xref="A2.SS1.3.p2.1.m1.2.3.3.1"></times><ci id="A2.SS1.3.p2.1.m1.2.3.3.2.cmml" xref="A2.SS1.3.p2.1.m1.2.3.3.2">𝐹</ci><ci id="A2.SS1.3.p2.1.m1.2.2.cmml" xref="A2.SS1.3.p2.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.1.m1.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.1.m1.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>. Equivalently <math alttext="P(\tau;\tau)-\pi_{L}=0" class="ltx_Math" display="inline" id="A2.SS1.3.p2.2.m2.2"><semantics id="A2.SS1.3.p2.2.m2.2a"><mrow id="A2.SS1.3.p2.2.m2.2.3" xref="A2.SS1.3.p2.2.m2.2.3.cmml"><mrow id="A2.SS1.3.p2.2.m2.2.3.2" xref="A2.SS1.3.p2.2.m2.2.3.2.cmml"><mrow id="A2.SS1.3.p2.2.m2.2.3.2.2" xref="A2.SS1.3.p2.2.m2.2.3.2.2.cmml"><mi id="A2.SS1.3.p2.2.m2.2.3.2.2.2" xref="A2.SS1.3.p2.2.m2.2.3.2.2.2.cmml">P</mi><mo id="A2.SS1.3.p2.2.m2.2.3.2.2.1" xref="A2.SS1.3.p2.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.2.m2.2.3.2.2.3.2" xref="A2.SS1.3.p2.2.m2.2.3.2.2.3.1.cmml"><mo id="A2.SS1.3.p2.2.m2.2.3.2.2.3.2.1" stretchy="false" xref="A2.SS1.3.p2.2.m2.2.3.2.2.3.1.cmml">(</mo><mi id="A2.SS1.3.p2.2.m2.1.1" xref="A2.SS1.3.p2.2.m2.1.1.cmml">τ</mi><mo id="A2.SS1.3.p2.2.m2.2.3.2.2.3.2.2" xref="A2.SS1.3.p2.2.m2.2.3.2.2.3.1.cmml">;</mo><mi id="A2.SS1.3.p2.2.m2.2.2" xref="A2.SS1.3.p2.2.m2.2.2.cmml">τ</mi><mo id="A2.SS1.3.p2.2.m2.2.3.2.2.3.2.3" stretchy="false" xref="A2.SS1.3.p2.2.m2.2.3.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.2.m2.2.3.2.1" xref="A2.SS1.3.p2.2.m2.2.3.2.1.cmml">−</mo><msub id="A2.SS1.3.p2.2.m2.2.3.2.3" xref="A2.SS1.3.p2.2.m2.2.3.2.3.cmml"><mi id="A2.SS1.3.p2.2.m2.2.3.2.3.2" xref="A2.SS1.3.p2.2.m2.2.3.2.3.2.cmml">π</mi><mi id="A2.SS1.3.p2.2.m2.2.3.2.3.3" xref="A2.SS1.3.p2.2.m2.2.3.2.3.3.cmml">L</mi></msub></mrow><mo id="A2.SS1.3.p2.2.m2.2.3.1" xref="A2.SS1.3.p2.2.m2.2.3.1.cmml">=</mo><mn id="A2.SS1.3.p2.2.m2.2.3.3" xref="A2.SS1.3.p2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.2.m2.2b"><apply id="A2.SS1.3.p2.2.m2.2.3.cmml" xref="A2.SS1.3.p2.2.m2.2.3"><eq id="A2.SS1.3.p2.2.m2.2.3.1.cmml" xref="A2.SS1.3.p2.2.m2.2.3.1"></eq><apply id="A2.SS1.3.p2.2.m2.2.3.2.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2"><minus id="A2.SS1.3.p2.2.m2.2.3.2.1.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.1"></minus><apply id="A2.SS1.3.p2.2.m2.2.3.2.2.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.2"><times id="A2.SS1.3.p2.2.m2.2.3.2.2.1.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.2.1"></times><ci id="A2.SS1.3.p2.2.m2.2.3.2.2.2.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.2.2">𝑃</ci><list id="A2.SS1.3.p2.2.m2.2.3.2.2.3.1.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.2.3.2"><ci id="A2.SS1.3.p2.2.m2.1.1.cmml" xref="A2.SS1.3.p2.2.m2.1.1">𝜏</ci><ci id="A2.SS1.3.p2.2.m2.2.2.cmml" xref="A2.SS1.3.p2.2.m2.2.2">𝜏</ci></list></apply><apply id="A2.SS1.3.p2.2.m2.2.3.2.3.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.3"><csymbol cd="ambiguous" id="A2.SS1.3.p2.2.m2.2.3.2.3.1.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.3">subscript</csymbol><ci id="A2.SS1.3.p2.2.m2.2.3.2.3.2.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.3.2">𝜋</ci><ci id="A2.SS1.3.p2.2.m2.2.3.2.3.3.cmml" xref="A2.SS1.3.p2.2.m2.2.3.2.3.3">𝐿</ci></apply></apply><cn id="A2.SS1.3.p2.2.m2.2.3.3.cmml" type="integer" xref="A2.SS1.3.p2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.2.m2.2c">P(\tau;\tau)-\pi_{L}=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.2.m2.2d">italic_P ( italic_τ ; italic_τ ) - italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = 0</annotation></semantics></math>, so (a) implies (b) and the single crossing is at <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.3.p2.3.m3.1"><semantics id="A2.SS1.3.p2.3.m3.1a"><mi id="A2.SS1.3.p2.3.m3.1.1" xref="A2.SS1.3.p2.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.3.m3.1b"><ci id="A2.SS1.3.p2.3.m3.1.1.cmml" xref="A2.SS1.3.p2.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.3.m3.1d">italic_τ</annotation></semantics></math>. Thus assume (b) holds and take some <math alttext="x>\tau" class="ltx_Math" display="inline" id="A2.SS1.3.p2.4.m4.1"><semantics id="A2.SS1.3.p2.4.m4.1a"><mrow id="A2.SS1.3.p2.4.m4.1.1" xref="A2.SS1.3.p2.4.m4.1.1.cmml"><mi id="A2.SS1.3.p2.4.m4.1.1.2" xref="A2.SS1.3.p2.4.m4.1.1.2.cmml">x</mi><mo id="A2.SS1.3.p2.4.m4.1.1.1" xref="A2.SS1.3.p2.4.m4.1.1.1.cmml">></mo><mi id="A2.SS1.3.p2.4.m4.1.1.3" xref="A2.SS1.3.p2.4.m4.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.4.m4.1b"><apply id="A2.SS1.3.p2.4.m4.1.1.cmml" xref="A2.SS1.3.p2.4.m4.1.1"><gt id="A2.SS1.3.p2.4.m4.1.1.1.cmml" xref="A2.SS1.3.p2.4.m4.1.1.1"></gt><ci id="A2.SS1.3.p2.4.m4.1.1.2.cmml" xref="A2.SS1.3.p2.4.m4.1.1.2">𝑥</ci><ci id="A2.SS1.3.p2.4.m4.1.1.3.cmml" xref="A2.SS1.3.p2.4.m4.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.4.m4.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.4.m4.1d">italic_x > italic_τ</annotation></semantics></math>. By single crossing, <math alttext="P(\tau;x)\leq P(\tau;\tau)=G(\tau)=F(\tau)=\pi_{L}" class="ltx_Math" display="inline" id="A2.SS1.3.p2.5.m5.6"><semantics id="A2.SS1.3.p2.5.m5.6a"><mrow id="A2.SS1.3.p2.5.m5.6.7" xref="A2.SS1.3.p2.5.m5.6.7.cmml"><mrow id="A2.SS1.3.p2.5.m5.6.7.2" xref="A2.SS1.3.p2.5.m5.6.7.2.cmml"><mi id="A2.SS1.3.p2.5.m5.6.7.2.2" xref="A2.SS1.3.p2.5.m5.6.7.2.2.cmml">P</mi><mo id="A2.SS1.3.p2.5.m5.6.7.2.1" xref="A2.SS1.3.p2.5.m5.6.7.2.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.2.3.2" xref="A2.SS1.3.p2.5.m5.6.7.2.3.1.cmml"><mo id="A2.SS1.3.p2.5.m5.6.7.2.3.2.1" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.2.3.1.cmml">(</mo><mi id="A2.SS1.3.p2.5.m5.1.1" xref="A2.SS1.3.p2.5.m5.1.1.cmml">τ</mi><mo id="A2.SS1.3.p2.5.m5.6.7.2.3.2.2" xref="A2.SS1.3.p2.5.m5.6.7.2.3.1.cmml">;</mo><mi id="A2.SS1.3.p2.5.m5.2.2" xref="A2.SS1.3.p2.5.m5.2.2.cmml">x</mi><mo id="A2.SS1.3.p2.5.m5.6.7.2.3.2.3" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.5.m5.6.7.3" xref="A2.SS1.3.p2.5.m5.6.7.3.cmml">≤</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.4" xref="A2.SS1.3.p2.5.m5.6.7.4.cmml"><mi id="A2.SS1.3.p2.5.m5.6.7.4.2" xref="A2.SS1.3.p2.5.m5.6.7.4.2.cmml">P</mi><mo id="A2.SS1.3.p2.5.m5.6.7.4.1" xref="A2.SS1.3.p2.5.m5.6.7.4.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.4.3.2" xref="A2.SS1.3.p2.5.m5.6.7.4.3.1.cmml"><mo id="A2.SS1.3.p2.5.m5.6.7.4.3.2.1" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.4.3.1.cmml">(</mo><mi id="A2.SS1.3.p2.5.m5.3.3" xref="A2.SS1.3.p2.5.m5.3.3.cmml">τ</mi><mo id="A2.SS1.3.p2.5.m5.6.7.4.3.2.2" xref="A2.SS1.3.p2.5.m5.6.7.4.3.1.cmml">;</mo><mi id="A2.SS1.3.p2.5.m5.4.4" xref="A2.SS1.3.p2.5.m5.4.4.cmml">τ</mi><mo id="A2.SS1.3.p2.5.m5.6.7.4.3.2.3" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.4.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.5.m5.6.7.5" xref="A2.SS1.3.p2.5.m5.6.7.5.cmml">=</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.6" xref="A2.SS1.3.p2.5.m5.6.7.6.cmml"><mi id="A2.SS1.3.p2.5.m5.6.7.6.2" xref="A2.SS1.3.p2.5.m5.6.7.6.2.cmml">G</mi><mo id="A2.SS1.3.p2.5.m5.6.7.6.1" xref="A2.SS1.3.p2.5.m5.6.7.6.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.6.3.2" xref="A2.SS1.3.p2.5.m5.6.7.6.cmml"><mo id="A2.SS1.3.p2.5.m5.6.7.6.3.2.1" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.6.cmml">(</mo><mi id="A2.SS1.3.p2.5.m5.5.5" xref="A2.SS1.3.p2.5.m5.5.5.cmml">τ</mi><mo id="A2.SS1.3.p2.5.m5.6.7.6.3.2.2" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.6.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.5.m5.6.7.7" xref="A2.SS1.3.p2.5.m5.6.7.7.cmml">=</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.8" xref="A2.SS1.3.p2.5.m5.6.7.8.cmml"><mi id="A2.SS1.3.p2.5.m5.6.7.8.2" xref="A2.SS1.3.p2.5.m5.6.7.8.2.cmml">F</mi><mo id="A2.SS1.3.p2.5.m5.6.7.8.1" xref="A2.SS1.3.p2.5.m5.6.7.8.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.5.m5.6.7.8.3.2" xref="A2.SS1.3.p2.5.m5.6.7.8.cmml"><mo id="A2.SS1.3.p2.5.m5.6.7.8.3.2.1" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.8.cmml">(</mo><mi id="A2.SS1.3.p2.5.m5.6.6" xref="A2.SS1.3.p2.5.m5.6.6.cmml">τ</mi><mo id="A2.SS1.3.p2.5.m5.6.7.8.3.2.2" stretchy="false" xref="A2.SS1.3.p2.5.m5.6.7.8.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.5.m5.6.7.9" xref="A2.SS1.3.p2.5.m5.6.7.9.cmml">=</mo><msub id="A2.SS1.3.p2.5.m5.6.7.10" xref="A2.SS1.3.p2.5.m5.6.7.10.cmml"><mi id="A2.SS1.3.p2.5.m5.6.7.10.2" xref="A2.SS1.3.p2.5.m5.6.7.10.2.cmml">π</mi><mi id="A2.SS1.3.p2.5.m5.6.7.10.3" xref="A2.SS1.3.p2.5.m5.6.7.10.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.5.m5.6b"><apply id="A2.SS1.3.p2.5.m5.6.7.cmml" xref="A2.SS1.3.p2.5.m5.6.7"><and id="A2.SS1.3.p2.5.m5.6.7a.cmml" xref="A2.SS1.3.p2.5.m5.6.7"></and><apply id="A2.SS1.3.p2.5.m5.6.7b.cmml" xref="A2.SS1.3.p2.5.m5.6.7"><leq id="A2.SS1.3.p2.5.m5.6.7.3.cmml" xref="A2.SS1.3.p2.5.m5.6.7.3"></leq><apply id="A2.SS1.3.p2.5.m5.6.7.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.2"><times id="A2.SS1.3.p2.5.m5.6.7.2.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.2.1"></times><ci id="A2.SS1.3.p2.5.m5.6.7.2.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.2.2">𝑃</ci><list id="A2.SS1.3.p2.5.m5.6.7.2.3.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.2.3.2"><ci id="A2.SS1.3.p2.5.m5.1.1.cmml" xref="A2.SS1.3.p2.5.m5.1.1">𝜏</ci><ci id="A2.SS1.3.p2.5.m5.2.2.cmml" xref="A2.SS1.3.p2.5.m5.2.2">𝑥</ci></list></apply><apply id="A2.SS1.3.p2.5.m5.6.7.4.cmml" xref="A2.SS1.3.p2.5.m5.6.7.4"><times id="A2.SS1.3.p2.5.m5.6.7.4.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.4.1"></times><ci id="A2.SS1.3.p2.5.m5.6.7.4.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.4.2">𝑃</ci><list id="A2.SS1.3.p2.5.m5.6.7.4.3.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.4.3.2"><ci id="A2.SS1.3.p2.5.m5.3.3.cmml" xref="A2.SS1.3.p2.5.m5.3.3">𝜏</ci><ci id="A2.SS1.3.p2.5.m5.4.4.cmml" xref="A2.SS1.3.p2.5.m5.4.4">𝜏</ci></list></apply></apply><apply id="A2.SS1.3.p2.5.m5.6.7c.cmml" xref="A2.SS1.3.p2.5.m5.6.7"><eq id="A2.SS1.3.p2.5.m5.6.7.5.cmml" xref="A2.SS1.3.p2.5.m5.6.7.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.3.p2.5.m5.6.7.4.cmml" id="A2.SS1.3.p2.5.m5.6.7d.cmml" xref="A2.SS1.3.p2.5.m5.6.7"></share><apply id="A2.SS1.3.p2.5.m5.6.7.6.cmml" xref="A2.SS1.3.p2.5.m5.6.7.6"><times id="A2.SS1.3.p2.5.m5.6.7.6.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.6.1"></times><ci id="A2.SS1.3.p2.5.m5.6.7.6.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.6.2">𝐺</ci><ci id="A2.SS1.3.p2.5.m5.5.5.cmml" xref="A2.SS1.3.p2.5.m5.5.5">𝜏</ci></apply></apply><apply id="A2.SS1.3.p2.5.m5.6.7e.cmml" xref="A2.SS1.3.p2.5.m5.6.7"><eq id="A2.SS1.3.p2.5.m5.6.7.7.cmml" xref="A2.SS1.3.p2.5.m5.6.7.7"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.3.p2.5.m5.6.7.6.cmml" id="A2.SS1.3.p2.5.m5.6.7f.cmml" xref="A2.SS1.3.p2.5.m5.6.7"></share><apply id="A2.SS1.3.p2.5.m5.6.7.8.cmml" xref="A2.SS1.3.p2.5.m5.6.7.8"><times id="A2.SS1.3.p2.5.m5.6.7.8.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.8.1"></times><ci id="A2.SS1.3.p2.5.m5.6.7.8.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.8.2">𝐹</ci><ci id="A2.SS1.3.p2.5.m5.6.6.cmml" xref="A2.SS1.3.p2.5.m5.6.6">𝜏</ci></apply></apply><apply id="A2.SS1.3.p2.5.m5.6.7g.cmml" xref="A2.SS1.3.p2.5.m5.6.7"><eq id="A2.SS1.3.p2.5.m5.6.7.9.cmml" xref="A2.SS1.3.p2.5.m5.6.7.9"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.3.p2.5.m5.6.7.8.cmml" id="A2.SS1.3.p2.5.m5.6.7h.cmml" xref="A2.SS1.3.p2.5.m5.6.7"></share><apply id="A2.SS1.3.p2.5.m5.6.7.10.cmml" xref="A2.SS1.3.p2.5.m5.6.7.10"><csymbol cd="ambiguous" id="A2.SS1.3.p2.5.m5.6.7.10.1.cmml" xref="A2.SS1.3.p2.5.m5.6.7.10">subscript</csymbol><ci id="A2.SS1.3.p2.5.m5.6.7.10.2.cmml" xref="A2.SS1.3.p2.5.m5.6.7.10.2">𝜋</ci><ci id="A2.SS1.3.p2.5.m5.6.7.10.3.cmml" xref="A2.SS1.3.p2.5.m5.6.7.10.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.5.m5.6c">P(\tau;x)\leq P(\tau;\tau)=G(\tau)=F(\tau)=\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.5.m5.6d">italic_P ( italic_τ ; italic_x ) ≤ italic_P ( italic_τ ; italic_τ ) = italic_G ( italic_τ ) = italic_F ( italic_τ ) = italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, if <math alttext="x\leq\tau" class="ltx_Math" display="inline" id="A2.SS1.3.p2.6.m6.1"><semantics id="A2.SS1.3.p2.6.m6.1a"><mrow id="A2.SS1.3.p2.6.m6.1.1" xref="A2.SS1.3.p2.6.m6.1.1.cmml"><mi id="A2.SS1.3.p2.6.m6.1.1.2" xref="A2.SS1.3.p2.6.m6.1.1.2.cmml">x</mi><mo id="A2.SS1.3.p2.6.m6.1.1.1" xref="A2.SS1.3.p2.6.m6.1.1.1.cmml">≤</mo><mi id="A2.SS1.3.p2.6.m6.1.1.3" xref="A2.SS1.3.p2.6.m6.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.6.m6.1b"><apply id="A2.SS1.3.p2.6.m6.1.1.cmml" xref="A2.SS1.3.p2.6.m6.1.1"><leq id="A2.SS1.3.p2.6.m6.1.1.1.cmml" xref="A2.SS1.3.p2.6.m6.1.1.1"></leq><ci id="A2.SS1.3.p2.6.m6.1.1.2.cmml" xref="A2.SS1.3.p2.6.m6.1.1.2">𝑥</ci><ci id="A2.SS1.3.p2.6.m6.1.1.3.cmml" xref="A2.SS1.3.p2.6.m6.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.6.m6.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.6.m6.1d">italic_x ≤ italic_τ</annotation></semantics></math>, <math alttext="P(\tau;x)\geq P(\tau;\tau)=\pi_{L}" class="ltx_Math" display="inline" id="A2.SS1.3.p2.7.m7.4"><semantics id="A2.SS1.3.p2.7.m7.4a"><mrow id="A2.SS1.3.p2.7.m7.4.5" xref="A2.SS1.3.p2.7.m7.4.5.cmml"><mrow id="A2.SS1.3.p2.7.m7.4.5.2" xref="A2.SS1.3.p2.7.m7.4.5.2.cmml"><mi id="A2.SS1.3.p2.7.m7.4.5.2.2" xref="A2.SS1.3.p2.7.m7.4.5.2.2.cmml">P</mi><mo id="A2.SS1.3.p2.7.m7.4.5.2.1" xref="A2.SS1.3.p2.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.7.m7.4.5.2.3.2" xref="A2.SS1.3.p2.7.m7.4.5.2.3.1.cmml"><mo id="A2.SS1.3.p2.7.m7.4.5.2.3.2.1" stretchy="false" xref="A2.SS1.3.p2.7.m7.4.5.2.3.1.cmml">(</mo><mi id="A2.SS1.3.p2.7.m7.1.1" xref="A2.SS1.3.p2.7.m7.1.1.cmml">τ</mi><mo id="A2.SS1.3.p2.7.m7.4.5.2.3.2.2" xref="A2.SS1.3.p2.7.m7.4.5.2.3.1.cmml">;</mo><mi id="A2.SS1.3.p2.7.m7.2.2" xref="A2.SS1.3.p2.7.m7.2.2.cmml">x</mi><mo id="A2.SS1.3.p2.7.m7.4.5.2.3.2.3" stretchy="false" xref="A2.SS1.3.p2.7.m7.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.7.m7.4.5.3" xref="A2.SS1.3.p2.7.m7.4.5.3.cmml">≥</mo><mrow id="A2.SS1.3.p2.7.m7.4.5.4" xref="A2.SS1.3.p2.7.m7.4.5.4.cmml"><mi id="A2.SS1.3.p2.7.m7.4.5.4.2" xref="A2.SS1.3.p2.7.m7.4.5.4.2.cmml">P</mi><mo id="A2.SS1.3.p2.7.m7.4.5.4.1" xref="A2.SS1.3.p2.7.m7.4.5.4.1.cmml">⁢</mo><mrow id="A2.SS1.3.p2.7.m7.4.5.4.3.2" xref="A2.SS1.3.p2.7.m7.4.5.4.3.1.cmml"><mo id="A2.SS1.3.p2.7.m7.4.5.4.3.2.1" stretchy="false" xref="A2.SS1.3.p2.7.m7.4.5.4.3.1.cmml">(</mo><mi id="A2.SS1.3.p2.7.m7.3.3" xref="A2.SS1.3.p2.7.m7.3.3.cmml">τ</mi><mo id="A2.SS1.3.p2.7.m7.4.5.4.3.2.2" xref="A2.SS1.3.p2.7.m7.4.5.4.3.1.cmml">;</mo><mi id="A2.SS1.3.p2.7.m7.4.4" xref="A2.SS1.3.p2.7.m7.4.4.cmml">τ</mi><mo id="A2.SS1.3.p2.7.m7.4.5.4.3.2.3" stretchy="false" xref="A2.SS1.3.p2.7.m7.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.3.p2.7.m7.4.5.5" xref="A2.SS1.3.p2.7.m7.4.5.5.cmml">=</mo><msub id="A2.SS1.3.p2.7.m7.4.5.6" xref="A2.SS1.3.p2.7.m7.4.5.6.cmml"><mi id="A2.SS1.3.p2.7.m7.4.5.6.2" xref="A2.SS1.3.p2.7.m7.4.5.6.2.cmml">π</mi><mi id="A2.SS1.3.p2.7.m7.4.5.6.3" xref="A2.SS1.3.p2.7.m7.4.5.6.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.7.m7.4b"><apply id="A2.SS1.3.p2.7.m7.4.5.cmml" xref="A2.SS1.3.p2.7.m7.4.5"><and id="A2.SS1.3.p2.7.m7.4.5a.cmml" xref="A2.SS1.3.p2.7.m7.4.5"></and><apply id="A2.SS1.3.p2.7.m7.4.5b.cmml" xref="A2.SS1.3.p2.7.m7.4.5"><geq id="A2.SS1.3.p2.7.m7.4.5.3.cmml" xref="A2.SS1.3.p2.7.m7.4.5.3"></geq><apply id="A2.SS1.3.p2.7.m7.4.5.2.cmml" xref="A2.SS1.3.p2.7.m7.4.5.2"><times id="A2.SS1.3.p2.7.m7.4.5.2.1.cmml" xref="A2.SS1.3.p2.7.m7.4.5.2.1"></times><ci id="A2.SS1.3.p2.7.m7.4.5.2.2.cmml" xref="A2.SS1.3.p2.7.m7.4.5.2.2">𝑃</ci><list id="A2.SS1.3.p2.7.m7.4.5.2.3.1.cmml" xref="A2.SS1.3.p2.7.m7.4.5.2.3.2"><ci id="A2.SS1.3.p2.7.m7.1.1.cmml" xref="A2.SS1.3.p2.7.m7.1.1">𝜏</ci><ci id="A2.SS1.3.p2.7.m7.2.2.cmml" xref="A2.SS1.3.p2.7.m7.2.2">𝑥</ci></list></apply><apply id="A2.SS1.3.p2.7.m7.4.5.4.cmml" xref="A2.SS1.3.p2.7.m7.4.5.4"><times id="A2.SS1.3.p2.7.m7.4.5.4.1.cmml" xref="A2.SS1.3.p2.7.m7.4.5.4.1"></times><ci id="A2.SS1.3.p2.7.m7.4.5.4.2.cmml" xref="A2.SS1.3.p2.7.m7.4.5.4.2">𝑃</ci><list id="A2.SS1.3.p2.7.m7.4.5.4.3.1.cmml" xref="A2.SS1.3.p2.7.m7.4.5.4.3.2"><ci id="A2.SS1.3.p2.7.m7.3.3.cmml" xref="A2.SS1.3.p2.7.m7.3.3">𝜏</ci><ci id="A2.SS1.3.p2.7.m7.4.4.cmml" xref="A2.SS1.3.p2.7.m7.4.4">𝜏</ci></list></apply></apply><apply id="A2.SS1.3.p2.7.m7.4.5c.cmml" xref="A2.SS1.3.p2.7.m7.4.5"><eq id="A2.SS1.3.p2.7.m7.4.5.5.cmml" xref="A2.SS1.3.p2.7.m7.4.5.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.3.p2.7.m7.4.5.4.cmml" id="A2.SS1.3.p2.7.m7.4.5d.cmml" xref="A2.SS1.3.p2.7.m7.4.5"></share><apply id="A2.SS1.3.p2.7.m7.4.5.6.cmml" xref="A2.SS1.3.p2.7.m7.4.5.6"><csymbol cd="ambiguous" id="A2.SS1.3.p2.7.m7.4.5.6.1.cmml" xref="A2.SS1.3.p2.7.m7.4.5.6">subscript</csymbol><ci id="A2.SS1.3.p2.7.m7.4.5.6.2.cmml" xref="A2.SS1.3.p2.7.m7.4.5.6.2">𝜋</ci><ci id="A2.SS1.3.p2.7.m7.4.5.6.3.cmml" xref="A2.SS1.3.p2.7.m7.4.5.6.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.7.m7.4c">P(\tau;x)\geq P(\tau;\tau)=\pi_{L}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.7.m7.4d">italic_P ( italic_τ ; italic_x ) ≥ italic_P ( italic_τ ; italic_τ ) = italic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. This establishes Equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E11" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">11</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S3.E12" title="In 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">12</span></a>), so <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.3.p2.8.m8.1"><semantics id="A2.SS1.3.p2.8.m8.1a"><mi id="A2.SS1.3.p2.8.m8.1.1" xref="A2.SS1.3.p2.8.m8.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.3.p2.8.m8.1b"><ci id="A2.SS1.3.p2.8.m8.1.1.cmml" xref="A2.SS1.3.p2.8.m8.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.3.p2.8.m8.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.3.p2.8.m8.1d">italic_τ</annotation></semantics></math> is a threshold equilibrum. ∎</p> </div> </div> <div class="ltx_proof" id="A2.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition4" title="Proposition 4. ‣ 3.2 Dynamics ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>.</h6> <div class="ltx_para" id="A2.SS1.4.p1"> <p class="ltx_p" id="A2.SS1.4.p1.14">If <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.4.p1.1.m1.1"><semantics id="A2.SS1.4.p1.1.m1.1a"><mi id="A2.SS1.4.p1.1.m1.1.1" xref="A2.SS1.4.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.1.m1.1b"><ci id="A2.SS1.4.p1.1.m1.1.1.cmml" xref="A2.SS1.4.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.1.m1.1d">italic_τ</annotation></semantics></math> is the current threshold strategy that all other agents are following, an agent who receives signal <math alttext="x" class="ltx_Math" display="inline" id="A2.SS1.4.p1.2.m2.1"><semantics id="A2.SS1.4.p1.2.m2.1a"><mi id="A2.SS1.4.p1.2.m2.1.1" xref="A2.SS1.4.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.2.m2.1b"><ci id="A2.SS1.4.p1.2.m2.1.1.cmml" xref="A2.SS1.4.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.2.m2.1d">italic_x</annotation></semantics></math> will best respond with <math alttext="H" class="ltx_Math" display="inline" id="A2.SS1.4.p1.3.m3.1"><semantics id="A2.SS1.4.p1.3.m3.1a"><mi id="A2.SS1.4.p1.3.m3.1.1" xref="A2.SS1.4.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.3.m3.1b"><ci id="A2.SS1.4.p1.3.m3.1.1.cmml" xref="A2.SS1.4.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.3.m3.1d">italic_H</annotation></semantics></math> if <math alttext="P(\tau;x)\leq F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.4.p1.4.m4.3"><semantics id="A2.SS1.4.p1.4.m4.3a"><mrow id="A2.SS1.4.p1.4.m4.3.4" xref="A2.SS1.4.p1.4.m4.3.4.cmml"><mrow id="A2.SS1.4.p1.4.m4.3.4.2" xref="A2.SS1.4.p1.4.m4.3.4.2.cmml"><mi id="A2.SS1.4.p1.4.m4.3.4.2.2" xref="A2.SS1.4.p1.4.m4.3.4.2.2.cmml">P</mi><mo id="A2.SS1.4.p1.4.m4.3.4.2.1" xref="A2.SS1.4.p1.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.4.m4.3.4.2.3.2" xref="A2.SS1.4.p1.4.m4.3.4.2.3.1.cmml"><mo id="A2.SS1.4.p1.4.m4.3.4.2.3.2.1" stretchy="false" xref="A2.SS1.4.p1.4.m4.3.4.2.3.1.cmml">(</mo><mi id="A2.SS1.4.p1.4.m4.1.1" xref="A2.SS1.4.p1.4.m4.1.1.cmml">τ</mi><mo id="A2.SS1.4.p1.4.m4.3.4.2.3.2.2" xref="A2.SS1.4.p1.4.m4.3.4.2.3.1.cmml">;</mo><mi id="A2.SS1.4.p1.4.m4.2.2" xref="A2.SS1.4.p1.4.m4.2.2.cmml">x</mi><mo id="A2.SS1.4.p1.4.m4.3.4.2.3.2.3" stretchy="false" xref="A2.SS1.4.p1.4.m4.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.4.p1.4.m4.3.4.1" xref="A2.SS1.4.p1.4.m4.3.4.1.cmml">≤</mo><mrow id="A2.SS1.4.p1.4.m4.3.4.3" xref="A2.SS1.4.p1.4.m4.3.4.3.cmml"><mi id="A2.SS1.4.p1.4.m4.3.4.3.2" xref="A2.SS1.4.p1.4.m4.3.4.3.2.cmml">F</mi><mo id="A2.SS1.4.p1.4.m4.3.4.3.1" xref="A2.SS1.4.p1.4.m4.3.4.3.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.4.m4.3.4.3.3.2" xref="A2.SS1.4.p1.4.m4.3.4.3.cmml"><mo id="A2.SS1.4.p1.4.m4.3.4.3.3.2.1" stretchy="false" xref="A2.SS1.4.p1.4.m4.3.4.3.cmml">(</mo><mi id="A2.SS1.4.p1.4.m4.3.3" xref="A2.SS1.4.p1.4.m4.3.3.cmml">τ</mi><mo id="A2.SS1.4.p1.4.m4.3.4.3.3.2.2" stretchy="false" xref="A2.SS1.4.p1.4.m4.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.4.m4.3b"><apply id="A2.SS1.4.p1.4.m4.3.4.cmml" xref="A2.SS1.4.p1.4.m4.3.4"><leq id="A2.SS1.4.p1.4.m4.3.4.1.cmml" xref="A2.SS1.4.p1.4.m4.3.4.1"></leq><apply id="A2.SS1.4.p1.4.m4.3.4.2.cmml" xref="A2.SS1.4.p1.4.m4.3.4.2"><times id="A2.SS1.4.p1.4.m4.3.4.2.1.cmml" xref="A2.SS1.4.p1.4.m4.3.4.2.1"></times><ci id="A2.SS1.4.p1.4.m4.3.4.2.2.cmml" xref="A2.SS1.4.p1.4.m4.3.4.2.2">𝑃</ci><list id="A2.SS1.4.p1.4.m4.3.4.2.3.1.cmml" xref="A2.SS1.4.p1.4.m4.3.4.2.3.2"><ci id="A2.SS1.4.p1.4.m4.1.1.cmml" xref="A2.SS1.4.p1.4.m4.1.1">𝜏</ci><ci id="A2.SS1.4.p1.4.m4.2.2.cmml" xref="A2.SS1.4.p1.4.m4.2.2">𝑥</ci></list></apply><apply id="A2.SS1.4.p1.4.m4.3.4.3.cmml" xref="A2.SS1.4.p1.4.m4.3.4.3"><times id="A2.SS1.4.p1.4.m4.3.4.3.1.cmml" xref="A2.SS1.4.p1.4.m4.3.4.3.1"></times><ci id="A2.SS1.4.p1.4.m4.3.4.3.2.cmml" xref="A2.SS1.4.p1.4.m4.3.4.3.2">𝐹</ci><ci id="A2.SS1.4.p1.4.m4.3.3.cmml" xref="A2.SS1.4.p1.4.m4.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.4.m4.3c">P(\tau;x)\leq F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.4.m4.3d">italic_P ( italic_τ ; italic_x ) ≤ italic_F ( italic_τ )</annotation></semantics></math>, and <math alttext="L" class="ltx_Math" display="inline" id="A2.SS1.4.p1.5.m5.1"><semantics id="A2.SS1.4.p1.5.m5.1a"><mi id="A2.SS1.4.p1.5.m5.1.1" xref="A2.SS1.4.p1.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.5.m5.1b"><ci id="A2.SS1.4.p1.5.m5.1.1.cmml" xref="A2.SS1.4.p1.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.5.m5.1d">italic_L</annotation></semantics></math> otherwise. Since <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="A2.SS1.4.p1.6.m6.2"><semantics id="A2.SS1.4.p1.6.m6.2a"><mrow id="A2.SS1.4.p1.6.m6.2.3" xref="A2.SS1.4.p1.6.m6.2.3.cmml"><mi id="A2.SS1.4.p1.6.m6.2.3.2" xref="A2.SS1.4.p1.6.m6.2.3.2.cmml">P</mi><mo id="A2.SS1.4.p1.6.m6.2.3.1" xref="A2.SS1.4.p1.6.m6.2.3.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.6.m6.2.3.3.2" xref="A2.SS1.4.p1.6.m6.2.3.3.1.cmml"><mo id="A2.SS1.4.p1.6.m6.2.3.3.2.1" stretchy="false" xref="A2.SS1.4.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="A2.SS1.4.p1.6.m6.1.1" xref="A2.SS1.4.p1.6.m6.1.1.cmml">τ</mi><mo id="A2.SS1.4.p1.6.m6.2.3.3.2.2" xref="A2.SS1.4.p1.6.m6.2.3.3.1.cmml">;</mo><mi id="A2.SS1.4.p1.6.m6.2.2" xref="A2.SS1.4.p1.6.m6.2.2.cmml">x</mi><mo id="A2.SS1.4.p1.6.m6.2.3.3.2.3" stretchy="false" xref="A2.SS1.4.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.6.m6.2b"><apply id="A2.SS1.4.p1.6.m6.2.3.cmml" xref="A2.SS1.4.p1.6.m6.2.3"><times id="A2.SS1.4.p1.6.m6.2.3.1.cmml" xref="A2.SS1.4.p1.6.m6.2.3.1"></times><ci id="A2.SS1.4.p1.6.m6.2.3.2.cmml" xref="A2.SS1.4.p1.6.m6.2.3.2">𝑃</ci><list id="A2.SS1.4.p1.6.m6.2.3.3.1.cmml" xref="A2.SS1.4.p1.6.m6.2.3.3.2"><ci id="A2.SS1.4.p1.6.m6.1.1.cmml" xref="A2.SS1.4.p1.6.m6.1.1">𝜏</ci><ci id="A2.SS1.4.p1.6.m6.2.2.cmml" xref="A2.SS1.4.p1.6.m6.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.6.m6.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.6.m6.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> is strictly decreasing and continuous over <math alttext="x" class="ltx_Math" display="inline" id="A2.SS1.4.p1.7.m7.1"><semantics id="A2.SS1.4.p1.7.m7.1a"><mi id="A2.SS1.4.p1.7.m7.1.1" xref="A2.SS1.4.p1.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.7.m7.1b"><ci id="A2.SS1.4.p1.7.m7.1.1.cmml" xref="A2.SS1.4.p1.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.7.m7.1d">italic_x</annotation></semantics></math>, there will be a unique point <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.4.p1.8.m8.1"><semantics id="A2.SS1.4.p1.8.m8.1a"><mover accent="true" id="A2.SS1.4.p1.8.m8.1.1" xref="A2.SS1.4.p1.8.m8.1.1.cmml"><mi id="A2.SS1.4.p1.8.m8.1.1.2" xref="A2.SS1.4.p1.8.m8.1.1.2.cmml">τ</mi><mo id="A2.SS1.4.p1.8.m8.1.1.1" xref="A2.SS1.4.p1.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.8.m8.1b"><apply id="A2.SS1.4.p1.8.m8.1.1.cmml" xref="A2.SS1.4.p1.8.m8.1.1"><ci id="A2.SS1.4.p1.8.m8.1.1.1.cmml" xref="A2.SS1.4.p1.8.m8.1.1.1">^</ci><ci id="A2.SS1.4.p1.8.m8.1.1.2.cmml" xref="A2.SS1.4.p1.8.m8.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.8.m8.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.8.m8.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> such that <math alttext="P(\tau;\hat{\tau})=F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.4.p1.9.m9.3"><semantics id="A2.SS1.4.p1.9.m9.3a"><mrow id="A2.SS1.4.p1.9.m9.3.4" xref="A2.SS1.4.p1.9.m9.3.4.cmml"><mrow id="A2.SS1.4.p1.9.m9.3.4.2" xref="A2.SS1.4.p1.9.m9.3.4.2.cmml"><mi id="A2.SS1.4.p1.9.m9.3.4.2.2" xref="A2.SS1.4.p1.9.m9.3.4.2.2.cmml">P</mi><mo id="A2.SS1.4.p1.9.m9.3.4.2.1" xref="A2.SS1.4.p1.9.m9.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.9.m9.3.4.2.3.2" xref="A2.SS1.4.p1.9.m9.3.4.2.3.1.cmml"><mo id="A2.SS1.4.p1.9.m9.3.4.2.3.2.1" stretchy="false" xref="A2.SS1.4.p1.9.m9.3.4.2.3.1.cmml">(</mo><mi id="A2.SS1.4.p1.9.m9.1.1" xref="A2.SS1.4.p1.9.m9.1.1.cmml">τ</mi><mo id="A2.SS1.4.p1.9.m9.3.4.2.3.2.2" xref="A2.SS1.4.p1.9.m9.3.4.2.3.1.cmml">;</mo><mover accent="true" id="A2.SS1.4.p1.9.m9.2.2" xref="A2.SS1.4.p1.9.m9.2.2.cmml"><mi id="A2.SS1.4.p1.9.m9.2.2.2" xref="A2.SS1.4.p1.9.m9.2.2.2.cmml">τ</mi><mo id="A2.SS1.4.p1.9.m9.2.2.1" xref="A2.SS1.4.p1.9.m9.2.2.1.cmml">^</mo></mover><mo id="A2.SS1.4.p1.9.m9.3.4.2.3.2.3" stretchy="false" xref="A2.SS1.4.p1.9.m9.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.4.p1.9.m9.3.4.1" xref="A2.SS1.4.p1.9.m9.3.4.1.cmml">=</mo><mrow id="A2.SS1.4.p1.9.m9.3.4.3" xref="A2.SS1.4.p1.9.m9.3.4.3.cmml"><mi id="A2.SS1.4.p1.9.m9.3.4.3.2" xref="A2.SS1.4.p1.9.m9.3.4.3.2.cmml">F</mi><mo id="A2.SS1.4.p1.9.m9.3.4.3.1" xref="A2.SS1.4.p1.9.m9.3.4.3.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.9.m9.3.4.3.3.2" xref="A2.SS1.4.p1.9.m9.3.4.3.cmml"><mo id="A2.SS1.4.p1.9.m9.3.4.3.3.2.1" stretchy="false" xref="A2.SS1.4.p1.9.m9.3.4.3.cmml">(</mo><mi id="A2.SS1.4.p1.9.m9.3.3" xref="A2.SS1.4.p1.9.m9.3.3.cmml">τ</mi><mo id="A2.SS1.4.p1.9.m9.3.4.3.3.2.2" stretchy="false" xref="A2.SS1.4.p1.9.m9.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.9.m9.3b"><apply id="A2.SS1.4.p1.9.m9.3.4.cmml" xref="A2.SS1.4.p1.9.m9.3.4"><eq id="A2.SS1.4.p1.9.m9.3.4.1.cmml" xref="A2.SS1.4.p1.9.m9.3.4.1"></eq><apply id="A2.SS1.4.p1.9.m9.3.4.2.cmml" xref="A2.SS1.4.p1.9.m9.3.4.2"><times id="A2.SS1.4.p1.9.m9.3.4.2.1.cmml" xref="A2.SS1.4.p1.9.m9.3.4.2.1"></times><ci id="A2.SS1.4.p1.9.m9.3.4.2.2.cmml" xref="A2.SS1.4.p1.9.m9.3.4.2.2">𝑃</ci><list id="A2.SS1.4.p1.9.m9.3.4.2.3.1.cmml" xref="A2.SS1.4.p1.9.m9.3.4.2.3.2"><ci id="A2.SS1.4.p1.9.m9.1.1.cmml" xref="A2.SS1.4.p1.9.m9.1.1">𝜏</ci><apply id="A2.SS1.4.p1.9.m9.2.2.cmml" xref="A2.SS1.4.p1.9.m9.2.2"><ci id="A2.SS1.4.p1.9.m9.2.2.1.cmml" xref="A2.SS1.4.p1.9.m9.2.2.1">^</ci><ci id="A2.SS1.4.p1.9.m9.2.2.2.cmml" xref="A2.SS1.4.p1.9.m9.2.2.2">𝜏</ci></apply></list></apply><apply id="A2.SS1.4.p1.9.m9.3.4.3.cmml" xref="A2.SS1.4.p1.9.m9.3.4.3"><times id="A2.SS1.4.p1.9.m9.3.4.3.1.cmml" xref="A2.SS1.4.p1.9.m9.3.4.3.1"></times><ci id="A2.SS1.4.p1.9.m9.3.4.3.2.cmml" xref="A2.SS1.4.p1.9.m9.3.4.3.2">𝐹</ci><ci id="A2.SS1.4.p1.9.m9.3.3.cmml" xref="A2.SS1.4.p1.9.m9.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.9.m9.3c">P(\tau;\hat{\tau})=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.9.m9.3d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = italic_F ( italic_τ )</annotation></semantics></math>, with <math alttext="P(\tau;x)\geq F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.4.p1.10.m10.3"><semantics id="A2.SS1.4.p1.10.m10.3a"><mrow id="A2.SS1.4.p1.10.m10.3.4" xref="A2.SS1.4.p1.10.m10.3.4.cmml"><mrow id="A2.SS1.4.p1.10.m10.3.4.2" xref="A2.SS1.4.p1.10.m10.3.4.2.cmml"><mi id="A2.SS1.4.p1.10.m10.3.4.2.2" xref="A2.SS1.4.p1.10.m10.3.4.2.2.cmml">P</mi><mo id="A2.SS1.4.p1.10.m10.3.4.2.1" xref="A2.SS1.4.p1.10.m10.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.10.m10.3.4.2.3.2" xref="A2.SS1.4.p1.10.m10.3.4.2.3.1.cmml"><mo id="A2.SS1.4.p1.10.m10.3.4.2.3.2.1" stretchy="false" xref="A2.SS1.4.p1.10.m10.3.4.2.3.1.cmml">(</mo><mi id="A2.SS1.4.p1.10.m10.1.1" xref="A2.SS1.4.p1.10.m10.1.1.cmml">τ</mi><mo id="A2.SS1.4.p1.10.m10.3.4.2.3.2.2" xref="A2.SS1.4.p1.10.m10.3.4.2.3.1.cmml">;</mo><mi id="A2.SS1.4.p1.10.m10.2.2" xref="A2.SS1.4.p1.10.m10.2.2.cmml">x</mi><mo id="A2.SS1.4.p1.10.m10.3.4.2.3.2.3" stretchy="false" xref="A2.SS1.4.p1.10.m10.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.4.p1.10.m10.3.4.1" xref="A2.SS1.4.p1.10.m10.3.4.1.cmml">≥</mo><mrow id="A2.SS1.4.p1.10.m10.3.4.3" xref="A2.SS1.4.p1.10.m10.3.4.3.cmml"><mi id="A2.SS1.4.p1.10.m10.3.4.3.2" xref="A2.SS1.4.p1.10.m10.3.4.3.2.cmml">F</mi><mo id="A2.SS1.4.p1.10.m10.3.4.3.1" xref="A2.SS1.4.p1.10.m10.3.4.3.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.10.m10.3.4.3.3.2" xref="A2.SS1.4.p1.10.m10.3.4.3.cmml"><mo id="A2.SS1.4.p1.10.m10.3.4.3.3.2.1" stretchy="false" xref="A2.SS1.4.p1.10.m10.3.4.3.cmml">(</mo><mi id="A2.SS1.4.p1.10.m10.3.3" xref="A2.SS1.4.p1.10.m10.3.3.cmml">τ</mi><mo id="A2.SS1.4.p1.10.m10.3.4.3.3.2.2" stretchy="false" xref="A2.SS1.4.p1.10.m10.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.10.m10.3b"><apply id="A2.SS1.4.p1.10.m10.3.4.cmml" xref="A2.SS1.4.p1.10.m10.3.4"><geq id="A2.SS1.4.p1.10.m10.3.4.1.cmml" xref="A2.SS1.4.p1.10.m10.3.4.1"></geq><apply id="A2.SS1.4.p1.10.m10.3.4.2.cmml" xref="A2.SS1.4.p1.10.m10.3.4.2"><times id="A2.SS1.4.p1.10.m10.3.4.2.1.cmml" xref="A2.SS1.4.p1.10.m10.3.4.2.1"></times><ci id="A2.SS1.4.p1.10.m10.3.4.2.2.cmml" xref="A2.SS1.4.p1.10.m10.3.4.2.2">𝑃</ci><list id="A2.SS1.4.p1.10.m10.3.4.2.3.1.cmml" xref="A2.SS1.4.p1.10.m10.3.4.2.3.2"><ci id="A2.SS1.4.p1.10.m10.1.1.cmml" xref="A2.SS1.4.p1.10.m10.1.1">𝜏</ci><ci id="A2.SS1.4.p1.10.m10.2.2.cmml" xref="A2.SS1.4.p1.10.m10.2.2">𝑥</ci></list></apply><apply id="A2.SS1.4.p1.10.m10.3.4.3.cmml" xref="A2.SS1.4.p1.10.m10.3.4.3"><times id="A2.SS1.4.p1.10.m10.3.4.3.1.cmml" xref="A2.SS1.4.p1.10.m10.3.4.3.1"></times><ci id="A2.SS1.4.p1.10.m10.3.4.3.2.cmml" xref="A2.SS1.4.p1.10.m10.3.4.3.2">𝐹</ci><ci id="A2.SS1.4.p1.10.m10.3.3.cmml" xref="A2.SS1.4.p1.10.m10.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.10.m10.3c">P(\tau;x)\geq F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.10.m10.3d">italic_P ( italic_τ ; italic_x ) ≥ italic_F ( italic_τ )</annotation></semantics></math> for <math alttext="x\leq\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.4.p1.11.m11.1"><semantics id="A2.SS1.4.p1.11.m11.1a"><mrow id="A2.SS1.4.p1.11.m11.1.1" xref="A2.SS1.4.p1.11.m11.1.1.cmml"><mi id="A2.SS1.4.p1.11.m11.1.1.2" xref="A2.SS1.4.p1.11.m11.1.1.2.cmml">x</mi><mo id="A2.SS1.4.p1.11.m11.1.1.1" xref="A2.SS1.4.p1.11.m11.1.1.1.cmml">≤</mo><mover accent="true" id="A2.SS1.4.p1.11.m11.1.1.3" xref="A2.SS1.4.p1.11.m11.1.1.3.cmml"><mi id="A2.SS1.4.p1.11.m11.1.1.3.2" xref="A2.SS1.4.p1.11.m11.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.4.p1.11.m11.1.1.3.1" xref="A2.SS1.4.p1.11.m11.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.11.m11.1b"><apply id="A2.SS1.4.p1.11.m11.1.1.cmml" xref="A2.SS1.4.p1.11.m11.1.1"><leq id="A2.SS1.4.p1.11.m11.1.1.1.cmml" xref="A2.SS1.4.p1.11.m11.1.1.1"></leq><ci id="A2.SS1.4.p1.11.m11.1.1.2.cmml" xref="A2.SS1.4.p1.11.m11.1.1.2">𝑥</ci><apply id="A2.SS1.4.p1.11.m11.1.1.3.cmml" xref="A2.SS1.4.p1.11.m11.1.1.3"><ci id="A2.SS1.4.p1.11.m11.1.1.3.1.cmml" xref="A2.SS1.4.p1.11.m11.1.1.3.1">^</ci><ci id="A2.SS1.4.p1.11.m11.1.1.3.2.cmml" xref="A2.SS1.4.p1.11.m11.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.11.m11.1c">x\leq\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.11.m11.1d">italic_x ≤ over^ start_ARG italic_τ end_ARG</annotation></semantics></math> and <math alttext="P(\tau;x)\leq F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.4.p1.12.m12.3"><semantics id="A2.SS1.4.p1.12.m12.3a"><mrow id="A2.SS1.4.p1.12.m12.3.4" xref="A2.SS1.4.p1.12.m12.3.4.cmml"><mrow id="A2.SS1.4.p1.12.m12.3.4.2" xref="A2.SS1.4.p1.12.m12.3.4.2.cmml"><mi id="A2.SS1.4.p1.12.m12.3.4.2.2" xref="A2.SS1.4.p1.12.m12.3.4.2.2.cmml">P</mi><mo id="A2.SS1.4.p1.12.m12.3.4.2.1" xref="A2.SS1.4.p1.12.m12.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.12.m12.3.4.2.3.2" xref="A2.SS1.4.p1.12.m12.3.4.2.3.1.cmml"><mo id="A2.SS1.4.p1.12.m12.3.4.2.3.2.1" stretchy="false" xref="A2.SS1.4.p1.12.m12.3.4.2.3.1.cmml">(</mo><mi id="A2.SS1.4.p1.12.m12.1.1" xref="A2.SS1.4.p1.12.m12.1.1.cmml">τ</mi><mo id="A2.SS1.4.p1.12.m12.3.4.2.3.2.2" xref="A2.SS1.4.p1.12.m12.3.4.2.3.1.cmml">;</mo><mi id="A2.SS1.4.p1.12.m12.2.2" xref="A2.SS1.4.p1.12.m12.2.2.cmml">x</mi><mo id="A2.SS1.4.p1.12.m12.3.4.2.3.2.3" stretchy="false" xref="A2.SS1.4.p1.12.m12.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.4.p1.12.m12.3.4.1" xref="A2.SS1.4.p1.12.m12.3.4.1.cmml">≤</mo><mrow id="A2.SS1.4.p1.12.m12.3.4.3" xref="A2.SS1.4.p1.12.m12.3.4.3.cmml"><mi id="A2.SS1.4.p1.12.m12.3.4.3.2" xref="A2.SS1.4.p1.12.m12.3.4.3.2.cmml">F</mi><mo id="A2.SS1.4.p1.12.m12.3.4.3.1" xref="A2.SS1.4.p1.12.m12.3.4.3.1.cmml">⁢</mo><mrow id="A2.SS1.4.p1.12.m12.3.4.3.3.2" xref="A2.SS1.4.p1.12.m12.3.4.3.cmml"><mo id="A2.SS1.4.p1.12.m12.3.4.3.3.2.1" stretchy="false" xref="A2.SS1.4.p1.12.m12.3.4.3.cmml">(</mo><mi id="A2.SS1.4.p1.12.m12.3.3" xref="A2.SS1.4.p1.12.m12.3.3.cmml">τ</mi><mo id="A2.SS1.4.p1.12.m12.3.4.3.3.2.2" stretchy="false" xref="A2.SS1.4.p1.12.m12.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.12.m12.3b"><apply id="A2.SS1.4.p1.12.m12.3.4.cmml" xref="A2.SS1.4.p1.12.m12.3.4"><leq id="A2.SS1.4.p1.12.m12.3.4.1.cmml" xref="A2.SS1.4.p1.12.m12.3.4.1"></leq><apply id="A2.SS1.4.p1.12.m12.3.4.2.cmml" xref="A2.SS1.4.p1.12.m12.3.4.2"><times id="A2.SS1.4.p1.12.m12.3.4.2.1.cmml" xref="A2.SS1.4.p1.12.m12.3.4.2.1"></times><ci id="A2.SS1.4.p1.12.m12.3.4.2.2.cmml" xref="A2.SS1.4.p1.12.m12.3.4.2.2">𝑃</ci><list id="A2.SS1.4.p1.12.m12.3.4.2.3.1.cmml" xref="A2.SS1.4.p1.12.m12.3.4.2.3.2"><ci id="A2.SS1.4.p1.12.m12.1.1.cmml" xref="A2.SS1.4.p1.12.m12.1.1">𝜏</ci><ci id="A2.SS1.4.p1.12.m12.2.2.cmml" xref="A2.SS1.4.p1.12.m12.2.2">𝑥</ci></list></apply><apply id="A2.SS1.4.p1.12.m12.3.4.3.cmml" xref="A2.SS1.4.p1.12.m12.3.4.3"><times id="A2.SS1.4.p1.12.m12.3.4.3.1.cmml" xref="A2.SS1.4.p1.12.m12.3.4.3.1"></times><ci id="A2.SS1.4.p1.12.m12.3.4.3.2.cmml" xref="A2.SS1.4.p1.12.m12.3.4.3.2">𝐹</ci><ci id="A2.SS1.4.p1.12.m12.3.3.cmml" xref="A2.SS1.4.p1.12.m12.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.12.m12.3c">P(\tau;x)\leq F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.12.m12.3d">italic_P ( italic_τ ; italic_x ) ≤ italic_F ( italic_τ )</annotation></semantics></math> for <math alttext="x>\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.4.p1.13.m13.1"><semantics id="A2.SS1.4.p1.13.m13.1a"><mrow id="A2.SS1.4.p1.13.m13.1.1" xref="A2.SS1.4.p1.13.m13.1.1.cmml"><mi id="A2.SS1.4.p1.13.m13.1.1.2" xref="A2.SS1.4.p1.13.m13.1.1.2.cmml">x</mi><mo id="A2.SS1.4.p1.13.m13.1.1.1" xref="A2.SS1.4.p1.13.m13.1.1.1.cmml">></mo><mover accent="true" id="A2.SS1.4.p1.13.m13.1.1.3" xref="A2.SS1.4.p1.13.m13.1.1.3.cmml"><mi id="A2.SS1.4.p1.13.m13.1.1.3.2" xref="A2.SS1.4.p1.13.m13.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.4.p1.13.m13.1.1.3.1" xref="A2.SS1.4.p1.13.m13.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.13.m13.1b"><apply id="A2.SS1.4.p1.13.m13.1.1.cmml" xref="A2.SS1.4.p1.13.m13.1.1"><gt id="A2.SS1.4.p1.13.m13.1.1.1.cmml" xref="A2.SS1.4.p1.13.m13.1.1.1"></gt><ci id="A2.SS1.4.p1.13.m13.1.1.2.cmml" xref="A2.SS1.4.p1.13.m13.1.1.2">𝑥</ci><apply id="A2.SS1.4.p1.13.m13.1.1.3.cmml" xref="A2.SS1.4.p1.13.m13.1.1.3"><ci id="A2.SS1.4.p1.13.m13.1.1.3.1.cmml" xref="A2.SS1.4.p1.13.m13.1.1.3.1">^</ci><ci id="A2.SS1.4.p1.13.m13.1.1.3.2.cmml" xref="A2.SS1.4.p1.13.m13.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.13.m13.1c">x>\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.13.m13.1d">italic_x > over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. This threshold <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.4.p1.14.m14.1"><semantics id="A2.SS1.4.p1.14.m14.1a"><mover accent="true" id="A2.SS1.4.p1.14.m14.1.1" xref="A2.SS1.4.p1.14.m14.1.1.cmml"><mi id="A2.SS1.4.p1.14.m14.1.1.2" xref="A2.SS1.4.p1.14.m14.1.1.2.cmml">τ</mi><mo id="A2.SS1.4.p1.14.m14.1.1.1" xref="A2.SS1.4.p1.14.m14.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A2.SS1.4.p1.14.m14.1b"><apply id="A2.SS1.4.p1.14.m14.1.1.cmml" xref="A2.SS1.4.p1.14.m14.1.1"><ci id="A2.SS1.4.p1.14.m14.1.1.1.cmml" xref="A2.SS1.4.p1.14.m14.1.1.1">^</ci><ci id="A2.SS1.4.p1.14.m14.1.1.2.cmml" xref="A2.SS1.4.p1.14.m14.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.4.p1.14.m14.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.4.p1.14.m14.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> thus corresponds to the best response. ∎</p> </div> </div> <div class="ltx_proof" id="A2.SS1.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem4" title="Theorem 4. ‣ 3.2 Dynamics ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>.</h6> <div class="ltx_para" id="A2.SS1.5.p1"> <p class="ltx_p" id="A2.SS1.5.p1.5">Consider the dynamics at time step <math alttext="t" class="ltx_Math" display="inline" id="A2.SS1.5.p1.1.m1.1"><semantics id="A2.SS1.5.p1.1.m1.1a"><mi id="A2.SS1.5.p1.1.m1.1.1" xref="A2.SS1.5.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.5.p1.1.m1.1b"><ci id="A2.SS1.5.p1.1.m1.1.1.cmml" xref="A2.SS1.5.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.5.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.5.p1.1.m1.1d">italic_t</annotation></semantics></math>, with the current threshold <math alttext="\tau(t)" class="ltx_Math" display="inline" id="A2.SS1.5.p1.2.m2.1"><semantics id="A2.SS1.5.p1.2.m2.1a"><mrow id="A2.SS1.5.p1.2.m2.1.2" xref="A2.SS1.5.p1.2.m2.1.2.cmml"><mi id="A2.SS1.5.p1.2.m2.1.2.2" xref="A2.SS1.5.p1.2.m2.1.2.2.cmml">τ</mi><mo id="A2.SS1.5.p1.2.m2.1.2.1" xref="A2.SS1.5.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="A2.SS1.5.p1.2.m2.1.2.3.2" xref="A2.SS1.5.p1.2.m2.1.2.cmml"><mo id="A2.SS1.5.p1.2.m2.1.2.3.2.1" stretchy="false" xref="A2.SS1.5.p1.2.m2.1.2.cmml">(</mo><mi id="A2.SS1.5.p1.2.m2.1.1" xref="A2.SS1.5.p1.2.m2.1.1.cmml">t</mi><mo id="A2.SS1.5.p1.2.m2.1.2.3.2.2" stretchy="false" xref="A2.SS1.5.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.5.p1.2.m2.1b"><apply id="A2.SS1.5.p1.2.m2.1.2.cmml" xref="A2.SS1.5.p1.2.m2.1.2"><times id="A2.SS1.5.p1.2.m2.1.2.1.cmml" xref="A2.SS1.5.p1.2.m2.1.2.1"></times><ci id="A2.SS1.5.p1.2.m2.1.2.2.cmml" xref="A2.SS1.5.p1.2.m2.1.2.2">𝜏</ci><ci id="A2.SS1.5.p1.2.m2.1.1.cmml" xref="A2.SS1.5.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.5.p1.2.m2.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.5.p1.2.m2.1d">italic_τ ( italic_t )</annotation></semantics></math>. Since we are considering a fixed time <math alttext="t" class="ltx_Math" display="inline" id="A2.SS1.5.p1.3.m3.1"><semantics id="A2.SS1.5.p1.3.m3.1a"><mi id="A2.SS1.5.p1.3.m3.1.1" xref="A2.SS1.5.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.5.p1.3.m3.1b"><ci id="A2.SS1.5.p1.3.m3.1.1.cmml" xref="A2.SS1.5.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.5.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.5.p1.3.m3.1d">italic_t</annotation></semantics></math>, we refer to <math alttext="\tau(t)" class="ltx_Math" display="inline" id="A2.SS1.5.p1.4.m4.1"><semantics id="A2.SS1.5.p1.4.m4.1a"><mrow id="A2.SS1.5.p1.4.m4.1.2" xref="A2.SS1.5.p1.4.m4.1.2.cmml"><mi id="A2.SS1.5.p1.4.m4.1.2.2" xref="A2.SS1.5.p1.4.m4.1.2.2.cmml">τ</mi><mo id="A2.SS1.5.p1.4.m4.1.2.1" xref="A2.SS1.5.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="A2.SS1.5.p1.4.m4.1.2.3.2" xref="A2.SS1.5.p1.4.m4.1.2.cmml"><mo id="A2.SS1.5.p1.4.m4.1.2.3.2.1" stretchy="false" xref="A2.SS1.5.p1.4.m4.1.2.cmml">(</mo><mi id="A2.SS1.5.p1.4.m4.1.1" xref="A2.SS1.5.p1.4.m4.1.1.cmml">t</mi><mo id="A2.SS1.5.p1.4.m4.1.2.3.2.2" stretchy="false" xref="A2.SS1.5.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.5.p1.4.m4.1b"><apply id="A2.SS1.5.p1.4.m4.1.2.cmml" xref="A2.SS1.5.p1.4.m4.1.2"><times id="A2.SS1.5.p1.4.m4.1.2.1.cmml" xref="A2.SS1.5.p1.4.m4.1.2.1"></times><ci id="A2.SS1.5.p1.4.m4.1.2.2.cmml" xref="A2.SS1.5.p1.4.m4.1.2.2">𝜏</ci><ci id="A2.SS1.5.p1.4.m4.1.1.cmml" xref="A2.SS1.5.p1.4.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.5.p1.4.m4.1c">\tau(t)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.5.p1.4.m4.1d">italic_τ ( italic_t )</annotation></semantics></math> as <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.5.p1.5.m5.1"><semantics id="A2.SS1.5.p1.5.m5.1a"><mi id="A2.SS1.5.p1.5.m5.1.1" xref="A2.SS1.5.p1.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.5.p1.5.m5.1b"><ci id="A2.SS1.5.p1.5.m5.1.1.cmml" xref="A2.SS1.5.p1.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.5.p1.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.5.p1.5.m5.1d">italic_τ</annotation></semantics></math> throughout the proof.</p> </div> <div class="ltx_para" id="A2.SS1.6.p2"> <p class="ltx_p" id="A2.SS1.6.p2.11">We consider what happens when an agent receives a signal exactly at <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.6.p2.1.m1.1"><semantics id="A2.SS1.6.p2.1.m1.1a"><mi id="A2.SS1.6.p2.1.m1.1.1" xref="A2.SS1.6.p2.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.1.m1.1b"><ci id="A2.SS1.6.p2.1.m1.1.1.cmml" xref="A2.SS1.6.p2.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.1.m1.1d">italic_τ</annotation></semantics></math>. There are three cases to consider. In the first, <math alttext="G(\tau)=F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.6.p2.2.m2.2"><semantics id="A2.SS1.6.p2.2.m2.2a"><mrow id="A2.SS1.6.p2.2.m2.2.3" xref="A2.SS1.6.p2.2.m2.2.3.cmml"><mrow id="A2.SS1.6.p2.2.m2.2.3.2" xref="A2.SS1.6.p2.2.m2.2.3.2.cmml"><mi id="A2.SS1.6.p2.2.m2.2.3.2.2" xref="A2.SS1.6.p2.2.m2.2.3.2.2.cmml">G</mi><mo id="A2.SS1.6.p2.2.m2.2.3.2.1" xref="A2.SS1.6.p2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.6.p2.2.m2.2.3.2.3.2" xref="A2.SS1.6.p2.2.m2.2.3.2.cmml"><mo id="A2.SS1.6.p2.2.m2.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.6.p2.2.m2.2.3.2.cmml">(</mo><mi id="A2.SS1.6.p2.2.m2.1.1" xref="A2.SS1.6.p2.2.m2.1.1.cmml">τ</mi><mo id="A2.SS1.6.p2.2.m2.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.6.p2.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.6.p2.2.m2.2.3.1" xref="A2.SS1.6.p2.2.m2.2.3.1.cmml">=</mo><mrow id="A2.SS1.6.p2.2.m2.2.3.3" xref="A2.SS1.6.p2.2.m2.2.3.3.cmml"><mi id="A2.SS1.6.p2.2.m2.2.3.3.2" xref="A2.SS1.6.p2.2.m2.2.3.3.2.cmml">F</mi><mo id="A2.SS1.6.p2.2.m2.2.3.3.1" xref="A2.SS1.6.p2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.6.p2.2.m2.2.3.3.3.2" xref="A2.SS1.6.p2.2.m2.2.3.3.cmml"><mo id="A2.SS1.6.p2.2.m2.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.6.p2.2.m2.2.3.3.cmml">(</mo><mi id="A2.SS1.6.p2.2.m2.2.2" xref="A2.SS1.6.p2.2.m2.2.2.cmml">τ</mi><mo id="A2.SS1.6.p2.2.m2.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.6.p2.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.2.m2.2b"><apply id="A2.SS1.6.p2.2.m2.2.3.cmml" xref="A2.SS1.6.p2.2.m2.2.3"><eq id="A2.SS1.6.p2.2.m2.2.3.1.cmml" xref="A2.SS1.6.p2.2.m2.2.3.1"></eq><apply id="A2.SS1.6.p2.2.m2.2.3.2.cmml" xref="A2.SS1.6.p2.2.m2.2.3.2"><times id="A2.SS1.6.p2.2.m2.2.3.2.1.cmml" xref="A2.SS1.6.p2.2.m2.2.3.2.1"></times><ci id="A2.SS1.6.p2.2.m2.2.3.2.2.cmml" xref="A2.SS1.6.p2.2.m2.2.3.2.2">𝐺</ci><ci id="A2.SS1.6.p2.2.m2.1.1.cmml" xref="A2.SS1.6.p2.2.m2.1.1">𝜏</ci></apply><apply id="A2.SS1.6.p2.2.m2.2.3.3.cmml" xref="A2.SS1.6.p2.2.m2.2.3.3"><times id="A2.SS1.6.p2.2.m2.2.3.3.1.cmml" xref="A2.SS1.6.p2.2.m2.2.3.3.1"></times><ci id="A2.SS1.6.p2.2.m2.2.3.3.2.cmml" xref="A2.SS1.6.p2.2.m2.2.3.3.2">𝐹</ci><ci id="A2.SS1.6.p2.2.m2.2.2.cmml" xref="A2.SS1.6.p2.2.m2.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.2.m2.2c">G(\tau)=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.2.m2.2d">italic_G ( italic_τ ) = italic_F ( italic_τ )</annotation></semantics></math>; then the expected utility of reporting <math alttext="L" class="ltx_Math" display="inline" id="A2.SS1.6.p2.3.m3.1"><semantics id="A2.SS1.6.p2.3.m3.1a"><mi id="A2.SS1.6.p2.3.m3.1.1" xref="A2.SS1.6.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.3.m3.1b"><ci id="A2.SS1.6.p2.3.m3.1.1.cmml" xref="A2.SS1.6.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.3.m3.1d">italic_L</annotation></semantics></math> is <math alttext="\Pr[X^{\prime}\leq\tau\mid X=\tau]-\Pr[X^{\prime}\leq\tau]=0" class="ltx_Math" display="inline" id="A2.SS1.6.p2.4.m4.4"><semantics id="A2.SS1.6.p2.4.m4.4a"><mrow id="A2.SS1.6.p2.4.m4.4.4" xref="A2.SS1.6.p2.4.m4.4.4.cmml"><mrow id="A2.SS1.6.p2.4.m4.4.4.2" xref="A2.SS1.6.p2.4.m4.4.4.2.cmml"><mrow id="A2.SS1.6.p2.4.m4.3.3.1.1.1" xref="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml"><mi id="A2.SS1.6.p2.4.m4.1.1" xref="A2.SS1.6.p2.4.m4.1.1.cmml">Pr</mi><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1a" xref="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml">⁡</mo><mrow id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1" xref="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml"><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.2" stretchy="false" xref="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml">[</mo><mrow id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.cmml"><msup id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.cmml"><mi id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.2" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.2.cmml">X</mi><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.3" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.3" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.3.cmml">≤</mo><mrow id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.cmml"><mi id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.2" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.2.cmml">τ</mi><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.1" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.1.cmml">∣</mo><mi id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.3" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.5" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.5.cmml">=</mo><mi id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.6" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.6.cmml">τ</mi></mrow><mo id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.3" stretchy="false" xref="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml">]</mo></mrow></mrow><mo id="A2.SS1.6.p2.4.m4.4.4.2.3" xref="A2.SS1.6.p2.4.m4.4.4.2.3.cmml">−</mo><mrow id="A2.SS1.6.p2.4.m4.4.4.2.2.1" xref="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml"><mi id="A2.SS1.6.p2.4.m4.2.2" xref="A2.SS1.6.p2.4.m4.2.2.cmml">Pr</mi><mo id="A2.SS1.6.p2.4.m4.4.4.2.2.1a" xref="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml">⁡</mo><mrow id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1" xref="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml"><mo id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.2" stretchy="false" xref="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml">[</mo><mrow id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.cmml"><msup id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.cmml"><mi id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.2" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.2.cmml">X</mi><mo id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.3" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.1" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.1.cmml">≤</mo><mi id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.3" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.3.cmml">τ</mi></mrow><mo id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.3" stretchy="false" xref="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml">]</mo></mrow></mrow></mrow><mo id="A2.SS1.6.p2.4.m4.4.4.3" xref="A2.SS1.6.p2.4.m4.4.4.3.cmml">=</mo><mn id="A2.SS1.6.p2.4.m4.4.4.4" xref="A2.SS1.6.p2.4.m4.4.4.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.4.m4.4b"><apply id="A2.SS1.6.p2.4.m4.4.4.cmml" xref="A2.SS1.6.p2.4.m4.4.4"><eq id="A2.SS1.6.p2.4.m4.4.4.3.cmml" xref="A2.SS1.6.p2.4.m4.4.4.3"></eq><apply id="A2.SS1.6.p2.4.m4.4.4.2.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2"><minus id="A2.SS1.6.p2.4.m4.4.4.2.3.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.3"></minus><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.2.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1"><ci id="A2.SS1.6.p2.4.m4.1.1.cmml" xref="A2.SS1.6.p2.4.m4.1.1">Pr</ci><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1"><and id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1a.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1"></and><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1b.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1"><leq id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.3.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.3"></leq><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.1.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2">superscript</csymbol><ci id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.2.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.2">𝑋</ci><ci id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.3.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.2.3">′</ci></apply><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4"><csymbol cd="latexml" id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.1.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.1">conditional</csymbol><ci id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.2.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.2">𝜏</ci><ci id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.3.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1c.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1"><eq id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.5.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.4.cmml" id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1d.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1"></share><ci id="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.6.cmml" xref="A2.SS1.6.p2.4.m4.3.3.1.1.1.1.1.6">𝜏</ci></apply></apply></apply><apply id="A2.SS1.6.p2.4.m4.4.4.2.2.2.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1"><ci id="A2.SS1.6.p2.4.m4.2.2.cmml" xref="A2.SS1.6.p2.4.m4.2.2">Pr</ci><apply id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1"><leq id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.1.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.1"></leq><apply id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.1.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2">superscript</csymbol><ci id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.2.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.2">𝑋</ci><ci id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.3.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.2.3">′</ci></apply><ci id="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.3.cmml" xref="A2.SS1.6.p2.4.m4.4.4.2.2.1.1.1.3">𝜏</ci></apply></apply></apply><cn id="A2.SS1.6.p2.4.m4.4.4.4.cmml" type="integer" xref="A2.SS1.6.p2.4.m4.4.4.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.4.m4.4c">\Pr[X^{\prime}\leq\tau\mid X=\tau]-\Pr[X^{\prime}\leq\tau]=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.4.m4.4d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_τ ] - roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ] = 0</annotation></semantics></math>, while the expected utility of reporting <math alttext="H" class="ltx_Math" display="inline" id="A2.SS1.6.p2.5.m5.1"><semantics id="A2.SS1.6.p2.5.m5.1a"><mi id="A2.SS1.6.p2.5.m5.1.1" xref="A2.SS1.6.p2.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.5.m5.1b"><ci id="A2.SS1.6.p2.5.m5.1.1.cmml" xref="A2.SS1.6.p2.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.5.m5.1d">italic_H</annotation></semantics></math> is <math alttext="\Pr[X^{\prime}>\tau\mid X=\tau]-\Pr[X^{\prime}>\tau]=0" class="ltx_Math" display="inline" id="A2.SS1.6.p2.6.m6.4"><semantics id="A2.SS1.6.p2.6.m6.4a"><mrow id="A2.SS1.6.p2.6.m6.4.4" xref="A2.SS1.6.p2.6.m6.4.4.cmml"><mrow id="A2.SS1.6.p2.6.m6.4.4.2" xref="A2.SS1.6.p2.6.m6.4.4.2.cmml"><mrow id="A2.SS1.6.p2.6.m6.3.3.1.1.1" xref="A2.SS1.6.p2.6.m6.3.3.1.1.2.cmml"><mi id="A2.SS1.6.p2.6.m6.1.1" xref="A2.SS1.6.p2.6.m6.1.1.cmml">Pr</mi><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1a" xref="A2.SS1.6.p2.6.m6.3.3.1.1.2.cmml">⁡</mo><mrow id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1" xref="A2.SS1.6.p2.6.m6.3.3.1.1.2.cmml"><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.2" stretchy="false" xref="A2.SS1.6.p2.6.m6.3.3.1.1.2.cmml">[</mo><mrow id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.cmml"><msup id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2.cmml"><mi id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2.2" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2.2.cmml">X</mi><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2.3" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.3" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.3.cmml">></mo><mrow id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.cmml"><mi id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.2" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.2.cmml">τ</mi><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.1" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.1.cmml">∣</mo><mi id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.3" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.5" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.5.cmml">=</mo><mi id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.6" xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.6.cmml">τ</mi></mrow><mo id="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.3" stretchy="false" xref="A2.SS1.6.p2.6.m6.3.3.1.1.2.cmml">]</mo></mrow></mrow><mo id="A2.SS1.6.p2.6.m6.4.4.2.3" xref="A2.SS1.6.p2.6.m6.4.4.2.3.cmml">−</mo><mrow id="A2.SS1.6.p2.6.m6.4.4.2.2.1" xref="A2.SS1.6.p2.6.m6.4.4.2.2.2.cmml"><mi id="A2.SS1.6.p2.6.m6.2.2" xref="A2.SS1.6.p2.6.m6.2.2.cmml">Pr</mi><mo id="A2.SS1.6.p2.6.m6.4.4.2.2.1a" xref="A2.SS1.6.p2.6.m6.4.4.2.2.2.cmml">⁡</mo><mrow id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1" xref="A2.SS1.6.p2.6.m6.4.4.2.2.2.cmml"><mo id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.2" stretchy="false" xref="A2.SS1.6.p2.6.m6.4.4.2.2.2.cmml">[</mo><mrow 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xref="A2.SS1.6.p2.6.m6.3.3.1.1.1.1.1.6">𝜏</ci></apply></apply></apply><apply id="A2.SS1.6.p2.6.m6.4.4.2.2.2.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1"><ci id="A2.SS1.6.p2.6.m6.2.2.cmml" xref="A2.SS1.6.p2.6.m6.2.2">Pr</ci><apply id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1"><gt id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.1.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.1"></gt><apply id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.1.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2">superscript</csymbol><ci id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.2.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.2">𝑋</ci><ci id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.3.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.2.3">′</ci></apply><ci id="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.3.cmml" xref="A2.SS1.6.p2.6.m6.4.4.2.2.1.1.1.3">𝜏</ci></apply></apply></apply><cn id="A2.SS1.6.p2.6.m6.4.4.4.cmml" type="integer" xref="A2.SS1.6.p2.6.m6.4.4.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.6.m6.4c">\Pr[X^{\prime}>\tau\mid X=\tau]-\Pr[X^{\prime}>\tau]=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.6.m6.4d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ ∣ italic_X = italic_τ ] - roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_τ ] = 0</annotation></semantics></math>; thus the agent is indifferent between reporting <math alttext="H" class="ltx_Math" display="inline" id="A2.SS1.6.p2.7.m7.1"><semantics id="A2.SS1.6.p2.7.m7.1a"><mi id="A2.SS1.6.p2.7.m7.1.1" xref="A2.SS1.6.p2.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.7.m7.1b"><ci id="A2.SS1.6.p2.7.m7.1.1.cmml" xref="A2.SS1.6.p2.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.7.m7.1d">italic_H</annotation></semantics></math> or <math alttext="L" class="ltx_Math" display="inline" id="A2.SS1.6.p2.8.m8.1"><semantics id="A2.SS1.6.p2.8.m8.1a"><mi id="A2.SS1.6.p2.8.m8.1.1" xref="A2.SS1.6.p2.8.m8.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.8.m8.1b"><ci id="A2.SS1.6.p2.8.m8.1.1.cmml" xref="A2.SS1.6.p2.8.m8.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.8.m8.1c">L</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.8.m8.1d">italic_L</annotation></semantics></math>. Moreover, the best response <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.6.p2.9.m9.1"><semantics id="A2.SS1.6.p2.9.m9.1a"><mover accent="true" id="A2.SS1.6.p2.9.m9.1.1" xref="A2.SS1.6.p2.9.m9.1.1.cmml"><mi id="A2.SS1.6.p2.9.m9.1.1.2" xref="A2.SS1.6.p2.9.m9.1.1.2.cmml">τ</mi><mo id="A2.SS1.6.p2.9.m9.1.1.1" xref="A2.SS1.6.p2.9.m9.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.9.m9.1b"><apply id="A2.SS1.6.p2.9.m9.1.1.cmml" xref="A2.SS1.6.p2.9.m9.1.1"><ci id="A2.SS1.6.p2.9.m9.1.1.1.cmml" xref="A2.SS1.6.p2.9.m9.1.1.1">^</ci><ci id="A2.SS1.6.p2.9.m9.1.1.2.cmml" xref="A2.SS1.6.p2.9.m9.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.9.m9.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.9.m9.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> is exactly <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.6.p2.10.m10.1"><semantics id="A2.SS1.6.p2.10.m10.1a"><mi id="A2.SS1.6.p2.10.m10.1.1" xref="A2.SS1.6.p2.10.m10.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.10.m10.1b"><ci id="A2.SS1.6.p2.10.m10.1.1.cmml" xref="A2.SS1.6.p2.10.m10.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.10.m10.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.10.m10.1d">italic_τ</annotation></semantics></math>. It follows the system is at an equilibrium since <math alttext="\dot{\tau}=\hat{\tau}-\tau=0" class="ltx_Math" display="inline" id="A2.SS1.6.p2.11.m11.1"><semantics id="A2.SS1.6.p2.11.m11.1a"><mrow id="A2.SS1.6.p2.11.m11.1.1" xref="A2.SS1.6.p2.11.m11.1.1.cmml"><mover accent="true" id="A2.SS1.6.p2.11.m11.1.1.2" xref="A2.SS1.6.p2.11.m11.1.1.2.cmml"><mi id="A2.SS1.6.p2.11.m11.1.1.2.2" xref="A2.SS1.6.p2.11.m11.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.6.p2.11.m11.1.1.2.1" xref="A2.SS1.6.p2.11.m11.1.1.2.1.cmml">˙</mo></mover><mo id="A2.SS1.6.p2.11.m11.1.1.3" xref="A2.SS1.6.p2.11.m11.1.1.3.cmml">=</mo><mrow id="A2.SS1.6.p2.11.m11.1.1.4" xref="A2.SS1.6.p2.11.m11.1.1.4.cmml"><mover accent="true" id="A2.SS1.6.p2.11.m11.1.1.4.2" xref="A2.SS1.6.p2.11.m11.1.1.4.2.cmml"><mi id="A2.SS1.6.p2.11.m11.1.1.4.2.2" xref="A2.SS1.6.p2.11.m11.1.1.4.2.2.cmml">τ</mi><mo id="A2.SS1.6.p2.11.m11.1.1.4.2.1" xref="A2.SS1.6.p2.11.m11.1.1.4.2.1.cmml">^</mo></mover><mo id="A2.SS1.6.p2.11.m11.1.1.4.1" xref="A2.SS1.6.p2.11.m11.1.1.4.1.cmml">−</mo><mi id="A2.SS1.6.p2.11.m11.1.1.4.3" xref="A2.SS1.6.p2.11.m11.1.1.4.3.cmml">τ</mi></mrow><mo id="A2.SS1.6.p2.11.m11.1.1.5" xref="A2.SS1.6.p2.11.m11.1.1.5.cmml">=</mo><mn id="A2.SS1.6.p2.11.m11.1.1.6" xref="A2.SS1.6.p2.11.m11.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.6.p2.11.m11.1b"><apply id="A2.SS1.6.p2.11.m11.1.1.cmml" xref="A2.SS1.6.p2.11.m11.1.1"><and id="A2.SS1.6.p2.11.m11.1.1a.cmml" xref="A2.SS1.6.p2.11.m11.1.1"></and><apply id="A2.SS1.6.p2.11.m11.1.1b.cmml" xref="A2.SS1.6.p2.11.m11.1.1"><eq id="A2.SS1.6.p2.11.m11.1.1.3.cmml" xref="A2.SS1.6.p2.11.m11.1.1.3"></eq><apply id="A2.SS1.6.p2.11.m11.1.1.2.cmml" xref="A2.SS1.6.p2.11.m11.1.1.2"><ci id="A2.SS1.6.p2.11.m11.1.1.2.1.cmml" xref="A2.SS1.6.p2.11.m11.1.1.2.1">˙</ci><ci id="A2.SS1.6.p2.11.m11.1.1.2.2.cmml" xref="A2.SS1.6.p2.11.m11.1.1.2.2">𝜏</ci></apply><apply id="A2.SS1.6.p2.11.m11.1.1.4.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4"><minus id="A2.SS1.6.p2.11.m11.1.1.4.1.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4.1"></minus><apply id="A2.SS1.6.p2.11.m11.1.1.4.2.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4.2"><ci id="A2.SS1.6.p2.11.m11.1.1.4.2.1.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4.2.1">^</ci><ci id="A2.SS1.6.p2.11.m11.1.1.4.2.2.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4.2.2">𝜏</ci></apply><ci id="A2.SS1.6.p2.11.m11.1.1.4.3.cmml" xref="A2.SS1.6.p2.11.m11.1.1.4.3">𝜏</ci></apply></apply><apply id="A2.SS1.6.p2.11.m11.1.1c.cmml" xref="A2.SS1.6.p2.11.m11.1.1"><eq id="A2.SS1.6.p2.11.m11.1.1.5.cmml" xref="A2.SS1.6.p2.11.m11.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.6.p2.11.m11.1.1.4.cmml" id="A2.SS1.6.p2.11.m11.1.1d.cmml" xref="A2.SS1.6.p2.11.m11.1.1"></share><cn id="A2.SS1.6.p2.11.m11.1.1.6.cmml" type="integer" xref="A2.SS1.6.p2.11.m11.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.6.p2.11.m11.1c">\dot{\tau}=\hat{\tau}-\tau=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.6.p2.11.m11.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS1.7.p3"> <p class="ltx_p" id="A2.SS1.7.p3.12">In the second case, <math alttext="G(\tau)>F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.7.p3.1.m1.2"><semantics id="A2.SS1.7.p3.1.m1.2a"><mrow id="A2.SS1.7.p3.1.m1.2.3" xref="A2.SS1.7.p3.1.m1.2.3.cmml"><mrow id="A2.SS1.7.p3.1.m1.2.3.2" xref="A2.SS1.7.p3.1.m1.2.3.2.cmml"><mi id="A2.SS1.7.p3.1.m1.2.3.2.2" xref="A2.SS1.7.p3.1.m1.2.3.2.2.cmml">G</mi><mo id="A2.SS1.7.p3.1.m1.2.3.2.1" xref="A2.SS1.7.p3.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.1.m1.2.3.2.3.2" xref="A2.SS1.7.p3.1.m1.2.3.2.cmml"><mo id="A2.SS1.7.p3.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.7.p3.1.m1.2.3.2.cmml">(</mo><mi id="A2.SS1.7.p3.1.m1.1.1" xref="A2.SS1.7.p3.1.m1.1.1.cmml">τ</mi><mo id="A2.SS1.7.p3.1.m1.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.7.p3.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.1.m1.2.3.1" xref="A2.SS1.7.p3.1.m1.2.3.1.cmml">></mo><mrow id="A2.SS1.7.p3.1.m1.2.3.3" xref="A2.SS1.7.p3.1.m1.2.3.3.cmml"><mi id="A2.SS1.7.p3.1.m1.2.3.3.2" xref="A2.SS1.7.p3.1.m1.2.3.3.2.cmml">F</mi><mo id="A2.SS1.7.p3.1.m1.2.3.3.1" xref="A2.SS1.7.p3.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.1.m1.2.3.3.3.2" xref="A2.SS1.7.p3.1.m1.2.3.3.cmml"><mo id="A2.SS1.7.p3.1.m1.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.7.p3.1.m1.2.3.3.cmml">(</mo><mi id="A2.SS1.7.p3.1.m1.2.2" xref="A2.SS1.7.p3.1.m1.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.1.m1.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.7.p3.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.1.m1.2b"><apply id="A2.SS1.7.p3.1.m1.2.3.cmml" xref="A2.SS1.7.p3.1.m1.2.3"><gt id="A2.SS1.7.p3.1.m1.2.3.1.cmml" xref="A2.SS1.7.p3.1.m1.2.3.1"></gt><apply id="A2.SS1.7.p3.1.m1.2.3.2.cmml" xref="A2.SS1.7.p3.1.m1.2.3.2"><times id="A2.SS1.7.p3.1.m1.2.3.2.1.cmml" xref="A2.SS1.7.p3.1.m1.2.3.2.1"></times><ci id="A2.SS1.7.p3.1.m1.2.3.2.2.cmml" xref="A2.SS1.7.p3.1.m1.2.3.2.2">𝐺</ci><ci id="A2.SS1.7.p3.1.m1.1.1.cmml" xref="A2.SS1.7.p3.1.m1.1.1">𝜏</ci></apply><apply id="A2.SS1.7.p3.1.m1.2.3.3.cmml" xref="A2.SS1.7.p3.1.m1.2.3.3"><times id="A2.SS1.7.p3.1.m1.2.3.3.1.cmml" xref="A2.SS1.7.p3.1.m1.2.3.3.1"></times><ci id="A2.SS1.7.p3.1.m1.2.3.3.2.cmml" xref="A2.SS1.7.p3.1.m1.2.3.3.2">𝐹</ci><ci id="A2.SS1.7.p3.1.m1.2.2.cmml" xref="A2.SS1.7.p3.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.1.m1.2c">G(\tau)>F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.1.m1.2d">italic_G ( italic_τ ) > italic_F ( italic_τ )</annotation></semantics></math>. Then reporting <math alttext="L" class="ltx_Math" display="inline" id="A2.SS1.7.p3.2.m2.1"><semantics id="A2.SS1.7.p3.2.m2.1a"><mi id="A2.SS1.7.p3.2.m2.1.1" xref="A2.SS1.7.p3.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.2.m2.1b"><ci id="A2.SS1.7.p3.2.m2.1.1.cmml" xref="A2.SS1.7.p3.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.2.m2.1d">italic_L</annotation></semantics></math> is strictly preferred to reporting <math alttext="H" class="ltx_Math" display="inline" id="A2.SS1.7.p3.3.m3.1"><semantics id="A2.SS1.7.p3.3.m3.1a"><mi id="A2.SS1.7.p3.3.m3.1.1" xref="A2.SS1.7.p3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.3.m3.1b"><ci id="A2.SS1.7.p3.3.m3.1.1.cmml" xref="A2.SS1.7.p3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.3.m3.1d">italic_H</annotation></semantics></math>. Since <math alttext="P(\tau;\hat{\tau})=F(\tau)<G(\tau)=P(\tau;\tau)" class="ltx_Math" display="inline" id="A2.SS1.7.p3.4.m4.6"><semantics id="A2.SS1.7.p3.4.m4.6a"><mrow id="A2.SS1.7.p3.4.m4.6.7" xref="A2.SS1.7.p3.4.m4.6.7.cmml"><mrow id="A2.SS1.7.p3.4.m4.6.7.2" xref="A2.SS1.7.p3.4.m4.6.7.2.cmml"><mi id="A2.SS1.7.p3.4.m4.6.7.2.2" xref="A2.SS1.7.p3.4.m4.6.7.2.2.cmml">P</mi><mo id="A2.SS1.7.p3.4.m4.6.7.2.1" xref="A2.SS1.7.p3.4.m4.6.7.2.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.2.3.2" xref="A2.SS1.7.p3.4.m4.6.7.2.3.1.cmml"><mo id="A2.SS1.7.p3.4.m4.6.7.2.3.2.1" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.2.3.1.cmml">(</mo><mi id="A2.SS1.7.p3.4.m4.1.1" xref="A2.SS1.7.p3.4.m4.1.1.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.6.7.2.3.2.2" xref="A2.SS1.7.p3.4.m4.6.7.2.3.1.cmml">;</mo><mover accent="true" id="A2.SS1.7.p3.4.m4.2.2" xref="A2.SS1.7.p3.4.m4.2.2.cmml"><mi id="A2.SS1.7.p3.4.m4.2.2.2" xref="A2.SS1.7.p3.4.m4.2.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.2.2.1" xref="A2.SS1.7.p3.4.m4.2.2.1.cmml">^</mo></mover><mo id="A2.SS1.7.p3.4.m4.6.7.2.3.2.3" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.4.m4.6.7.3" xref="A2.SS1.7.p3.4.m4.6.7.3.cmml">=</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.4" xref="A2.SS1.7.p3.4.m4.6.7.4.cmml"><mi id="A2.SS1.7.p3.4.m4.6.7.4.2" xref="A2.SS1.7.p3.4.m4.6.7.4.2.cmml">F</mi><mo id="A2.SS1.7.p3.4.m4.6.7.4.1" xref="A2.SS1.7.p3.4.m4.6.7.4.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.4.3.2" xref="A2.SS1.7.p3.4.m4.6.7.4.cmml"><mo id="A2.SS1.7.p3.4.m4.6.7.4.3.2.1" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.4.cmml">(</mo><mi id="A2.SS1.7.p3.4.m4.3.3" xref="A2.SS1.7.p3.4.m4.3.3.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.6.7.4.3.2.2" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.4.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.4.m4.6.7.5" xref="A2.SS1.7.p3.4.m4.6.7.5.cmml"><</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.6" xref="A2.SS1.7.p3.4.m4.6.7.6.cmml"><mi id="A2.SS1.7.p3.4.m4.6.7.6.2" xref="A2.SS1.7.p3.4.m4.6.7.6.2.cmml">G</mi><mo id="A2.SS1.7.p3.4.m4.6.7.6.1" xref="A2.SS1.7.p3.4.m4.6.7.6.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.6.3.2" xref="A2.SS1.7.p3.4.m4.6.7.6.cmml"><mo id="A2.SS1.7.p3.4.m4.6.7.6.3.2.1" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.6.cmml">(</mo><mi id="A2.SS1.7.p3.4.m4.4.4" xref="A2.SS1.7.p3.4.m4.4.4.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.6.7.6.3.2.2" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.6.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.4.m4.6.7.7" xref="A2.SS1.7.p3.4.m4.6.7.7.cmml">=</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.8" xref="A2.SS1.7.p3.4.m4.6.7.8.cmml"><mi id="A2.SS1.7.p3.4.m4.6.7.8.2" xref="A2.SS1.7.p3.4.m4.6.7.8.2.cmml">P</mi><mo id="A2.SS1.7.p3.4.m4.6.7.8.1" xref="A2.SS1.7.p3.4.m4.6.7.8.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.4.m4.6.7.8.3.2" xref="A2.SS1.7.p3.4.m4.6.7.8.3.1.cmml"><mo id="A2.SS1.7.p3.4.m4.6.7.8.3.2.1" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.8.3.1.cmml">(</mo><mi id="A2.SS1.7.p3.4.m4.5.5" xref="A2.SS1.7.p3.4.m4.5.5.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.6.7.8.3.2.2" xref="A2.SS1.7.p3.4.m4.6.7.8.3.1.cmml">;</mo><mi id="A2.SS1.7.p3.4.m4.6.6" xref="A2.SS1.7.p3.4.m4.6.6.cmml">τ</mi><mo id="A2.SS1.7.p3.4.m4.6.7.8.3.2.3" stretchy="false" xref="A2.SS1.7.p3.4.m4.6.7.8.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.4.m4.6b"><apply id="A2.SS1.7.p3.4.m4.6.7.cmml" xref="A2.SS1.7.p3.4.m4.6.7"><and id="A2.SS1.7.p3.4.m4.6.7a.cmml" xref="A2.SS1.7.p3.4.m4.6.7"></and><apply id="A2.SS1.7.p3.4.m4.6.7b.cmml" xref="A2.SS1.7.p3.4.m4.6.7"><eq id="A2.SS1.7.p3.4.m4.6.7.3.cmml" xref="A2.SS1.7.p3.4.m4.6.7.3"></eq><apply id="A2.SS1.7.p3.4.m4.6.7.2.cmml" xref="A2.SS1.7.p3.4.m4.6.7.2"><times id="A2.SS1.7.p3.4.m4.6.7.2.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.2.1"></times><ci id="A2.SS1.7.p3.4.m4.6.7.2.2.cmml" xref="A2.SS1.7.p3.4.m4.6.7.2.2">𝑃</ci><list id="A2.SS1.7.p3.4.m4.6.7.2.3.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.2.3.2"><ci id="A2.SS1.7.p3.4.m4.1.1.cmml" xref="A2.SS1.7.p3.4.m4.1.1">𝜏</ci><apply id="A2.SS1.7.p3.4.m4.2.2.cmml" xref="A2.SS1.7.p3.4.m4.2.2"><ci id="A2.SS1.7.p3.4.m4.2.2.1.cmml" xref="A2.SS1.7.p3.4.m4.2.2.1">^</ci><ci id="A2.SS1.7.p3.4.m4.2.2.2.cmml" xref="A2.SS1.7.p3.4.m4.2.2.2">𝜏</ci></apply></list></apply><apply id="A2.SS1.7.p3.4.m4.6.7.4.cmml" xref="A2.SS1.7.p3.4.m4.6.7.4"><times id="A2.SS1.7.p3.4.m4.6.7.4.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.4.1"></times><ci id="A2.SS1.7.p3.4.m4.6.7.4.2.cmml" xref="A2.SS1.7.p3.4.m4.6.7.4.2">𝐹</ci><ci id="A2.SS1.7.p3.4.m4.3.3.cmml" xref="A2.SS1.7.p3.4.m4.3.3">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.4.m4.6.7c.cmml" xref="A2.SS1.7.p3.4.m4.6.7"><lt id="A2.SS1.7.p3.4.m4.6.7.5.cmml" xref="A2.SS1.7.p3.4.m4.6.7.5"></lt><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.4.m4.6.7.4.cmml" id="A2.SS1.7.p3.4.m4.6.7d.cmml" xref="A2.SS1.7.p3.4.m4.6.7"></share><apply id="A2.SS1.7.p3.4.m4.6.7.6.cmml" xref="A2.SS1.7.p3.4.m4.6.7.6"><times id="A2.SS1.7.p3.4.m4.6.7.6.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.6.1"></times><ci id="A2.SS1.7.p3.4.m4.6.7.6.2.cmml" xref="A2.SS1.7.p3.4.m4.6.7.6.2">𝐺</ci><ci id="A2.SS1.7.p3.4.m4.4.4.cmml" xref="A2.SS1.7.p3.4.m4.4.4">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.4.m4.6.7e.cmml" xref="A2.SS1.7.p3.4.m4.6.7"><eq id="A2.SS1.7.p3.4.m4.6.7.7.cmml" xref="A2.SS1.7.p3.4.m4.6.7.7"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.4.m4.6.7.6.cmml" id="A2.SS1.7.p3.4.m4.6.7f.cmml" xref="A2.SS1.7.p3.4.m4.6.7"></share><apply id="A2.SS1.7.p3.4.m4.6.7.8.cmml" xref="A2.SS1.7.p3.4.m4.6.7.8"><times id="A2.SS1.7.p3.4.m4.6.7.8.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.8.1"></times><ci id="A2.SS1.7.p3.4.m4.6.7.8.2.cmml" xref="A2.SS1.7.p3.4.m4.6.7.8.2">𝑃</ci><list id="A2.SS1.7.p3.4.m4.6.7.8.3.1.cmml" xref="A2.SS1.7.p3.4.m4.6.7.8.3.2"><ci id="A2.SS1.7.p3.4.m4.5.5.cmml" xref="A2.SS1.7.p3.4.m4.5.5">𝜏</ci><ci id="A2.SS1.7.p3.4.m4.6.6.cmml" xref="A2.SS1.7.p3.4.m4.6.6">𝜏</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.4.m4.6c">P(\tau;\hat{\tau})=F(\tau)<G(\tau)=P(\tau;\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.4.m4.6d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = italic_F ( italic_τ ) < italic_G ( italic_τ ) = italic_P ( italic_τ ; italic_τ )</annotation></semantics></math>, by decreasing monotonicity we have <math alttext="\hat{\tau}>\tau" class="ltx_Math" display="inline" id="A2.SS1.7.p3.5.m5.1"><semantics id="A2.SS1.7.p3.5.m5.1a"><mrow id="A2.SS1.7.p3.5.m5.1.1" xref="A2.SS1.7.p3.5.m5.1.1.cmml"><mover accent="true" id="A2.SS1.7.p3.5.m5.1.1.2" xref="A2.SS1.7.p3.5.m5.1.1.2.cmml"><mi id="A2.SS1.7.p3.5.m5.1.1.2.2" xref="A2.SS1.7.p3.5.m5.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.5.m5.1.1.2.1" xref="A2.SS1.7.p3.5.m5.1.1.2.1.cmml">^</mo></mover><mo id="A2.SS1.7.p3.5.m5.1.1.1" xref="A2.SS1.7.p3.5.m5.1.1.1.cmml">></mo><mi id="A2.SS1.7.p3.5.m5.1.1.3" xref="A2.SS1.7.p3.5.m5.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.5.m5.1b"><apply id="A2.SS1.7.p3.5.m5.1.1.cmml" xref="A2.SS1.7.p3.5.m5.1.1"><gt id="A2.SS1.7.p3.5.m5.1.1.1.cmml" xref="A2.SS1.7.p3.5.m5.1.1.1"></gt><apply id="A2.SS1.7.p3.5.m5.1.1.2.cmml" xref="A2.SS1.7.p3.5.m5.1.1.2"><ci id="A2.SS1.7.p3.5.m5.1.1.2.1.cmml" xref="A2.SS1.7.p3.5.m5.1.1.2.1">^</ci><ci id="A2.SS1.7.p3.5.m5.1.1.2.2.cmml" xref="A2.SS1.7.p3.5.m5.1.1.2.2">𝜏</ci></apply><ci id="A2.SS1.7.p3.5.m5.1.1.3.cmml" xref="A2.SS1.7.p3.5.m5.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.5.m5.1c">\hat{\tau}>\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.5.m5.1d">over^ start_ARG italic_τ end_ARG > italic_τ</annotation></semantics></math>. Thus <math alttext="\dot{\tau}=\hat{\tau}-\tau>0" class="ltx_Math" display="inline" id="A2.SS1.7.p3.6.m6.1"><semantics id="A2.SS1.7.p3.6.m6.1a"><mrow id="A2.SS1.7.p3.6.m6.1.1" xref="A2.SS1.7.p3.6.m6.1.1.cmml"><mover accent="true" id="A2.SS1.7.p3.6.m6.1.1.2" xref="A2.SS1.7.p3.6.m6.1.1.2.cmml"><mi id="A2.SS1.7.p3.6.m6.1.1.2.2" xref="A2.SS1.7.p3.6.m6.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.6.m6.1.1.2.1" xref="A2.SS1.7.p3.6.m6.1.1.2.1.cmml">˙</mo></mover><mo id="A2.SS1.7.p3.6.m6.1.1.3" xref="A2.SS1.7.p3.6.m6.1.1.3.cmml">=</mo><mrow id="A2.SS1.7.p3.6.m6.1.1.4" xref="A2.SS1.7.p3.6.m6.1.1.4.cmml"><mover accent="true" id="A2.SS1.7.p3.6.m6.1.1.4.2" xref="A2.SS1.7.p3.6.m6.1.1.4.2.cmml"><mi id="A2.SS1.7.p3.6.m6.1.1.4.2.2" xref="A2.SS1.7.p3.6.m6.1.1.4.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.6.m6.1.1.4.2.1" xref="A2.SS1.7.p3.6.m6.1.1.4.2.1.cmml">^</mo></mover><mo id="A2.SS1.7.p3.6.m6.1.1.4.1" xref="A2.SS1.7.p3.6.m6.1.1.4.1.cmml">−</mo><mi id="A2.SS1.7.p3.6.m6.1.1.4.3" xref="A2.SS1.7.p3.6.m6.1.1.4.3.cmml">τ</mi></mrow><mo id="A2.SS1.7.p3.6.m6.1.1.5" xref="A2.SS1.7.p3.6.m6.1.1.5.cmml">></mo><mn id="A2.SS1.7.p3.6.m6.1.1.6" xref="A2.SS1.7.p3.6.m6.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.6.m6.1b"><apply id="A2.SS1.7.p3.6.m6.1.1.cmml" xref="A2.SS1.7.p3.6.m6.1.1"><and id="A2.SS1.7.p3.6.m6.1.1a.cmml" xref="A2.SS1.7.p3.6.m6.1.1"></and><apply id="A2.SS1.7.p3.6.m6.1.1b.cmml" xref="A2.SS1.7.p3.6.m6.1.1"><eq id="A2.SS1.7.p3.6.m6.1.1.3.cmml" xref="A2.SS1.7.p3.6.m6.1.1.3"></eq><apply id="A2.SS1.7.p3.6.m6.1.1.2.cmml" xref="A2.SS1.7.p3.6.m6.1.1.2"><ci id="A2.SS1.7.p3.6.m6.1.1.2.1.cmml" xref="A2.SS1.7.p3.6.m6.1.1.2.1">˙</ci><ci id="A2.SS1.7.p3.6.m6.1.1.2.2.cmml" xref="A2.SS1.7.p3.6.m6.1.1.2.2">𝜏</ci></apply><apply id="A2.SS1.7.p3.6.m6.1.1.4.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4"><minus id="A2.SS1.7.p3.6.m6.1.1.4.1.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4.1"></minus><apply id="A2.SS1.7.p3.6.m6.1.1.4.2.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4.2"><ci id="A2.SS1.7.p3.6.m6.1.1.4.2.1.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4.2.1">^</ci><ci id="A2.SS1.7.p3.6.m6.1.1.4.2.2.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4.2.2">𝜏</ci></apply><ci id="A2.SS1.7.p3.6.m6.1.1.4.3.cmml" xref="A2.SS1.7.p3.6.m6.1.1.4.3">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.6.m6.1.1c.cmml" xref="A2.SS1.7.p3.6.m6.1.1"><gt id="A2.SS1.7.p3.6.m6.1.1.5.cmml" xref="A2.SS1.7.p3.6.m6.1.1.5"></gt><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.6.m6.1.1.4.cmml" id="A2.SS1.7.p3.6.m6.1.1d.cmml" xref="A2.SS1.7.p3.6.m6.1.1"></share><cn id="A2.SS1.7.p3.6.m6.1.1.6.cmml" type="integer" xref="A2.SS1.7.p3.6.m6.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.6.m6.1c">\dot{\tau}=\hat{\tau}-\tau>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.6.m6.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ > 0</annotation></semantics></math>. In the third case, <math alttext="G(\tau)<F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.7.p3.7.m7.2"><semantics id="A2.SS1.7.p3.7.m7.2a"><mrow id="A2.SS1.7.p3.7.m7.2.3" xref="A2.SS1.7.p3.7.m7.2.3.cmml"><mrow id="A2.SS1.7.p3.7.m7.2.3.2" xref="A2.SS1.7.p3.7.m7.2.3.2.cmml"><mi id="A2.SS1.7.p3.7.m7.2.3.2.2" xref="A2.SS1.7.p3.7.m7.2.3.2.2.cmml">G</mi><mo id="A2.SS1.7.p3.7.m7.2.3.2.1" xref="A2.SS1.7.p3.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.7.m7.2.3.2.3.2" xref="A2.SS1.7.p3.7.m7.2.3.2.cmml"><mo id="A2.SS1.7.p3.7.m7.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.7.p3.7.m7.2.3.2.cmml">(</mo><mi id="A2.SS1.7.p3.7.m7.1.1" xref="A2.SS1.7.p3.7.m7.1.1.cmml">τ</mi><mo id="A2.SS1.7.p3.7.m7.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.7.p3.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.7.m7.2.3.1" xref="A2.SS1.7.p3.7.m7.2.3.1.cmml"><</mo><mrow id="A2.SS1.7.p3.7.m7.2.3.3" xref="A2.SS1.7.p3.7.m7.2.3.3.cmml"><mi id="A2.SS1.7.p3.7.m7.2.3.3.2" xref="A2.SS1.7.p3.7.m7.2.3.3.2.cmml">F</mi><mo id="A2.SS1.7.p3.7.m7.2.3.3.1" xref="A2.SS1.7.p3.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.7.m7.2.3.3.3.2" xref="A2.SS1.7.p3.7.m7.2.3.3.cmml"><mo id="A2.SS1.7.p3.7.m7.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.7.p3.7.m7.2.3.3.cmml">(</mo><mi id="A2.SS1.7.p3.7.m7.2.2" xref="A2.SS1.7.p3.7.m7.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.7.m7.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.7.p3.7.m7.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.7.m7.2b"><apply id="A2.SS1.7.p3.7.m7.2.3.cmml" xref="A2.SS1.7.p3.7.m7.2.3"><lt id="A2.SS1.7.p3.7.m7.2.3.1.cmml" xref="A2.SS1.7.p3.7.m7.2.3.1"></lt><apply id="A2.SS1.7.p3.7.m7.2.3.2.cmml" xref="A2.SS1.7.p3.7.m7.2.3.2"><times id="A2.SS1.7.p3.7.m7.2.3.2.1.cmml" xref="A2.SS1.7.p3.7.m7.2.3.2.1"></times><ci id="A2.SS1.7.p3.7.m7.2.3.2.2.cmml" xref="A2.SS1.7.p3.7.m7.2.3.2.2">𝐺</ci><ci id="A2.SS1.7.p3.7.m7.1.1.cmml" xref="A2.SS1.7.p3.7.m7.1.1">𝜏</ci></apply><apply id="A2.SS1.7.p3.7.m7.2.3.3.cmml" xref="A2.SS1.7.p3.7.m7.2.3.3"><times id="A2.SS1.7.p3.7.m7.2.3.3.1.cmml" xref="A2.SS1.7.p3.7.m7.2.3.3.1"></times><ci id="A2.SS1.7.p3.7.m7.2.3.3.2.cmml" xref="A2.SS1.7.p3.7.m7.2.3.3.2">𝐹</ci><ci id="A2.SS1.7.p3.7.m7.2.2.cmml" xref="A2.SS1.7.p3.7.m7.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.7.m7.2c">G(\tau)<F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.7.m7.2d">italic_G ( italic_τ ) < italic_F ( italic_τ )</annotation></semantics></math>. Then reporting <math alttext="H" class="ltx_Math" display="inline" id="A2.SS1.7.p3.8.m8.1"><semantics id="A2.SS1.7.p3.8.m8.1a"><mi id="A2.SS1.7.p3.8.m8.1.1" xref="A2.SS1.7.p3.8.m8.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.8.m8.1b"><ci id="A2.SS1.7.p3.8.m8.1.1.cmml" xref="A2.SS1.7.p3.8.m8.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.8.m8.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.8.m8.1d">italic_H</annotation></semantics></math> is strictly preferred to reporting <math alttext="L" class="ltx_Math" display="inline" id="A2.SS1.7.p3.9.m9.1"><semantics id="A2.SS1.7.p3.9.m9.1a"><mi id="A2.SS1.7.p3.9.m9.1.1" xref="A2.SS1.7.p3.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.9.m9.1b"><ci id="A2.SS1.7.p3.9.m9.1.1.cmml" xref="A2.SS1.7.p3.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.9.m9.1d">italic_L</annotation></semantics></math>. Since <math alttext="P(\tau;\hat{\tau})=F(\tau)>G(\tau)=P(\tau;\tau)" class="ltx_Math" display="inline" id="A2.SS1.7.p3.10.m10.6"><semantics id="A2.SS1.7.p3.10.m10.6a"><mrow id="A2.SS1.7.p3.10.m10.6.7" xref="A2.SS1.7.p3.10.m10.6.7.cmml"><mrow id="A2.SS1.7.p3.10.m10.6.7.2" xref="A2.SS1.7.p3.10.m10.6.7.2.cmml"><mi id="A2.SS1.7.p3.10.m10.6.7.2.2" xref="A2.SS1.7.p3.10.m10.6.7.2.2.cmml">P</mi><mo id="A2.SS1.7.p3.10.m10.6.7.2.1" xref="A2.SS1.7.p3.10.m10.6.7.2.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.2.3.2" xref="A2.SS1.7.p3.10.m10.6.7.2.3.1.cmml"><mo id="A2.SS1.7.p3.10.m10.6.7.2.3.2.1" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.2.3.1.cmml">(</mo><mi id="A2.SS1.7.p3.10.m10.1.1" xref="A2.SS1.7.p3.10.m10.1.1.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.6.7.2.3.2.2" xref="A2.SS1.7.p3.10.m10.6.7.2.3.1.cmml">;</mo><mover accent="true" id="A2.SS1.7.p3.10.m10.2.2" xref="A2.SS1.7.p3.10.m10.2.2.cmml"><mi id="A2.SS1.7.p3.10.m10.2.2.2" xref="A2.SS1.7.p3.10.m10.2.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.2.2.1" xref="A2.SS1.7.p3.10.m10.2.2.1.cmml">^</mo></mover><mo id="A2.SS1.7.p3.10.m10.6.7.2.3.2.3" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.10.m10.6.7.3" xref="A2.SS1.7.p3.10.m10.6.7.3.cmml">=</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.4" xref="A2.SS1.7.p3.10.m10.6.7.4.cmml"><mi id="A2.SS1.7.p3.10.m10.6.7.4.2" xref="A2.SS1.7.p3.10.m10.6.7.4.2.cmml">F</mi><mo id="A2.SS1.7.p3.10.m10.6.7.4.1" xref="A2.SS1.7.p3.10.m10.6.7.4.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.4.3.2" xref="A2.SS1.7.p3.10.m10.6.7.4.cmml"><mo id="A2.SS1.7.p3.10.m10.6.7.4.3.2.1" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.4.cmml">(</mo><mi id="A2.SS1.7.p3.10.m10.3.3" xref="A2.SS1.7.p3.10.m10.3.3.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.6.7.4.3.2.2" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.4.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.10.m10.6.7.5" xref="A2.SS1.7.p3.10.m10.6.7.5.cmml">></mo><mrow id="A2.SS1.7.p3.10.m10.6.7.6" xref="A2.SS1.7.p3.10.m10.6.7.6.cmml"><mi id="A2.SS1.7.p3.10.m10.6.7.6.2" xref="A2.SS1.7.p3.10.m10.6.7.6.2.cmml">G</mi><mo id="A2.SS1.7.p3.10.m10.6.7.6.1" xref="A2.SS1.7.p3.10.m10.6.7.6.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.6.3.2" xref="A2.SS1.7.p3.10.m10.6.7.6.cmml"><mo id="A2.SS1.7.p3.10.m10.6.7.6.3.2.1" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.6.cmml">(</mo><mi id="A2.SS1.7.p3.10.m10.4.4" xref="A2.SS1.7.p3.10.m10.4.4.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.6.7.6.3.2.2" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.6.cmml">)</mo></mrow></mrow><mo id="A2.SS1.7.p3.10.m10.6.7.7" xref="A2.SS1.7.p3.10.m10.6.7.7.cmml">=</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.8" xref="A2.SS1.7.p3.10.m10.6.7.8.cmml"><mi id="A2.SS1.7.p3.10.m10.6.7.8.2" xref="A2.SS1.7.p3.10.m10.6.7.8.2.cmml">P</mi><mo id="A2.SS1.7.p3.10.m10.6.7.8.1" xref="A2.SS1.7.p3.10.m10.6.7.8.1.cmml">⁢</mo><mrow id="A2.SS1.7.p3.10.m10.6.7.8.3.2" xref="A2.SS1.7.p3.10.m10.6.7.8.3.1.cmml"><mo id="A2.SS1.7.p3.10.m10.6.7.8.3.2.1" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.8.3.1.cmml">(</mo><mi id="A2.SS1.7.p3.10.m10.5.5" xref="A2.SS1.7.p3.10.m10.5.5.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.6.7.8.3.2.2" xref="A2.SS1.7.p3.10.m10.6.7.8.3.1.cmml">;</mo><mi id="A2.SS1.7.p3.10.m10.6.6" xref="A2.SS1.7.p3.10.m10.6.6.cmml">τ</mi><mo id="A2.SS1.7.p3.10.m10.6.7.8.3.2.3" stretchy="false" xref="A2.SS1.7.p3.10.m10.6.7.8.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.10.m10.6b"><apply id="A2.SS1.7.p3.10.m10.6.7.cmml" xref="A2.SS1.7.p3.10.m10.6.7"><and id="A2.SS1.7.p3.10.m10.6.7a.cmml" xref="A2.SS1.7.p3.10.m10.6.7"></and><apply id="A2.SS1.7.p3.10.m10.6.7b.cmml" xref="A2.SS1.7.p3.10.m10.6.7"><eq id="A2.SS1.7.p3.10.m10.6.7.3.cmml" xref="A2.SS1.7.p3.10.m10.6.7.3"></eq><apply id="A2.SS1.7.p3.10.m10.6.7.2.cmml" xref="A2.SS1.7.p3.10.m10.6.7.2"><times id="A2.SS1.7.p3.10.m10.6.7.2.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.2.1"></times><ci id="A2.SS1.7.p3.10.m10.6.7.2.2.cmml" xref="A2.SS1.7.p3.10.m10.6.7.2.2">𝑃</ci><list id="A2.SS1.7.p3.10.m10.6.7.2.3.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.2.3.2"><ci id="A2.SS1.7.p3.10.m10.1.1.cmml" xref="A2.SS1.7.p3.10.m10.1.1">𝜏</ci><apply id="A2.SS1.7.p3.10.m10.2.2.cmml" xref="A2.SS1.7.p3.10.m10.2.2"><ci id="A2.SS1.7.p3.10.m10.2.2.1.cmml" xref="A2.SS1.7.p3.10.m10.2.2.1">^</ci><ci id="A2.SS1.7.p3.10.m10.2.2.2.cmml" xref="A2.SS1.7.p3.10.m10.2.2.2">𝜏</ci></apply></list></apply><apply id="A2.SS1.7.p3.10.m10.6.7.4.cmml" xref="A2.SS1.7.p3.10.m10.6.7.4"><times id="A2.SS1.7.p3.10.m10.6.7.4.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.4.1"></times><ci id="A2.SS1.7.p3.10.m10.6.7.4.2.cmml" xref="A2.SS1.7.p3.10.m10.6.7.4.2">𝐹</ci><ci id="A2.SS1.7.p3.10.m10.3.3.cmml" xref="A2.SS1.7.p3.10.m10.3.3">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.10.m10.6.7c.cmml" xref="A2.SS1.7.p3.10.m10.6.7"><gt id="A2.SS1.7.p3.10.m10.6.7.5.cmml" xref="A2.SS1.7.p3.10.m10.6.7.5"></gt><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.10.m10.6.7.4.cmml" id="A2.SS1.7.p3.10.m10.6.7d.cmml" xref="A2.SS1.7.p3.10.m10.6.7"></share><apply id="A2.SS1.7.p3.10.m10.6.7.6.cmml" xref="A2.SS1.7.p3.10.m10.6.7.6"><times id="A2.SS1.7.p3.10.m10.6.7.6.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.6.1"></times><ci id="A2.SS1.7.p3.10.m10.6.7.6.2.cmml" xref="A2.SS1.7.p3.10.m10.6.7.6.2">𝐺</ci><ci id="A2.SS1.7.p3.10.m10.4.4.cmml" xref="A2.SS1.7.p3.10.m10.4.4">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.10.m10.6.7e.cmml" xref="A2.SS1.7.p3.10.m10.6.7"><eq id="A2.SS1.7.p3.10.m10.6.7.7.cmml" xref="A2.SS1.7.p3.10.m10.6.7.7"></eq><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.10.m10.6.7.6.cmml" id="A2.SS1.7.p3.10.m10.6.7f.cmml" xref="A2.SS1.7.p3.10.m10.6.7"></share><apply id="A2.SS1.7.p3.10.m10.6.7.8.cmml" xref="A2.SS1.7.p3.10.m10.6.7.8"><times id="A2.SS1.7.p3.10.m10.6.7.8.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.8.1"></times><ci id="A2.SS1.7.p3.10.m10.6.7.8.2.cmml" xref="A2.SS1.7.p3.10.m10.6.7.8.2">𝑃</ci><list id="A2.SS1.7.p3.10.m10.6.7.8.3.1.cmml" xref="A2.SS1.7.p3.10.m10.6.7.8.3.2"><ci id="A2.SS1.7.p3.10.m10.5.5.cmml" xref="A2.SS1.7.p3.10.m10.5.5">𝜏</ci><ci id="A2.SS1.7.p3.10.m10.6.6.cmml" xref="A2.SS1.7.p3.10.m10.6.6">𝜏</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.10.m10.6c">P(\tau;\hat{\tau})=F(\tau)>G(\tau)=P(\tau;\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.10.m10.6d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = italic_F ( italic_τ ) > italic_G ( italic_τ ) = italic_P ( italic_τ ; italic_τ )</annotation></semantics></math>, by decreasing monotonicity we have <math alttext="\tau>\hat{\tau}" class="ltx_Math" display="inline" id="A2.SS1.7.p3.11.m11.1"><semantics id="A2.SS1.7.p3.11.m11.1a"><mrow id="A2.SS1.7.p3.11.m11.1.1" xref="A2.SS1.7.p3.11.m11.1.1.cmml"><mi id="A2.SS1.7.p3.11.m11.1.1.2" xref="A2.SS1.7.p3.11.m11.1.1.2.cmml">τ</mi><mo id="A2.SS1.7.p3.11.m11.1.1.1" xref="A2.SS1.7.p3.11.m11.1.1.1.cmml">></mo><mover accent="true" id="A2.SS1.7.p3.11.m11.1.1.3" xref="A2.SS1.7.p3.11.m11.1.1.3.cmml"><mi id="A2.SS1.7.p3.11.m11.1.1.3.2" xref="A2.SS1.7.p3.11.m11.1.1.3.2.cmml">τ</mi><mo id="A2.SS1.7.p3.11.m11.1.1.3.1" xref="A2.SS1.7.p3.11.m11.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.11.m11.1b"><apply id="A2.SS1.7.p3.11.m11.1.1.cmml" xref="A2.SS1.7.p3.11.m11.1.1"><gt id="A2.SS1.7.p3.11.m11.1.1.1.cmml" xref="A2.SS1.7.p3.11.m11.1.1.1"></gt><ci id="A2.SS1.7.p3.11.m11.1.1.2.cmml" xref="A2.SS1.7.p3.11.m11.1.1.2">𝜏</ci><apply id="A2.SS1.7.p3.11.m11.1.1.3.cmml" xref="A2.SS1.7.p3.11.m11.1.1.3"><ci id="A2.SS1.7.p3.11.m11.1.1.3.1.cmml" xref="A2.SS1.7.p3.11.m11.1.1.3.1">^</ci><ci id="A2.SS1.7.p3.11.m11.1.1.3.2.cmml" xref="A2.SS1.7.p3.11.m11.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.11.m11.1c">\tau>\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.11.m11.1d">italic_τ > over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. Thus <math alttext="\dot{\tau}=\hat{\tau}-\tau<0" class="ltx_Math" display="inline" id="A2.SS1.7.p3.12.m12.1"><semantics id="A2.SS1.7.p3.12.m12.1a"><mrow id="A2.SS1.7.p3.12.m12.1.1" xref="A2.SS1.7.p3.12.m12.1.1.cmml"><mover accent="true" id="A2.SS1.7.p3.12.m12.1.1.2" xref="A2.SS1.7.p3.12.m12.1.1.2.cmml"><mi id="A2.SS1.7.p3.12.m12.1.1.2.2" xref="A2.SS1.7.p3.12.m12.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.12.m12.1.1.2.1" xref="A2.SS1.7.p3.12.m12.1.1.2.1.cmml">˙</mo></mover><mo id="A2.SS1.7.p3.12.m12.1.1.3" xref="A2.SS1.7.p3.12.m12.1.1.3.cmml">=</mo><mrow id="A2.SS1.7.p3.12.m12.1.1.4" xref="A2.SS1.7.p3.12.m12.1.1.4.cmml"><mover accent="true" id="A2.SS1.7.p3.12.m12.1.1.4.2" xref="A2.SS1.7.p3.12.m12.1.1.4.2.cmml"><mi id="A2.SS1.7.p3.12.m12.1.1.4.2.2" xref="A2.SS1.7.p3.12.m12.1.1.4.2.2.cmml">τ</mi><mo id="A2.SS1.7.p3.12.m12.1.1.4.2.1" xref="A2.SS1.7.p3.12.m12.1.1.4.2.1.cmml">^</mo></mover><mo id="A2.SS1.7.p3.12.m12.1.1.4.1" xref="A2.SS1.7.p3.12.m12.1.1.4.1.cmml">−</mo><mi id="A2.SS1.7.p3.12.m12.1.1.4.3" xref="A2.SS1.7.p3.12.m12.1.1.4.3.cmml">τ</mi></mrow><mo id="A2.SS1.7.p3.12.m12.1.1.5" xref="A2.SS1.7.p3.12.m12.1.1.5.cmml"><</mo><mn id="A2.SS1.7.p3.12.m12.1.1.6" xref="A2.SS1.7.p3.12.m12.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.7.p3.12.m12.1b"><apply id="A2.SS1.7.p3.12.m12.1.1.cmml" xref="A2.SS1.7.p3.12.m12.1.1"><and id="A2.SS1.7.p3.12.m12.1.1a.cmml" xref="A2.SS1.7.p3.12.m12.1.1"></and><apply id="A2.SS1.7.p3.12.m12.1.1b.cmml" xref="A2.SS1.7.p3.12.m12.1.1"><eq id="A2.SS1.7.p3.12.m12.1.1.3.cmml" xref="A2.SS1.7.p3.12.m12.1.1.3"></eq><apply id="A2.SS1.7.p3.12.m12.1.1.2.cmml" xref="A2.SS1.7.p3.12.m12.1.1.2"><ci id="A2.SS1.7.p3.12.m12.1.1.2.1.cmml" xref="A2.SS1.7.p3.12.m12.1.1.2.1">˙</ci><ci id="A2.SS1.7.p3.12.m12.1.1.2.2.cmml" xref="A2.SS1.7.p3.12.m12.1.1.2.2">𝜏</ci></apply><apply id="A2.SS1.7.p3.12.m12.1.1.4.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4"><minus id="A2.SS1.7.p3.12.m12.1.1.4.1.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4.1"></minus><apply id="A2.SS1.7.p3.12.m12.1.1.4.2.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4.2"><ci id="A2.SS1.7.p3.12.m12.1.1.4.2.1.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4.2.1">^</ci><ci id="A2.SS1.7.p3.12.m12.1.1.4.2.2.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4.2.2">𝜏</ci></apply><ci id="A2.SS1.7.p3.12.m12.1.1.4.3.cmml" xref="A2.SS1.7.p3.12.m12.1.1.4.3">𝜏</ci></apply></apply><apply id="A2.SS1.7.p3.12.m12.1.1c.cmml" xref="A2.SS1.7.p3.12.m12.1.1"><lt id="A2.SS1.7.p3.12.m12.1.1.5.cmml" xref="A2.SS1.7.p3.12.m12.1.1.5"></lt><share href="https://arxiv.org/html/2503.16280v1#A2.SS1.7.p3.12.m12.1.1.4.cmml" id="A2.SS1.7.p3.12.m12.1.1d.cmml" xref="A2.SS1.7.p3.12.m12.1.1"></share><cn id="A2.SS1.7.p3.12.m12.1.1.6.cmml" type="integer" xref="A2.SS1.7.p3.12.m12.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.7.p3.12.m12.1c">\dot{\tau}=\hat{\tau}-\tau<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.7.p3.12.m12.1d">over˙ start_ARG italic_τ end_ARG = over^ start_ARG italic_τ end_ARG - italic_τ < 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.SS1.8.p4"> <p class="ltx_p" id="A2.SS1.8.p4.10">If <math alttext="G(\tau)-F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.8.p4.1.m1.2"><semantics id="A2.SS1.8.p4.1.m1.2a"><mrow id="A2.SS1.8.p4.1.m1.2.3" xref="A2.SS1.8.p4.1.m1.2.3.cmml"><mrow id="A2.SS1.8.p4.1.m1.2.3.2" xref="A2.SS1.8.p4.1.m1.2.3.2.cmml"><mi id="A2.SS1.8.p4.1.m1.2.3.2.2" xref="A2.SS1.8.p4.1.m1.2.3.2.2.cmml">G</mi><mo id="A2.SS1.8.p4.1.m1.2.3.2.1" xref="A2.SS1.8.p4.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.1.m1.2.3.2.3.2" xref="A2.SS1.8.p4.1.m1.2.3.2.cmml"><mo id="A2.SS1.8.p4.1.m1.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.8.p4.1.m1.2.3.2.cmml">(</mo><mi id="A2.SS1.8.p4.1.m1.1.1" xref="A2.SS1.8.p4.1.m1.1.1.cmml">τ</mi><mo id="A2.SS1.8.p4.1.m1.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.8.p4.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.8.p4.1.m1.2.3.1" xref="A2.SS1.8.p4.1.m1.2.3.1.cmml">−</mo><mrow id="A2.SS1.8.p4.1.m1.2.3.3" xref="A2.SS1.8.p4.1.m1.2.3.3.cmml"><mi id="A2.SS1.8.p4.1.m1.2.3.3.2" xref="A2.SS1.8.p4.1.m1.2.3.3.2.cmml">F</mi><mo id="A2.SS1.8.p4.1.m1.2.3.3.1" xref="A2.SS1.8.p4.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.1.m1.2.3.3.3.2" xref="A2.SS1.8.p4.1.m1.2.3.3.cmml"><mo id="A2.SS1.8.p4.1.m1.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.8.p4.1.m1.2.3.3.cmml">(</mo><mi id="A2.SS1.8.p4.1.m1.2.2" xref="A2.SS1.8.p4.1.m1.2.2.cmml">τ</mi><mo id="A2.SS1.8.p4.1.m1.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.8.p4.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.1.m1.2b"><apply id="A2.SS1.8.p4.1.m1.2.3.cmml" xref="A2.SS1.8.p4.1.m1.2.3"><minus id="A2.SS1.8.p4.1.m1.2.3.1.cmml" xref="A2.SS1.8.p4.1.m1.2.3.1"></minus><apply id="A2.SS1.8.p4.1.m1.2.3.2.cmml" xref="A2.SS1.8.p4.1.m1.2.3.2"><times id="A2.SS1.8.p4.1.m1.2.3.2.1.cmml" xref="A2.SS1.8.p4.1.m1.2.3.2.1"></times><ci id="A2.SS1.8.p4.1.m1.2.3.2.2.cmml" xref="A2.SS1.8.p4.1.m1.2.3.2.2">𝐺</ci><ci id="A2.SS1.8.p4.1.m1.1.1.cmml" xref="A2.SS1.8.p4.1.m1.1.1">𝜏</ci></apply><apply id="A2.SS1.8.p4.1.m1.2.3.3.cmml" xref="A2.SS1.8.p4.1.m1.2.3.3"><times id="A2.SS1.8.p4.1.m1.2.3.3.1.cmml" xref="A2.SS1.8.p4.1.m1.2.3.3.1"></times><ci id="A2.SS1.8.p4.1.m1.2.3.3.2.cmml" xref="A2.SS1.8.p4.1.m1.2.3.3.2">𝐹</ci><ci id="A2.SS1.8.p4.1.m1.2.2.cmml" xref="A2.SS1.8.p4.1.m1.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.1.m1.2c">G(\tau)-F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.1.m1.2d">italic_G ( italic_τ ) - italic_F ( italic_τ )</annotation></semantics></math> is strictly decreasing at <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.8.p4.2.m2.1"><semantics id="A2.SS1.8.p4.2.m2.1a"><msup id="A2.SS1.8.p4.2.m2.1.1" xref="A2.SS1.8.p4.2.m2.1.1.cmml"><mi id="A2.SS1.8.p4.2.m2.1.1.2" xref="A2.SS1.8.p4.2.m2.1.1.2.cmml">τ</mi><mo id="A2.SS1.8.p4.2.m2.1.1.3" xref="A2.SS1.8.p4.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.2.m2.1b"><apply id="A2.SS1.8.p4.2.m2.1.1.cmml" xref="A2.SS1.8.p4.2.m2.1.1"><csymbol cd="ambiguous" id="A2.SS1.8.p4.2.m2.1.1.1.cmml" xref="A2.SS1.8.p4.2.m2.1.1">superscript</csymbol><ci id="A2.SS1.8.p4.2.m2.1.1.2.cmml" xref="A2.SS1.8.p4.2.m2.1.1.2">𝜏</ci><times id="A2.SS1.8.p4.2.m2.1.1.3.cmml" xref="A2.SS1.8.p4.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.2.m2.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.2.m2.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, then at a sufficiently close perturbed point <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.8.p4.3.m3.1"><semantics id="A2.SS1.8.p4.3.m3.1a"><mi id="A2.SS1.8.p4.3.m3.1.1" xref="A2.SS1.8.p4.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.3.m3.1b"><ci id="A2.SS1.8.p4.3.m3.1.1.cmml" xref="A2.SS1.8.p4.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.3.m3.1d">italic_τ</annotation></semantics></math> to the left of <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.8.p4.4.m4.1"><semantics id="A2.SS1.8.p4.4.m4.1a"><msup id="A2.SS1.8.p4.4.m4.1.1" xref="A2.SS1.8.p4.4.m4.1.1.cmml"><mi id="A2.SS1.8.p4.4.m4.1.1.2" xref="A2.SS1.8.p4.4.m4.1.1.2.cmml">τ</mi><mo id="A2.SS1.8.p4.4.m4.1.1.3" xref="A2.SS1.8.p4.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.4.m4.1b"><apply id="A2.SS1.8.p4.4.m4.1.1.cmml" xref="A2.SS1.8.p4.4.m4.1.1"><csymbol cd="ambiguous" id="A2.SS1.8.p4.4.m4.1.1.1.cmml" xref="A2.SS1.8.p4.4.m4.1.1">superscript</csymbol><ci id="A2.SS1.8.p4.4.m4.1.1.2.cmml" xref="A2.SS1.8.p4.4.m4.1.1.2">𝜏</ci><times id="A2.SS1.8.p4.4.m4.1.1.3.cmml" xref="A2.SS1.8.p4.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.4.m4.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.4.m4.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we have <math alttext="G(\tau)>F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.8.p4.5.m5.2"><semantics id="A2.SS1.8.p4.5.m5.2a"><mrow id="A2.SS1.8.p4.5.m5.2.3" xref="A2.SS1.8.p4.5.m5.2.3.cmml"><mrow id="A2.SS1.8.p4.5.m5.2.3.2" xref="A2.SS1.8.p4.5.m5.2.3.2.cmml"><mi id="A2.SS1.8.p4.5.m5.2.3.2.2" xref="A2.SS1.8.p4.5.m5.2.3.2.2.cmml">G</mi><mo id="A2.SS1.8.p4.5.m5.2.3.2.1" xref="A2.SS1.8.p4.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.5.m5.2.3.2.3.2" xref="A2.SS1.8.p4.5.m5.2.3.2.cmml"><mo id="A2.SS1.8.p4.5.m5.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.8.p4.5.m5.2.3.2.cmml">(</mo><mi id="A2.SS1.8.p4.5.m5.1.1" xref="A2.SS1.8.p4.5.m5.1.1.cmml">τ</mi><mo id="A2.SS1.8.p4.5.m5.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.8.p4.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.8.p4.5.m5.2.3.1" xref="A2.SS1.8.p4.5.m5.2.3.1.cmml">></mo><mrow id="A2.SS1.8.p4.5.m5.2.3.3" xref="A2.SS1.8.p4.5.m5.2.3.3.cmml"><mi id="A2.SS1.8.p4.5.m5.2.3.3.2" xref="A2.SS1.8.p4.5.m5.2.3.3.2.cmml">F</mi><mo id="A2.SS1.8.p4.5.m5.2.3.3.1" xref="A2.SS1.8.p4.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.5.m5.2.3.3.3.2" xref="A2.SS1.8.p4.5.m5.2.3.3.cmml"><mo id="A2.SS1.8.p4.5.m5.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.8.p4.5.m5.2.3.3.cmml">(</mo><mi id="A2.SS1.8.p4.5.m5.2.2" xref="A2.SS1.8.p4.5.m5.2.2.cmml">τ</mi><mo id="A2.SS1.8.p4.5.m5.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.8.p4.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.5.m5.2b"><apply id="A2.SS1.8.p4.5.m5.2.3.cmml" xref="A2.SS1.8.p4.5.m5.2.3"><gt id="A2.SS1.8.p4.5.m5.2.3.1.cmml" xref="A2.SS1.8.p4.5.m5.2.3.1"></gt><apply id="A2.SS1.8.p4.5.m5.2.3.2.cmml" xref="A2.SS1.8.p4.5.m5.2.3.2"><times id="A2.SS1.8.p4.5.m5.2.3.2.1.cmml" xref="A2.SS1.8.p4.5.m5.2.3.2.1"></times><ci id="A2.SS1.8.p4.5.m5.2.3.2.2.cmml" xref="A2.SS1.8.p4.5.m5.2.3.2.2">𝐺</ci><ci id="A2.SS1.8.p4.5.m5.1.1.cmml" xref="A2.SS1.8.p4.5.m5.1.1">𝜏</ci></apply><apply id="A2.SS1.8.p4.5.m5.2.3.3.cmml" xref="A2.SS1.8.p4.5.m5.2.3.3"><times id="A2.SS1.8.p4.5.m5.2.3.3.1.cmml" xref="A2.SS1.8.p4.5.m5.2.3.3.1"></times><ci id="A2.SS1.8.p4.5.m5.2.3.3.2.cmml" xref="A2.SS1.8.p4.5.m5.2.3.3.2">𝐹</ci><ci id="A2.SS1.8.p4.5.m5.2.2.cmml" xref="A2.SS1.8.p4.5.m5.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.5.m5.2c">G(\tau)>F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.5.m5.2d">italic_G ( italic_τ ) > italic_F ( italic_τ )</annotation></semantics></math> and <math alttext="\dot{\tau}>0" class="ltx_Math" display="inline" id="A2.SS1.8.p4.6.m6.1"><semantics id="A2.SS1.8.p4.6.m6.1a"><mrow id="A2.SS1.8.p4.6.m6.1.1" xref="A2.SS1.8.p4.6.m6.1.1.cmml"><mover accent="true" id="A2.SS1.8.p4.6.m6.1.1.2" xref="A2.SS1.8.p4.6.m6.1.1.2.cmml"><mi id="A2.SS1.8.p4.6.m6.1.1.2.2" xref="A2.SS1.8.p4.6.m6.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.8.p4.6.m6.1.1.2.1" xref="A2.SS1.8.p4.6.m6.1.1.2.1.cmml">˙</mo></mover><mo id="A2.SS1.8.p4.6.m6.1.1.1" xref="A2.SS1.8.p4.6.m6.1.1.1.cmml">></mo><mn id="A2.SS1.8.p4.6.m6.1.1.3" xref="A2.SS1.8.p4.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.6.m6.1b"><apply id="A2.SS1.8.p4.6.m6.1.1.cmml" xref="A2.SS1.8.p4.6.m6.1.1"><gt id="A2.SS1.8.p4.6.m6.1.1.1.cmml" xref="A2.SS1.8.p4.6.m6.1.1.1"></gt><apply id="A2.SS1.8.p4.6.m6.1.1.2.cmml" xref="A2.SS1.8.p4.6.m6.1.1.2"><ci id="A2.SS1.8.p4.6.m6.1.1.2.1.cmml" xref="A2.SS1.8.p4.6.m6.1.1.2.1">˙</ci><ci id="A2.SS1.8.p4.6.m6.1.1.2.2.cmml" xref="A2.SS1.8.p4.6.m6.1.1.2.2">𝜏</ci></apply><cn id="A2.SS1.8.p4.6.m6.1.1.3.cmml" type="integer" xref="A2.SS1.8.p4.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.6.m6.1c">\dot{\tau}>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.6.m6.1d">over˙ start_ARG italic_τ end_ARG > 0</annotation></semantics></math>; and at a perturbed point <math alttext="\tau" class="ltx_Math" display="inline" id="A2.SS1.8.p4.7.m7.1"><semantics id="A2.SS1.8.p4.7.m7.1a"><mi id="A2.SS1.8.p4.7.m7.1.1" xref="A2.SS1.8.p4.7.m7.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.7.m7.1b"><ci id="A2.SS1.8.p4.7.m7.1.1.cmml" xref="A2.SS1.8.p4.7.m7.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.7.m7.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.7.m7.1d">italic_τ</annotation></semantics></math> to the right of <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS1.8.p4.8.m8.1"><semantics id="A2.SS1.8.p4.8.m8.1a"><msup id="A2.SS1.8.p4.8.m8.1.1" xref="A2.SS1.8.p4.8.m8.1.1.cmml"><mi id="A2.SS1.8.p4.8.m8.1.1.2" xref="A2.SS1.8.p4.8.m8.1.1.2.cmml">τ</mi><mo id="A2.SS1.8.p4.8.m8.1.1.3" xref="A2.SS1.8.p4.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.8.m8.1b"><apply id="A2.SS1.8.p4.8.m8.1.1.cmml" xref="A2.SS1.8.p4.8.m8.1.1"><csymbol cd="ambiguous" id="A2.SS1.8.p4.8.m8.1.1.1.cmml" xref="A2.SS1.8.p4.8.m8.1.1">superscript</csymbol><ci id="A2.SS1.8.p4.8.m8.1.1.2.cmml" xref="A2.SS1.8.p4.8.m8.1.1.2">𝜏</ci><times id="A2.SS1.8.p4.8.m8.1.1.3.cmml" xref="A2.SS1.8.p4.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.8.m8.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.8.m8.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we have <math alttext="G(\tau)<F(\tau)" class="ltx_Math" display="inline" id="A2.SS1.8.p4.9.m9.2"><semantics id="A2.SS1.8.p4.9.m9.2a"><mrow id="A2.SS1.8.p4.9.m9.2.3" xref="A2.SS1.8.p4.9.m9.2.3.cmml"><mrow id="A2.SS1.8.p4.9.m9.2.3.2" xref="A2.SS1.8.p4.9.m9.2.3.2.cmml"><mi id="A2.SS1.8.p4.9.m9.2.3.2.2" xref="A2.SS1.8.p4.9.m9.2.3.2.2.cmml">G</mi><mo id="A2.SS1.8.p4.9.m9.2.3.2.1" xref="A2.SS1.8.p4.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.9.m9.2.3.2.3.2" xref="A2.SS1.8.p4.9.m9.2.3.2.cmml"><mo id="A2.SS1.8.p4.9.m9.2.3.2.3.2.1" stretchy="false" xref="A2.SS1.8.p4.9.m9.2.3.2.cmml">(</mo><mi id="A2.SS1.8.p4.9.m9.1.1" xref="A2.SS1.8.p4.9.m9.1.1.cmml">τ</mi><mo id="A2.SS1.8.p4.9.m9.2.3.2.3.2.2" stretchy="false" xref="A2.SS1.8.p4.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS1.8.p4.9.m9.2.3.1" xref="A2.SS1.8.p4.9.m9.2.3.1.cmml"><</mo><mrow id="A2.SS1.8.p4.9.m9.2.3.3" xref="A2.SS1.8.p4.9.m9.2.3.3.cmml"><mi id="A2.SS1.8.p4.9.m9.2.3.3.2" xref="A2.SS1.8.p4.9.m9.2.3.3.2.cmml">F</mi><mo id="A2.SS1.8.p4.9.m9.2.3.3.1" xref="A2.SS1.8.p4.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS1.8.p4.9.m9.2.3.3.3.2" xref="A2.SS1.8.p4.9.m9.2.3.3.cmml"><mo id="A2.SS1.8.p4.9.m9.2.3.3.3.2.1" stretchy="false" xref="A2.SS1.8.p4.9.m9.2.3.3.cmml">(</mo><mi id="A2.SS1.8.p4.9.m9.2.2" xref="A2.SS1.8.p4.9.m9.2.2.cmml">τ</mi><mo id="A2.SS1.8.p4.9.m9.2.3.3.3.2.2" stretchy="false" xref="A2.SS1.8.p4.9.m9.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.9.m9.2b"><apply id="A2.SS1.8.p4.9.m9.2.3.cmml" xref="A2.SS1.8.p4.9.m9.2.3"><lt id="A2.SS1.8.p4.9.m9.2.3.1.cmml" xref="A2.SS1.8.p4.9.m9.2.3.1"></lt><apply id="A2.SS1.8.p4.9.m9.2.3.2.cmml" xref="A2.SS1.8.p4.9.m9.2.3.2"><times id="A2.SS1.8.p4.9.m9.2.3.2.1.cmml" xref="A2.SS1.8.p4.9.m9.2.3.2.1"></times><ci id="A2.SS1.8.p4.9.m9.2.3.2.2.cmml" xref="A2.SS1.8.p4.9.m9.2.3.2.2">𝐺</ci><ci id="A2.SS1.8.p4.9.m9.1.1.cmml" xref="A2.SS1.8.p4.9.m9.1.1">𝜏</ci></apply><apply id="A2.SS1.8.p4.9.m9.2.3.3.cmml" xref="A2.SS1.8.p4.9.m9.2.3.3"><times id="A2.SS1.8.p4.9.m9.2.3.3.1.cmml" xref="A2.SS1.8.p4.9.m9.2.3.3.1"></times><ci id="A2.SS1.8.p4.9.m9.2.3.3.2.cmml" xref="A2.SS1.8.p4.9.m9.2.3.3.2">𝐹</ci><ci id="A2.SS1.8.p4.9.m9.2.2.cmml" xref="A2.SS1.8.p4.9.m9.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.9.m9.2c">G(\tau)<F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.9.m9.2d">italic_G ( italic_τ ) < italic_F ( italic_τ )</annotation></semantics></math> and <math alttext="\dot{\tau}<0" class="ltx_Math" display="inline" id="A2.SS1.8.p4.10.m10.1"><semantics id="A2.SS1.8.p4.10.m10.1a"><mrow id="A2.SS1.8.p4.10.m10.1.1" xref="A2.SS1.8.p4.10.m10.1.1.cmml"><mover accent="true" id="A2.SS1.8.p4.10.m10.1.1.2" xref="A2.SS1.8.p4.10.m10.1.1.2.cmml"><mi id="A2.SS1.8.p4.10.m10.1.1.2.2" xref="A2.SS1.8.p4.10.m10.1.1.2.2.cmml">τ</mi><mo id="A2.SS1.8.p4.10.m10.1.1.2.1" xref="A2.SS1.8.p4.10.m10.1.1.2.1.cmml">˙</mo></mover><mo id="A2.SS1.8.p4.10.m10.1.1.1" xref="A2.SS1.8.p4.10.m10.1.1.1.cmml"><</mo><mn id="A2.SS1.8.p4.10.m10.1.1.3" xref="A2.SS1.8.p4.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.8.p4.10.m10.1b"><apply id="A2.SS1.8.p4.10.m10.1.1.cmml" xref="A2.SS1.8.p4.10.m10.1.1"><lt id="A2.SS1.8.p4.10.m10.1.1.1.cmml" xref="A2.SS1.8.p4.10.m10.1.1.1"></lt><apply id="A2.SS1.8.p4.10.m10.1.1.2.cmml" xref="A2.SS1.8.p4.10.m10.1.1.2"><ci id="A2.SS1.8.p4.10.m10.1.1.2.1.cmml" xref="A2.SS1.8.p4.10.m10.1.1.2.1">˙</ci><ci id="A2.SS1.8.p4.10.m10.1.1.2.2.cmml" xref="A2.SS1.8.p4.10.m10.1.1.2.2">𝜏</ci></apply><cn id="A2.SS1.8.p4.10.m10.1.1.3.cmml" type="integer" xref="A2.SS1.8.p4.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.8.p4.10.m10.1c">\dot{\tau}<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS1.8.p4.10.m10.1d">over˙ start_ARG italic_τ end_ARG < 0</annotation></semantics></math>. Stability follows. The same logic implies instability in the strictly increasing case. ∎</p> </div> </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>Equilibrium Characterization Generalization</h3> <div class="ltx_para" id="A2.SS2.p1"> <p class="ltx_p" id="A2.SS2.p1.2">As in OA, while we do not expect all settings to behave as symmetrically as the Gaussian model in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a>, we can still characterize existence and location of finite threshold equilibria based on the shapes of the functions <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.p1.1.m1.1"><semantics id="A2.SS2.p1.1.m1.1a"><mi id="A2.SS2.p1.1.m1.1.1" xref="A2.SS2.p1.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.1.m1.1b"><ci id="A2.SS2.p1.1.m1.1.1.cmml" xref="A2.SS2.p1.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.1.m1.1d">italic_F</annotation></semantics></math> and <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p1.2.m2.1d">italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_condition" id="Thmcondition2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcondition2.1.1.1">Condition 2</span></span><span class="ltx_text ltx_font_bold" id="Thmcondition2.2.2">.</span> </h6> <div class="ltx_para" id="Thmcondition2.p1"> <p class="ltx_p" id="Thmcondition2.p1.7"><span class="ltx_text ltx_font_italic" id="Thmcondition2.p1.7.7">Let <math alttext="I=[a,b]\subset\mathbb{R}" class="ltx_Math" display="inline" id="Thmcondition2.p1.1.1.m1.2"><semantics id="Thmcondition2.p1.1.1.m1.2a"><mrow id="Thmcondition2.p1.1.1.m1.2.3" xref="Thmcondition2.p1.1.1.m1.2.3.cmml"><mi id="Thmcondition2.p1.1.1.m1.2.3.2" xref="Thmcondition2.p1.1.1.m1.2.3.2.cmml">I</mi><mo id="Thmcondition2.p1.1.1.m1.2.3.3" xref="Thmcondition2.p1.1.1.m1.2.3.3.cmml">=</mo><mrow id="Thmcondition2.p1.1.1.m1.2.3.4.2" xref="Thmcondition2.p1.1.1.m1.2.3.4.1.cmml"><mo id="Thmcondition2.p1.1.1.m1.2.3.4.2.1" stretchy="false" xref="Thmcondition2.p1.1.1.m1.2.3.4.1.cmml">[</mo><mi id="Thmcondition2.p1.1.1.m1.1.1" xref="Thmcondition2.p1.1.1.m1.1.1.cmml">a</mi><mo id="Thmcondition2.p1.1.1.m1.2.3.4.2.2" xref="Thmcondition2.p1.1.1.m1.2.3.4.1.cmml">,</mo><mi id="Thmcondition2.p1.1.1.m1.2.2" xref="Thmcondition2.p1.1.1.m1.2.2.cmml">b</mi><mo id="Thmcondition2.p1.1.1.m1.2.3.4.2.3" stretchy="false" xref="Thmcondition2.p1.1.1.m1.2.3.4.1.cmml">]</mo></mrow><mo id="Thmcondition2.p1.1.1.m1.2.3.5" xref="Thmcondition2.p1.1.1.m1.2.3.5.cmml">⊂</mo><mi id="Thmcondition2.p1.1.1.m1.2.3.6" xref="Thmcondition2.p1.1.1.m1.2.3.6.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.1.1.m1.2b"><apply id="Thmcondition2.p1.1.1.m1.2.3.cmml" xref="Thmcondition2.p1.1.1.m1.2.3"><and id="Thmcondition2.p1.1.1.m1.2.3a.cmml" xref="Thmcondition2.p1.1.1.m1.2.3"></and><apply id="Thmcondition2.p1.1.1.m1.2.3b.cmml" xref="Thmcondition2.p1.1.1.m1.2.3"><eq id="Thmcondition2.p1.1.1.m1.2.3.3.cmml" xref="Thmcondition2.p1.1.1.m1.2.3.3"></eq><ci id="Thmcondition2.p1.1.1.m1.2.3.2.cmml" xref="Thmcondition2.p1.1.1.m1.2.3.2">𝐼</ci><interval closure="closed" id="Thmcondition2.p1.1.1.m1.2.3.4.1.cmml" xref="Thmcondition2.p1.1.1.m1.2.3.4.2"><ci id="Thmcondition2.p1.1.1.m1.1.1.cmml" xref="Thmcondition2.p1.1.1.m1.1.1">𝑎</ci><ci id="Thmcondition2.p1.1.1.m1.2.2.cmml" xref="Thmcondition2.p1.1.1.m1.2.2">𝑏</ci></interval></apply><apply id="Thmcondition2.p1.1.1.m1.2.3c.cmml" xref="Thmcondition2.p1.1.1.m1.2.3"><subset id="Thmcondition2.p1.1.1.m1.2.3.5.cmml" xref="Thmcondition2.p1.1.1.m1.2.3.5"></subset><share href="https://arxiv.org/html/2503.16280v1#Thmcondition2.p1.1.1.m1.2.3.4.cmml" id="Thmcondition2.p1.1.1.m1.2.3d.cmml" xref="Thmcondition2.p1.1.1.m1.2.3"></share><ci id="Thmcondition2.p1.1.1.m1.2.3.6.cmml" xref="Thmcondition2.p1.1.1.m1.2.3.6">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.1.1.m1.2c">I=[a,b]\subset\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.1.1.m1.2d">italic_I = [ italic_a , italic_b ] ⊂ blackboard_R</annotation></semantics></math> be an interval. Upon seeing signal <math alttext="x\leq a" class="ltx_Math" display="inline" id="Thmcondition2.p1.2.2.m2.1"><semantics id="Thmcondition2.p1.2.2.m2.1a"><mrow id="Thmcondition2.p1.2.2.m2.1.1" xref="Thmcondition2.p1.2.2.m2.1.1.cmml"><mi id="Thmcondition2.p1.2.2.m2.1.1.2" xref="Thmcondition2.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmcondition2.p1.2.2.m2.1.1.1" xref="Thmcondition2.p1.2.2.m2.1.1.1.cmml">≤</mo><mi id="Thmcondition2.p1.2.2.m2.1.1.3" xref="Thmcondition2.p1.2.2.m2.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.2.2.m2.1b"><apply id="Thmcondition2.p1.2.2.m2.1.1.cmml" xref="Thmcondition2.p1.2.2.m2.1.1"><leq id="Thmcondition2.p1.2.2.m2.1.1.1.cmml" xref="Thmcondition2.p1.2.2.m2.1.1.1"></leq><ci id="Thmcondition2.p1.2.2.m2.1.1.2.cmml" xref="Thmcondition2.p1.2.2.m2.1.1.2">𝑥</ci><ci id="Thmcondition2.p1.2.2.m2.1.1.3.cmml" xref="Thmcondition2.p1.2.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.2.2.m2.1c">x\leq a</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.2.2.m2.1d">italic_x ≤ italic_a</annotation></semantics></math>, then relative to the prior, an agent believes it more likely that another signal will be <em class="ltx_emph ltx_font_upright" id="Thmcondition2.p1.7.7.1">less</em> than <math alttext="x" class="ltx_Math" display="inline" id="Thmcondition2.p1.3.3.m3.1"><semantics id="Thmcondition2.p1.3.3.m3.1a"><mi id="Thmcondition2.p1.3.3.m3.1.1" xref="Thmcondition2.p1.3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.3.3.m3.1b"><ci id="Thmcondition2.p1.3.3.m3.1.1.cmml" xref="Thmcondition2.p1.3.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.3.3.m3.1d">italic_x</annotation></semantics></math>. Meanwhile, upon seeing signal <math alttext="x\geq b" class="ltx_Math" display="inline" id="Thmcondition2.p1.4.4.m4.1"><semantics id="Thmcondition2.p1.4.4.m4.1a"><mrow id="Thmcondition2.p1.4.4.m4.1.1" xref="Thmcondition2.p1.4.4.m4.1.1.cmml"><mi id="Thmcondition2.p1.4.4.m4.1.1.2" xref="Thmcondition2.p1.4.4.m4.1.1.2.cmml">x</mi><mo id="Thmcondition2.p1.4.4.m4.1.1.1" xref="Thmcondition2.p1.4.4.m4.1.1.1.cmml">≥</mo><mi id="Thmcondition2.p1.4.4.m4.1.1.3" xref="Thmcondition2.p1.4.4.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.4.4.m4.1b"><apply id="Thmcondition2.p1.4.4.m4.1.1.cmml" xref="Thmcondition2.p1.4.4.m4.1.1"><geq id="Thmcondition2.p1.4.4.m4.1.1.1.cmml" xref="Thmcondition2.p1.4.4.m4.1.1.1"></geq><ci id="Thmcondition2.p1.4.4.m4.1.1.2.cmml" xref="Thmcondition2.p1.4.4.m4.1.1.2">𝑥</ci><ci id="Thmcondition2.p1.4.4.m4.1.1.3.cmml" xref="Thmcondition2.p1.4.4.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.4.4.m4.1c">x\geq b</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.4.4.m4.1d">italic_x ≥ italic_b</annotation></semantics></math>, then relative to the prior, an agent believes it more likely that another signal will be <em class="ltx_emph ltx_font_upright" id="Thmcondition2.p1.7.7.2">greater</em> than <math alttext="x" class="ltx_Math" display="inline" id="Thmcondition2.p1.5.5.m5.1"><semantics id="Thmcondition2.p1.5.5.m5.1a"><mi id="Thmcondition2.p1.5.5.m5.1.1" xref="Thmcondition2.p1.5.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.5.5.m5.1b"><ci id="Thmcondition2.p1.5.5.m5.1.1.cmml" xref="Thmcondition2.p1.5.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.5.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.5.5.m5.1d">italic_x</annotation></semantics></math>. Formally, <math alttext="x\leq a\implies P(x^{\prime}\leq x\mid x)>P(x^{\prime}\leq x)" class="ltx_Math" display="inline" id="Thmcondition2.p1.6.6.m6.2"><semantics id="Thmcondition2.p1.6.6.m6.2a"><mrow id="Thmcondition2.p1.6.6.m6.2.2" xref="Thmcondition2.p1.6.6.m6.2.2.cmml"><mi id="Thmcondition2.p1.6.6.m6.2.2.4" xref="Thmcondition2.p1.6.6.m6.2.2.4.cmml">x</mi><mo id="Thmcondition2.p1.6.6.m6.2.2.5" xref="Thmcondition2.p1.6.6.m6.2.2.5.cmml">≤</mo><mi id="Thmcondition2.p1.6.6.m6.2.2.6" xref="Thmcondition2.p1.6.6.m6.2.2.6.cmml">a</mi><mo id="Thmcondition2.p1.6.6.m6.2.2.7" stretchy="false" xref="Thmcondition2.p1.6.6.m6.2.2.7.cmml">⟹</mo><mrow id="Thmcondition2.p1.6.6.m6.1.1.1" xref="Thmcondition2.p1.6.6.m6.1.1.1.cmml"><mi id="Thmcondition2.p1.6.6.m6.1.1.1.3" xref="Thmcondition2.p1.6.6.m6.1.1.1.3.cmml">P</mi><mo id="Thmcondition2.p1.6.6.m6.1.1.1.2" xref="Thmcondition2.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="Thmcondition2.p1.6.6.m6.1.1.1.1.1" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.cmml"><msup id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2.cmml"><mi id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2.2" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2.2.cmml">x</mi><mo id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2.3" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.1" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.1.cmml">≤</mo><mrow id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.cmml"><mi id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.2" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.2.cmml">x</mi><mo id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.1" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.3" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.3.3.cmml">x</mi></mrow></mrow><mo id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmcondition2.p1.6.6.m6.2.2.8" xref="Thmcondition2.p1.6.6.m6.2.2.8.cmml">></mo><mrow id="Thmcondition2.p1.6.6.m6.2.2.2" xref="Thmcondition2.p1.6.6.m6.2.2.2.cmml"><mi id="Thmcondition2.p1.6.6.m6.2.2.2.3" xref="Thmcondition2.p1.6.6.m6.2.2.2.3.cmml">P</mi><mo id="Thmcondition2.p1.6.6.m6.2.2.2.2" xref="Thmcondition2.p1.6.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="Thmcondition2.p1.6.6.m6.2.2.2.1.1" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.cmml"><mo id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.2" stretchy="false" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.cmml"><msup id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2.cmml"><mi id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2.2" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2.2.cmml">x</mi><mo id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2.3" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.1" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.1.cmml">≤</mo><mi id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.3" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.3.cmml">x</mi></mrow><mo id="Thmcondition2.p1.6.6.m6.2.2.2.1.1.3" stretchy="false" xref="Thmcondition2.p1.6.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmcondition2.p1.6.6.m6.2b"><apply id="Thmcondition2.p1.6.6.m6.2.2.cmml" xref="Thmcondition2.p1.6.6.m6.2.2"><and id="Thmcondition2.p1.6.6.m6.2.2a.cmml" xref="Thmcondition2.p1.6.6.m6.2.2"></and><apply id="Thmcondition2.p1.6.6.m6.2.2b.cmml" xref="Thmcondition2.p1.6.6.m6.2.2"><leq id="Thmcondition2.p1.6.6.m6.2.2.5.cmml" xref="Thmcondition2.p1.6.6.m6.2.2.5"></leq><ci id="Thmcondition2.p1.6.6.m6.2.2.4.cmml" xref="Thmcondition2.p1.6.6.m6.2.2.4">𝑥</ci><ci id="Thmcondition2.p1.6.6.m6.2.2.6.cmml" xref="Thmcondition2.p1.6.6.m6.2.2.6">𝑎</ci></apply><apply id="Thmcondition2.p1.6.6.m6.2.2c.cmml" xref="Thmcondition2.p1.6.6.m6.2.2"><implies id="Thmcondition2.p1.6.6.m6.2.2.7.cmml" xref="Thmcondition2.p1.6.6.m6.2.2.7"></implies><share href="https://arxiv.org/html/2503.16280v1#Thmcondition2.p1.6.6.m6.2.2.6.cmml" id="Thmcondition2.p1.6.6.m6.2.2d.cmml" xref="Thmcondition2.p1.6.6.m6.2.2"></share><apply id="Thmcondition2.p1.6.6.m6.1.1.1.cmml" xref="Thmcondition2.p1.6.6.m6.1.1.1"><times id="Thmcondition2.p1.6.6.m6.1.1.1.2.cmml" xref="Thmcondition2.p1.6.6.m6.1.1.1.2"></times><ci id="Thmcondition2.p1.6.6.m6.1.1.1.3.cmml" xref="Thmcondition2.p1.6.6.m6.1.1.1.3">𝑃</ci><apply id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="Thmcondition2.p1.6.6.m6.1.1.1.1.1"><leq id="Thmcondition2.p1.6.6.m6.1.1.1.1.1.1.1.cmml" 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xref="Thmcondition2.p1.7.7.m7.2.2"><gt id="Thmcondition2.p1.7.7.m7.2.2.8.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.8"></gt><share href="https://arxiv.org/html/2503.16280v1#Thmcondition2.p1.7.7.m7.1.1.1.cmml" id="Thmcondition2.p1.7.7.m7.2.2f.cmml" xref="Thmcondition2.p1.7.7.m7.2.2"></share><apply id="Thmcondition2.p1.7.7.m7.2.2.2.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2"><times id="Thmcondition2.p1.7.7.m7.2.2.2.2.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.2"></times><ci id="Thmcondition2.p1.7.7.m7.2.2.2.3.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.3">𝑃</ci><apply id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1"><geq id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.1.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.1"></geq><apply id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.1.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2">superscript</csymbol><ci id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.2.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.2">𝑥</ci><ci id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.3.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.2.3">′</ci></apply><ci id="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.3.cmml" xref="Thmcondition2.p1.7.7.m7.2.2.2.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcondition2.p1.7.7.m7.2c">x\geq b\implies P(x^{\prime}\geq x\mid x)>P(x^{\prime}\geq x)</annotation><annotation encoding="application/x-llamapun" id="Thmcondition2.p1.7.7.m7.2d">italic_x ≥ italic_b ⟹ italic_P ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_x ∣ italic_x ) > italic_P ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_x )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem8.1.1.1">Theorem 8</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem8.p1"> <p class="ltx_p" id="Thmtheorem8.p1.5">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, with <math alttext="\Pr[X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="Thmtheorem8.p1.1.m1.2"><semantics id="Thmtheorem8.p1.1.m1.2a"><mrow id="Thmtheorem8.p1.1.m1.2.2.1" xref="Thmtheorem8.p1.1.m1.2.2.2.cmml"><mi id="Thmtheorem8.p1.1.m1.1.1" xref="Thmtheorem8.p1.1.m1.1.1.cmml">Pr</mi><mo id="Thmtheorem8.p1.1.m1.2.2.1a" xref="Thmtheorem8.p1.1.m1.2.2.2.cmml">⁡</mo><mrow id="Thmtheorem8.p1.1.m1.2.2.1.1" xref="Thmtheorem8.p1.1.m1.2.2.2.cmml"><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.2" stretchy="false" xref="Thmtheorem8.p1.1.m1.2.2.2.cmml">[</mo><mrow id="Thmtheorem8.p1.1.m1.2.2.1.1.1" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.cmml"><msup id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.cmml"><mi id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.2" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.2.cmml">X</mi><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.3" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.1.3" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.3.cmml">≤</mo><mrow id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.cmml"><mi id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.2" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.2.cmml">τ</mi><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.1" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.1.cmml">∣</mo><mi id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.3" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.3.cmml">X</mi></mrow><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.1.5" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.5.cmml">=</mo><mi id="Thmtheorem8.p1.1.m1.2.2.1.1.1.6" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.6.cmml">x</mi></mrow><mo id="Thmtheorem8.p1.1.m1.2.2.1.1.3" stretchy="false" xref="Thmtheorem8.p1.1.m1.2.2.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.1.m1.2b"><apply id="Thmtheorem8.p1.1.m1.2.2.2.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1"><ci id="Thmtheorem8.p1.1.m1.1.1.cmml" xref="Thmtheorem8.p1.1.m1.1.1">Pr</ci><apply id="Thmtheorem8.p1.1.m1.2.2.1.1.1.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1"><and id="Thmtheorem8.p1.1.m1.2.2.1.1.1a.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1"></and><apply id="Thmtheorem8.p1.1.m1.2.2.1.1.1b.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1"><leq id="Thmtheorem8.p1.1.m1.2.2.1.1.1.3.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.3"></leq><apply id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.1.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.2">𝑋</ci><ci id="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.3.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.2.3">′</ci></apply><apply id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4"><csymbol cd="latexml" id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.1.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.1">conditional</csymbol><ci id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.2.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.2">𝜏</ci><ci id="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.3.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="Thmtheorem8.p1.1.m1.2.2.1.1.1c.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1"><eq id="Thmtheorem8.p1.1.m1.2.2.1.1.1.5.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#Thmtheorem8.p1.1.m1.2.2.1.1.1.4.cmml" id="Thmtheorem8.p1.1.m1.2.2.1.1.1d.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1"></share><ci id="Thmtheorem8.p1.1.m1.2.2.1.1.1.6.cmml" xref="Thmtheorem8.p1.1.m1.2.2.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.1.m1.2c">\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math> monotone decreasing and continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmtheorem8.p1.2.m2.1"><semantics id="Thmtheorem8.p1.2.m2.1a"><mi id="Thmtheorem8.p1.2.m2.1.1" xref="Thmtheorem8.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.2.m2.1b"><ci id="Thmtheorem8.p1.2.m2.1.1.cmml" xref="Thmtheorem8.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.2.m2.1d">italic_x</annotation></semantics></math>, and <math alttext="F" class="ltx_Math" display="inline" id="Thmtheorem8.p1.3.m3.1"><semantics id="Thmtheorem8.p1.3.m3.1a"><mi id="Thmtheorem8.p1.3.m3.1.1" xref="Thmtheorem8.p1.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.3.m3.1b"><ci id="Thmtheorem8.p1.3.m3.1.1.cmml" xref="Thmtheorem8.p1.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.3.m3.1d">italic_F</annotation></semantics></math> and <math alttext="G" class="ltx_Math" display="inline" id="Thmtheorem8.p1.4.m4.1"><semantics id="Thmtheorem8.p1.4.m4.1a"><mi id="Thmtheorem8.p1.4.m4.1.1" xref="Thmtheorem8.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.4.m4.1b"><ci id="Thmtheorem8.p1.4.m4.1.1.cmml" xref="Thmtheorem8.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.4.m4.1d">italic_G</annotation></semantics></math> continuous. Then there exists an equilibrium <math alttext="\tau^{*}\in I" class="ltx_Math" display="inline" id="Thmtheorem8.p1.5.m5.1"><semantics id="Thmtheorem8.p1.5.m5.1a"><mrow id="Thmtheorem8.p1.5.m5.1.1" xref="Thmtheorem8.p1.5.m5.1.1.cmml"><msup id="Thmtheorem8.p1.5.m5.1.1.2" xref="Thmtheorem8.p1.5.m5.1.1.2.cmml"><mi id="Thmtheorem8.p1.5.m5.1.1.2.2" xref="Thmtheorem8.p1.5.m5.1.1.2.2.cmml">τ</mi><mo id="Thmtheorem8.p1.5.m5.1.1.2.3" xref="Thmtheorem8.p1.5.m5.1.1.2.3.cmml">∗</mo></msup><mo id="Thmtheorem8.p1.5.m5.1.1.1" xref="Thmtheorem8.p1.5.m5.1.1.1.cmml">∈</mo><mi id="Thmtheorem8.p1.5.m5.1.1.3" xref="Thmtheorem8.p1.5.m5.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem8.p1.5.m5.1b"><apply id="Thmtheorem8.p1.5.m5.1.1.cmml" xref="Thmtheorem8.p1.5.m5.1.1"><in id="Thmtheorem8.p1.5.m5.1.1.1.cmml" xref="Thmtheorem8.p1.5.m5.1.1.1"></in><apply id="Thmtheorem8.p1.5.m5.1.1.2.cmml" xref="Thmtheorem8.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmtheorem8.p1.5.m5.1.1.2.1.cmml" xref="Thmtheorem8.p1.5.m5.1.1.2">superscript</csymbol><ci id="Thmtheorem8.p1.5.m5.1.1.2.2.cmml" xref="Thmtheorem8.p1.5.m5.1.1.2.2">𝜏</ci><times id="Thmtheorem8.p1.5.m5.1.1.2.3.cmml" xref="Thmtheorem8.p1.5.m5.1.1.2.3"></times></apply><ci id="Thmtheorem8.p1.5.m5.1.1.3.cmml" xref="Thmtheorem8.p1.5.m5.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem8.p1.5.m5.1c">\tau^{*}\in I</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem8.p1.5.m5.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math>.</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.p1.9">Take the function <math alttext="H(x)=G(x)-F(x)" 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" xref="A2.SS2.1.p1.1.m1.3.4.cmml"><mrow id="A2.SS2.1.p1.1.m1.3.4.2" xref="A2.SS2.1.p1.1.m1.3.4.2.cmml"><mi id="A2.SS2.1.p1.1.m1.3.4.2.2" xref="A2.SS2.1.p1.1.m1.3.4.2.2.cmml">H</mi><mo id="A2.SS2.1.p1.1.m1.3.4.2.1" xref="A2.SS2.1.p1.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.1.m1.3.4.2.3.2" xref="A2.SS2.1.p1.1.m1.3.4.2.cmml"><mo id="A2.SS2.1.p1.1.m1.3.4.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.2.cmml">(</mo><mi id="A2.SS2.1.p1.1.m1.1.1" xref="A2.SS2.1.p1.1.m1.1.1.cmml">x</mi><mo id="A2.SS2.1.p1.1.m1.3.4.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.1.m1.3.4.1" xref="A2.SS2.1.p1.1.m1.3.4.1.cmml">=</mo><mrow id="A2.SS2.1.p1.1.m1.3.4.3" xref="A2.SS2.1.p1.1.m1.3.4.3.cmml"><mrow id="A2.SS2.1.p1.1.m1.3.4.3.2" xref="A2.SS2.1.p1.1.m1.3.4.3.2.cmml"><mi id="A2.SS2.1.p1.1.m1.3.4.3.2.2" xref="A2.SS2.1.p1.1.m1.3.4.3.2.2.cmml">G</mi><mo id="A2.SS2.1.p1.1.m1.3.4.3.2.1" xref="A2.SS2.1.p1.1.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.1.m1.3.4.3.2.3.2" xref="A2.SS2.1.p1.1.m1.3.4.3.2.cmml"><mo id="A2.SS2.1.p1.1.m1.3.4.3.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.3.2.cmml">(</mo><mi id="A2.SS2.1.p1.1.m1.2.2" xref="A2.SS2.1.p1.1.m1.2.2.cmml">x</mi><mo id="A2.SS2.1.p1.1.m1.3.4.3.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.1.m1.3.4.3.1" xref="A2.SS2.1.p1.1.m1.3.4.3.1.cmml">−</mo><mrow id="A2.SS2.1.p1.1.m1.3.4.3.3" xref="A2.SS2.1.p1.1.m1.3.4.3.3.cmml"><mi id="A2.SS2.1.p1.1.m1.3.4.3.3.2" xref="A2.SS2.1.p1.1.m1.3.4.3.3.2.cmml">F</mi><mo id="A2.SS2.1.p1.1.m1.3.4.3.3.1" xref="A2.SS2.1.p1.1.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.1.m1.3.4.3.3.3.2" xref="A2.SS2.1.p1.1.m1.3.4.3.3.cmml"><mo id="A2.SS2.1.p1.1.m1.3.4.3.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.3.3.cmml">(</mo><mi id="A2.SS2.1.p1.1.m1.3.3" xref="A2.SS2.1.p1.1.m1.3.3.cmml">x</mi><mo id="A2.SS2.1.p1.1.m1.3.4.3.3.3.2.2" stretchy="false" xref="A2.SS2.1.p1.1.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.1.m1.3b"><apply id="A2.SS2.1.p1.1.m1.3.4.cmml" xref="A2.SS2.1.p1.1.m1.3.4"><eq id="A2.SS2.1.p1.1.m1.3.4.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.1"></eq><apply id="A2.SS2.1.p1.1.m1.3.4.2.cmml" xref="A2.SS2.1.p1.1.m1.3.4.2"><times id="A2.SS2.1.p1.1.m1.3.4.2.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.2.1"></times><ci id="A2.SS2.1.p1.1.m1.3.4.2.2.cmml" xref="A2.SS2.1.p1.1.m1.3.4.2.2">𝐻</ci><ci id="A2.SS2.1.p1.1.m1.1.1.cmml" xref="A2.SS2.1.p1.1.m1.1.1">𝑥</ci></apply><apply id="A2.SS2.1.p1.1.m1.3.4.3.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3"><minus id="A2.SS2.1.p1.1.m1.3.4.3.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.1"></minus><apply id="A2.SS2.1.p1.1.m1.3.4.3.2.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.2"><times id="A2.SS2.1.p1.1.m1.3.4.3.2.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.2.1"></times><ci id="A2.SS2.1.p1.1.m1.3.4.3.2.2.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.2.2">𝐺</ci><ci id="A2.SS2.1.p1.1.m1.2.2.cmml" xref="A2.SS2.1.p1.1.m1.2.2">𝑥</ci></apply><apply id="A2.SS2.1.p1.1.m1.3.4.3.3.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.3"><times id="A2.SS2.1.p1.1.m1.3.4.3.3.1.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.3.1"></times><ci id="A2.SS2.1.p1.1.m1.3.4.3.3.2.cmml" xref="A2.SS2.1.p1.1.m1.3.4.3.3.2">𝐹</ci><ci id="A2.SS2.1.p1.1.m1.3.3.cmml" xref="A2.SS2.1.p1.1.m1.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.1.m1.3c">H(x)=G(x)-F(x)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.1.m1.3d">italic_H ( italic_x ) = italic_G ( italic_x ) - italic_F ( italic_x )</annotation></semantics></math>. Then Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> implies <math alttext="H(a)=G(a)-F(a)>0" class="ltx_Math" display="inline" id="A2.SS2.1.p1.2.m2.3"><semantics id="A2.SS2.1.p1.2.m2.3a"><mrow id="A2.SS2.1.p1.2.m2.3.4" xref="A2.SS2.1.p1.2.m2.3.4.cmml"><mrow id="A2.SS2.1.p1.2.m2.3.4.2" xref="A2.SS2.1.p1.2.m2.3.4.2.cmml"><mi id="A2.SS2.1.p1.2.m2.3.4.2.2" xref="A2.SS2.1.p1.2.m2.3.4.2.2.cmml">H</mi><mo id="A2.SS2.1.p1.2.m2.3.4.2.1" xref="A2.SS2.1.p1.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.2.m2.3.4.2.3.2" xref="A2.SS2.1.p1.2.m2.3.4.2.cmml"><mo id="A2.SS2.1.p1.2.m2.3.4.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.2.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.m2.3.4.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.2.m2.3.4.3" xref="A2.SS2.1.p1.2.m2.3.4.3.cmml">=</mo><mrow id="A2.SS2.1.p1.2.m2.3.4.4" xref="A2.SS2.1.p1.2.m2.3.4.4.cmml"><mrow id="A2.SS2.1.p1.2.m2.3.4.4.2" xref="A2.SS2.1.p1.2.m2.3.4.4.2.cmml"><mi id="A2.SS2.1.p1.2.m2.3.4.4.2.2" xref="A2.SS2.1.p1.2.m2.3.4.4.2.2.cmml">G</mi><mo id="A2.SS2.1.p1.2.m2.3.4.4.2.1" xref="A2.SS2.1.p1.2.m2.3.4.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.2.m2.3.4.4.2.3.2" xref="A2.SS2.1.p1.2.m2.3.4.4.2.cmml"><mo id="A2.SS2.1.p1.2.m2.3.4.4.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.4.2.cmml">(</mo><mi id="A2.SS2.1.p1.2.m2.2.2" xref="A2.SS2.1.p1.2.m2.2.2.cmml">a</mi><mo id="A2.SS2.1.p1.2.m2.3.4.4.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.2.m2.3.4.4.1" xref="A2.SS2.1.p1.2.m2.3.4.4.1.cmml">−</mo><mrow id="A2.SS2.1.p1.2.m2.3.4.4.3" xref="A2.SS2.1.p1.2.m2.3.4.4.3.cmml"><mi id="A2.SS2.1.p1.2.m2.3.4.4.3.2" xref="A2.SS2.1.p1.2.m2.3.4.4.3.2.cmml">F</mi><mo id="A2.SS2.1.p1.2.m2.3.4.4.3.1" xref="A2.SS2.1.p1.2.m2.3.4.4.3.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.2.m2.3.4.4.3.3.2" xref="A2.SS2.1.p1.2.m2.3.4.4.3.cmml"><mo id="A2.SS2.1.p1.2.m2.3.4.4.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.4.3.cmml">(</mo><mi id="A2.SS2.1.p1.2.m2.3.3" xref="A2.SS2.1.p1.2.m2.3.3.cmml">a</mi><mo id="A2.SS2.1.p1.2.m2.3.4.4.3.3.2.2" stretchy="false" xref="A2.SS2.1.p1.2.m2.3.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="A2.SS2.1.p1.2.m2.3.4.5" xref="A2.SS2.1.p1.2.m2.3.4.5.cmml">></mo><mn id="A2.SS2.1.p1.2.m2.3.4.6" xref="A2.SS2.1.p1.2.m2.3.4.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.2.m2.3b"><apply id="A2.SS2.1.p1.2.m2.3.4.cmml" xref="A2.SS2.1.p1.2.m2.3.4"><and id="A2.SS2.1.p1.2.m2.3.4a.cmml" xref="A2.SS2.1.p1.2.m2.3.4"></and><apply id="A2.SS2.1.p1.2.m2.3.4b.cmml" xref="A2.SS2.1.p1.2.m2.3.4"><eq id="A2.SS2.1.p1.2.m2.3.4.3.cmml" xref="A2.SS2.1.p1.2.m2.3.4.3"></eq><apply id="A2.SS2.1.p1.2.m2.3.4.2.cmml" xref="A2.SS2.1.p1.2.m2.3.4.2"><times id="A2.SS2.1.p1.2.m2.3.4.2.1.cmml" xref="A2.SS2.1.p1.2.m2.3.4.2.1"></times><ci id="A2.SS2.1.p1.2.m2.3.4.2.2.cmml" xref="A2.SS2.1.p1.2.m2.3.4.2.2">𝐻</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 id="A2.SS2.1.p1.2.m2.3.4.4.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4"><minus id="A2.SS2.1.p1.2.m2.3.4.4.1.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.1"></minus><apply id="A2.SS2.1.p1.2.m2.3.4.4.2.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.2"><times id="A2.SS2.1.p1.2.m2.3.4.4.2.1.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.2.1"></times><ci id="A2.SS2.1.p1.2.m2.3.4.4.2.2.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.2.2">𝐺</ci><ci id="A2.SS2.1.p1.2.m2.2.2.cmml" xref="A2.SS2.1.p1.2.m2.2.2">𝑎</ci></apply><apply id="A2.SS2.1.p1.2.m2.3.4.4.3.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.3"><times id="A2.SS2.1.p1.2.m2.3.4.4.3.1.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.3.1"></times><ci id="A2.SS2.1.p1.2.m2.3.4.4.3.2.cmml" xref="A2.SS2.1.p1.2.m2.3.4.4.3.2">𝐹</ci><ci id="A2.SS2.1.p1.2.m2.3.3.cmml" xref="A2.SS2.1.p1.2.m2.3.3">𝑎</ci></apply></apply></apply><apply id="A2.SS2.1.p1.2.m2.3.4c.cmml" xref="A2.SS2.1.p1.2.m2.3.4"><gt id="A2.SS2.1.p1.2.m2.3.4.5.cmml" xref="A2.SS2.1.p1.2.m2.3.4.5"></gt><share href="https://arxiv.org/html/2503.16280v1#A2.SS2.1.p1.2.m2.3.4.4.cmml" id="A2.SS2.1.p1.2.m2.3.4d.cmml" xref="A2.SS2.1.p1.2.m2.3.4"></share><cn id="A2.SS2.1.p1.2.m2.3.4.6.cmml" type="integer" xref="A2.SS2.1.p1.2.m2.3.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.2.m2.3c">H(a)=G(a)-F(a)>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.2.m2.3d">italic_H ( italic_a ) = italic_G ( italic_a ) - italic_F ( italic_a ) > 0</annotation></semantics></math> and <math alttext="H(b)=G(b)-F(b)<0" class="ltx_Math" display="inline" id="A2.SS2.1.p1.3.m3.3"><semantics id="A2.SS2.1.p1.3.m3.3a"><mrow id="A2.SS2.1.p1.3.m3.3.4" xref="A2.SS2.1.p1.3.m3.3.4.cmml"><mrow id="A2.SS2.1.p1.3.m3.3.4.2" xref="A2.SS2.1.p1.3.m3.3.4.2.cmml"><mi id="A2.SS2.1.p1.3.m3.3.4.2.2" xref="A2.SS2.1.p1.3.m3.3.4.2.2.cmml">H</mi><mo id="A2.SS2.1.p1.3.m3.3.4.2.1" xref="A2.SS2.1.p1.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.3.m3.3.4.2.3.2" xref="A2.SS2.1.p1.3.m3.3.4.2.cmml"><mo id="A2.SS2.1.p1.3.m3.3.4.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.2.cmml">(</mo><mi id="A2.SS2.1.p1.3.m3.1.1" xref="A2.SS2.1.p1.3.m3.1.1.cmml">b</mi><mo id="A2.SS2.1.p1.3.m3.3.4.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.3.m3.3.4.3" xref="A2.SS2.1.p1.3.m3.3.4.3.cmml">=</mo><mrow id="A2.SS2.1.p1.3.m3.3.4.4" xref="A2.SS2.1.p1.3.m3.3.4.4.cmml"><mrow id="A2.SS2.1.p1.3.m3.3.4.4.2" xref="A2.SS2.1.p1.3.m3.3.4.4.2.cmml"><mi id="A2.SS2.1.p1.3.m3.3.4.4.2.2" xref="A2.SS2.1.p1.3.m3.3.4.4.2.2.cmml">G</mi><mo id="A2.SS2.1.p1.3.m3.3.4.4.2.1" xref="A2.SS2.1.p1.3.m3.3.4.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.3.m3.3.4.4.2.3.2" xref="A2.SS2.1.p1.3.m3.3.4.4.2.cmml"><mo id="A2.SS2.1.p1.3.m3.3.4.4.2.3.2.1" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.4.2.cmml">(</mo><mi id="A2.SS2.1.p1.3.m3.2.2" xref="A2.SS2.1.p1.3.m3.2.2.cmml">b</mi><mo id="A2.SS2.1.p1.3.m3.3.4.4.2.3.2.2" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.3.m3.3.4.4.1" xref="A2.SS2.1.p1.3.m3.3.4.4.1.cmml">−</mo><mrow id="A2.SS2.1.p1.3.m3.3.4.4.3" xref="A2.SS2.1.p1.3.m3.3.4.4.3.cmml"><mi id="A2.SS2.1.p1.3.m3.3.4.4.3.2" xref="A2.SS2.1.p1.3.m3.3.4.4.3.2.cmml">F</mi><mo id="A2.SS2.1.p1.3.m3.3.4.4.3.1" xref="A2.SS2.1.p1.3.m3.3.4.4.3.1.cmml">⁢</mo><mrow id="A2.SS2.1.p1.3.m3.3.4.4.3.3.2" xref="A2.SS2.1.p1.3.m3.3.4.4.3.cmml"><mo id="A2.SS2.1.p1.3.m3.3.4.4.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.4.3.cmml">(</mo><mi id="A2.SS2.1.p1.3.m3.3.3" xref="A2.SS2.1.p1.3.m3.3.3.cmml">b</mi><mo id="A2.SS2.1.p1.3.m3.3.4.4.3.3.2.2" stretchy="false" xref="A2.SS2.1.p1.3.m3.3.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="A2.SS2.1.p1.3.m3.3.4.5" xref="A2.SS2.1.p1.3.m3.3.4.5.cmml"><</mo><mn id="A2.SS2.1.p1.3.m3.3.4.6" xref="A2.SS2.1.p1.3.m3.3.4.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.3.m3.3b"><apply id="A2.SS2.1.p1.3.m3.3.4.cmml" xref="A2.SS2.1.p1.3.m3.3.4"><and id="A2.SS2.1.p1.3.m3.3.4a.cmml" xref="A2.SS2.1.p1.3.m3.3.4"></and><apply id="A2.SS2.1.p1.3.m3.3.4b.cmml" xref="A2.SS2.1.p1.3.m3.3.4"><eq id="A2.SS2.1.p1.3.m3.3.4.3.cmml" xref="A2.SS2.1.p1.3.m3.3.4.3"></eq><apply id="A2.SS2.1.p1.3.m3.3.4.2.cmml" xref="A2.SS2.1.p1.3.m3.3.4.2"><times id="A2.SS2.1.p1.3.m3.3.4.2.1.cmml" xref="A2.SS2.1.p1.3.m3.3.4.2.1"></times><ci id="A2.SS2.1.p1.3.m3.3.4.2.2.cmml" xref="A2.SS2.1.p1.3.m3.3.4.2.2">𝐻</ci><ci id="A2.SS2.1.p1.3.m3.1.1.cmml" xref="A2.SS2.1.p1.3.m3.1.1">𝑏</ci></apply><apply id="A2.SS2.1.p1.3.m3.3.4.4.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4"><minus id="A2.SS2.1.p1.3.m3.3.4.4.1.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.1"></minus><apply id="A2.SS2.1.p1.3.m3.3.4.4.2.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.2"><times id="A2.SS2.1.p1.3.m3.3.4.4.2.1.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.2.1"></times><ci id="A2.SS2.1.p1.3.m3.3.4.4.2.2.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.2.2">𝐺</ci><ci id="A2.SS2.1.p1.3.m3.2.2.cmml" xref="A2.SS2.1.p1.3.m3.2.2">𝑏</ci></apply><apply id="A2.SS2.1.p1.3.m3.3.4.4.3.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.3"><times id="A2.SS2.1.p1.3.m3.3.4.4.3.1.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.3.1"></times><ci id="A2.SS2.1.p1.3.m3.3.4.4.3.2.cmml" xref="A2.SS2.1.p1.3.m3.3.4.4.3.2">𝐹</ci><ci id="A2.SS2.1.p1.3.m3.3.3.cmml" xref="A2.SS2.1.p1.3.m3.3.3">𝑏</ci></apply></apply></apply><apply id="A2.SS2.1.p1.3.m3.3.4c.cmml" xref="A2.SS2.1.p1.3.m3.3.4"><lt id="A2.SS2.1.p1.3.m3.3.4.5.cmml" xref="A2.SS2.1.p1.3.m3.3.4.5"></lt><share href="https://arxiv.org/html/2503.16280v1#A2.SS2.1.p1.3.m3.3.4.4.cmml" id="A2.SS2.1.p1.3.m3.3.4d.cmml" xref="A2.SS2.1.p1.3.m3.3.4"></share><cn id="A2.SS2.1.p1.3.m3.3.4.6.cmml" type="integer" xref="A2.SS2.1.p1.3.m3.3.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.3.m3.3c">H(b)=G(b)-F(b)<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.3.m3.3d">italic_H ( italic_b ) = italic_G ( italic_b ) - italic_F ( italic_b ) < 0</annotation></semantics></math>. Since <math alttext="G" class="ltx_Math" display="inline" id="A2.SS2.1.p1.4.m4.1"><semantics id="A2.SS2.1.p1.4.m4.1a"><mi id="A2.SS2.1.p1.4.m4.1.1" xref="A2.SS2.1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.4.m4.1b"><ci id="A2.SS2.1.p1.4.m4.1.1.cmml" xref="A2.SS2.1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.4.m4.1d">italic_G</annotation></semantics></math> and <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.1.p1.5.m5.1"><semantics id="A2.SS2.1.p1.5.m5.1a"><mi id="A2.SS2.1.p1.5.m5.1.1" xref="A2.SS2.1.p1.5.m5.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.5.m5.1b"><ci id="A2.SS2.1.p1.5.m5.1.1.cmml" xref="A2.SS2.1.p1.5.m5.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.5.m5.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.5.m5.1d">italic_F</annotation></semantics></math> are continuous, <math alttext="H" class="ltx_Math" display="inline" id="A2.SS2.1.p1.6.m6.1"><semantics id="A2.SS2.1.p1.6.m6.1a"><mi id="A2.SS2.1.p1.6.m6.1.1" xref="A2.SS2.1.p1.6.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.6.m6.1b"><ci id="A2.SS2.1.p1.6.m6.1.1.cmml" xref="A2.SS2.1.p1.6.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.6.m6.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.6.m6.1d">italic_H</annotation></semantics></math> is continuous; thus we can apply the Intermediate Value Theorem to conclude there exists a point <math alttext="\tau^{*}\in[a,b]" class="ltx_Math" display="inline" id="A2.SS2.1.p1.7.m7.2"><semantics id="A2.SS2.1.p1.7.m7.2a"><mrow id="A2.SS2.1.p1.7.m7.2.3" xref="A2.SS2.1.p1.7.m7.2.3.cmml"><msup id="A2.SS2.1.p1.7.m7.2.3.2" xref="A2.SS2.1.p1.7.m7.2.3.2.cmml"><mi id="A2.SS2.1.p1.7.m7.2.3.2.2" xref="A2.SS2.1.p1.7.m7.2.3.2.2.cmml">τ</mi><mo id="A2.SS2.1.p1.7.m7.2.3.2.3" xref="A2.SS2.1.p1.7.m7.2.3.2.3.cmml">∗</mo></msup><mo id="A2.SS2.1.p1.7.m7.2.3.1" xref="A2.SS2.1.p1.7.m7.2.3.1.cmml">∈</mo><mrow id="A2.SS2.1.p1.7.m7.2.3.3.2" xref="A2.SS2.1.p1.7.m7.2.3.3.1.cmml"><mo id="A2.SS2.1.p1.7.m7.2.3.3.2.1" stretchy="false" xref="A2.SS2.1.p1.7.m7.2.3.3.1.cmml">[</mo><mi id="A2.SS2.1.p1.7.m7.1.1" xref="A2.SS2.1.p1.7.m7.1.1.cmml">a</mi><mo id="A2.SS2.1.p1.7.m7.2.3.3.2.2" xref="A2.SS2.1.p1.7.m7.2.3.3.1.cmml">,</mo><mi id="A2.SS2.1.p1.7.m7.2.2" xref="A2.SS2.1.p1.7.m7.2.2.cmml">b</mi><mo id="A2.SS2.1.p1.7.m7.2.3.3.2.3" stretchy="false" xref="A2.SS2.1.p1.7.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.7.m7.2b"><apply id="A2.SS2.1.p1.7.m7.2.3.cmml" xref="A2.SS2.1.p1.7.m7.2.3"><in id="A2.SS2.1.p1.7.m7.2.3.1.cmml" xref="A2.SS2.1.p1.7.m7.2.3.1"></in><apply id="A2.SS2.1.p1.7.m7.2.3.2.cmml" xref="A2.SS2.1.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="A2.SS2.1.p1.7.m7.2.3.2.1.cmml" xref="A2.SS2.1.p1.7.m7.2.3.2">superscript</csymbol><ci id="A2.SS2.1.p1.7.m7.2.3.2.2.cmml" xref="A2.SS2.1.p1.7.m7.2.3.2.2">𝜏</ci><times id="A2.SS2.1.p1.7.m7.2.3.2.3.cmml" xref="A2.SS2.1.p1.7.m7.2.3.2.3"></times></apply><interval closure="closed" id="A2.SS2.1.p1.7.m7.2.3.3.1.cmml" xref="A2.SS2.1.p1.7.m7.2.3.3.2"><ci id="A2.SS2.1.p1.7.m7.1.1.cmml" xref="A2.SS2.1.p1.7.m7.1.1">𝑎</ci><ci id="A2.SS2.1.p1.7.m7.2.2.cmml" xref="A2.SS2.1.p1.7.m7.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.7.m7.2c">\tau^{*}\in[a,b]</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.7.m7.2d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ [ italic_a , italic_b ]</annotation></semantics></math> such that <math alttext="H(\tau^{*})=0" class="ltx_Math" display="inline" id="A2.SS2.1.p1.8.m8.1"><semantics id="A2.SS2.1.p1.8.m8.1a"><mrow id="A2.SS2.1.p1.8.m8.1.1" xref="A2.SS2.1.p1.8.m8.1.1.cmml"><mrow id="A2.SS2.1.p1.8.m8.1.1.1" xref="A2.SS2.1.p1.8.m8.1.1.1.cmml"><mi id="A2.SS2.1.p1.8.m8.1.1.1.3" xref="A2.SS2.1.p1.8.m8.1.1.1.3.cmml">H</mi><mo id="A2.SS2.1.p1.8.m8.1.1.1.2" xref="A2.SS2.1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="A2.SS2.1.p1.8.m8.1.1.1.1.1" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="A2.SS2.1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.2" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.2.cmml">τ</mi><mo id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.3" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="A2.SS2.1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.SS2.1.p1.8.m8.1.1.2" xref="A2.SS2.1.p1.8.m8.1.1.2.cmml">=</mo><mn id="A2.SS2.1.p1.8.m8.1.1.3" xref="A2.SS2.1.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.8.m8.1b"><apply id="A2.SS2.1.p1.8.m8.1.1.cmml" xref="A2.SS2.1.p1.8.m8.1.1"><eq id="A2.SS2.1.p1.8.m8.1.1.2.cmml" xref="A2.SS2.1.p1.8.m8.1.1.2"></eq><apply id="A2.SS2.1.p1.8.m8.1.1.1.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1"><times id="A2.SS2.1.p1.8.m8.1.1.1.2.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.2"></times><ci id="A2.SS2.1.p1.8.m8.1.1.1.3.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.3">𝐻</ci><apply id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1">superscript</csymbol><ci id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.2">𝜏</ci><times id="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="A2.SS2.1.p1.8.m8.1.1.1.1.1.1.3"></times></apply></apply><cn id="A2.SS2.1.p1.8.m8.1.1.3.cmml" type="integer" xref="A2.SS2.1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.8.m8.1c">H(\tau^{*})=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.8.m8.1d">italic_H ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>. Then <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS2.1.p1.9.m9.1"><semantics id="A2.SS2.1.p1.9.m9.1a"><msup id="A2.SS2.1.p1.9.m9.1.1" xref="A2.SS2.1.p1.9.m9.1.1.cmml"><mi id="A2.SS2.1.p1.9.m9.1.1.2" xref="A2.SS2.1.p1.9.m9.1.1.2.cmml">τ</mi><mo id="A2.SS2.1.p1.9.m9.1.1.3" xref="A2.SS2.1.p1.9.m9.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.1.p1.9.m9.1b"><apply id="A2.SS2.1.p1.9.m9.1.1.cmml" xref="A2.SS2.1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A2.SS2.1.p1.9.m9.1.1.1.cmml" xref="A2.SS2.1.p1.9.m9.1.1">superscript</csymbol><ci id="A2.SS2.1.p1.9.m9.1.1.2.cmml" xref="A2.SS2.1.p1.9.m9.1.1.2">𝜏</ci><times id="A2.SS2.1.p1.9.m9.1.1.3.cmml" xref="A2.SS2.1.p1.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.1.p1.9.m9.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.1.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an equilibrium by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem3" title="Theorem 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="A2.SS2.p2"> <p class="ltx_p" id="A2.SS2.p2.3">If Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> holds and <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.p2.1.m1.1"><semantics id="A2.SS2.p2.1.m1.1a"><mi id="A2.SS2.p2.1.m1.1.1" xref="A2.SS2.p2.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.1.m1.1b"><ci id="A2.SS2.p2.1.m1.1.1.cmml" xref="A2.SS2.p2.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.1.m1.1d">italic_F</annotation></semantics></math> is symmetric about a point (its median) in the interval <math alttext="I" class="ltx_Math" display="inline" id="A2.SS2.p2.2.m2.1"><semantics id="A2.SS2.p2.2.m2.1a"><mi id="A2.SS2.p2.2.m2.1.1" xref="A2.SS2.p2.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.2.m2.1b"><ci id="A2.SS2.p2.2.m2.1.1.cmml" xref="A2.SS2.p2.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.2.m2.1d">italic_I</annotation></semantics></math>, as in the Gaussian model in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.SS4" title="2.4 A Gaussian Model ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2.4</span></a>, it is reasonable to expect that <math alttext="G" class="ltx_Math" display="inline" id="A2.SS2.p2.3.m3.1"><semantics id="A2.SS2.p2.3.m3.1a"><mi id="A2.SS2.p2.3.m3.1.1" xref="A2.SS2.p2.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.3.m3.1b"><ci id="A2.SS2.p2.3.m3.1.1.cmml" xref="A2.SS2.p2.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p2.3.m3.1d">italic_G</annotation></semantics></math> is symmetric about the same point and an equilibrium occurs at the median. However, we can also characterize how equilibria change relative to the median in skewed settings.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition9.1.1.1">Proposition 9</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition9.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition9.p1"> <p class="ltx_p" id="Thmproposition9.p1.12">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, with <math alttext="\Pr[X^{\prime}\leq\tau\mid X=x]" class="ltx_Math" display="inline" id="Thmproposition9.p1.1.m1.2"><semantics id="Thmproposition9.p1.1.m1.2a"><mrow id="Thmproposition9.p1.1.m1.2.2.1" xref="Thmproposition9.p1.1.m1.2.2.2.cmml"><mi id="Thmproposition9.p1.1.m1.1.1" xref="Thmproposition9.p1.1.m1.1.1.cmml">Pr</mi><mo id="Thmproposition9.p1.1.m1.2.2.1a" xref="Thmproposition9.p1.1.m1.2.2.2.cmml">⁡</mo><mrow id="Thmproposition9.p1.1.m1.2.2.1.1" xref="Thmproposition9.p1.1.m1.2.2.2.cmml"><mo id="Thmproposition9.p1.1.m1.2.2.1.1.2" stretchy="false" xref="Thmproposition9.p1.1.m1.2.2.2.cmml">[</mo><mrow id="Thmproposition9.p1.1.m1.2.2.1.1.1" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.cmml"><msup id="Thmproposition9.p1.1.m1.2.2.1.1.1.2" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2.cmml"><mi id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.2" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.3" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition9.p1.1.m1.2.2.1.1.1.3" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.3.cmml">≤</mo><mrow id="Thmproposition9.p1.1.m1.2.2.1.1.1.4" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.cmml"><mi id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.2" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.2.cmml">τ</mi><mo id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.1" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.1.cmml">∣</mo><mi id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.3" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.3.cmml">X</mi></mrow><mo id="Thmproposition9.p1.1.m1.2.2.1.1.1.5" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.5.cmml">=</mo><mi id="Thmproposition9.p1.1.m1.2.2.1.1.1.6" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.6.cmml">x</mi></mrow><mo id="Thmproposition9.p1.1.m1.2.2.1.1.3" stretchy="false" xref="Thmproposition9.p1.1.m1.2.2.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.1.m1.2b"><apply id="Thmproposition9.p1.1.m1.2.2.2.cmml" xref="Thmproposition9.p1.1.m1.2.2.1"><ci id="Thmproposition9.p1.1.m1.1.1.cmml" xref="Thmproposition9.p1.1.m1.1.1">Pr</ci><apply id="Thmproposition9.p1.1.m1.2.2.1.1.1.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1"><and id="Thmproposition9.p1.1.m1.2.2.1.1.1a.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1"></and><apply id="Thmproposition9.p1.1.m1.2.2.1.1.1b.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1"><leq id="Thmproposition9.p1.1.m1.2.2.1.1.1.3.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.3"></leq><apply id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.1.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2.2">𝑋</ci><ci id="Thmproposition9.p1.1.m1.2.2.1.1.1.2.3.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.2.3">′</ci></apply><apply id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4"><csymbol cd="latexml" id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.1.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.1">conditional</csymbol><ci id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.2.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.2">𝜏</ci><ci id="Thmproposition9.p1.1.m1.2.2.1.1.1.4.3.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.4.3">𝑋</ci></apply></apply><apply id="Thmproposition9.p1.1.m1.2.2.1.1.1c.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1"><eq id="Thmproposition9.p1.1.m1.2.2.1.1.1.5.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#Thmproposition9.p1.1.m1.2.2.1.1.1.4.cmml" id="Thmproposition9.p1.1.m1.2.2.1.1.1d.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1"></share><ci id="Thmproposition9.p1.1.m1.2.2.1.1.1.6.cmml" xref="Thmproposition9.p1.1.m1.2.2.1.1.1.6">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.1.m1.2c">\Pr[X^{\prime}\leq\tau\mid X=x]</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X = italic_x ]</annotation></semantics></math> monotone decreasing and continuous in <math alttext="x" class="ltx_Math" display="inline" id="Thmproposition9.p1.2.m2.1"><semantics id="Thmproposition9.p1.2.m2.1a"><mi id="Thmproposition9.p1.2.m2.1.1" xref="Thmproposition9.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.2.m2.1b"><ci id="Thmproposition9.p1.2.m2.1.1.cmml" xref="Thmproposition9.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.2.m2.1d">italic_x</annotation></semantics></math>, and <math alttext="G" class="ltx_Math" display="inline" id="Thmproposition9.p1.3.m3.1"><semantics id="Thmproposition9.p1.3.m3.1a"><mi id="Thmproposition9.p1.3.m3.1.1" xref="Thmproposition9.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.3.m3.1b"><ci id="Thmproposition9.p1.3.m3.1.1.cmml" xref="Thmproposition9.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.3.m3.1d">italic_G</annotation></semantics></math> and <math alttext="F" class="ltx_Math" display="inline" id="Thmproposition9.p1.4.m4.1"><semantics id="Thmproposition9.p1.4.m4.1a"><mi id="Thmproposition9.p1.4.m4.1.1" xref="Thmproposition9.p1.4.m4.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.4.m4.1b"><ci id="Thmproposition9.p1.4.m4.1.1.cmml" xref="Thmproposition9.p1.4.m4.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.4.m4.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.4.m4.1d">italic_F</annotation></semantics></math> continuous. Let <math alttext="m" class="ltx_Math" display="inline" id="Thmproposition9.p1.5.m5.1"><semantics id="Thmproposition9.p1.5.m5.1a"><mi id="Thmproposition9.p1.5.m5.1.1" xref="Thmproposition9.p1.5.m5.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.5.m5.1b"><ci id="Thmproposition9.p1.5.m5.1.1.cmml" xref="Thmproposition9.p1.5.m5.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.5.m5.1c">m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.5.m5.1d">italic_m</annotation></semantics></math> be the median of <math alttext="F" class="ltx_Math" display="inline" id="Thmproposition9.p1.6.m6.1"><semantics id="Thmproposition9.p1.6.m6.1a"><mi id="Thmproposition9.p1.6.m6.1.1" xref="Thmproposition9.p1.6.m6.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.6.m6.1b"><ci id="Thmproposition9.p1.6.m6.1.1.cmml" xref="Thmproposition9.p1.6.m6.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.6.m6.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.6.m6.1d">italic_F</annotation></semantics></math>, and consider running the DG mechanism. Then if <math alttext="\Pr[X^{\prime}\leq m\mid m]>1/2" class="ltx_Math" display="inline" id="Thmproposition9.p1.7.m7.2"><semantics id="Thmproposition9.p1.7.m7.2a"><mrow id="Thmproposition9.p1.7.m7.2.2" xref="Thmproposition9.p1.7.m7.2.2.cmml"><mrow id="Thmproposition9.p1.7.m7.2.2.1.1" xref="Thmproposition9.p1.7.m7.2.2.1.2.cmml"><mi id="Thmproposition9.p1.7.m7.1.1" xref="Thmproposition9.p1.7.m7.1.1.cmml">Pr</mi><mo id="Thmproposition9.p1.7.m7.2.2.1.1a" xref="Thmproposition9.p1.7.m7.2.2.1.2.cmml">⁡</mo><mrow id="Thmproposition9.p1.7.m7.2.2.1.1.1" xref="Thmproposition9.p1.7.m7.2.2.1.2.cmml"><mo id="Thmproposition9.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="Thmproposition9.p1.7.m7.2.2.1.2.cmml">[</mo><mrow id="Thmproposition9.p1.7.m7.2.2.1.1.1.1" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.cmml"><msup id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.cmml"><mi id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.2" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.3" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.1" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.cmml"><mi id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.2" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.1" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.3" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="Thmproposition9.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="Thmproposition9.p1.7.m7.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="Thmproposition9.p1.7.m7.2.2.2" xref="Thmproposition9.p1.7.m7.2.2.2.cmml">></mo><mrow id="Thmproposition9.p1.7.m7.2.2.3" xref="Thmproposition9.p1.7.m7.2.2.3.cmml"><mn id="Thmproposition9.p1.7.m7.2.2.3.2" xref="Thmproposition9.p1.7.m7.2.2.3.2.cmml">1</mn><mo id="Thmproposition9.p1.7.m7.2.2.3.1" xref="Thmproposition9.p1.7.m7.2.2.3.1.cmml">/</mo><mn id="Thmproposition9.p1.7.m7.2.2.3.3" xref="Thmproposition9.p1.7.m7.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.7.m7.2b"><apply id="Thmproposition9.p1.7.m7.2.2.cmml" xref="Thmproposition9.p1.7.m7.2.2"><gt id="Thmproposition9.p1.7.m7.2.2.2.cmml" xref="Thmproposition9.p1.7.m7.2.2.2"></gt><apply id="Thmproposition9.p1.7.m7.2.2.1.2.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1"><ci id="Thmproposition9.p1.7.m7.1.1.cmml" xref="Thmproposition9.p1.7.m7.1.1">Pr</ci><apply id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1"><leq id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.1"></leq><apply id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.1.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2">superscript</csymbol><ci id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.2.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.2">𝑋</ci><ci id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.3.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.2.3">′</ci></apply><apply id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3"><csymbol cd="latexml" id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.1.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.2.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.3.cmml" xref="Thmproposition9.p1.7.m7.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="Thmproposition9.p1.7.m7.2.2.3.cmml" xref="Thmproposition9.p1.7.m7.2.2.3"><divide id="Thmproposition9.p1.7.m7.2.2.3.1.cmml" xref="Thmproposition9.p1.7.m7.2.2.3.1"></divide><cn id="Thmproposition9.p1.7.m7.2.2.3.2.cmml" type="integer" xref="Thmproposition9.p1.7.m7.2.2.3.2">1</cn><cn id="Thmproposition9.p1.7.m7.2.2.3.3.cmml" type="integer" xref="Thmproposition9.p1.7.m7.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.7.m7.2c">\Pr[X^{\prime}\leq m\mid m]>1/2</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.7.m7.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] > 1 / 2</annotation></semantics></math>, there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmproposition9.p1.8.m8.1"><semantics id="Thmproposition9.p1.8.m8.1a"><msup id="Thmproposition9.p1.8.m8.1.1" xref="Thmproposition9.p1.8.m8.1.1.cmml"><mi id="Thmproposition9.p1.8.m8.1.1.2" xref="Thmproposition9.p1.8.m8.1.1.2.cmml">τ</mi><mo id="Thmproposition9.p1.8.m8.1.1.3" xref="Thmproposition9.p1.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.8.m8.1b"><apply id="Thmproposition9.p1.8.m8.1.1.cmml" xref="Thmproposition9.p1.8.m8.1.1"><csymbol cd="ambiguous" id="Thmproposition9.p1.8.m8.1.1.1.cmml" xref="Thmproposition9.p1.8.m8.1.1">superscript</csymbol><ci id="Thmproposition9.p1.8.m8.1.1.2.cmml" xref="Thmproposition9.p1.8.m8.1.1.2">𝜏</ci><times id="Thmproposition9.p1.8.m8.1.1.3.cmml" xref="Thmproposition9.p1.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.8.m8.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.8.m8.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\tau^{*}>m" class="ltx_Math" display="inline" id="Thmproposition9.p1.9.m9.1"><semantics id="Thmproposition9.p1.9.m9.1a"><mrow id="Thmproposition9.p1.9.m9.1.1" xref="Thmproposition9.p1.9.m9.1.1.cmml"><msup id="Thmproposition9.p1.9.m9.1.1.2" xref="Thmproposition9.p1.9.m9.1.1.2.cmml"><mi id="Thmproposition9.p1.9.m9.1.1.2.2" xref="Thmproposition9.p1.9.m9.1.1.2.2.cmml">τ</mi><mo id="Thmproposition9.p1.9.m9.1.1.2.3" xref="Thmproposition9.p1.9.m9.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition9.p1.9.m9.1.1.1" xref="Thmproposition9.p1.9.m9.1.1.1.cmml">></mo><mi id="Thmproposition9.p1.9.m9.1.1.3" xref="Thmproposition9.p1.9.m9.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.9.m9.1b"><apply id="Thmproposition9.p1.9.m9.1.1.cmml" xref="Thmproposition9.p1.9.m9.1.1"><gt id="Thmproposition9.p1.9.m9.1.1.1.cmml" xref="Thmproposition9.p1.9.m9.1.1.1"></gt><apply id="Thmproposition9.p1.9.m9.1.1.2.cmml" xref="Thmproposition9.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="Thmproposition9.p1.9.m9.1.1.2.1.cmml" xref="Thmproposition9.p1.9.m9.1.1.2">superscript</csymbol><ci id="Thmproposition9.p1.9.m9.1.1.2.2.cmml" xref="Thmproposition9.p1.9.m9.1.1.2.2">𝜏</ci><times id="Thmproposition9.p1.9.m9.1.1.2.3.cmml" xref="Thmproposition9.p1.9.m9.1.1.2.3"></times></apply><ci id="Thmproposition9.p1.9.m9.1.1.3.cmml" xref="Thmproposition9.p1.9.m9.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.9.m9.1c">\tau^{*}>m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT > italic_m</annotation></semantics></math>. Similarly, if <math alttext="\Pr[X^{\prime}\leq m\mid m]<1/2" class="ltx_Math" display="inline" id="Thmproposition9.p1.10.m10.2"><semantics id="Thmproposition9.p1.10.m10.2a"><mrow id="Thmproposition9.p1.10.m10.2.2" xref="Thmproposition9.p1.10.m10.2.2.cmml"><mrow id="Thmproposition9.p1.10.m10.2.2.1.1" xref="Thmproposition9.p1.10.m10.2.2.1.2.cmml"><mi id="Thmproposition9.p1.10.m10.1.1" xref="Thmproposition9.p1.10.m10.1.1.cmml">Pr</mi><mo id="Thmproposition9.p1.10.m10.2.2.1.1a" xref="Thmproposition9.p1.10.m10.2.2.1.2.cmml">⁡</mo><mrow id="Thmproposition9.p1.10.m10.2.2.1.1.1" xref="Thmproposition9.p1.10.m10.2.2.1.2.cmml"><mo id="Thmproposition9.p1.10.m10.2.2.1.1.1.2" stretchy="false" xref="Thmproposition9.p1.10.m10.2.2.1.2.cmml">[</mo><mrow id="Thmproposition9.p1.10.m10.2.2.1.1.1.1" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.cmml"><msup id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.cmml"><mi id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.2" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.3" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.1" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.cmml"><mi id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.2" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.2.cmml">m</mi><mo id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.1" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.1.cmml">∣</mo><mi id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.3" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="Thmproposition9.p1.10.m10.2.2.1.1.1.3" stretchy="false" xref="Thmproposition9.p1.10.m10.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="Thmproposition9.p1.10.m10.2.2.2" xref="Thmproposition9.p1.10.m10.2.2.2.cmml"><</mo><mrow id="Thmproposition9.p1.10.m10.2.2.3" xref="Thmproposition9.p1.10.m10.2.2.3.cmml"><mn id="Thmproposition9.p1.10.m10.2.2.3.2" xref="Thmproposition9.p1.10.m10.2.2.3.2.cmml">1</mn><mo id="Thmproposition9.p1.10.m10.2.2.3.1" xref="Thmproposition9.p1.10.m10.2.2.3.1.cmml">/</mo><mn id="Thmproposition9.p1.10.m10.2.2.3.3" xref="Thmproposition9.p1.10.m10.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.10.m10.2b"><apply id="Thmproposition9.p1.10.m10.2.2.cmml" xref="Thmproposition9.p1.10.m10.2.2"><lt id="Thmproposition9.p1.10.m10.2.2.2.cmml" xref="Thmproposition9.p1.10.m10.2.2.2"></lt><apply id="Thmproposition9.p1.10.m10.2.2.1.2.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1"><ci id="Thmproposition9.p1.10.m10.1.1.cmml" xref="Thmproposition9.p1.10.m10.1.1">Pr</ci><apply id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1"><leq id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.1"></leq><apply id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.1.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2">superscript</csymbol><ci id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.2.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.2">𝑋</ci><ci id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.3.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.2.3">′</ci></apply><apply id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3"><csymbol cd="latexml" id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.1.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.2.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.2">𝑚</ci><ci id="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.3.cmml" xref="Thmproposition9.p1.10.m10.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="Thmproposition9.p1.10.m10.2.2.3.cmml" xref="Thmproposition9.p1.10.m10.2.2.3"><divide id="Thmproposition9.p1.10.m10.2.2.3.1.cmml" xref="Thmproposition9.p1.10.m10.2.2.3.1"></divide><cn id="Thmproposition9.p1.10.m10.2.2.3.2.cmml" type="integer" xref="Thmproposition9.p1.10.m10.2.2.3.2">1</cn><cn id="Thmproposition9.p1.10.m10.2.2.3.3.cmml" type="integer" xref="Thmproposition9.p1.10.m10.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.10.m10.2c">\Pr[X^{\prime}\leq m\mid m]<1/2</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.10.m10.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] < 1 / 2</annotation></semantics></math>, there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="Thmproposition9.p1.11.m11.1"><semantics id="Thmproposition9.p1.11.m11.1a"><msup id="Thmproposition9.p1.11.m11.1.1" xref="Thmproposition9.p1.11.m11.1.1.cmml"><mi id="Thmproposition9.p1.11.m11.1.1.2" xref="Thmproposition9.p1.11.m11.1.1.2.cmml">τ</mi><mo id="Thmproposition9.p1.11.m11.1.1.3" xref="Thmproposition9.p1.11.m11.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.11.m11.1b"><apply id="Thmproposition9.p1.11.m11.1.1.cmml" xref="Thmproposition9.p1.11.m11.1.1"><csymbol cd="ambiguous" id="Thmproposition9.p1.11.m11.1.1.1.cmml" xref="Thmproposition9.p1.11.m11.1.1">superscript</csymbol><ci id="Thmproposition9.p1.11.m11.1.1.2.cmml" xref="Thmproposition9.p1.11.m11.1.1.2">𝜏</ci><times id="Thmproposition9.p1.11.m11.1.1.3.cmml" xref="Thmproposition9.p1.11.m11.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.11.m11.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.11.m11.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\tau^{*}<m" class="ltx_Math" display="inline" id="Thmproposition9.p1.12.m12.1"><semantics id="Thmproposition9.p1.12.m12.1a"><mrow id="Thmproposition9.p1.12.m12.1.1" xref="Thmproposition9.p1.12.m12.1.1.cmml"><msup id="Thmproposition9.p1.12.m12.1.1.2" xref="Thmproposition9.p1.12.m12.1.1.2.cmml"><mi id="Thmproposition9.p1.12.m12.1.1.2.2" xref="Thmproposition9.p1.12.m12.1.1.2.2.cmml">τ</mi><mo id="Thmproposition9.p1.12.m12.1.1.2.3" xref="Thmproposition9.p1.12.m12.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition9.p1.12.m12.1.1.1" xref="Thmproposition9.p1.12.m12.1.1.1.cmml"><</mo><mi id="Thmproposition9.p1.12.m12.1.1.3" xref="Thmproposition9.p1.12.m12.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition9.p1.12.m12.1b"><apply id="Thmproposition9.p1.12.m12.1.1.cmml" xref="Thmproposition9.p1.12.m12.1.1"><lt id="Thmproposition9.p1.12.m12.1.1.1.cmml" xref="Thmproposition9.p1.12.m12.1.1.1"></lt><apply id="Thmproposition9.p1.12.m12.1.1.2.cmml" xref="Thmproposition9.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="Thmproposition9.p1.12.m12.1.1.2.1.cmml" xref="Thmproposition9.p1.12.m12.1.1.2">superscript</csymbol><ci id="Thmproposition9.p1.12.m12.1.1.2.2.cmml" xref="Thmproposition9.p1.12.m12.1.1.2.2">𝜏</ci><times id="Thmproposition9.p1.12.m12.1.1.2.3.cmml" xref="Thmproposition9.p1.12.m12.1.1.2.3"></times></apply><ci id="Thmproposition9.p1.12.m12.1.1.3.cmml" xref="Thmproposition9.p1.12.m12.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition9.p1.12.m12.1c">\tau^{*}<m</annotation><annotation encoding="application/x-llamapun" id="Thmproposition9.p1.12.m12.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT < italic_m</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="A2.SS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A2.SS2.2.p1"> <p class="ltx_p" id="A2.SS2.2.p1.19">First, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem8" title="Theorem 8. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a>, we know at least one equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS2.2.p1.1.m1.1"><semantics id="A2.SS2.2.p1.1.m1.1a"><msup id="A2.SS2.2.p1.1.m1.1.1" xref="A2.SS2.2.p1.1.m1.1.1.cmml"><mi id="A2.SS2.2.p1.1.m1.1.1.2" xref="A2.SS2.2.p1.1.m1.1.1.2.cmml">τ</mi><mo id="A2.SS2.2.p1.1.m1.1.1.3" xref="A2.SS2.2.p1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.1.m1.1b"><apply id="A2.SS2.2.p1.1.m1.1.1.cmml" xref="A2.SS2.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.2.p1.1.m1.1.1.1.cmml" xref="A2.SS2.2.p1.1.m1.1.1">superscript</csymbol><ci id="A2.SS2.2.p1.1.m1.1.1.2.cmml" xref="A2.SS2.2.p1.1.m1.1.1.2">𝜏</ci><times id="A2.SS2.2.p1.1.m1.1.1.3.cmml" xref="A2.SS2.2.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.1.m1.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.1.m1.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> exists in the interval <math alttext="I" class="ltx_Math" display="inline" id="A2.SS2.2.p1.2.m2.1"><semantics id="A2.SS2.2.p1.2.m2.1a"><mi id="A2.SS2.2.p1.2.m2.1.1" xref="A2.SS2.2.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.2.m2.1b"><ci id="A2.SS2.2.p1.2.m2.1.1.cmml" xref="A2.SS2.2.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.2.m2.1d">italic_I</annotation></semantics></math>. Now assume that <math alttext="\Pr[X^{\prime}\leq m\mid m]>1/2" class="ltx_Math" display="inline" id="A2.SS2.2.p1.3.m3.2"><semantics id="A2.SS2.2.p1.3.m3.2a"><mrow id="A2.SS2.2.p1.3.m3.2.2" xref="A2.SS2.2.p1.3.m3.2.2.cmml"><mrow id="A2.SS2.2.p1.3.m3.2.2.1.1" xref="A2.SS2.2.p1.3.m3.2.2.1.2.cmml"><mi id="A2.SS2.2.p1.3.m3.1.1" xref="A2.SS2.2.p1.3.m3.1.1.cmml">Pr</mi><mo id="A2.SS2.2.p1.3.m3.2.2.1.1a" xref="A2.SS2.2.p1.3.m3.2.2.1.2.cmml">⁡</mo><mrow id="A2.SS2.2.p1.3.m3.2.2.1.1.1" xref="A2.SS2.2.p1.3.m3.2.2.1.2.cmml"><mo id="A2.SS2.2.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="A2.SS2.2.p1.3.m3.2.2.1.2.cmml">[</mo><mrow id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.cmml"><msup id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.cmml"><mi id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.2" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.3" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.1" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.1.cmml">≤</mo><mrow id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.cmml"><mi id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.2" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.2.cmml">m</mi><mo id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.1" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.1.cmml">∣</mo><mi id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.3" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="A2.SS2.2.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="A2.SS2.2.p1.3.m3.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="A2.SS2.2.p1.3.m3.2.2.2" xref="A2.SS2.2.p1.3.m3.2.2.2.cmml">></mo><mrow id="A2.SS2.2.p1.3.m3.2.2.3" xref="A2.SS2.2.p1.3.m3.2.2.3.cmml"><mn id="A2.SS2.2.p1.3.m3.2.2.3.2" xref="A2.SS2.2.p1.3.m3.2.2.3.2.cmml">1</mn><mo id="A2.SS2.2.p1.3.m3.2.2.3.1" xref="A2.SS2.2.p1.3.m3.2.2.3.1.cmml">/</mo><mn id="A2.SS2.2.p1.3.m3.2.2.3.3" xref="A2.SS2.2.p1.3.m3.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.3.m3.2b"><apply id="A2.SS2.2.p1.3.m3.2.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2"><gt id="A2.SS2.2.p1.3.m3.2.2.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2.2"></gt><apply id="A2.SS2.2.p1.3.m3.2.2.1.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1"><ci id="A2.SS2.2.p1.3.m3.1.1.cmml" xref="A2.SS2.2.p1.3.m3.1.1">Pr</ci><apply id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1"><leq id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.1"></leq><apply id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.1.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2">superscript</csymbol><ci id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.2">𝑋</ci><ci id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.3.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.2.3">′</ci></apply><apply id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3"><csymbol cd="latexml" id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.1.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.2.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.2">𝑚</ci><ci id="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.3.cmml" xref="A2.SS2.2.p1.3.m3.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="A2.SS2.2.p1.3.m3.2.2.3.cmml" xref="A2.SS2.2.p1.3.m3.2.2.3"><divide id="A2.SS2.2.p1.3.m3.2.2.3.1.cmml" xref="A2.SS2.2.p1.3.m3.2.2.3.1"></divide><cn id="A2.SS2.2.p1.3.m3.2.2.3.2.cmml" type="integer" xref="A2.SS2.2.p1.3.m3.2.2.3.2">1</cn><cn id="A2.SS2.2.p1.3.m3.2.2.3.3.cmml" type="integer" xref="A2.SS2.2.p1.3.m3.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.3.m3.2c">\Pr[X^{\prime}\leq m\mid m]>1/2</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.3.m3.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] > 1 / 2</annotation></semantics></math>. In other words, since <math alttext="F(m)=1/2" class="ltx_Math" display="inline" id="A2.SS2.2.p1.4.m4.1"><semantics id="A2.SS2.2.p1.4.m4.1a"><mrow id="A2.SS2.2.p1.4.m4.1.2" xref="A2.SS2.2.p1.4.m4.1.2.cmml"><mrow id="A2.SS2.2.p1.4.m4.1.2.2" xref="A2.SS2.2.p1.4.m4.1.2.2.cmml"><mi id="A2.SS2.2.p1.4.m4.1.2.2.2" xref="A2.SS2.2.p1.4.m4.1.2.2.2.cmml">F</mi><mo id="A2.SS2.2.p1.4.m4.1.2.2.1" xref="A2.SS2.2.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.4.m4.1.2.2.3.2" xref="A2.SS2.2.p1.4.m4.1.2.2.cmml"><mo id="A2.SS2.2.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.4.m4.1.2.2.cmml">(</mo><mi id="A2.SS2.2.p1.4.m4.1.1" xref="A2.SS2.2.p1.4.m4.1.1.cmml">m</mi><mo id="A2.SS2.2.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.4.m4.1.2.1" xref="A2.SS2.2.p1.4.m4.1.2.1.cmml">=</mo><mrow id="A2.SS2.2.p1.4.m4.1.2.3" xref="A2.SS2.2.p1.4.m4.1.2.3.cmml"><mn id="A2.SS2.2.p1.4.m4.1.2.3.2" xref="A2.SS2.2.p1.4.m4.1.2.3.2.cmml">1</mn><mo id="A2.SS2.2.p1.4.m4.1.2.3.1" xref="A2.SS2.2.p1.4.m4.1.2.3.1.cmml">/</mo><mn id="A2.SS2.2.p1.4.m4.1.2.3.3" xref="A2.SS2.2.p1.4.m4.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.4.m4.1b"><apply id="A2.SS2.2.p1.4.m4.1.2.cmml" xref="A2.SS2.2.p1.4.m4.1.2"><eq id="A2.SS2.2.p1.4.m4.1.2.1.cmml" xref="A2.SS2.2.p1.4.m4.1.2.1"></eq><apply id="A2.SS2.2.p1.4.m4.1.2.2.cmml" xref="A2.SS2.2.p1.4.m4.1.2.2"><times id="A2.SS2.2.p1.4.m4.1.2.2.1.cmml" xref="A2.SS2.2.p1.4.m4.1.2.2.1"></times><ci id="A2.SS2.2.p1.4.m4.1.2.2.2.cmml" xref="A2.SS2.2.p1.4.m4.1.2.2.2">𝐹</ci><ci id="A2.SS2.2.p1.4.m4.1.1.cmml" xref="A2.SS2.2.p1.4.m4.1.1">𝑚</ci></apply><apply id="A2.SS2.2.p1.4.m4.1.2.3.cmml" xref="A2.SS2.2.p1.4.m4.1.2.3"><divide id="A2.SS2.2.p1.4.m4.1.2.3.1.cmml" xref="A2.SS2.2.p1.4.m4.1.2.3.1"></divide><cn id="A2.SS2.2.p1.4.m4.1.2.3.2.cmml" type="integer" xref="A2.SS2.2.p1.4.m4.1.2.3.2">1</cn><cn id="A2.SS2.2.p1.4.m4.1.2.3.3.cmml" type="integer" xref="A2.SS2.2.p1.4.m4.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.4.m4.1c">F(m)=1/2</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.4.m4.1d">italic_F ( italic_m ) = 1 / 2</annotation></semantics></math>, <math alttext="G(m)>F(m)" class="ltx_Math" display="inline" id="A2.SS2.2.p1.5.m5.2"><semantics id="A2.SS2.2.p1.5.m5.2a"><mrow id="A2.SS2.2.p1.5.m5.2.3" xref="A2.SS2.2.p1.5.m5.2.3.cmml"><mrow id="A2.SS2.2.p1.5.m5.2.3.2" xref="A2.SS2.2.p1.5.m5.2.3.2.cmml"><mi id="A2.SS2.2.p1.5.m5.2.3.2.2" xref="A2.SS2.2.p1.5.m5.2.3.2.2.cmml">G</mi><mo id="A2.SS2.2.p1.5.m5.2.3.2.1" xref="A2.SS2.2.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.5.m5.2.3.2.3.2" xref="A2.SS2.2.p1.5.m5.2.3.2.cmml"><mo id="A2.SS2.2.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.5.m5.2.3.2.cmml">(</mo><mi id="A2.SS2.2.p1.5.m5.1.1" xref="A2.SS2.2.p1.5.m5.1.1.cmml">m</mi><mo id="A2.SS2.2.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.5.m5.2.3.1" xref="A2.SS2.2.p1.5.m5.2.3.1.cmml">></mo><mrow id="A2.SS2.2.p1.5.m5.2.3.3" xref="A2.SS2.2.p1.5.m5.2.3.3.cmml"><mi id="A2.SS2.2.p1.5.m5.2.3.3.2" xref="A2.SS2.2.p1.5.m5.2.3.3.2.cmml">F</mi><mo id="A2.SS2.2.p1.5.m5.2.3.3.1" xref="A2.SS2.2.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.5.m5.2.3.3.3.2" xref="A2.SS2.2.p1.5.m5.2.3.3.cmml"><mo id="A2.SS2.2.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="A2.SS2.2.p1.5.m5.2.3.3.cmml">(</mo><mi id="A2.SS2.2.p1.5.m5.2.2" xref="A2.SS2.2.p1.5.m5.2.2.cmml">m</mi><mo id="A2.SS2.2.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="A2.SS2.2.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.5.m5.2b"><apply id="A2.SS2.2.p1.5.m5.2.3.cmml" xref="A2.SS2.2.p1.5.m5.2.3"><gt id="A2.SS2.2.p1.5.m5.2.3.1.cmml" xref="A2.SS2.2.p1.5.m5.2.3.1"></gt><apply id="A2.SS2.2.p1.5.m5.2.3.2.cmml" xref="A2.SS2.2.p1.5.m5.2.3.2"><times id="A2.SS2.2.p1.5.m5.2.3.2.1.cmml" xref="A2.SS2.2.p1.5.m5.2.3.2.1"></times><ci id="A2.SS2.2.p1.5.m5.2.3.2.2.cmml" xref="A2.SS2.2.p1.5.m5.2.3.2.2">𝐺</ci><ci id="A2.SS2.2.p1.5.m5.1.1.cmml" xref="A2.SS2.2.p1.5.m5.1.1">𝑚</ci></apply><apply id="A2.SS2.2.p1.5.m5.2.3.3.cmml" xref="A2.SS2.2.p1.5.m5.2.3.3"><times id="A2.SS2.2.p1.5.m5.2.3.3.1.cmml" xref="A2.SS2.2.p1.5.m5.2.3.3.1"></times><ci id="A2.SS2.2.p1.5.m5.2.3.3.2.cmml" xref="A2.SS2.2.p1.5.m5.2.3.3.2">𝐹</ci><ci id="A2.SS2.2.p1.5.m5.2.2.cmml" xref="A2.SS2.2.p1.5.m5.2.2">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.5.m5.2c">G(m)>F(m)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.5.m5.2d">italic_G ( italic_m ) > italic_F ( italic_m )</annotation></semantics></math>. It follows under Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> that either <math alttext="m<a" class="ltx_Math" display="inline" id="A2.SS2.2.p1.6.m6.1"><semantics id="A2.SS2.2.p1.6.m6.1a"><mrow id="A2.SS2.2.p1.6.m6.1.1" xref="A2.SS2.2.p1.6.m6.1.1.cmml"><mi id="A2.SS2.2.p1.6.m6.1.1.2" xref="A2.SS2.2.p1.6.m6.1.1.2.cmml">m</mi><mo id="A2.SS2.2.p1.6.m6.1.1.1" xref="A2.SS2.2.p1.6.m6.1.1.1.cmml"><</mo><mi id="A2.SS2.2.p1.6.m6.1.1.3" xref="A2.SS2.2.p1.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.6.m6.1b"><apply id="A2.SS2.2.p1.6.m6.1.1.cmml" xref="A2.SS2.2.p1.6.m6.1.1"><lt id="A2.SS2.2.p1.6.m6.1.1.1.cmml" xref="A2.SS2.2.p1.6.m6.1.1.1"></lt><ci id="A2.SS2.2.p1.6.m6.1.1.2.cmml" xref="A2.SS2.2.p1.6.m6.1.1.2">𝑚</ci><ci id="A2.SS2.2.p1.6.m6.1.1.3.cmml" xref="A2.SS2.2.p1.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.6.m6.1c">m<a</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.6.m6.1d">italic_m < italic_a</annotation></semantics></math> or <math alttext="m\in I" class="ltx_Math" display="inline" id="A2.SS2.2.p1.7.m7.1"><semantics id="A2.SS2.2.p1.7.m7.1a"><mrow id="A2.SS2.2.p1.7.m7.1.1" xref="A2.SS2.2.p1.7.m7.1.1.cmml"><mi id="A2.SS2.2.p1.7.m7.1.1.2" xref="A2.SS2.2.p1.7.m7.1.1.2.cmml">m</mi><mo id="A2.SS2.2.p1.7.m7.1.1.1" xref="A2.SS2.2.p1.7.m7.1.1.1.cmml">∈</mo><mi id="A2.SS2.2.p1.7.m7.1.1.3" xref="A2.SS2.2.p1.7.m7.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.7.m7.1b"><apply id="A2.SS2.2.p1.7.m7.1.1.cmml" xref="A2.SS2.2.p1.7.m7.1.1"><in id="A2.SS2.2.p1.7.m7.1.1.1.cmml" xref="A2.SS2.2.p1.7.m7.1.1.1"></in><ci id="A2.SS2.2.p1.7.m7.1.1.2.cmml" xref="A2.SS2.2.p1.7.m7.1.1.2">𝑚</ci><ci id="A2.SS2.2.p1.7.m7.1.1.3.cmml" xref="A2.SS2.2.p1.7.m7.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.7.m7.1c">m\in I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.7.m7.1d">italic_m ∈ italic_I</annotation></semantics></math>. If <math alttext="m<a" class="ltx_Math" display="inline" id="A2.SS2.2.p1.8.m8.1"><semantics id="A2.SS2.2.p1.8.m8.1a"><mrow id="A2.SS2.2.p1.8.m8.1.1" xref="A2.SS2.2.p1.8.m8.1.1.cmml"><mi id="A2.SS2.2.p1.8.m8.1.1.2" xref="A2.SS2.2.p1.8.m8.1.1.2.cmml">m</mi><mo id="A2.SS2.2.p1.8.m8.1.1.1" xref="A2.SS2.2.p1.8.m8.1.1.1.cmml"><</mo><mi id="A2.SS2.2.p1.8.m8.1.1.3" xref="A2.SS2.2.p1.8.m8.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.8.m8.1b"><apply id="A2.SS2.2.p1.8.m8.1.1.cmml" xref="A2.SS2.2.p1.8.m8.1.1"><lt id="A2.SS2.2.p1.8.m8.1.1.1.cmml" xref="A2.SS2.2.p1.8.m8.1.1.1"></lt><ci id="A2.SS2.2.p1.8.m8.1.1.2.cmml" xref="A2.SS2.2.p1.8.m8.1.1.2">𝑚</ci><ci id="A2.SS2.2.p1.8.m8.1.1.3.cmml" xref="A2.SS2.2.p1.8.m8.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.8.m8.1c">m<a</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.8.m8.1d">italic_m < italic_a</annotation></semantics></math>, then by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem8" title="Theorem 8. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a> there exists an equilibrium <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS2.2.p1.9.m9.1"><semantics id="A2.SS2.2.p1.9.m9.1a"><msup id="A2.SS2.2.p1.9.m9.1.1" xref="A2.SS2.2.p1.9.m9.1.1.cmml"><mi id="A2.SS2.2.p1.9.m9.1.1.2" xref="A2.SS2.2.p1.9.m9.1.1.2.cmml">τ</mi><mo id="A2.SS2.2.p1.9.m9.1.1.3" xref="A2.SS2.2.p1.9.m9.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.9.m9.1b"><apply id="A2.SS2.2.p1.9.m9.1.1.cmml" xref="A2.SS2.2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A2.SS2.2.p1.9.m9.1.1.1.cmml" xref="A2.SS2.2.p1.9.m9.1.1">superscript</csymbol><ci id="A2.SS2.2.p1.9.m9.1.1.2.cmml" xref="A2.SS2.2.p1.9.m9.1.1.2">𝜏</ci><times id="A2.SS2.2.p1.9.m9.1.1.3.cmml" xref="A2.SS2.2.p1.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.9.m9.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.9.m9.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> in the interval <math alttext="I" class="ltx_Math" display="inline" id="A2.SS2.2.p1.10.m10.1"><semantics id="A2.SS2.2.p1.10.m10.1a"><mi id="A2.SS2.2.p1.10.m10.1.1" xref="A2.SS2.2.p1.10.m10.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.10.m10.1b"><ci id="A2.SS2.2.p1.10.m10.1.1.cmml" xref="A2.SS2.2.p1.10.m10.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.10.m10.1c">I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.10.m10.1d">italic_I</annotation></semantics></math>; <math alttext="\tau^{*}\geq a>m" class="ltx_Math" display="inline" id="A2.SS2.2.p1.11.m11.1"><semantics id="A2.SS2.2.p1.11.m11.1a"><mrow id="A2.SS2.2.p1.11.m11.1.1" xref="A2.SS2.2.p1.11.m11.1.1.cmml"><msup id="A2.SS2.2.p1.11.m11.1.1.2" xref="A2.SS2.2.p1.11.m11.1.1.2.cmml"><mi id="A2.SS2.2.p1.11.m11.1.1.2.2" xref="A2.SS2.2.p1.11.m11.1.1.2.2.cmml">τ</mi><mo id="A2.SS2.2.p1.11.m11.1.1.2.3" xref="A2.SS2.2.p1.11.m11.1.1.2.3.cmml">∗</mo></msup><mo id="A2.SS2.2.p1.11.m11.1.1.3" xref="A2.SS2.2.p1.11.m11.1.1.3.cmml">≥</mo><mi id="A2.SS2.2.p1.11.m11.1.1.4" xref="A2.SS2.2.p1.11.m11.1.1.4.cmml">a</mi><mo id="A2.SS2.2.p1.11.m11.1.1.5" xref="A2.SS2.2.p1.11.m11.1.1.5.cmml">></mo><mi id="A2.SS2.2.p1.11.m11.1.1.6" xref="A2.SS2.2.p1.11.m11.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.11.m11.1b"><apply id="A2.SS2.2.p1.11.m11.1.1.cmml" xref="A2.SS2.2.p1.11.m11.1.1"><and id="A2.SS2.2.p1.11.m11.1.1a.cmml" xref="A2.SS2.2.p1.11.m11.1.1"></and><apply id="A2.SS2.2.p1.11.m11.1.1b.cmml" xref="A2.SS2.2.p1.11.m11.1.1"><geq id="A2.SS2.2.p1.11.m11.1.1.3.cmml" xref="A2.SS2.2.p1.11.m11.1.1.3"></geq><apply id="A2.SS2.2.p1.11.m11.1.1.2.cmml" xref="A2.SS2.2.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.2.p1.11.m11.1.1.2.1.cmml" xref="A2.SS2.2.p1.11.m11.1.1.2">superscript</csymbol><ci id="A2.SS2.2.p1.11.m11.1.1.2.2.cmml" xref="A2.SS2.2.p1.11.m11.1.1.2.2">𝜏</ci><times id="A2.SS2.2.p1.11.m11.1.1.2.3.cmml" xref="A2.SS2.2.p1.11.m11.1.1.2.3"></times></apply><ci id="A2.SS2.2.p1.11.m11.1.1.4.cmml" xref="A2.SS2.2.p1.11.m11.1.1.4">𝑎</ci></apply><apply id="A2.SS2.2.p1.11.m11.1.1c.cmml" xref="A2.SS2.2.p1.11.m11.1.1"><gt id="A2.SS2.2.p1.11.m11.1.1.5.cmml" xref="A2.SS2.2.p1.11.m11.1.1.5"></gt><share href="https://arxiv.org/html/2503.16280v1#A2.SS2.2.p1.11.m11.1.1.4.cmml" id="A2.SS2.2.p1.11.m11.1.1d.cmml" xref="A2.SS2.2.p1.11.m11.1.1"></share><ci id="A2.SS2.2.p1.11.m11.1.1.6.cmml" xref="A2.SS2.2.p1.11.m11.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.11.m11.1c">\tau^{*}\geq a>m</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.11.m11.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≥ italic_a > italic_m</annotation></semantics></math>, so the result follows. Otherwise, consider the case where <math alttext="m\in I" class="ltx_Math" display="inline" id="A2.SS2.2.p1.12.m12.1"><semantics id="A2.SS2.2.p1.12.m12.1a"><mrow id="A2.SS2.2.p1.12.m12.1.1" xref="A2.SS2.2.p1.12.m12.1.1.cmml"><mi id="A2.SS2.2.p1.12.m12.1.1.2" xref="A2.SS2.2.p1.12.m12.1.1.2.cmml">m</mi><mo id="A2.SS2.2.p1.12.m12.1.1.1" xref="A2.SS2.2.p1.12.m12.1.1.1.cmml">∈</mo><mi id="A2.SS2.2.p1.12.m12.1.1.3" xref="A2.SS2.2.p1.12.m12.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.12.m12.1b"><apply id="A2.SS2.2.p1.12.m12.1.1.cmml" xref="A2.SS2.2.p1.12.m12.1.1"><in id="A2.SS2.2.p1.12.m12.1.1.1.cmml" xref="A2.SS2.2.p1.12.m12.1.1.1"></in><ci id="A2.SS2.2.p1.12.m12.1.1.2.cmml" xref="A2.SS2.2.p1.12.m12.1.1.2">𝑚</ci><ci id="A2.SS2.2.p1.12.m12.1.1.3.cmml" xref="A2.SS2.2.p1.12.m12.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.12.m12.1c">m\in I</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.12.m12.1d">italic_m ∈ italic_I</annotation></semantics></math>. Then define <math alttext="H(x)=G(x)-F(x)" class="ltx_Math" display="inline" id="A2.SS2.2.p1.13.m13.3"><semantics id="A2.SS2.2.p1.13.m13.3a"><mrow id="A2.SS2.2.p1.13.m13.3.4" xref="A2.SS2.2.p1.13.m13.3.4.cmml"><mrow id="A2.SS2.2.p1.13.m13.3.4.2" xref="A2.SS2.2.p1.13.m13.3.4.2.cmml"><mi id="A2.SS2.2.p1.13.m13.3.4.2.2" xref="A2.SS2.2.p1.13.m13.3.4.2.2.cmml">H</mi><mo id="A2.SS2.2.p1.13.m13.3.4.2.1" xref="A2.SS2.2.p1.13.m13.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.13.m13.3.4.2.3.2" xref="A2.SS2.2.p1.13.m13.3.4.2.cmml"><mo id="A2.SS2.2.p1.13.m13.3.4.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.2.cmml">(</mo><mi id="A2.SS2.2.p1.13.m13.1.1" xref="A2.SS2.2.p1.13.m13.1.1.cmml">x</mi><mo id="A2.SS2.2.p1.13.m13.3.4.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.13.m13.3.4.1" xref="A2.SS2.2.p1.13.m13.3.4.1.cmml">=</mo><mrow id="A2.SS2.2.p1.13.m13.3.4.3" xref="A2.SS2.2.p1.13.m13.3.4.3.cmml"><mrow id="A2.SS2.2.p1.13.m13.3.4.3.2" xref="A2.SS2.2.p1.13.m13.3.4.3.2.cmml"><mi id="A2.SS2.2.p1.13.m13.3.4.3.2.2" xref="A2.SS2.2.p1.13.m13.3.4.3.2.2.cmml">G</mi><mo id="A2.SS2.2.p1.13.m13.3.4.3.2.1" xref="A2.SS2.2.p1.13.m13.3.4.3.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.13.m13.3.4.3.2.3.2" xref="A2.SS2.2.p1.13.m13.3.4.3.2.cmml"><mo id="A2.SS2.2.p1.13.m13.3.4.3.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.3.2.cmml">(</mo><mi id="A2.SS2.2.p1.13.m13.2.2" xref="A2.SS2.2.p1.13.m13.2.2.cmml">x</mi><mo id="A2.SS2.2.p1.13.m13.3.4.3.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.13.m13.3.4.3.1" xref="A2.SS2.2.p1.13.m13.3.4.3.1.cmml">−</mo><mrow id="A2.SS2.2.p1.13.m13.3.4.3.3" xref="A2.SS2.2.p1.13.m13.3.4.3.3.cmml"><mi id="A2.SS2.2.p1.13.m13.3.4.3.3.2" xref="A2.SS2.2.p1.13.m13.3.4.3.3.2.cmml">F</mi><mo id="A2.SS2.2.p1.13.m13.3.4.3.3.1" xref="A2.SS2.2.p1.13.m13.3.4.3.3.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.13.m13.3.4.3.3.3.2" xref="A2.SS2.2.p1.13.m13.3.4.3.3.cmml"><mo id="A2.SS2.2.p1.13.m13.3.4.3.3.3.2.1" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.3.3.cmml">(</mo><mi id="A2.SS2.2.p1.13.m13.3.3" xref="A2.SS2.2.p1.13.m13.3.3.cmml">x</mi><mo id="A2.SS2.2.p1.13.m13.3.4.3.3.3.2.2" stretchy="false" xref="A2.SS2.2.p1.13.m13.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.13.m13.3b"><apply id="A2.SS2.2.p1.13.m13.3.4.cmml" xref="A2.SS2.2.p1.13.m13.3.4"><eq id="A2.SS2.2.p1.13.m13.3.4.1.cmml" xref="A2.SS2.2.p1.13.m13.3.4.1"></eq><apply id="A2.SS2.2.p1.13.m13.3.4.2.cmml" xref="A2.SS2.2.p1.13.m13.3.4.2"><times id="A2.SS2.2.p1.13.m13.3.4.2.1.cmml" xref="A2.SS2.2.p1.13.m13.3.4.2.1"></times><ci id="A2.SS2.2.p1.13.m13.3.4.2.2.cmml" xref="A2.SS2.2.p1.13.m13.3.4.2.2">𝐻</ci><ci id="A2.SS2.2.p1.13.m13.1.1.cmml" xref="A2.SS2.2.p1.13.m13.1.1">𝑥</ci></apply><apply id="A2.SS2.2.p1.13.m13.3.4.3.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3"><minus id="A2.SS2.2.p1.13.m13.3.4.3.1.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.1"></minus><apply id="A2.SS2.2.p1.13.m13.3.4.3.2.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.2"><times id="A2.SS2.2.p1.13.m13.3.4.3.2.1.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.2.1"></times><ci id="A2.SS2.2.p1.13.m13.3.4.3.2.2.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.2.2">𝐺</ci><ci id="A2.SS2.2.p1.13.m13.2.2.cmml" xref="A2.SS2.2.p1.13.m13.2.2">𝑥</ci></apply><apply id="A2.SS2.2.p1.13.m13.3.4.3.3.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.3"><times id="A2.SS2.2.p1.13.m13.3.4.3.3.1.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.3.1"></times><ci id="A2.SS2.2.p1.13.m13.3.4.3.3.2.cmml" xref="A2.SS2.2.p1.13.m13.3.4.3.3.2">𝐹</ci><ci id="A2.SS2.2.p1.13.m13.3.3.cmml" xref="A2.SS2.2.p1.13.m13.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.13.m13.3c">H(x)=G(x)-F(x)</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.13.m13.3d">italic_H ( italic_x ) = italic_G ( italic_x ) - italic_F ( italic_x )</annotation></semantics></math>. <math alttext="H(m)>0" class="ltx_Math" display="inline" id="A2.SS2.2.p1.14.m14.1"><semantics id="A2.SS2.2.p1.14.m14.1a"><mrow id="A2.SS2.2.p1.14.m14.1.2" xref="A2.SS2.2.p1.14.m14.1.2.cmml"><mrow id="A2.SS2.2.p1.14.m14.1.2.2" xref="A2.SS2.2.p1.14.m14.1.2.2.cmml"><mi id="A2.SS2.2.p1.14.m14.1.2.2.2" xref="A2.SS2.2.p1.14.m14.1.2.2.2.cmml">H</mi><mo id="A2.SS2.2.p1.14.m14.1.2.2.1" xref="A2.SS2.2.p1.14.m14.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.14.m14.1.2.2.3.2" xref="A2.SS2.2.p1.14.m14.1.2.2.cmml"><mo id="A2.SS2.2.p1.14.m14.1.2.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.14.m14.1.2.2.cmml">(</mo><mi id="A2.SS2.2.p1.14.m14.1.1" xref="A2.SS2.2.p1.14.m14.1.1.cmml">m</mi><mo id="A2.SS2.2.p1.14.m14.1.2.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.14.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.14.m14.1.2.1" xref="A2.SS2.2.p1.14.m14.1.2.1.cmml">></mo><mn id="A2.SS2.2.p1.14.m14.1.2.3" xref="A2.SS2.2.p1.14.m14.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.14.m14.1b"><apply id="A2.SS2.2.p1.14.m14.1.2.cmml" xref="A2.SS2.2.p1.14.m14.1.2"><gt id="A2.SS2.2.p1.14.m14.1.2.1.cmml" xref="A2.SS2.2.p1.14.m14.1.2.1"></gt><apply id="A2.SS2.2.p1.14.m14.1.2.2.cmml" xref="A2.SS2.2.p1.14.m14.1.2.2"><times id="A2.SS2.2.p1.14.m14.1.2.2.1.cmml" xref="A2.SS2.2.p1.14.m14.1.2.2.1"></times><ci id="A2.SS2.2.p1.14.m14.1.2.2.2.cmml" xref="A2.SS2.2.p1.14.m14.1.2.2.2">𝐻</ci><ci id="A2.SS2.2.p1.14.m14.1.1.cmml" xref="A2.SS2.2.p1.14.m14.1.1">𝑚</ci></apply><cn id="A2.SS2.2.p1.14.m14.1.2.3.cmml" type="integer" xref="A2.SS2.2.p1.14.m14.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.14.m14.1c">H(m)>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.14.m14.1d">italic_H ( italic_m ) > 0</annotation></semantics></math>, while under Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, <math alttext="H(b)<0" class="ltx_Math" display="inline" id="A2.SS2.2.p1.15.m15.1"><semantics id="A2.SS2.2.p1.15.m15.1a"><mrow id="A2.SS2.2.p1.15.m15.1.2" xref="A2.SS2.2.p1.15.m15.1.2.cmml"><mrow id="A2.SS2.2.p1.15.m15.1.2.2" xref="A2.SS2.2.p1.15.m15.1.2.2.cmml"><mi id="A2.SS2.2.p1.15.m15.1.2.2.2" xref="A2.SS2.2.p1.15.m15.1.2.2.2.cmml">H</mi><mo id="A2.SS2.2.p1.15.m15.1.2.2.1" xref="A2.SS2.2.p1.15.m15.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS2.2.p1.15.m15.1.2.2.3.2" xref="A2.SS2.2.p1.15.m15.1.2.2.cmml"><mo id="A2.SS2.2.p1.15.m15.1.2.2.3.2.1" stretchy="false" xref="A2.SS2.2.p1.15.m15.1.2.2.cmml">(</mo><mi id="A2.SS2.2.p1.15.m15.1.1" xref="A2.SS2.2.p1.15.m15.1.1.cmml">b</mi><mo id="A2.SS2.2.p1.15.m15.1.2.2.3.2.2" stretchy="false" xref="A2.SS2.2.p1.15.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS2.2.p1.15.m15.1.2.1" xref="A2.SS2.2.p1.15.m15.1.2.1.cmml"><</mo><mn id="A2.SS2.2.p1.15.m15.1.2.3" xref="A2.SS2.2.p1.15.m15.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.15.m15.1b"><apply id="A2.SS2.2.p1.15.m15.1.2.cmml" xref="A2.SS2.2.p1.15.m15.1.2"><lt id="A2.SS2.2.p1.15.m15.1.2.1.cmml" xref="A2.SS2.2.p1.15.m15.1.2.1"></lt><apply id="A2.SS2.2.p1.15.m15.1.2.2.cmml" xref="A2.SS2.2.p1.15.m15.1.2.2"><times id="A2.SS2.2.p1.15.m15.1.2.2.1.cmml" xref="A2.SS2.2.p1.15.m15.1.2.2.1"></times><ci id="A2.SS2.2.p1.15.m15.1.2.2.2.cmml" xref="A2.SS2.2.p1.15.m15.1.2.2.2">𝐻</ci><ci id="A2.SS2.2.p1.15.m15.1.1.cmml" xref="A2.SS2.2.p1.15.m15.1.1">𝑏</ci></apply><cn id="A2.SS2.2.p1.15.m15.1.2.3.cmml" type="integer" xref="A2.SS2.2.p1.15.m15.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.15.m15.1c">H(b)<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.15.m15.1d">italic_H ( italic_b ) < 0</annotation></semantics></math>. By the Intermediate Value Theorem, since <math alttext="H" class="ltx_Math" display="inline" id="A2.SS2.2.p1.16.m16.1"><semantics id="A2.SS2.2.p1.16.m16.1a"><mi id="A2.SS2.2.p1.16.m16.1.1" xref="A2.SS2.2.p1.16.m16.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.16.m16.1b"><ci id="A2.SS2.2.p1.16.m16.1.1.cmml" xref="A2.SS2.2.p1.16.m16.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.16.m16.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.16.m16.1d">italic_H</annotation></semantics></math> is continuous, there exists a point <math alttext="\tau^{*}\in[m,b]" class="ltx_Math" display="inline" id="A2.SS2.2.p1.17.m17.2"><semantics id="A2.SS2.2.p1.17.m17.2a"><mrow id="A2.SS2.2.p1.17.m17.2.3" xref="A2.SS2.2.p1.17.m17.2.3.cmml"><msup id="A2.SS2.2.p1.17.m17.2.3.2" xref="A2.SS2.2.p1.17.m17.2.3.2.cmml"><mi id="A2.SS2.2.p1.17.m17.2.3.2.2" xref="A2.SS2.2.p1.17.m17.2.3.2.2.cmml">τ</mi><mo id="A2.SS2.2.p1.17.m17.2.3.2.3" xref="A2.SS2.2.p1.17.m17.2.3.2.3.cmml">∗</mo></msup><mo id="A2.SS2.2.p1.17.m17.2.3.1" xref="A2.SS2.2.p1.17.m17.2.3.1.cmml">∈</mo><mrow id="A2.SS2.2.p1.17.m17.2.3.3.2" xref="A2.SS2.2.p1.17.m17.2.3.3.1.cmml"><mo id="A2.SS2.2.p1.17.m17.2.3.3.2.1" stretchy="false" xref="A2.SS2.2.p1.17.m17.2.3.3.1.cmml">[</mo><mi id="A2.SS2.2.p1.17.m17.1.1" xref="A2.SS2.2.p1.17.m17.1.1.cmml">m</mi><mo id="A2.SS2.2.p1.17.m17.2.3.3.2.2" xref="A2.SS2.2.p1.17.m17.2.3.3.1.cmml">,</mo><mi id="A2.SS2.2.p1.17.m17.2.2" xref="A2.SS2.2.p1.17.m17.2.2.cmml">b</mi><mo id="A2.SS2.2.p1.17.m17.2.3.3.2.3" stretchy="false" xref="A2.SS2.2.p1.17.m17.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.17.m17.2b"><apply id="A2.SS2.2.p1.17.m17.2.3.cmml" xref="A2.SS2.2.p1.17.m17.2.3"><in id="A2.SS2.2.p1.17.m17.2.3.1.cmml" xref="A2.SS2.2.p1.17.m17.2.3.1"></in><apply id="A2.SS2.2.p1.17.m17.2.3.2.cmml" xref="A2.SS2.2.p1.17.m17.2.3.2"><csymbol cd="ambiguous" id="A2.SS2.2.p1.17.m17.2.3.2.1.cmml" xref="A2.SS2.2.p1.17.m17.2.3.2">superscript</csymbol><ci id="A2.SS2.2.p1.17.m17.2.3.2.2.cmml" xref="A2.SS2.2.p1.17.m17.2.3.2.2">𝜏</ci><times id="A2.SS2.2.p1.17.m17.2.3.2.3.cmml" xref="A2.SS2.2.p1.17.m17.2.3.2.3"></times></apply><interval closure="closed" id="A2.SS2.2.p1.17.m17.2.3.3.1.cmml" xref="A2.SS2.2.p1.17.m17.2.3.3.2"><ci id="A2.SS2.2.p1.17.m17.1.1.cmml" xref="A2.SS2.2.p1.17.m17.1.1">𝑚</ci><ci id="A2.SS2.2.p1.17.m17.2.2.cmml" xref="A2.SS2.2.p1.17.m17.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.17.m17.2c">\tau^{*}\in[m,b]</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.17.m17.2d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ [ italic_m , italic_b ]</annotation></semantics></math> such that <math alttext="H(\tau^{*})=0" class="ltx_Math" display="inline" id="A2.SS2.2.p1.18.m18.1"><semantics id="A2.SS2.2.p1.18.m18.1a"><mrow id="A2.SS2.2.p1.18.m18.1.1" 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xref="A2.SS2.2.p1.18.m18.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.18.m18.1b"><apply id="A2.SS2.2.p1.18.m18.1.1.cmml" xref="A2.SS2.2.p1.18.m18.1.1"><eq id="A2.SS2.2.p1.18.m18.1.1.2.cmml" xref="A2.SS2.2.p1.18.m18.1.1.2"></eq><apply id="A2.SS2.2.p1.18.m18.1.1.1.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1"><times id="A2.SS2.2.p1.18.m18.1.1.1.2.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.2"></times><ci id="A2.SS2.2.p1.18.m18.1.1.1.3.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.3">𝐻</ci><apply id="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.1.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.1.1">superscript</csymbol><ci id="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.2.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.2">𝜏</ci><times id="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.3.cmml" xref="A2.SS2.2.p1.18.m18.1.1.1.1.1.1.3"></times></apply></apply><cn id="A2.SS2.2.p1.18.m18.1.1.3.cmml" type="integer" xref="A2.SS2.2.p1.18.m18.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.18.m18.1c">H(\tau^{*})=0</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.18.m18.1d">italic_H ( italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>. By our characterization in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem3" title="Theorem 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>, <math alttext="\tau^{*}" class="ltx_Math" display="inline" id="A2.SS2.2.p1.19.m19.1"><semantics id="A2.SS2.2.p1.19.m19.1a"><msup id="A2.SS2.2.p1.19.m19.1.1" xref="A2.SS2.2.p1.19.m19.1.1.cmml"><mi id="A2.SS2.2.p1.19.m19.1.1.2" xref="A2.SS2.2.p1.19.m19.1.1.2.cmml">τ</mi><mo id="A2.SS2.2.p1.19.m19.1.1.3" xref="A2.SS2.2.p1.19.m19.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="A2.SS2.2.p1.19.m19.1b"><apply id="A2.SS2.2.p1.19.m19.1.1.cmml" xref="A2.SS2.2.p1.19.m19.1.1"><csymbol cd="ambiguous" id="A2.SS2.2.p1.19.m19.1.1.1.cmml" xref="A2.SS2.2.p1.19.m19.1.1">superscript</csymbol><ci id="A2.SS2.2.p1.19.m19.1.1.2.cmml" xref="A2.SS2.2.p1.19.m19.1.1.2">𝜏</ci><times id="A2.SS2.2.p1.19.m19.1.1.3.cmml" xref="A2.SS2.2.p1.19.m19.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.2.p1.19.m19.1c">\tau^{*}</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.2.p1.19.m19.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an equilibrium.</p> </div> <div class="ltx_para" id="A2.SS2.3.p2"> <p class="ltx_p" id="A2.SS2.3.p2.1">A symmetric argument holds for when <math alttext="\Pr[X^{\prime}\leq m\mid m]<1/2" class="ltx_Math" display="inline" id="A2.SS2.3.p2.1.m1.2"><semantics id="A2.SS2.3.p2.1.m1.2a"><mrow id="A2.SS2.3.p2.1.m1.2.2" xref="A2.SS2.3.p2.1.m1.2.2.cmml"><mrow id="A2.SS2.3.p2.1.m1.2.2.1.1" xref="A2.SS2.3.p2.1.m1.2.2.1.2.cmml"><mi id="A2.SS2.3.p2.1.m1.1.1" xref="A2.SS2.3.p2.1.m1.1.1.cmml">Pr</mi><mo id="A2.SS2.3.p2.1.m1.2.2.1.1a" xref="A2.SS2.3.p2.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="A2.SS2.3.p2.1.m1.2.2.1.1.1" 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stretchy="false" xref="A2.SS2.3.p2.1.m1.2.2.1.2.cmml">]</mo></mrow></mrow><mo id="A2.SS2.3.p2.1.m1.2.2.2" xref="A2.SS2.3.p2.1.m1.2.2.2.cmml"><</mo><mrow id="A2.SS2.3.p2.1.m1.2.2.3" xref="A2.SS2.3.p2.1.m1.2.2.3.cmml"><mn id="A2.SS2.3.p2.1.m1.2.2.3.2" xref="A2.SS2.3.p2.1.m1.2.2.3.2.cmml">1</mn><mo id="A2.SS2.3.p2.1.m1.2.2.3.1" xref="A2.SS2.3.p2.1.m1.2.2.3.1.cmml">/</mo><mn id="A2.SS2.3.p2.1.m1.2.2.3.3" xref="A2.SS2.3.p2.1.m1.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.3.p2.1.m1.2b"><apply id="A2.SS2.3.p2.1.m1.2.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2"><lt id="A2.SS2.3.p2.1.m1.2.2.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2.2"></lt><apply id="A2.SS2.3.p2.1.m1.2.2.1.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1"><ci id="A2.SS2.3.p2.1.m1.1.1.cmml" xref="A2.SS2.3.p2.1.m1.1.1">Pr</ci><apply id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1"><leq id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.1"></leq><apply id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.1.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2">superscript</csymbol><ci id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.2">𝑋</ci><ci id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.3.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.2.3">′</ci></apply><apply id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3"><csymbol cd="latexml" id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.1.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.1">conditional</csymbol><ci id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.2.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.2">𝑚</ci><ci id="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.3.cmml" xref="A2.SS2.3.p2.1.m1.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply></apply><apply id="A2.SS2.3.p2.1.m1.2.2.3.cmml" xref="A2.SS2.3.p2.1.m1.2.2.3"><divide id="A2.SS2.3.p2.1.m1.2.2.3.1.cmml" xref="A2.SS2.3.p2.1.m1.2.2.3.1"></divide><cn id="A2.SS2.3.p2.1.m1.2.2.3.2.cmml" type="integer" xref="A2.SS2.3.p2.1.m1.2.2.3.2">1</cn><cn id="A2.SS2.3.p2.1.m1.2.2.3.3.cmml" type="integer" xref="A2.SS2.3.p2.1.m1.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.3.p2.1.m1.2c">\Pr[X^{\prime}\leq m\mid m]<1/2</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.3.p2.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ] < 1 / 2</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="A2.SS2.p3"> <p class="ltx_p" id="A2.SS2.p3.4">In unimodal, continuous settings, we can often diagnose skewness of a distribution based on the difference between its mean and median. In a right-skewed distribution, it is often the case that the bulk of high probability outcomes lie below the median, so that conditioning on the median essentially amplifies the impact of these outcomes. We would therefore expect <math alttext="\Pr[X^{\prime}\leq m\mid m]" class="ltx_Math" display="inline" id="A2.SS2.p3.1.m1.2"><semantics id="A2.SS2.p3.1.m1.2a"><mrow id="A2.SS2.p3.1.m1.2.2.1" xref="A2.SS2.p3.1.m1.2.2.2.cmml"><mi id="A2.SS2.p3.1.m1.1.1" xref="A2.SS2.p3.1.m1.1.1.cmml">Pr</mi><mo id="A2.SS2.p3.1.m1.2.2.1a" xref="A2.SS2.p3.1.m1.2.2.2.cmml">⁡</mo><mrow id="A2.SS2.p3.1.m1.2.2.1.1" xref="A2.SS2.p3.1.m1.2.2.2.cmml"><mo id="A2.SS2.p3.1.m1.2.2.1.1.2" stretchy="false" xref="A2.SS2.p3.1.m1.2.2.2.cmml">[</mo><mrow id="A2.SS2.p3.1.m1.2.2.1.1.1" xref="A2.SS2.p3.1.m1.2.2.1.1.1.cmml"><msup id="A2.SS2.p3.1.m1.2.2.1.1.1.2" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2.cmml"><mi id="A2.SS2.p3.1.m1.2.2.1.1.1.2.2" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2.2.cmml">X</mi><mo id="A2.SS2.p3.1.m1.2.2.1.1.1.2.3" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="A2.SS2.p3.1.m1.2.2.1.1.1.1" xref="A2.SS2.p3.1.m1.2.2.1.1.1.1.cmml">≤</mo><mrow id="A2.SS2.p3.1.m1.2.2.1.1.1.3" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.cmml"><mi id="A2.SS2.p3.1.m1.2.2.1.1.1.3.2" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.2.cmml">m</mi><mo id="A2.SS2.p3.1.m1.2.2.1.1.1.3.1" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.1.cmml">∣</mo><mi id="A2.SS2.p3.1.m1.2.2.1.1.1.3.3" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.3.cmml">m</mi></mrow></mrow><mo id="A2.SS2.p3.1.m1.2.2.1.1.3" stretchy="false" xref="A2.SS2.p3.1.m1.2.2.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p3.1.m1.2b"><apply id="A2.SS2.p3.1.m1.2.2.2.cmml" xref="A2.SS2.p3.1.m1.2.2.1"><ci id="A2.SS2.p3.1.m1.1.1.cmml" xref="A2.SS2.p3.1.m1.1.1">Pr</ci><apply id="A2.SS2.p3.1.m1.2.2.1.1.1.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1"><leq id="A2.SS2.p3.1.m1.2.2.1.1.1.1.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.1"></leq><apply id="A2.SS2.p3.1.m1.2.2.1.1.1.2.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS2.p3.1.m1.2.2.1.1.1.2.1.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2">superscript</csymbol><ci id="A2.SS2.p3.1.m1.2.2.1.1.1.2.2.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2.2">𝑋</ci><ci id="A2.SS2.p3.1.m1.2.2.1.1.1.2.3.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.2.3">′</ci></apply><apply id="A2.SS2.p3.1.m1.2.2.1.1.1.3.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3"><csymbol cd="latexml" id="A2.SS2.p3.1.m1.2.2.1.1.1.3.1.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.1">conditional</csymbol><ci id="A2.SS2.p3.1.m1.2.2.1.1.1.3.2.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.2">𝑚</ci><ci id="A2.SS2.p3.1.m1.2.2.1.1.1.3.3.cmml" xref="A2.SS2.p3.1.m1.2.2.1.1.1.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p3.1.m1.2c">\Pr[X^{\prime}\leq m\mid m]</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p3.1.m1.2d">roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ∣ italic_m ]</annotation></semantics></math> to <em class="ltx_emph ltx_font_italic" id="A2.SS2.p3.4.1">increase</em> relative to 1/2. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition9" title="Proposition 9. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a>, equilibria drift to the right of <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.p3.2.m2.1"><semantics id="A2.SS2.p3.2.m2.1a"><mi id="A2.SS2.p3.2.m2.1.1" xref="A2.SS2.p3.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p3.2.m2.1b"><ci id="A2.SS2.p3.2.m2.1.1.cmml" xref="A2.SS2.p3.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p3.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p3.2.m2.1d">italic_F</annotation></semantics></math>’s median. A similar argument holds for left-skewed distributions: there is more high probability mass concentrated to the right of the median, so conditioning on the median should <em class="ltx_emph ltx_font_italic" id="A2.SS2.p3.4.2">decrease</em> the probability of a signal being below the median. Thus we would expect by Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition9" title="Proposition 9. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> that when <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.p3.3.m3.1"><semantics id="A2.SS2.p3.3.m3.1a"><mi id="A2.SS2.p3.3.m3.1.1" xref="A2.SS2.p3.3.m3.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p3.3.m3.1b"><ci id="A2.SS2.p3.3.m3.1.1.cmml" xref="A2.SS2.p3.3.m3.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p3.3.m3.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p3.3.m3.1d">italic_F</annotation></semantics></math> is left-skewed, equilibria drift to the left of <math alttext="F" class="ltx_Math" display="inline" id="A2.SS2.p3.4.m4.1"><semantics id="A2.SS2.p3.4.m4.1a"><mi id="A2.SS2.p3.4.m4.1.1" xref="A2.SS2.p3.4.m4.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p3.4.m4.1b"><ci id="A2.SS2.p3.4.m4.1.1.cmml" xref="A2.SS2.p3.4.m4.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p3.4.m4.1c">F</annotation><annotation encoding="application/x-llamapun" id="A2.SS2.p3.4.m4.1d">italic_F</annotation></semantics></math>’s median.</p> </div> </section> <section class="ltx_subsection" id="A2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.3 </span>Dynamics Generalization</h3> <div class="ltx_para" id="A2.SS3.p1"> <p class="ltx_p" id="A2.SS3.p1.1">Under reasonable behavior in the tails, we expect existence of a stable nontrivial equilibria, while the uninformative equilibria remain unstable.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition10.1.1.1">Proposition 10</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition10.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition10.p1"> <p class="ltx_p" id="Thmproposition10.p1.5">Let the agent signal structure satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> for interval <math alttext="I" class="ltx_Math" display="inline" id="Thmproposition10.p1.1.m1.1"><semantics id="Thmproposition10.p1.1.m1.1a"><mi id="Thmproposition10.p1.1.m1.1.1" xref="Thmproposition10.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="Thmproposition10.p1.1.m1.1b"><ci id="Thmproposition10.p1.1.m1.1.1.cmml" xref="Thmproposition10.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition10.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="Thmproposition10.p1.1.m1.1d">italic_I</annotation></semantics></math>, and such that <math alttext="F" class="ltx_Math" display="inline" id="Thmproposition10.p1.2.m2.1"><semantics id="Thmproposition10.p1.2.m2.1a"><mi id="Thmproposition10.p1.2.m2.1.1" xref="Thmproposition10.p1.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="Thmproposition10.p1.2.m2.1b"><ci id="Thmproposition10.p1.2.m2.1.1.cmml" xref="Thmproposition10.p1.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition10.p1.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="Thmproposition10.p1.2.m2.1d">italic_F</annotation></semantics></math> and <math alttext="G" class="ltx_Math" display="inline" id="Thmproposition10.p1.3.m3.1"><semantics id="Thmproposition10.p1.3.m3.1a"><mi id="Thmproposition10.p1.3.m3.1.1" xref="Thmproposition10.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmproposition10.p1.3.m3.1b"><ci id="Thmproposition10.p1.3.m3.1.1.cmml" xref="Thmproposition10.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition10.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmproposition10.p1.3.m3.1d">italic_G</annotation></semantics></math> are continuous, and differentiable. Then there exists an equilibrium <math alttext="\tau^{*}\in I" class="ltx_Math" display="inline" id="Thmproposition10.p1.4.m4.1"><semantics id="Thmproposition10.p1.4.m4.1a"><mrow id="Thmproposition10.p1.4.m4.1.1" xref="Thmproposition10.p1.4.m4.1.1.cmml"><msup id="Thmproposition10.p1.4.m4.1.1.2" xref="Thmproposition10.p1.4.m4.1.1.2.cmml"><mi id="Thmproposition10.p1.4.m4.1.1.2.2" xref="Thmproposition10.p1.4.m4.1.1.2.2.cmml">τ</mi><mo id="Thmproposition10.p1.4.m4.1.1.2.3" xref="Thmproposition10.p1.4.m4.1.1.2.3.cmml">∗</mo></msup><mo id="Thmproposition10.p1.4.m4.1.1.1" xref="Thmproposition10.p1.4.m4.1.1.1.cmml">∈</mo><mi id="Thmproposition10.p1.4.m4.1.1.3" xref="Thmproposition10.p1.4.m4.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition10.p1.4.m4.1b"><apply id="Thmproposition10.p1.4.m4.1.1.cmml" xref="Thmproposition10.p1.4.m4.1.1"><in id="Thmproposition10.p1.4.m4.1.1.1.cmml" xref="Thmproposition10.p1.4.m4.1.1.1"></in><apply id="Thmproposition10.p1.4.m4.1.1.2.cmml" xref="Thmproposition10.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmproposition10.p1.4.m4.1.1.2.1.cmml" xref="Thmproposition10.p1.4.m4.1.1.2">superscript</csymbol><ci id="Thmproposition10.p1.4.m4.1.1.2.2.cmml" xref="Thmproposition10.p1.4.m4.1.1.2.2">𝜏</ci><times id="Thmproposition10.p1.4.m4.1.1.2.3.cmml" xref="Thmproposition10.p1.4.m4.1.1.2.3"></times></apply><ci id="Thmproposition10.p1.4.m4.1.1.3.cmml" xref="Thmproposition10.p1.4.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition10.p1.4.m4.1c">\tau^{*}\in I</annotation><annotation encoding="application/x-llamapun" id="Thmproposition10.p1.4.m4.1d">italic_τ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math> which is locally stable, while the equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="Thmproposition10.p1.5.m5.1"><semantics id="Thmproposition10.p1.5.m5.1a"><mrow id="Thmproposition10.p1.5.m5.1.1" xref="Thmproposition10.p1.5.m5.1.1.cmml"><mo id="Thmproposition10.p1.5.m5.1.1a" xref="Thmproposition10.p1.5.m5.1.1.cmml">±</mo><mi id="Thmproposition10.p1.5.m5.1.1.2" mathvariant="normal" xref="Thmproposition10.p1.5.m5.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition10.p1.5.m5.1b"><apply id="Thmproposition10.p1.5.m5.1.1.cmml" xref="Thmproposition10.p1.5.m5.1.1"><csymbol cd="latexml" id="Thmproposition10.p1.5.m5.1.1.1.cmml" xref="Thmproposition10.p1.5.m5.1.1">plus-or-minus</csymbol><infinity id="Thmproposition10.p1.5.m5.1.1.2.cmml" xref="Thmproposition10.p1.5.m5.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition10.p1.5.m5.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="Thmproposition10.p1.5.m5.1d">± ∞</annotation></semantics></math> are unstable.</p> </div> </div> <div class="ltx_proof" id="A2.SS3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A2.SS3.1.p1"> <p class="ltx_p" id="A2.SS3.1.p1.12">We know already from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem8" title="Theorem 8. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">8</span></a> that at least one equilibrium exists in the interval <math alttext="I" class="ltx_Math" display="inline" id="A2.SS3.1.p1.1.m1.1"><semantics id="A2.SS3.1.p1.1.m1.1a"><mi id="A2.SS3.1.p1.1.m1.1.1" xref="A2.SS3.1.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.1.m1.1b"><ci id="A2.SS3.1.p1.1.m1.1.1.cmml" xref="A2.SS3.1.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.1.m1.1d">italic_I</annotation></semantics></math>. Again, we let <math alttext="H(x)=G(x)-F(x)" class="ltx_Math" display="inline" id="A2.SS3.1.p1.2.m2.3"><semantics id="A2.SS3.1.p1.2.m2.3a"><mrow id="A2.SS3.1.p1.2.m2.3.4" xref="A2.SS3.1.p1.2.m2.3.4.cmml"><mrow id="A2.SS3.1.p1.2.m2.3.4.2" xref="A2.SS3.1.p1.2.m2.3.4.2.cmml"><mi id="A2.SS3.1.p1.2.m2.3.4.2.2" xref="A2.SS3.1.p1.2.m2.3.4.2.2.cmml">H</mi><mo id="A2.SS3.1.p1.2.m2.3.4.2.1" xref="A2.SS3.1.p1.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.2.m2.3.4.2.3.2" xref="A2.SS3.1.p1.2.m2.3.4.2.cmml"><mo id="A2.SS3.1.p1.2.m2.3.4.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.2.cmml">(</mo><mi id="A2.SS3.1.p1.2.m2.1.1" xref="A2.SS3.1.p1.2.m2.1.1.cmml">x</mi><mo id="A2.SS3.1.p1.2.m2.3.4.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.2.m2.3.4.1" xref="A2.SS3.1.p1.2.m2.3.4.1.cmml">=</mo><mrow id="A2.SS3.1.p1.2.m2.3.4.3" xref="A2.SS3.1.p1.2.m2.3.4.3.cmml"><mrow id="A2.SS3.1.p1.2.m2.3.4.3.2" xref="A2.SS3.1.p1.2.m2.3.4.3.2.cmml"><mi id="A2.SS3.1.p1.2.m2.3.4.3.2.2" xref="A2.SS3.1.p1.2.m2.3.4.3.2.2.cmml">G</mi><mo id="A2.SS3.1.p1.2.m2.3.4.3.2.1" xref="A2.SS3.1.p1.2.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.2.m2.3.4.3.2.3.2" xref="A2.SS3.1.p1.2.m2.3.4.3.2.cmml"><mo id="A2.SS3.1.p1.2.m2.3.4.3.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.3.2.cmml">(</mo><mi id="A2.SS3.1.p1.2.m2.2.2" xref="A2.SS3.1.p1.2.m2.2.2.cmml">x</mi><mo id="A2.SS3.1.p1.2.m2.3.4.3.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.2.m2.3.4.3.1" xref="A2.SS3.1.p1.2.m2.3.4.3.1.cmml">−</mo><mrow id="A2.SS3.1.p1.2.m2.3.4.3.3" xref="A2.SS3.1.p1.2.m2.3.4.3.3.cmml"><mi id="A2.SS3.1.p1.2.m2.3.4.3.3.2" xref="A2.SS3.1.p1.2.m2.3.4.3.3.2.cmml">F</mi><mo id="A2.SS3.1.p1.2.m2.3.4.3.3.1" xref="A2.SS3.1.p1.2.m2.3.4.3.3.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.2.m2.3.4.3.3.3.2" xref="A2.SS3.1.p1.2.m2.3.4.3.3.cmml"><mo id="A2.SS3.1.p1.2.m2.3.4.3.3.3.2.1" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.3.3.cmml">(</mo><mi id="A2.SS3.1.p1.2.m2.3.3" xref="A2.SS3.1.p1.2.m2.3.3.cmml">x</mi><mo id="A2.SS3.1.p1.2.m2.3.4.3.3.3.2.2" stretchy="false" xref="A2.SS3.1.p1.2.m2.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.2.m2.3b"><apply id="A2.SS3.1.p1.2.m2.3.4.cmml" xref="A2.SS3.1.p1.2.m2.3.4"><eq id="A2.SS3.1.p1.2.m2.3.4.1.cmml" xref="A2.SS3.1.p1.2.m2.3.4.1"></eq><apply id="A2.SS3.1.p1.2.m2.3.4.2.cmml" xref="A2.SS3.1.p1.2.m2.3.4.2"><times id="A2.SS3.1.p1.2.m2.3.4.2.1.cmml" xref="A2.SS3.1.p1.2.m2.3.4.2.1"></times><ci id="A2.SS3.1.p1.2.m2.3.4.2.2.cmml" xref="A2.SS3.1.p1.2.m2.3.4.2.2">𝐻</ci><ci id="A2.SS3.1.p1.2.m2.1.1.cmml" xref="A2.SS3.1.p1.2.m2.1.1">𝑥</ci></apply><apply id="A2.SS3.1.p1.2.m2.3.4.3.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3"><minus id="A2.SS3.1.p1.2.m2.3.4.3.1.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.1"></minus><apply id="A2.SS3.1.p1.2.m2.3.4.3.2.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.2"><times id="A2.SS3.1.p1.2.m2.3.4.3.2.1.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.2.1"></times><ci id="A2.SS3.1.p1.2.m2.3.4.3.2.2.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.2.2">𝐺</ci><ci id="A2.SS3.1.p1.2.m2.2.2.cmml" xref="A2.SS3.1.p1.2.m2.2.2">𝑥</ci></apply><apply id="A2.SS3.1.p1.2.m2.3.4.3.3.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.3"><times id="A2.SS3.1.p1.2.m2.3.4.3.3.1.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.3.1"></times><ci id="A2.SS3.1.p1.2.m2.3.4.3.3.2.cmml" xref="A2.SS3.1.p1.2.m2.3.4.3.3.2">𝐹</ci><ci id="A2.SS3.1.p1.2.m2.3.3.cmml" xref="A2.SS3.1.p1.2.m2.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.2.m2.3c">H(x)=G(x)-F(x)</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.2.m2.3d">italic_H ( italic_x ) = italic_G ( italic_x ) - italic_F ( italic_x )</annotation></semantics></math>. Note first that since <math alttext="H(a)>0" class="ltx_Math" display="inline" id="A2.SS3.1.p1.3.m3.1"><semantics id="A2.SS3.1.p1.3.m3.1a"><mrow id="A2.SS3.1.p1.3.m3.1.2" xref="A2.SS3.1.p1.3.m3.1.2.cmml"><mrow id="A2.SS3.1.p1.3.m3.1.2.2" xref="A2.SS3.1.p1.3.m3.1.2.2.cmml"><mi id="A2.SS3.1.p1.3.m3.1.2.2.2" xref="A2.SS3.1.p1.3.m3.1.2.2.2.cmml">H</mi><mo id="A2.SS3.1.p1.3.m3.1.2.2.1" xref="A2.SS3.1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.3.m3.1.2.2.3.2" xref="A2.SS3.1.p1.3.m3.1.2.2.cmml"><mo id="A2.SS3.1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.3.m3.1.2.2.cmml">(</mo><mi id="A2.SS3.1.p1.3.m3.1.1" xref="A2.SS3.1.p1.3.m3.1.1.cmml">a</mi><mo id="A2.SS3.1.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.3.m3.1.2.1" xref="A2.SS3.1.p1.3.m3.1.2.1.cmml">></mo><mn id="A2.SS3.1.p1.3.m3.1.2.3" xref="A2.SS3.1.p1.3.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.3.m3.1b"><apply id="A2.SS3.1.p1.3.m3.1.2.cmml" xref="A2.SS3.1.p1.3.m3.1.2"><gt id="A2.SS3.1.p1.3.m3.1.2.1.cmml" xref="A2.SS3.1.p1.3.m3.1.2.1"></gt><apply id="A2.SS3.1.p1.3.m3.1.2.2.cmml" xref="A2.SS3.1.p1.3.m3.1.2.2"><times id="A2.SS3.1.p1.3.m3.1.2.2.1.cmml" xref="A2.SS3.1.p1.3.m3.1.2.2.1"></times><ci id="A2.SS3.1.p1.3.m3.1.2.2.2.cmml" xref="A2.SS3.1.p1.3.m3.1.2.2.2">𝐻</ci><ci id="A2.SS3.1.p1.3.m3.1.1.cmml" xref="A2.SS3.1.p1.3.m3.1.1">𝑎</ci></apply><cn id="A2.SS3.1.p1.3.m3.1.2.3.cmml" type="integer" xref="A2.SS3.1.p1.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.3.m3.1c">H(a)>0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.3.m3.1d">italic_H ( italic_a ) > 0</annotation></semantics></math> and <math alttext="H(b)<0" class="ltx_Math" display="inline" id="A2.SS3.1.p1.4.m4.1"><semantics id="A2.SS3.1.p1.4.m4.1a"><mrow id="A2.SS3.1.p1.4.m4.1.2" xref="A2.SS3.1.p1.4.m4.1.2.cmml"><mrow id="A2.SS3.1.p1.4.m4.1.2.2" xref="A2.SS3.1.p1.4.m4.1.2.2.cmml"><mi id="A2.SS3.1.p1.4.m4.1.2.2.2" xref="A2.SS3.1.p1.4.m4.1.2.2.2.cmml">H</mi><mo id="A2.SS3.1.p1.4.m4.1.2.2.1" xref="A2.SS3.1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.4.m4.1.2.2.3.2" xref="A2.SS3.1.p1.4.m4.1.2.2.cmml"><mo id="A2.SS3.1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.4.m4.1.2.2.cmml">(</mo><mi id="A2.SS3.1.p1.4.m4.1.1" xref="A2.SS3.1.p1.4.m4.1.1.cmml">b</mi><mo id="A2.SS3.1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.4.m4.1.2.1" xref="A2.SS3.1.p1.4.m4.1.2.1.cmml"><</mo><mn id="A2.SS3.1.p1.4.m4.1.2.3" xref="A2.SS3.1.p1.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.4.m4.1b"><apply id="A2.SS3.1.p1.4.m4.1.2.cmml" xref="A2.SS3.1.p1.4.m4.1.2"><lt id="A2.SS3.1.p1.4.m4.1.2.1.cmml" xref="A2.SS3.1.p1.4.m4.1.2.1"></lt><apply id="A2.SS3.1.p1.4.m4.1.2.2.cmml" xref="A2.SS3.1.p1.4.m4.1.2.2"><times id="A2.SS3.1.p1.4.m4.1.2.2.1.cmml" xref="A2.SS3.1.p1.4.m4.1.2.2.1"></times><ci id="A2.SS3.1.p1.4.m4.1.2.2.2.cmml" xref="A2.SS3.1.p1.4.m4.1.2.2.2">𝐻</ci><ci id="A2.SS3.1.p1.4.m4.1.1.cmml" xref="A2.SS3.1.p1.4.m4.1.1">𝑏</ci></apply><cn id="A2.SS3.1.p1.4.m4.1.2.3.cmml" type="integer" xref="A2.SS3.1.p1.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.4.m4.1c">H(b)<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.4.m4.1d">italic_H ( italic_b ) < 0</annotation></semantics></math>, by continuity and differentiability <math alttext="H" class="ltx_Math" display="inline" id="A2.SS3.1.p1.5.m5.1"><semantics id="A2.SS3.1.p1.5.m5.1a"><mi id="A2.SS3.1.p1.5.m5.1.1" xref="A2.SS3.1.p1.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.5.m5.1b"><ci id="A2.SS3.1.p1.5.m5.1.1.cmml" xref="A2.SS3.1.p1.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.5.m5.1d">italic_H</annotation></semantics></math> must cross 0 at some point <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="A2.SS3.1.p1.6.m6.1"><semantics id="A2.SS3.1.p1.6.m6.1a"><msub id="A2.SS3.1.p1.6.m6.1.1" xref="A2.SS3.1.p1.6.m6.1.1.cmml"><mi id="A2.SS3.1.p1.6.m6.1.1.2" xref="A2.SS3.1.p1.6.m6.1.1.2.cmml">τ</mi><mn id="A2.SS3.1.p1.6.m6.1.1.3" xref="A2.SS3.1.p1.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.6.m6.1b"><apply id="A2.SS3.1.p1.6.m6.1.1.cmml" xref="A2.SS3.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A2.SS3.1.p1.6.m6.1.1.1.cmml" xref="A2.SS3.1.p1.6.m6.1.1">subscript</csymbol><ci id="A2.SS3.1.p1.6.m6.1.1.2.cmml" xref="A2.SS3.1.p1.6.m6.1.1.2">𝜏</ci><cn id="A2.SS3.1.p1.6.m6.1.1.3.cmml" type="integer" xref="A2.SS3.1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.6.m6.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.6.m6.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="H^{\prime}(\tau_{0})<0" class="ltx_Math" display="inline" id="A2.SS3.1.p1.7.m7.1"><semantics id="A2.SS3.1.p1.7.m7.1a"><mrow id="A2.SS3.1.p1.7.m7.1.1" xref="A2.SS3.1.p1.7.m7.1.1.cmml"><mrow id="A2.SS3.1.p1.7.m7.1.1.1" xref="A2.SS3.1.p1.7.m7.1.1.1.cmml"><msup id="A2.SS3.1.p1.7.m7.1.1.1.3" xref="A2.SS3.1.p1.7.m7.1.1.1.3.cmml"><mi id="A2.SS3.1.p1.7.m7.1.1.1.3.2" xref="A2.SS3.1.p1.7.m7.1.1.1.3.2.cmml">H</mi><mo id="A2.SS3.1.p1.7.m7.1.1.1.3.3" xref="A2.SS3.1.p1.7.m7.1.1.1.3.3.cmml">′</mo></msup><mo id="A2.SS3.1.p1.7.m7.1.1.1.2" xref="A2.SS3.1.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="A2.SS3.1.p1.7.m7.1.1.1.1.1" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="A2.SS3.1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.2" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.3" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="A2.SS3.1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.7.m7.1.1.2" xref="A2.SS3.1.p1.7.m7.1.1.2.cmml"><</mo><mn id="A2.SS3.1.p1.7.m7.1.1.3" xref="A2.SS3.1.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.7.m7.1b"><apply id="A2.SS3.1.p1.7.m7.1.1.cmml" xref="A2.SS3.1.p1.7.m7.1.1"><lt id="A2.SS3.1.p1.7.m7.1.1.2.cmml" xref="A2.SS3.1.p1.7.m7.1.1.2"></lt><apply id="A2.SS3.1.p1.7.m7.1.1.1.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1"><times id="A2.SS3.1.p1.7.m7.1.1.1.2.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.2"></times><apply id="A2.SS3.1.p1.7.m7.1.1.1.3.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS3.1.p1.7.m7.1.1.1.3.1.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.3">superscript</csymbol><ci id="A2.SS3.1.p1.7.m7.1.1.1.3.2.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.3.2">𝐻</ci><ci id="A2.SS3.1.p1.7.m7.1.1.1.3.3.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.3.3">′</ci></apply><apply id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1">subscript</csymbol><ci id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.2">𝜏</ci><cn id="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.SS3.1.p1.7.m7.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="A2.SS3.1.p1.7.m7.1.1.3.cmml" type="integer" xref="A2.SS3.1.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.7.m7.1c">H^{\prime}(\tau_{0})<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.7.m7.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < 0</annotation></semantics></math>. Assume not: then <math alttext="H(x)\geq 0" class="ltx_Math" display="inline" id="A2.SS3.1.p1.8.m8.1"><semantics id="A2.SS3.1.p1.8.m8.1a"><mrow id="A2.SS3.1.p1.8.m8.1.2" xref="A2.SS3.1.p1.8.m8.1.2.cmml"><mrow id="A2.SS3.1.p1.8.m8.1.2.2" xref="A2.SS3.1.p1.8.m8.1.2.2.cmml"><mi id="A2.SS3.1.p1.8.m8.1.2.2.2" xref="A2.SS3.1.p1.8.m8.1.2.2.2.cmml">H</mi><mo id="A2.SS3.1.p1.8.m8.1.2.2.1" xref="A2.SS3.1.p1.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.8.m8.1.2.2.3.2" xref="A2.SS3.1.p1.8.m8.1.2.2.cmml"><mo id="A2.SS3.1.p1.8.m8.1.2.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.8.m8.1.2.2.cmml">(</mo><mi id="A2.SS3.1.p1.8.m8.1.1" xref="A2.SS3.1.p1.8.m8.1.1.cmml">x</mi><mo id="A2.SS3.1.p1.8.m8.1.2.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.8.m8.1.2.1" xref="A2.SS3.1.p1.8.m8.1.2.1.cmml">≥</mo><mn id="A2.SS3.1.p1.8.m8.1.2.3" xref="A2.SS3.1.p1.8.m8.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.8.m8.1b"><apply id="A2.SS3.1.p1.8.m8.1.2.cmml" xref="A2.SS3.1.p1.8.m8.1.2"><geq id="A2.SS3.1.p1.8.m8.1.2.1.cmml" xref="A2.SS3.1.p1.8.m8.1.2.1"></geq><apply id="A2.SS3.1.p1.8.m8.1.2.2.cmml" xref="A2.SS3.1.p1.8.m8.1.2.2"><times id="A2.SS3.1.p1.8.m8.1.2.2.1.cmml" xref="A2.SS3.1.p1.8.m8.1.2.2.1"></times><ci id="A2.SS3.1.p1.8.m8.1.2.2.2.cmml" xref="A2.SS3.1.p1.8.m8.1.2.2.2">𝐻</ci><ci id="A2.SS3.1.p1.8.m8.1.1.cmml" xref="A2.SS3.1.p1.8.m8.1.1">𝑥</ci></apply><cn id="A2.SS3.1.p1.8.m8.1.2.3.cmml" type="integer" xref="A2.SS3.1.p1.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.8.m8.1c">H(x)\geq 0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.8.m8.1d">italic_H ( italic_x ) ≥ 0</annotation></semantics></math> for all <math alttext="x" class="ltx_Math" display="inline" id="A2.SS3.1.p1.9.m9.1"><semantics id="A2.SS3.1.p1.9.m9.1a"><mi id="A2.SS3.1.p1.9.m9.1.1" xref="A2.SS3.1.p1.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.9.m9.1b"><ci id="A2.SS3.1.p1.9.m9.1.1.cmml" xref="A2.SS3.1.p1.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.9.m9.1d">italic_x</annotation></semantics></math>, which contradicts the fact that <math alttext="H(x)<0" class="ltx_Math" display="inline" id="A2.SS3.1.p1.10.m10.1"><semantics id="A2.SS3.1.p1.10.m10.1a"><mrow id="A2.SS3.1.p1.10.m10.1.2" xref="A2.SS3.1.p1.10.m10.1.2.cmml"><mrow id="A2.SS3.1.p1.10.m10.1.2.2" xref="A2.SS3.1.p1.10.m10.1.2.2.cmml"><mi id="A2.SS3.1.p1.10.m10.1.2.2.2" xref="A2.SS3.1.p1.10.m10.1.2.2.2.cmml">H</mi><mo id="A2.SS3.1.p1.10.m10.1.2.2.1" xref="A2.SS3.1.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS3.1.p1.10.m10.1.2.2.3.2" xref="A2.SS3.1.p1.10.m10.1.2.2.cmml"><mo id="A2.SS3.1.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="A2.SS3.1.p1.10.m10.1.2.2.cmml">(</mo><mi id="A2.SS3.1.p1.10.m10.1.1" xref="A2.SS3.1.p1.10.m10.1.1.cmml">x</mi><mo id="A2.SS3.1.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="A2.SS3.1.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.1.p1.10.m10.1.2.1" xref="A2.SS3.1.p1.10.m10.1.2.1.cmml"><</mo><mn id="A2.SS3.1.p1.10.m10.1.2.3" xref="A2.SS3.1.p1.10.m10.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.10.m10.1b"><apply id="A2.SS3.1.p1.10.m10.1.2.cmml" xref="A2.SS3.1.p1.10.m10.1.2"><lt id="A2.SS3.1.p1.10.m10.1.2.1.cmml" xref="A2.SS3.1.p1.10.m10.1.2.1"></lt><apply id="A2.SS3.1.p1.10.m10.1.2.2.cmml" xref="A2.SS3.1.p1.10.m10.1.2.2"><times id="A2.SS3.1.p1.10.m10.1.2.2.1.cmml" xref="A2.SS3.1.p1.10.m10.1.2.2.1"></times><ci id="A2.SS3.1.p1.10.m10.1.2.2.2.cmml" xref="A2.SS3.1.p1.10.m10.1.2.2.2">𝐻</ci><ci id="A2.SS3.1.p1.10.m10.1.1.cmml" xref="A2.SS3.1.p1.10.m10.1.1">𝑥</ci></apply><cn id="A2.SS3.1.p1.10.m10.1.2.3.cmml" type="integer" xref="A2.SS3.1.p1.10.m10.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.10.m10.1c">H(x)<0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.10.m10.1d">italic_H ( italic_x ) < 0</annotation></semantics></math> for <math alttext="x\geq b" class="ltx_Math" display="inline" id="A2.SS3.1.p1.11.m11.1"><semantics id="A2.SS3.1.p1.11.m11.1a"><mrow id="A2.SS3.1.p1.11.m11.1.1" xref="A2.SS3.1.p1.11.m11.1.1.cmml"><mi id="A2.SS3.1.p1.11.m11.1.1.2" xref="A2.SS3.1.p1.11.m11.1.1.2.cmml">x</mi><mo id="A2.SS3.1.p1.11.m11.1.1.1" xref="A2.SS3.1.p1.11.m11.1.1.1.cmml">≥</mo><mi id="A2.SS3.1.p1.11.m11.1.1.3" xref="A2.SS3.1.p1.11.m11.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.11.m11.1b"><apply id="A2.SS3.1.p1.11.m11.1.1.cmml" xref="A2.SS3.1.p1.11.m11.1.1"><geq id="A2.SS3.1.p1.11.m11.1.1.1.cmml" xref="A2.SS3.1.p1.11.m11.1.1.1"></geq><ci id="A2.SS3.1.p1.11.m11.1.1.2.cmml" xref="A2.SS3.1.p1.11.m11.1.1.2">𝑥</ci><ci id="A2.SS3.1.p1.11.m11.1.1.3.cmml" xref="A2.SS3.1.p1.11.m11.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.11.m11.1c">x\geq b</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.11.m11.1d">italic_x ≥ italic_b</annotation></semantics></math>. It follows by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem4" title="Theorem 4. ‣ 3.2 Dynamics ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a> that <math alttext="\tau_{0}" class="ltx_Math" display="inline" id="A2.SS3.1.p1.12.m12.1"><semantics id="A2.SS3.1.p1.12.m12.1a"><msub id="A2.SS3.1.p1.12.m12.1.1" xref="A2.SS3.1.p1.12.m12.1.1.cmml"><mi id="A2.SS3.1.p1.12.m12.1.1.2" xref="A2.SS3.1.p1.12.m12.1.1.2.cmml">τ</mi><mn id="A2.SS3.1.p1.12.m12.1.1.3" xref="A2.SS3.1.p1.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.1.p1.12.m12.1b"><apply id="A2.SS3.1.p1.12.m12.1.1.cmml" xref="A2.SS3.1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="A2.SS3.1.p1.12.m12.1.1.1.cmml" xref="A2.SS3.1.p1.12.m12.1.1">subscript</csymbol><ci id="A2.SS3.1.p1.12.m12.1.1.2.cmml" xref="A2.SS3.1.p1.12.m12.1.1.2">𝜏</ci><cn id="A2.SS3.1.p1.12.m12.1.1.3.cmml" type="integer" xref="A2.SS3.1.p1.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.1.p1.12.m12.1c">\tau_{0}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.1.p1.12.m12.1d">italic_τ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a stable equilibrium.</p> </div> <div class="ltx_para" id="A2.SS3.2.p2"> <ol class="ltx_enumerate" id="A2.I1"> <li class="ltx_item" id="A2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="A2.I1.i1.p1"> <p class="ltx_p" id="A2.I1.i1.p1.9">Now, WLOG we pick the equilibrium <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.I1.i1.p1.1.m1.1"><semantics id="A2.I1.i1.p1.1.m1.1a"><msub id="A2.I1.i1.p1.1.m1.1.1" xref="A2.I1.i1.p1.1.m1.1.1.cmml"><mi id="A2.I1.i1.p1.1.m1.1.1.2" xref="A2.I1.i1.p1.1.m1.1.1.2.cmml">τ</mi><mn id="A2.I1.i1.p1.1.m1.1.1.3" xref="A2.I1.i1.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.1.m1.1b"><apply id="A2.I1.i1.p1.1.m1.1.1.cmml" xref="A2.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.I1.i1.p1.1.m1.1.1.1.cmml" xref="A2.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="A2.I1.i1.p1.1.m1.1.1.2.cmml" xref="A2.I1.i1.p1.1.m1.1.1.2">𝜏</ci><cn id="A2.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.1.m1.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.1.m1.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> which is closest to <math alttext="a" 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">a</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">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.2.m2.1d">italic_a</annotation></semantics></math>. By Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> and since <math alttext="H(x)" class="ltx_Math" display="inline" id="A2.I1.i1.p1.3.m3.1"><semantics id="A2.I1.i1.p1.3.m3.1a"><mrow id="A2.I1.i1.p1.3.m3.1.2" xref="A2.I1.i1.p1.3.m3.1.2.cmml"><mi id="A2.I1.i1.p1.3.m3.1.2.2" xref="A2.I1.i1.p1.3.m3.1.2.2.cmml">H</mi><mo id="A2.I1.i1.p1.3.m3.1.2.1" xref="A2.I1.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="A2.I1.i1.p1.3.m3.1.2.3.2" xref="A2.I1.i1.p1.3.m3.1.2.cmml"><mo id="A2.I1.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="A2.I1.i1.p1.3.m3.1.2.cmml">(</mo><mi id="A2.I1.i1.p1.3.m3.1.1" xref="A2.I1.i1.p1.3.m3.1.1.cmml">x</mi><mo id="A2.I1.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="A2.I1.i1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.3.m3.1b"><apply id="A2.I1.i1.p1.3.m3.1.2.cmml" xref="A2.I1.i1.p1.3.m3.1.2"><times id="A2.I1.i1.p1.3.m3.1.2.1.cmml" xref="A2.I1.i1.p1.3.m3.1.2.1"></times><ci id="A2.I1.i1.p1.3.m3.1.2.2.cmml" xref="A2.I1.i1.p1.3.m3.1.2.2">𝐻</ci><ci id="A2.I1.i1.p1.3.m3.1.1.cmml" xref="A2.I1.i1.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.3.m3.1c">H(x)</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.3.m3.1d">italic_H ( italic_x )</annotation></semantics></math> is continuous, <math alttext="H(x)>0" class="ltx_Math" display="inline" id="A2.I1.i1.p1.4.m4.1"><semantics id="A2.I1.i1.p1.4.m4.1a"><mrow id="A2.I1.i1.p1.4.m4.1.2" xref="A2.I1.i1.p1.4.m4.1.2.cmml"><mrow id="A2.I1.i1.p1.4.m4.1.2.2" xref="A2.I1.i1.p1.4.m4.1.2.2.cmml"><mi id="A2.I1.i1.p1.4.m4.1.2.2.2" xref="A2.I1.i1.p1.4.m4.1.2.2.2.cmml">H</mi><mo id="A2.I1.i1.p1.4.m4.1.2.2.1" xref="A2.I1.i1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="A2.I1.i1.p1.4.m4.1.2.2.3.2" xref="A2.I1.i1.p1.4.m4.1.2.2.cmml"><mo id="A2.I1.i1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="A2.I1.i1.p1.4.m4.1.2.2.cmml">(</mo><mi id="A2.I1.i1.p1.4.m4.1.1" xref="A2.I1.i1.p1.4.m4.1.1.cmml">x</mi><mo id="A2.I1.i1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="A2.I1.i1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.I1.i1.p1.4.m4.1.2.1" xref="A2.I1.i1.p1.4.m4.1.2.1.cmml">></mo><mn id="A2.I1.i1.p1.4.m4.1.2.3" xref="A2.I1.i1.p1.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.4.m4.1b"><apply id="A2.I1.i1.p1.4.m4.1.2.cmml" xref="A2.I1.i1.p1.4.m4.1.2"><gt id="A2.I1.i1.p1.4.m4.1.2.1.cmml" xref="A2.I1.i1.p1.4.m4.1.2.1"></gt><apply id="A2.I1.i1.p1.4.m4.1.2.2.cmml" xref="A2.I1.i1.p1.4.m4.1.2.2"><times id="A2.I1.i1.p1.4.m4.1.2.2.1.cmml" xref="A2.I1.i1.p1.4.m4.1.2.2.1"></times><ci id="A2.I1.i1.p1.4.m4.1.2.2.2.cmml" xref="A2.I1.i1.p1.4.m4.1.2.2.2">𝐻</ci><ci id="A2.I1.i1.p1.4.m4.1.1.cmml" xref="A2.I1.i1.p1.4.m4.1.1">𝑥</ci></apply><cn id="A2.I1.i1.p1.4.m4.1.2.3.cmml" type="integer" xref="A2.I1.i1.p1.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.4.m4.1c">H(x)>0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.4.m4.1d">italic_H ( italic_x ) > 0</annotation></semantics></math> for <math alttext="x<\tau_{1}" class="ltx_Math" display="inline" id="A2.I1.i1.p1.5.m5.1"><semantics id="A2.I1.i1.p1.5.m5.1a"><mrow id="A2.I1.i1.p1.5.m5.1.1" xref="A2.I1.i1.p1.5.m5.1.1.cmml"><mi id="A2.I1.i1.p1.5.m5.1.1.2" xref="A2.I1.i1.p1.5.m5.1.1.2.cmml">x</mi><mo id="A2.I1.i1.p1.5.m5.1.1.1" xref="A2.I1.i1.p1.5.m5.1.1.1.cmml"><</mo><msub id="A2.I1.i1.p1.5.m5.1.1.3" xref="A2.I1.i1.p1.5.m5.1.1.3.cmml"><mi id="A2.I1.i1.p1.5.m5.1.1.3.2" xref="A2.I1.i1.p1.5.m5.1.1.3.2.cmml">τ</mi><mn id="A2.I1.i1.p1.5.m5.1.1.3.3" xref="A2.I1.i1.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.5.m5.1b"><apply id="A2.I1.i1.p1.5.m5.1.1.cmml" xref="A2.I1.i1.p1.5.m5.1.1"><lt id="A2.I1.i1.p1.5.m5.1.1.1.cmml" xref="A2.I1.i1.p1.5.m5.1.1.1"></lt><ci id="A2.I1.i1.p1.5.m5.1.1.2.cmml" xref="A2.I1.i1.p1.5.m5.1.1.2">𝑥</ci><apply id="A2.I1.i1.p1.5.m5.1.1.3.cmml" xref="A2.I1.i1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i1.p1.5.m5.1.1.3.1.cmml" xref="A2.I1.i1.p1.5.m5.1.1.3">subscript</csymbol><ci id="A2.I1.i1.p1.5.m5.1.1.3.2.cmml" xref="A2.I1.i1.p1.5.m5.1.1.3.2">𝜏</ci><cn id="A2.I1.i1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="A2.I1.i1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.5.m5.1c">x<\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.5.m5.1d">italic_x < italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. It follows that <math alttext="H^{\prime}(\tau_{1})\leq 0" class="ltx_Math" display="inline" id="A2.I1.i1.p1.6.m6.1"><semantics id="A2.I1.i1.p1.6.m6.1a"><mrow id="A2.I1.i1.p1.6.m6.1.1" xref="A2.I1.i1.p1.6.m6.1.1.cmml"><mrow id="A2.I1.i1.p1.6.m6.1.1.1" xref="A2.I1.i1.p1.6.m6.1.1.1.cmml"><msup id="A2.I1.i1.p1.6.m6.1.1.1.3" xref="A2.I1.i1.p1.6.m6.1.1.1.3.cmml"><mi id="A2.I1.i1.p1.6.m6.1.1.1.3.2" xref="A2.I1.i1.p1.6.m6.1.1.1.3.2.cmml">H</mi><mo id="A2.I1.i1.p1.6.m6.1.1.1.3.3" xref="A2.I1.i1.p1.6.m6.1.1.1.3.3.cmml">′</mo></msup><mo id="A2.I1.i1.p1.6.m6.1.1.1.2" xref="A2.I1.i1.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="A2.I1.i1.p1.6.m6.1.1.1.1.1" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="A2.I1.i1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.2" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.3" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A2.I1.i1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.I1.i1.p1.6.m6.1.1.2" xref="A2.I1.i1.p1.6.m6.1.1.2.cmml">≤</mo><mn id="A2.I1.i1.p1.6.m6.1.1.3" xref="A2.I1.i1.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.6.m6.1b"><apply id="A2.I1.i1.p1.6.m6.1.1.cmml" xref="A2.I1.i1.p1.6.m6.1.1"><leq id="A2.I1.i1.p1.6.m6.1.1.2.cmml" xref="A2.I1.i1.p1.6.m6.1.1.2"></leq><apply id="A2.I1.i1.p1.6.m6.1.1.1.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1"><times id="A2.I1.i1.p1.6.m6.1.1.1.2.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.2"></times><apply id="A2.I1.i1.p1.6.m6.1.1.1.3.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i1.p1.6.m6.1.1.1.3.1.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.3">superscript</csymbol><ci id="A2.I1.i1.p1.6.m6.1.1.1.3.2.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.3.2">𝐻</ci><ci id="A2.I1.i1.p1.6.m6.1.1.1.3.3.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.3.3">′</ci></apply><apply id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.2">𝜏</ci><cn id="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.6.m6.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="A2.I1.i1.p1.6.m6.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.6.m6.1c">H^{\prime}(\tau_{1})\leq 0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.6.m6.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≤ 0</annotation></semantics></math>. Then <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.I1.i1.p1.7.m7.1"><semantics id="A2.I1.i1.p1.7.m7.1a"><msub id="A2.I1.i1.p1.7.m7.1.1" xref="A2.I1.i1.p1.7.m7.1.1.cmml"><mi id="A2.I1.i1.p1.7.m7.1.1.2" xref="A2.I1.i1.p1.7.m7.1.1.2.cmml">τ</mi><mn id="A2.I1.i1.p1.7.m7.1.1.3" xref="A2.I1.i1.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.7.m7.1b"><apply id="A2.I1.i1.p1.7.m7.1.1.cmml" xref="A2.I1.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A2.I1.i1.p1.7.m7.1.1.1.cmml" xref="A2.I1.i1.p1.7.m7.1.1">subscript</csymbol><ci id="A2.I1.i1.p1.7.m7.1.1.2.cmml" xref="A2.I1.i1.p1.7.m7.1.1.2">𝜏</ci><cn id="A2.I1.i1.p1.7.m7.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.7.m7.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.7.m7.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is stable, at least on the left (if <math alttext="H^{\prime}(\tau_{1})=0" class="ltx_Math" display="inline" id="A2.I1.i1.p1.8.m8.1"><semantics id="A2.I1.i1.p1.8.m8.1a"><mrow id="A2.I1.i1.p1.8.m8.1.1" xref="A2.I1.i1.p1.8.m8.1.1.cmml"><mrow id="A2.I1.i1.p1.8.m8.1.1.1" xref="A2.I1.i1.p1.8.m8.1.1.1.cmml"><msup id="A2.I1.i1.p1.8.m8.1.1.1.3" xref="A2.I1.i1.p1.8.m8.1.1.1.3.cmml"><mi id="A2.I1.i1.p1.8.m8.1.1.1.3.2" xref="A2.I1.i1.p1.8.m8.1.1.1.3.2.cmml">H</mi><mo id="A2.I1.i1.p1.8.m8.1.1.1.3.3" xref="A2.I1.i1.p1.8.m8.1.1.1.3.3.cmml">′</mo></msup><mo id="A2.I1.i1.p1.8.m8.1.1.1.2" xref="A2.I1.i1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="A2.I1.i1.p1.8.m8.1.1.1.1.1" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="A2.I1.i1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.2" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.3" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A2.I1.i1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.I1.i1.p1.8.m8.1.1.2" xref="A2.I1.i1.p1.8.m8.1.1.2.cmml">=</mo><mn id="A2.I1.i1.p1.8.m8.1.1.3" xref="A2.I1.i1.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.8.m8.1b"><apply id="A2.I1.i1.p1.8.m8.1.1.cmml" xref="A2.I1.i1.p1.8.m8.1.1"><eq id="A2.I1.i1.p1.8.m8.1.1.2.cmml" xref="A2.I1.i1.p1.8.m8.1.1.2"></eq><apply id="A2.I1.i1.p1.8.m8.1.1.1.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1"><times id="A2.I1.i1.p1.8.m8.1.1.1.2.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.2"></times><apply id="A2.I1.i1.p1.8.m8.1.1.1.3.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i1.p1.8.m8.1.1.1.3.1.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.3">superscript</csymbol><ci id="A2.I1.i1.p1.8.m8.1.1.1.3.2.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.3.2">𝐻</ci><ci id="A2.I1.i1.p1.8.m8.1.1.1.3.3.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.3.3">′</ci></apply><apply id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.2">𝜏</ci><cn id="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.8.m8.1.1.1.1.1.1.3">1</cn></apply></apply><cn id="A2.I1.i1.p1.8.m8.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.8.m8.1c">H^{\prime}(\tau_{1})=0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.8.m8.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, then <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.I1.i1.p1.9.m9.1"><semantics id="A2.I1.i1.p1.9.m9.1a"><msub id="A2.I1.i1.p1.9.m9.1.1" xref="A2.I1.i1.p1.9.m9.1.1.cmml"><mi id="A2.I1.i1.p1.9.m9.1.1.2" xref="A2.I1.i1.p1.9.m9.1.1.2.cmml">τ</mi><mn id="A2.I1.i1.p1.9.m9.1.1.3" xref="A2.I1.i1.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i1.p1.9.m9.1b"><apply id="A2.I1.i1.p1.9.m9.1.1.cmml" xref="A2.I1.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A2.I1.i1.p1.9.m9.1.1.1.cmml" xref="A2.I1.i1.p1.9.m9.1.1">subscript</csymbol><ci id="A2.I1.i1.p1.9.m9.1.1.2.cmml" xref="A2.I1.i1.p1.9.m9.1.1.2">𝜏</ci><cn id="A2.I1.i1.p1.9.m9.1.1.3.cmml" type="integer" xref="A2.I1.i1.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i1.p1.9.m9.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i1.p1.9.m9.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is stable on the left and unstable on the right).</p> </div> </li> <li class="ltx_item" id="A2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="A2.I1.i2.p1"> <p class="ltx_p" id="A2.I1.i2.p1.10">Pick the equilibrium <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A2.I1.i2.p1.1.m1.1"><semantics id="A2.I1.i2.p1.1.m1.1a"><msub id="A2.I1.i2.p1.1.m1.1.1" xref="A2.I1.i2.p1.1.m1.1.1.cmml"><mi id="A2.I1.i2.p1.1.m1.1.1.2" xref="A2.I1.i2.p1.1.m1.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.1.m1.1.1.3" xref="A2.I1.i2.p1.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.1.m1.1b"><apply id="A2.I1.i2.p1.1.m1.1.1.cmml" xref="A2.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.1.m1.1.1.1.cmml" xref="A2.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.1.m1.1.1.2.cmml" xref="A2.I1.i2.p1.1.m1.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.1.m1.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.1.m1.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> which is closest to <math alttext="b" 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">b</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">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.2.m2.1d">italic_b</annotation></semantics></math> (this could be the same as the <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.I1.i2.p1.3.m3.1"><semantics id="A2.I1.i2.p1.3.m3.1a"><msub id="A2.I1.i2.p1.3.m3.1.1" xref="A2.I1.i2.p1.3.m3.1.1.cmml"><mi id="A2.I1.i2.p1.3.m3.1.1.2" xref="A2.I1.i2.p1.3.m3.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.3.m3.1.1.3" xref="A2.I1.i2.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.3.m3.1b"><apply id="A2.I1.i2.p1.3.m3.1.1.cmml" xref="A2.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.3.m3.1.1.1.cmml" xref="A2.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.3.m3.1.1.2.cmml" xref="A2.I1.i2.p1.3.m3.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.3.m3.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.3.m3.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we chose in the previous step). By Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> and since <math alttext="H(x)" class="ltx_Math" display="inline" id="A2.I1.i2.p1.4.m4.1"><semantics id="A2.I1.i2.p1.4.m4.1a"><mrow id="A2.I1.i2.p1.4.m4.1.2" xref="A2.I1.i2.p1.4.m4.1.2.cmml"><mi id="A2.I1.i2.p1.4.m4.1.2.2" xref="A2.I1.i2.p1.4.m4.1.2.2.cmml">H</mi><mo id="A2.I1.i2.p1.4.m4.1.2.1" xref="A2.I1.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="A2.I1.i2.p1.4.m4.1.2.3.2" xref="A2.I1.i2.p1.4.m4.1.2.cmml"><mo id="A2.I1.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="A2.I1.i2.p1.4.m4.1.2.cmml">(</mo><mi id="A2.I1.i2.p1.4.m4.1.1" xref="A2.I1.i2.p1.4.m4.1.1.cmml">x</mi><mo id="A2.I1.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="A2.I1.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.4.m4.1b"><apply id="A2.I1.i2.p1.4.m4.1.2.cmml" xref="A2.I1.i2.p1.4.m4.1.2"><times id="A2.I1.i2.p1.4.m4.1.2.1.cmml" xref="A2.I1.i2.p1.4.m4.1.2.1"></times><ci id="A2.I1.i2.p1.4.m4.1.2.2.cmml" xref="A2.I1.i2.p1.4.m4.1.2.2">𝐻</ci><ci id="A2.I1.i2.p1.4.m4.1.1.cmml" xref="A2.I1.i2.p1.4.m4.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.4.m4.1c">H(x)</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.4.m4.1d">italic_H ( italic_x )</annotation></semantics></math> is continuous, <math alttext="H(x)<0" class="ltx_Math" display="inline" id="A2.I1.i2.p1.5.m5.1"><semantics id="A2.I1.i2.p1.5.m5.1a"><mrow id="A2.I1.i2.p1.5.m5.1.2" xref="A2.I1.i2.p1.5.m5.1.2.cmml"><mrow id="A2.I1.i2.p1.5.m5.1.2.2" xref="A2.I1.i2.p1.5.m5.1.2.2.cmml"><mi id="A2.I1.i2.p1.5.m5.1.2.2.2" xref="A2.I1.i2.p1.5.m5.1.2.2.2.cmml">H</mi><mo id="A2.I1.i2.p1.5.m5.1.2.2.1" xref="A2.I1.i2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="A2.I1.i2.p1.5.m5.1.2.2.3.2" xref="A2.I1.i2.p1.5.m5.1.2.2.cmml"><mo id="A2.I1.i2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="A2.I1.i2.p1.5.m5.1.2.2.cmml">(</mo><mi id="A2.I1.i2.p1.5.m5.1.1" xref="A2.I1.i2.p1.5.m5.1.1.cmml">x</mi><mo id="A2.I1.i2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="A2.I1.i2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.I1.i2.p1.5.m5.1.2.1" xref="A2.I1.i2.p1.5.m5.1.2.1.cmml"><</mo><mn id="A2.I1.i2.p1.5.m5.1.2.3" xref="A2.I1.i2.p1.5.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.5.m5.1b"><apply id="A2.I1.i2.p1.5.m5.1.2.cmml" xref="A2.I1.i2.p1.5.m5.1.2"><lt id="A2.I1.i2.p1.5.m5.1.2.1.cmml" xref="A2.I1.i2.p1.5.m5.1.2.1"></lt><apply id="A2.I1.i2.p1.5.m5.1.2.2.cmml" xref="A2.I1.i2.p1.5.m5.1.2.2"><times id="A2.I1.i2.p1.5.m5.1.2.2.1.cmml" xref="A2.I1.i2.p1.5.m5.1.2.2.1"></times><ci id="A2.I1.i2.p1.5.m5.1.2.2.2.cmml" xref="A2.I1.i2.p1.5.m5.1.2.2.2">𝐻</ci><ci id="A2.I1.i2.p1.5.m5.1.1.cmml" xref="A2.I1.i2.p1.5.m5.1.1">𝑥</ci></apply><cn id="A2.I1.i2.p1.5.m5.1.2.3.cmml" type="integer" xref="A2.I1.i2.p1.5.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.5.m5.1c">H(x)<0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.5.m5.1d">italic_H ( italic_x ) < 0</annotation></semantics></math> for <math alttext="x>\tau_{2}" class="ltx_Math" display="inline" id="A2.I1.i2.p1.6.m6.1"><semantics id="A2.I1.i2.p1.6.m6.1a"><mrow id="A2.I1.i2.p1.6.m6.1.1" xref="A2.I1.i2.p1.6.m6.1.1.cmml"><mi id="A2.I1.i2.p1.6.m6.1.1.2" xref="A2.I1.i2.p1.6.m6.1.1.2.cmml">x</mi><mo id="A2.I1.i2.p1.6.m6.1.1.1" xref="A2.I1.i2.p1.6.m6.1.1.1.cmml">></mo><msub id="A2.I1.i2.p1.6.m6.1.1.3" xref="A2.I1.i2.p1.6.m6.1.1.3.cmml"><mi id="A2.I1.i2.p1.6.m6.1.1.3.2" xref="A2.I1.i2.p1.6.m6.1.1.3.2.cmml">τ</mi><mn id="A2.I1.i2.p1.6.m6.1.1.3.3" xref="A2.I1.i2.p1.6.m6.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.6.m6.1b"><apply id="A2.I1.i2.p1.6.m6.1.1.cmml" xref="A2.I1.i2.p1.6.m6.1.1"><gt id="A2.I1.i2.p1.6.m6.1.1.1.cmml" xref="A2.I1.i2.p1.6.m6.1.1.1"></gt><ci id="A2.I1.i2.p1.6.m6.1.1.2.cmml" xref="A2.I1.i2.p1.6.m6.1.1.2">𝑥</ci><apply id="A2.I1.i2.p1.6.m6.1.1.3.cmml" xref="A2.I1.i2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i2.p1.6.m6.1.1.3.1.cmml" xref="A2.I1.i2.p1.6.m6.1.1.3">subscript</csymbol><ci id="A2.I1.i2.p1.6.m6.1.1.3.2.cmml" xref="A2.I1.i2.p1.6.m6.1.1.3.2">𝜏</ci><cn id="A2.I1.i2.p1.6.m6.1.1.3.3.cmml" type="integer" xref="A2.I1.i2.p1.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.6.m6.1c">x>\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.6.m6.1d">italic_x > italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. It follows taht <math alttext="H^{\prime}(\tau_{2})\leq 0" class="ltx_Math" display="inline" id="A2.I1.i2.p1.7.m7.1"><semantics id="A2.I1.i2.p1.7.m7.1a"><mrow id="A2.I1.i2.p1.7.m7.1.1" xref="A2.I1.i2.p1.7.m7.1.1.cmml"><mrow id="A2.I1.i2.p1.7.m7.1.1.1" xref="A2.I1.i2.p1.7.m7.1.1.1.cmml"><msup id="A2.I1.i2.p1.7.m7.1.1.1.3" xref="A2.I1.i2.p1.7.m7.1.1.1.3.cmml"><mi id="A2.I1.i2.p1.7.m7.1.1.1.3.2" xref="A2.I1.i2.p1.7.m7.1.1.1.3.2.cmml">H</mi><mo id="A2.I1.i2.p1.7.m7.1.1.1.3.3" xref="A2.I1.i2.p1.7.m7.1.1.1.3.3.cmml">′</mo></msup><mo id="A2.I1.i2.p1.7.m7.1.1.1.2" xref="A2.I1.i2.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="A2.I1.i2.p1.7.m7.1.1.1.1.1" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="A2.I1.i2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.2" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.3" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="A2.I1.i2.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.I1.i2.p1.7.m7.1.1.2" xref="A2.I1.i2.p1.7.m7.1.1.2.cmml">≤</mo><mn id="A2.I1.i2.p1.7.m7.1.1.3" xref="A2.I1.i2.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.7.m7.1b"><apply id="A2.I1.i2.p1.7.m7.1.1.cmml" xref="A2.I1.i2.p1.7.m7.1.1"><leq id="A2.I1.i2.p1.7.m7.1.1.2.cmml" xref="A2.I1.i2.p1.7.m7.1.1.2"></leq><apply id="A2.I1.i2.p1.7.m7.1.1.1.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1"><times id="A2.I1.i2.p1.7.m7.1.1.1.2.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.2"></times><apply id="A2.I1.i2.p1.7.m7.1.1.1.3.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i2.p1.7.m7.1.1.1.3.1.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.3">superscript</csymbol><ci id="A2.I1.i2.p1.7.m7.1.1.1.3.2.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.3.2">𝐻</ci><ci id="A2.I1.i2.p1.7.m7.1.1.1.3.3.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.3.3">′</ci></apply><apply id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.7.m7.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="A2.I1.i2.p1.7.m7.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.7.m7.1c">H^{\prime}(\tau_{2})\leq 0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.7.m7.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≤ 0</annotation></semantics></math>. Then <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A2.I1.i2.p1.8.m8.1"><semantics id="A2.I1.i2.p1.8.m8.1a"><msub id="A2.I1.i2.p1.8.m8.1.1" xref="A2.I1.i2.p1.8.m8.1.1.cmml"><mi id="A2.I1.i2.p1.8.m8.1.1.2" xref="A2.I1.i2.p1.8.m8.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.8.m8.1.1.3" xref="A2.I1.i2.p1.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.8.m8.1b"><apply id="A2.I1.i2.p1.8.m8.1.1.cmml" xref="A2.I1.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.8.m8.1.1.1.cmml" xref="A2.I1.i2.p1.8.m8.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.8.m8.1.1.2.cmml" xref="A2.I1.i2.p1.8.m8.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.8.m8.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.8.m8.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.8.m8.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is stable, at least on the right (if <math alttext="H^{\prime}(\tau_{2})=0" class="ltx_Math" display="inline" id="A2.I1.i2.p1.9.m9.1"><semantics id="A2.I1.i2.p1.9.m9.1a"><mrow id="A2.I1.i2.p1.9.m9.1.1" xref="A2.I1.i2.p1.9.m9.1.1.cmml"><mrow id="A2.I1.i2.p1.9.m9.1.1.1" xref="A2.I1.i2.p1.9.m9.1.1.1.cmml"><msup id="A2.I1.i2.p1.9.m9.1.1.1.3" xref="A2.I1.i2.p1.9.m9.1.1.1.3.cmml"><mi id="A2.I1.i2.p1.9.m9.1.1.1.3.2" xref="A2.I1.i2.p1.9.m9.1.1.1.3.2.cmml">H</mi><mo id="A2.I1.i2.p1.9.m9.1.1.1.3.3" xref="A2.I1.i2.p1.9.m9.1.1.1.3.3.cmml">′</mo></msup><mo id="A2.I1.i2.p1.9.m9.1.1.1.2" xref="A2.I1.i2.p1.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="A2.I1.i2.p1.9.m9.1.1.1.1.1" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml"><mo id="A2.I1.i2.p1.9.m9.1.1.1.1.1.2" stretchy="false" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml"><mi id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.2" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.3" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="A2.I1.i2.p1.9.m9.1.1.1.1.1.3" stretchy="false" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A2.I1.i2.p1.9.m9.1.1.2" xref="A2.I1.i2.p1.9.m9.1.1.2.cmml">=</mo><mn id="A2.I1.i2.p1.9.m9.1.1.3" xref="A2.I1.i2.p1.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.9.m9.1b"><apply id="A2.I1.i2.p1.9.m9.1.1.cmml" xref="A2.I1.i2.p1.9.m9.1.1"><eq id="A2.I1.i2.p1.9.m9.1.1.2.cmml" xref="A2.I1.i2.p1.9.m9.1.1.2"></eq><apply id="A2.I1.i2.p1.9.m9.1.1.1.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1"><times id="A2.I1.i2.p1.9.m9.1.1.1.2.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.2"></times><apply id="A2.I1.i2.p1.9.m9.1.1.1.3.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="A2.I1.i2.p1.9.m9.1.1.1.3.1.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.3">superscript</csymbol><ci id="A2.I1.i2.p1.9.m9.1.1.1.3.2.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.3.2">𝐻</ci><ci id="A2.I1.i2.p1.9.m9.1.1.1.3.3.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.3.3">′</ci></apply><apply id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.1.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.2.cmml" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.9.m9.1.1.1.1.1.1.3">2</cn></apply></apply><cn id="A2.I1.i2.p1.9.m9.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.9.m9.1c">H^{\prime}(\tau_{2})=0</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.9.m9.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, then <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A2.I1.i2.p1.10.m10.1"><semantics id="A2.I1.i2.p1.10.m10.1a"><msub id="A2.I1.i2.p1.10.m10.1.1" xref="A2.I1.i2.p1.10.m10.1.1.cmml"><mi id="A2.I1.i2.p1.10.m10.1.1.2" xref="A2.I1.i2.p1.10.m10.1.1.2.cmml">τ</mi><mn id="A2.I1.i2.p1.10.m10.1.1.3" xref="A2.I1.i2.p1.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.I1.i2.p1.10.m10.1b"><apply id="A2.I1.i2.p1.10.m10.1.1.cmml" xref="A2.I1.i2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="A2.I1.i2.p1.10.m10.1.1.1.cmml" xref="A2.I1.i2.p1.10.m10.1.1">subscript</csymbol><ci id="A2.I1.i2.p1.10.m10.1.1.2.cmml" xref="A2.I1.i2.p1.10.m10.1.1.2">𝜏</ci><cn id="A2.I1.i2.p1.10.m10.1.1.3.cmml" type="integer" xref="A2.I1.i2.p1.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.I1.i2.p1.10.m10.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.I1.i2.p1.10.m10.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is stable on the right and unstable on the left).</p> </div> </li> </ol> <p class="ltx_p" id="A2.SS3.2.p2.10">We know by Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> that <math alttext="H(x)\neq 0" class="ltx_Math" display="inline" id="A2.SS3.2.p2.1.m1.1"><semantics id="A2.SS3.2.p2.1.m1.1a"><mrow id="A2.SS3.2.p2.1.m1.1.2" xref="A2.SS3.2.p2.1.m1.1.2.cmml"><mrow id="A2.SS3.2.p2.1.m1.1.2.2" xref="A2.SS3.2.p2.1.m1.1.2.2.cmml"><mi id="A2.SS3.2.p2.1.m1.1.2.2.2" xref="A2.SS3.2.p2.1.m1.1.2.2.2.cmml">H</mi><mo id="A2.SS3.2.p2.1.m1.1.2.2.1" xref="A2.SS3.2.p2.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="A2.SS3.2.p2.1.m1.1.2.2.3.2" xref="A2.SS3.2.p2.1.m1.1.2.2.cmml"><mo id="A2.SS3.2.p2.1.m1.1.2.2.3.2.1" stretchy="false" xref="A2.SS3.2.p2.1.m1.1.2.2.cmml">(</mo><mi id="A2.SS3.2.p2.1.m1.1.1" xref="A2.SS3.2.p2.1.m1.1.1.cmml">x</mi><mo id="A2.SS3.2.p2.1.m1.1.2.2.3.2.2" stretchy="false" xref="A2.SS3.2.p2.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="A2.SS3.2.p2.1.m1.1.2.1" xref="A2.SS3.2.p2.1.m1.1.2.1.cmml">≠</mo><mn id="A2.SS3.2.p2.1.m1.1.2.3" xref="A2.SS3.2.p2.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.1.m1.1b"><apply id="A2.SS3.2.p2.1.m1.1.2.cmml" xref="A2.SS3.2.p2.1.m1.1.2"><neq id="A2.SS3.2.p2.1.m1.1.2.1.cmml" xref="A2.SS3.2.p2.1.m1.1.2.1"></neq><apply id="A2.SS3.2.p2.1.m1.1.2.2.cmml" xref="A2.SS3.2.p2.1.m1.1.2.2"><times id="A2.SS3.2.p2.1.m1.1.2.2.1.cmml" xref="A2.SS3.2.p2.1.m1.1.2.2.1"></times><ci id="A2.SS3.2.p2.1.m1.1.2.2.2.cmml" xref="A2.SS3.2.p2.1.m1.1.2.2.2">𝐻</ci><ci id="A2.SS3.2.p2.1.m1.1.1.cmml" xref="A2.SS3.2.p2.1.m1.1.1">𝑥</ci></apply><cn id="A2.SS3.2.p2.1.m1.1.2.3.cmml" type="integer" xref="A2.SS3.2.p2.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.1.m1.1c">H(x)\neq 0</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.1.m1.1d">italic_H ( italic_x ) ≠ 0</annotation></semantics></math> for <math alttext="x\notin[a,b]" class="ltx_Math" display="inline" id="A2.SS3.2.p2.2.m2.2"><semantics id="A2.SS3.2.p2.2.m2.2a"><mrow id="A2.SS3.2.p2.2.m2.2.3" xref="A2.SS3.2.p2.2.m2.2.3.cmml"><mi id="A2.SS3.2.p2.2.m2.2.3.2" xref="A2.SS3.2.p2.2.m2.2.3.2.cmml">x</mi><mo id="A2.SS3.2.p2.2.m2.2.3.1" xref="A2.SS3.2.p2.2.m2.2.3.1.cmml">∉</mo><mrow id="A2.SS3.2.p2.2.m2.2.3.3.2" xref="A2.SS3.2.p2.2.m2.2.3.3.1.cmml"><mo id="A2.SS3.2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="A2.SS3.2.p2.2.m2.2.3.3.1.cmml">[</mo><mi id="A2.SS3.2.p2.2.m2.1.1" xref="A2.SS3.2.p2.2.m2.1.1.cmml">a</mi><mo id="A2.SS3.2.p2.2.m2.2.3.3.2.2" xref="A2.SS3.2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="A2.SS3.2.p2.2.m2.2.2" xref="A2.SS3.2.p2.2.m2.2.2.cmml">b</mi><mo id="A2.SS3.2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="A2.SS3.2.p2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.2.m2.2b"><apply id="A2.SS3.2.p2.2.m2.2.3.cmml" xref="A2.SS3.2.p2.2.m2.2.3"><notin id="A2.SS3.2.p2.2.m2.2.3.1.cmml" xref="A2.SS3.2.p2.2.m2.2.3.1"></notin><ci id="A2.SS3.2.p2.2.m2.2.3.2.cmml" xref="A2.SS3.2.p2.2.m2.2.3.2">𝑥</ci><interval closure="closed" id="A2.SS3.2.p2.2.m2.2.3.3.1.cmml" xref="A2.SS3.2.p2.2.m2.2.3.3.2"><ci id="A2.SS3.2.p2.2.m2.1.1.cmml" xref="A2.SS3.2.p2.2.m2.1.1">𝑎</ci><ci id="A2.SS3.2.p2.2.m2.2.2.cmml" xref="A2.SS3.2.p2.2.m2.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.2.m2.2c">x\notin[a,b]</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.2.m2.2d">italic_x ∉ [ italic_a , italic_b ]</annotation></semantics></math>, so that no equilibria occur outside the interval <math alttext="I" class="ltx_Math" display="inline" id="A2.SS3.2.p2.3.m3.1"><semantics id="A2.SS3.2.p2.3.m3.1a"><mi id="A2.SS3.2.p2.3.m3.1.1" xref="A2.SS3.2.p2.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.3.m3.1b"><ci id="A2.SS3.2.p2.3.m3.1.1.cmml" xref="A2.SS3.2.p2.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.3.m3.1d">italic_I</annotation></semantics></math> other than <math alttext="\pm\infty" class="ltx_Math" display="inline" id="A2.SS3.2.p2.4.m4.1"><semantics id="A2.SS3.2.p2.4.m4.1a"><mrow id="A2.SS3.2.p2.4.m4.1.1" xref="A2.SS3.2.p2.4.m4.1.1.cmml"><mo id="A2.SS3.2.p2.4.m4.1.1a" xref="A2.SS3.2.p2.4.m4.1.1.cmml">±</mo><mi id="A2.SS3.2.p2.4.m4.1.1.2" mathvariant="normal" xref="A2.SS3.2.p2.4.m4.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.4.m4.1b"><apply id="A2.SS3.2.p2.4.m4.1.1.cmml" xref="A2.SS3.2.p2.4.m4.1.1"><csymbol cd="latexml" id="A2.SS3.2.p2.4.m4.1.1.1.cmml" xref="A2.SS3.2.p2.4.m4.1.1">plus-or-minus</csymbol><infinity id="A2.SS3.2.p2.4.m4.1.1.2.cmml" xref="A2.SS3.2.p2.4.m4.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.4.m4.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.4.m4.1d">± ∞</annotation></semantics></math>. Since both <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.SS3.2.p2.5.m5.1"><semantics id="A2.SS3.2.p2.5.m5.1a"><msub id="A2.SS3.2.p2.5.m5.1.1" xref="A2.SS3.2.p2.5.m5.1.1.cmml"><mi id="A2.SS3.2.p2.5.m5.1.1.2" xref="A2.SS3.2.p2.5.m5.1.1.2.cmml">τ</mi><mn id="A2.SS3.2.p2.5.m5.1.1.3" xref="A2.SS3.2.p2.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.5.m5.1b"><apply id="A2.SS3.2.p2.5.m5.1.1.cmml" xref="A2.SS3.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A2.SS3.2.p2.5.m5.1.1.1.cmml" xref="A2.SS3.2.p2.5.m5.1.1">subscript</csymbol><ci id="A2.SS3.2.p2.5.m5.1.1.2.cmml" xref="A2.SS3.2.p2.5.m5.1.1.2">𝜏</ci><cn id="A2.SS3.2.p2.5.m5.1.1.3.cmml" type="integer" xref="A2.SS3.2.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.5.m5.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.5.m5.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A2.SS3.2.p2.6.m6.1"><semantics id="A2.SS3.2.p2.6.m6.1a"><msub id="A2.SS3.2.p2.6.m6.1.1" xref="A2.SS3.2.p2.6.m6.1.1.cmml"><mi id="A2.SS3.2.p2.6.m6.1.1.2" xref="A2.SS3.2.p2.6.m6.1.1.2.cmml">τ</mi><mn id="A2.SS3.2.p2.6.m6.1.1.3" xref="A2.SS3.2.p2.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.6.m6.1b"><apply id="A2.SS3.2.p2.6.m6.1.1.cmml" xref="A2.SS3.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A2.SS3.2.p2.6.m6.1.1.1.cmml" xref="A2.SS3.2.p2.6.m6.1.1">subscript</csymbol><ci id="A2.SS3.2.p2.6.m6.1.1.2.cmml" xref="A2.SS3.2.p2.6.m6.1.1.2">𝜏</ci><cn id="A2.SS3.2.p2.6.m6.1.1.3.cmml" type="integer" xref="A2.SS3.2.p2.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.6.m6.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.6.m6.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> (or just <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.SS3.2.p2.7.m7.1"><semantics id="A2.SS3.2.p2.7.m7.1a"><msub id="A2.SS3.2.p2.7.m7.1.1" xref="A2.SS3.2.p2.7.m7.1.1.cmml"><mi id="A2.SS3.2.p2.7.m7.1.1.2" xref="A2.SS3.2.p2.7.m7.1.1.2.cmml">τ</mi><mn id="A2.SS3.2.p2.7.m7.1.1.3" xref="A2.SS3.2.p2.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.7.m7.1b"><apply id="A2.SS3.2.p2.7.m7.1.1.cmml" xref="A2.SS3.2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A2.SS3.2.p2.7.m7.1.1.1.cmml" xref="A2.SS3.2.p2.7.m7.1.1">subscript</csymbol><ci id="A2.SS3.2.p2.7.m7.1.1.2.cmml" xref="A2.SS3.2.p2.7.m7.1.1.2">𝜏</ci><cn id="A2.SS3.2.p2.7.m7.1.1.3.cmml" type="integer" xref="A2.SS3.2.p2.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.7.m7.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.7.m7.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, if they are the same equilibrium) are stable (at least to the left for <math alttext="\tau_{1}" class="ltx_Math" display="inline" id="A2.SS3.2.p2.8.m8.1"><semantics id="A2.SS3.2.p2.8.m8.1a"><msub id="A2.SS3.2.p2.8.m8.1.1" xref="A2.SS3.2.p2.8.m8.1.1.cmml"><mi id="A2.SS3.2.p2.8.m8.1.1.2" xref="A2.SS3.2.p2.8.m8.1.1.2.cmml">τ</mi><mn id="A2.SS3.2.p2.8.m8.1.1.3" xref="A2.SS3.2.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.8.m8.1b"><apply id="A2.SS3.2.p2.8.m8.1.1.cmml" xref="A2.SS3.2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="A2.SS3.2.p2.8.m8.1.1.1.cmml" xref="A2.SS3.2.p2.8.m8.1.1">subscript</csymbol><ci id="A2.SS3.2.p2.8.m8.1.1.2.cmml" xref="A2.SS3.2.p2.8.m8.1.1.2">𝜏</ci><cn id="A2.SS3.2.p2.8.m8.1.1.3.cmml" type="integer" xref="A2.SS3.2.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.8.m8.1c">\tau_{1}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.8.m8.1d">italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and right for <math alttext="\tau_{2}" class="ltx_Math" display="inline" id="A2.SS3.2.p2.9.m9.1"><semantics id="A2.SS3.2.p2.9.m9.1a"><msub id="A2.SS3.2.p2.9.m9.1.1" xref="A2.SS3.2.p2.9.m9.1.1.cmml"><mi id="A2.SS3.2.p2.9.m9.1.1.2" xref="A2.SS3.2.p2.9.m9.1.1.2.cmml">τ</mi><mn id="A2.SS3.2.p2.9.m9.1.1.3" xref="A2.SS3.2.p2.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.9.m9.1b"><apply id="A2.SS3.2.p2.9.m9.1.1.cmml" xref="A2.SS3.2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="A2.SS3.2.p2.9.m9.1.1.1.cmml" xref="A2.SS3.2.p2.9.m9.1.1">subscript</csymbol><ci id="A2.SS3.2.p2.9.m9.1.1.2.cmml" xref="A2.SS3.2.p2.9.m9.1.1.2">𝜏</ci><cn id="A2.SS3.2.p2.9.m9.1.1.3.cmml" type="integer" xref="A2.SS3.2.p2.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.9.m9.1c">\tau_{2}</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.9.m9.1d">italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>), it is topologically necessary that the uninformative equilibria <math alttext="\pm\infty" class="ltx_Math" display="inline" id="A2.SS3.2.p2.10.m10.1"><semantics id="A2.SS3.2.p2.10.m10.1a"><mrow id="A2.SS3.2.p2.10.m10.1.1" xref="A2.SS3.2.p2.10.m10.1.1.cmml"><mo id="A2.SS3.2.p2.10.m10.1.1a" xref="A2.SS3.2.p2.10.m10.1.1.cmml">±</mo><mi id="A2.SS3.2.p2.10.m10.1.1.2" mathvariant="normal" xref="A2.SS3.2.p2.10.m10.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS3.2.p2.10.m10.1b"><apply id="A2.SS3.2.p2.10.m10.1.1.cmml" xref="A2.SS3.2.p2.10.m10.1.1"><csymbol cd="latexml" id="A2.SS3.2.p2.10.m10.1.1.1.cmml" xref="A2.SS3.2.p2.10.m10.1.1">plus-or-minus</csymbol><infinity id="A2.SS3.2.p2.10.m10.1.1.2.cmml" xref="A2.SS3.2.p2.10.m10.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.2.p2.10.m10.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.2.p2.10.m10.1d">± ∞</annotation></semantics></math> are unstable. ∎</p> </div> </div> <div class="ltx_para" id="A2.SS3.p2"> <p class="ltx_p" id="A2.SS3.p2.1">In natural settings, we would expect Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> to hold. Specifically, conditioning on a small enough signal, one would expect the probability of other small signals to increase relative to the prior. Conditioning on a large signal, one would expect decreased probability of much smaller signals and thus a smaller value of <math alttext="G" class="ltx_Math" display="inline" id="A2.SS3.p2.1.m1.1"><semantics id="A2.SS3.p2.1.m1.1a"><mi id="A2.SS3.p2.1.m1.1.1" xref="A2.SS3.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A2.SS3.p2.1.m1.1b"><ci id="A2.SS3.p2.1.m1.1.1.cmml" xref="A2.SS3.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS3.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A2.SS3.p2.1.m1.1d">italic_G</annotation></semantics></math> relative to the prior. Ultimately, then, when conditioning on a signal leads to local sensitivity, we would expect the dynamics in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition10" title="Proposition 10. ‣ B.3 Dynamics Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">10</span></a> to hold. In particular, note that Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition10" title="Proposition 10. ‣ B.3 Dynamics Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">10</span></a> applies to the model in § <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S6" title="6 Experiments ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>, when the distributions are skewed. Even under complicated, multi-modal information structures, we expect tail behavior to often still satisfy Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcondition2" title="Condition 2. ‣ B.2 Equilibrium Characterization Generalization ‣ Appendix B Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a>, so that there exists a stable equilibrium in the interval.</p> </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>Omitted Proofs for DMI</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.1">As discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S4" title="4 Determinant-based Mutual Information (DMI) Mechanism ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">4</span></a>, the behavior of DMI under joint-task strategies, which allow agents to map all 4 signals to all 4 reports simultaneously, differs between the traditional binary signal model and our real-valued signal model. In particular, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> shows that truthfulness is a Bayes Nash among all joint-task strategies for the binary signal model, but below we show that this property fails for the real-valued signal model.</p> </div> <div class="ltx_para" id="A3.p2"> <p class="ltx_p" id="A3.p2.4">To get some intuition for these results, first consider the binary signal model and an agent who receives signals <math alttext="H,L,L,L" class="ltx_Math" display="inline" id="A3.p2.1.m1.4"><semantics id="A3.p2.1.m1.4a"><mrow id="A3.p2.1.m1.4.5.2" xref="A3.p2.1.m1.4.5.1.cmml"><mi id="A3.p2.1.m1.1.1" xref="A3.p2.1.m1.1.1.cmml">H</mi><mo id="A3.p2.1.m1.4.5.2.1" xref="A3.p2.1.m1.4.5.1.cmml">,</mo><mi id="A3.p2.1.m1.2.2" xref="A3.p2.1.m1.2.2.cmml">L</mi><mo id="A3.p2.1.m1.4.5.2.2" xref="A3.p2.1.m1.4.5.1.cmml">,</mo><mi id="A3.p2.1.m1.3.3" xref="A3.p2.1.m1.3.3.cmml">L</mi><mo id="A3.p2.1.m1.4.5.2.3" xref="A3.p2.1.m1.4.5.1.cmml">,</mo><mi id="A3.p2.1.m1.4.4" xref="A3.p2.1.m1.4.4.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.1.m1.4b"><list id="A3.p2.1.m1.4.5.1.cmml" xref="A3.p2.1.m1.4.5.2"><ci id="A3.p2.1.m1.1.1.cmml" xref="A3.p2.1.m1.1.1">𝐻</ci><ci id="A3.p2.1.m1.2.2.cmml" xref="A3.p2.1.m1.2.2">𝐿</ci><ci id="A3.p2.1.m1.3.3.cmml" xref="A3.p2.1.m1.3.3">𝐿</ci><ci id="A3.p2.1.m1.4.4.cmml" xref="A3.p2.1.m1.4.4">𝐿</ci></list></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.1.m1.4c">H,L,L,L</annotation><annotation encoding="application/x-llamapun" id="A3.p2.1.m1.4d">italic_H , italic_L , italic_L , italic_L</annotation></semantics></math>. From the form of the mechanism, if they report truthfully, their score will be zero deterministically, since <math alttext="M_{34}" class="ltx_Math" display="inline" id="A3.p2.2.m2.1"><semantics id="A3.p2.2.m2.1a"><msub id="A3.p2.2.m2.1.1" xref="A3.p2.2.m2.1.1.cmml"><mi id="A3.p2.2.m2.1.1.2" xref="A3.p2.2.m2.1.1.2.cmml">M</mi><mn id="A3.p2.2.m2.1.1.3" xref="A3.p2.2.m2.1.1.3.cmml">34</mn></msub><annotation-xml encoding="MathML-Content" id="A3.p2.2.m2.1b"><apply id="A3.p2.2.m2.1.1.cmml" xref="A3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A3.p2.2.m2.1.1.1.cmml" xref="A3.p2.2.m2.1.1">subscript</csymbol><ci id="A3.p2.2.m2.1.1.2.cmml" xref="A3.p2.2.m2.1.1.2">𝑀</ci><cn id="A3.p2.2.m2.1.1.3.cmml" type="integer" xref="A3.p2.2.m2.1.1.3">34</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.2.m2.1c">M_{34}</annotation><annotation encoding="application/x-llamapun" id="A3.p2.2.m2.1d">italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT</annotation></semantics></math> will have rank 1. Thus, they might consider deviating, say flipping the last <math alttext="L" class="ltx_Math" display="inline" id="A3.p2.3.m3.1"><semantics id="A3.p2.3.m3.1a"><mi id="A3.p2.3.m3.1.1" xref="A3.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A3.p2.3.m3.1b"><ci id="A3.p2.3.m3.1.1.cmml" xref="A3.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="A3.p2.3.m3.1d">italic_L</annotation></semantics></math> to <math alttext="H" class="ltx_Math" display="inline" id="A3.p2.4.m4.1"><semantics id="A3.p2.4.m4.1a"><mi id="A3.p2.4.m4.1.1" xref="A3.p2.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A3.p2.4.m4.1b"><ci id="A3.p2.4.m4.1.1.cmml" xref="A3.p2.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="A3.p2.4.m4.1d">italic_H</annotation></semantics></math>. It turns out doing so keeps their expected score at zero, since the other agent is equally likely to align or misalign with their reports. The consistency of the other agent’s strategies is crucial for this last fact; given any slight difference in their strategy for tasks 3 and 4, the first agent would deviate from truthfulness.</p> </div> <div class="ltx_para" id="A3.p3"> <p class="ltx_p" id="A3.p3.13">It is precisely this sort of imbalance that easily arises in the real-valued signal model. Consider the Gaussian model with <math alttext="a=b=1" class="ltx_Math" display="inline" id="A3.p3.1.m1.1"><semantics id="A3.p3.1.m1.1a"><mrow id="A3.p3.1.m1.1.1" xref="A3.p3.1.m1.1.1.cmml"><mi id="A3.p3.1.m1.1.1.2" xref="A3.p3.1.m1.1.1.2.cmml">a</mi><mo id="A3.p3.1.m1.1.1.3" xref="A3.p3.1.m1.1.1.3.cmml">=</mo><mi id="A3.p3.1.m1.1.1.4" xref="A3.p3.1.m1.1.1.4.cmml">b</mi><mo id="A3.p3.1.m1.1.1.5" xref="A3.p3.1.m1.1.1.5.cmml">=</mo><mn id="A3.p3.1.m1.1.1.6" xref="A3.p3.1.m1.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.1.m1.1b"><apply id="A3.p3.1.m1.1.1.cmml" xref="A3.p3.1.m1.1.1"><and id="A3.p3.1.m1.1.1a.cmml" xref="A3.p3.1.m1.1.1"></and><apply id="A3.p3.1.m1.1.1b.cmml" xref="A3.p3.1.m1.1.1"><eq id="A3.p3.1.m1.1.1.3.cmml" xref="A3.p3.1.m1.1.1.3"></eq><ci id="A3.p3.1.m1.1.1.2.cmml" xref="A3.p3.1.m1.1.1.2">𝑎</ci><ci id="A3.p3.1.m1.1.1.4.cmml" xref="A3.p3.1.m1.1.1.4">𝑏</ci></apply><apply id="A3.p3.1.m1.1.1c.cmml" xref="A3.p3.1.m1.1.1"><eq id="A3.p3.1.m1.1.1.5.cmml" xref="A3.p3.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A3.p3.1.m1.1.1.4.cmml" id="A3.p3.1.m1.1.1d.cmml" xref="A3.p3.1.m1.1.1"></share><cn id="A3.p3.1.m1.1.1.6.cmml" type="integer" xref="A3.p3.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.1.m1.1c">a=b=1</annotation><annotation encoding="application/x-llamapun" id="A3.p3.1.m1.1d">italic_a = italic_b = 1</annotation></semantics></math> and <math alttext="\rho=1/2" class="ltx_Math" display="inline" id="A3.p3.2.m2.1"><semantics id="A3.p3.2.m2.1a"><mrow id="A3.p3.2.m2.1.1" xref="A3.p3.2.m2.1.1.cmml"><mi id="A3.p3.2.m2.1.1.2" xref="A3.p3.2.m2.1.1.2.cmml">ρ</mi><mo id="A3.p3.2.m2.1.1.1" xref="A3.p3.2.m2.1.1.1.cmml">=</mo><mrow id="A3.p3.2.m2.1.1.3" xref="A3.p3.2.m2.1.1.3.cmml"><mn id="A3.p3.2.m2.1.1.3.2" xref="A3.p3.2.m2.1.1.3.2.cmml">1</mn><mo id="A3.p3.2.m2.1.1.3.1" xref="A3.p3.2.m2.1.1.3.1.cmml">/</mo><mn id="A3.p3.2.m2.1.1.3.3" xref="A3.p3.2.m2.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.2.m2.1b"><apply id="A3.p3.2.m2.1.1.cmml" xref="A3.p3.2.m2.1.1"><eq id="A3.p3.2.m2.1.1.1.cmml" xref="A3.p3.2.m2.1.1.1"></eq><ci id="A3.p3.2.m2.1.1.2.cmml" xref="A3.p3.2.m2.1.1.2">𝜌</ci><apply id="A3.p3.2.m2.1.1.3.cmml" xref="A3.p3.2.m2.1.1.3"><divide id="A3.p3.2.m2.1.1.3.1.cmml" xref="A3.p3.2.m2.1.1.3.1"></divide><cn id="A3.p3.2.m2.1.1.3.2.cmml" type="integer" xref="A3.p3.2.m2.1.1.3.2">1</cn><cn id="A3.p3.2.m2.1.1.3.3.cmml" type="integer" xref="A3.p3.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.2.m2.1c">\rho=1/2</annotation><annotation encoding="application/x-llamapun" id="A3.p3.2.m2.1d">italic_ρ = 1 / 2</annotation></semantics></math>, at the equilibrium threshold <math alttext="\tau=0" class="ltx_Math" display="inline" id="A3.p3.3.m3.1"><semantics id="A3.p3.3.m3.1a"><mrow id="A3.p3.3.m3.1.1" xref="A3.p3.3.m3.1.1.cmml"><mi id="A3.p3.3.m3.1.1.2" xref="A3.p3.3.m3.1.1.2.cmml">τ</mi><mo id="A3.p3.3.m3.1.1.1" xref="A3.p3.3.m3.1.1.1.cmml">=</mo><mn id="A3.p3.3.m3.1.1.3" xref="A3.p3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.3.m3.1b"><apply id="A3.p3.3.m3.1.1.cmml" xref="A3.p3.3.m3.1.1"><eq id="A3.p3.3.m3.1.1.1.cmml" xref="A3.p3.3.m3.1.1.1"></eq><ci id="A3.p3.3.m3.1.1.2.cmml" xref="A3.p3.3.m3.1.1.2">𝜏</ci><cn id="A3.p3.3.m3.1.1.3.cmml" type="integer" xref="A3.p3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.3.m3.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A3.p3.3.m3.1d">italic_τ = 0</annotation></semantics></math>. Suppose the agent receives <math alttext="x_{1}=10" class="ltx_Math" display="inline" id="A3.p3.4.m4.1"><semantics id="A3.p3.4.m4.1a"><mrow id="A3.p3.4.m4.1.1" xref="A3.p3.4.m4.1.1.cmml"><msub id="A3.p3.4.m4.1.1.2" xref="A3.p3.4.m4.1.1.2.cmml"><mi id="A3.p3.4.m4.1.1.2.2" xref="A3.p3.4.m4.1.1.2.2.cmml">x</mi><mn id="A3.p3.4.m4.1.1.2.3" xref="A3.p3.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="A3.p3.4.m4.1.1.1" xref="A3.p3.4.m4.1.1.1.cmml">=</mo><mn id="A3.p3.4.m4.1.1.3" xref="A3.p3.4.m4.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.4.m4.1b"><apply id="A3.p3.4.m4.1.1.cmml" xref="A3.p3.4.m4.1.1"><eq id="A3.p3.4.m4.1.1.1.cmml" xref="A3.p3.4.m4.1.1.1"></eq><apply id="A3.p3.4.m4.1.1.2.cmml" xref="A3.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="A3.p3.4.m4.1.1.2.1.cmml" xref="A3.p3.4.m4.1.1.2">subscript</csymbol><ci id="A3.p3.4.m4.1.1.2.2.cmml" xref="A3.p3.4.m4.1.1.2.2">𝑥</ci><cn id="A3.p3.4.m4.1.1.2.3.cmml" type="integer" xref="A3.p3.4.m4.1.1.2.3">1</cn></apply><cn id="A3.p3.4.m4.1.1.3.cmml" type="integer" xref="A3.p3.4.m4.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.4.m4.1c">x_{1}=10</annotation><annotation encoding="application/x-llamapun" id="A3.p3.4.m4.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 10</annotation></semantics></math>, <math alttext="x_{2}=-10" class="ltx_Math" display="inline" id="A3.p3.5.m5.1"><semantics id="A3.p3.5.m5.1a"><mrow id="A3.p3.5.m5.1.1" xref="A3.p3.5.m5.1.1.cmml"><msub id="A3.p3.5.m5.1.1.2" xref="A3.p3.5.m5.1.1.2.cmml"><mi id="A3.p3.5.m5.1.1.2.2" xref="A3.p3.5.m5.1.1.2.2.cmml">x</mi><mn id="A3.p3.5.m5.1.1.2.3" xref="A3.p3.5.m5.1.1.2.3.cmml">2</mn></msub><mo id="A3.p3.5.m5.1.1.1" xref="A3.p3.5.m5.1.1.1.cmml">=</mo><mrow id="A3.p3.5.m5.1.1.3" xref="A3.p3.5.m5.1.1.3.cmml"><mo id="A3.p3.5.m5.1.1.3a" xref="A3.p3.5.m5.1.1.3.cmml">−</mo><mn id="A3.p3.5.m5.1.1.3.2" xref="A3.p3.5.m5.1.1.3.2.cmml">10</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.5.m5.1b"><apply id="A3.p3.5.m5.1.1.cmml" xref="A3.p3.5.m5.1.1"><eq id="A3.p3.5.m5.1.1.1.cmml" xref="A3.p3.5.m5.1.1.1"></eq><apply id="A3.p3.5.m5.1.1.2.cmml" xref="A3.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="A3.p3.5.m5.1.1.2.1.cmml" xref="A3.p3.5.m5.1.1.2">subscript</csymbol><ci id="A3.p3.5.m5.1.1.2.2.cmml" xref="A3.p3.5.m5.1.1.2.2">𝑥</ci><cn id="A3.p3.5.m5.1.1.2.3.cmml" type="integer" xref="A3.p3.5.m5.1.1.2.3">2</cn></apply><apply id="A3.p3.5.m5.1.1.3.cmml" xref="A3.p3.5.m5.1.1.3"><minus id="A3.p3.5.m5.1.1.3.1.cmml" xref="A3.p3.5.m5.1.1.3"></minus><cn id="A3.p3.5.m5.1.1.3.2.cmml" type="integer" xref="A3.p3.5.m5.1.1.3.2">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.5.m5.1c">x_{2}=-10</annotation><annotation encoding="application/x-llamapun" id="A3.p3.5.m5.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - 10</annotation></semantics></math>, <math alttext="x_{3}=-10" class="ltx_Math" display="inline" id="A3.p3.6.m6.1"><semantics id="A3.p3.6.m6.1a"><mrow id="A3.p3.6.m6.1.1" xref="A3.p3.6.m6.1.1.cmml"><msub id="A3.p3.6.m6.1.1.2" xref="A3.p3.6.m6.1.1.2.cmml"><mi id="A3.p3.6.m6.1.1.2.2" xref="A3.p3.6.m6.1.1.2.2.cmml">x</mi><mn id="A3.p3.6.m6.1.1.2.3" xref="A3.p3.6.m6.1.1.2.3.cmml">3</mn></msub><mo id="A3.p3.6.m6.1.1.1" xref="A3.p3.6.m6.1.1.1.cmml">=</mo><mrow id="A3.p3.6.m6.1.1.3" xref="A3.p3.6.m6.1.1.3.cmml"><mo id="A3.p3.6.m6.1.1.3a" xref="A3.p3.6.m6.1.1.3.cmml">−</mo><mn id="A3.p3.6.m6.1.1.3.2" xref="A3.p3.6.m6.1.1.3.2.cmml">10</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.6.m6.1b"><apply id="A3.p3.6.m6.1.1.cmml" xref="A3.p3.6.m6.1.1"><eq id="A3.p3.6.m6.1.1.1.cmml" xref="A3.p3.6.m6.1.1.1"></eq><apply id="A3.p3.6.m6.1.1.2.cmml" xref="A3.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="A3.p3.6.m6.1.1.2.1.cmml" xref="A3.p3.6.m6.1.1.2">subscript</csymbol><ci id="A3.p3.6.m6.1.1.2.2.cmml" xref="A3.p3.6.m6.1.1.2.2">𝑥</ci><cn id="A3.p3.6.m6.1.1.2.3.cmml" type="integer" xref="A3.p3.6.m6.1.1.2.3">3</cn></apply><apply id="A3.p3.6.m6.1.1.3.cmml" xref="A3.p3.6.m6.1.1.3"><minus id="A3.p3.6.m6.1.1.3.1.cmml" xref="A3.p3.6.m6.1.1.3"></minus><cn id="A3.p3.6.m6.1.1.3.2.cmml" type="integer" xref="A3.p3.6.m6.1.1.3.2">10</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.6.m6.1c">x_{3}=-10</annotation><annotation encoding="application/x-llamapun" id="A3.p3.6.m6.1d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 10</annotation></semantics></math>, and <math alttext="x_{4}=-1" class="ltx_Math" display="inline" id="A3.p3.7.m7.1"><semantics id="A3.p3.7.m7.1a"><mrow id="A3.p3.7.m7.1.1" xref="A3.p3.7.m7.1.1.cmml"><msub id="A3.p3.7.m7.1.1.2" xref="A3.p3.7.m7.1.1.2.cmml"><mi id="A3.p3.7.m7.1.1.2.2" xref="A3.p3.7.m7.1.1.2.2.cmml">x</mi><mn id="A3.p3.7.m7.1.1.2.3" xref="A3.p3.7.m7.1.1.2.3.cmml">4</mn></msub><mo id="A3.p3.7.m7.1.1.1" xref="A3.p3.7.m7.1.1.1.cmml">=</mo><mrow id="A3.p3.7.m7.1.1.3" xref="A3.p3.7.m7.1.1.3.cmml"><mo id="A3.p3.7.m7.1.1.3a" xref="A3.p3.7.m7.1.1.3.cmml">−</mo><mn id="A3.p3.7.m7.1.1.3.2" xref="A3.p3.7.m7.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.7.m7.1b"><apply id="A3.p3.7.m7.1.1.cmml" xref="A3.p3.7.m7.1.1"><eq id="A3.p3.7.m7.1.1.1.cmml" xref="A3.p3.7.m7.1.1.1"></eq><apply id="A3.p3.7.m7.1.1.2.cmml" xref="A3.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="A3.p3.7.m7.1.1.2.1.cmml" xref="A3.p3.7.m7.1.1.2">subscript</csymbol><ci id="A3.p3.7.m7.1.1.2.2.cmml" xref="A3.p3.7.m7.1.1.2.2">𝑥</ci><cn id="A3.p3.7.m7.1.1.2.3.cmml" type="integer" xref="A3.p3.7.m7.1.1.2.3">4</cn></apply><apply id="A3.p3.7.m7.1.1.3.cmml" xref="A3.p3.7.m7.1.1.3"><minus id="A3.p3.7.m7.1.1.3.1.cmml" xref="A3.p3.7.m7.1.1.3"></minus><cn id="A3.p3.7.m7.1.1.3.2.cmml" type="integer" xref="A3.p3.7.m7.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.7.m7.1c">x_{4}=-1</annotation><annotation encoding="application/x-llamapun" id="A3.p3.7.m7.1d">italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = - 1</annotation></semantics></math>. Reporting truthfully, <math alttext="(H,L,L,L)" class="ltx_Math" display="inline" id="A3.p3.8.m8.4"><semantics id="A3.p3.8.m8.4a"><mrow id="A3.p3.8.m8.4.5.2" xref="A3.p3.8.m8.4.5.1.cmml"><mo id="A3.p3.8.m8.4.5.2.1" stretchy="false" xref="A3.p3.8.m8.4.5.1.cmml">(</mo><mi id="A3.p3.8.m8.1.1" xref="A3.p3.8.m8.1.1.cmml">H</mi><mo id="A3.p3.8.m8.4.5.2.2" xref="A3.p3.8.m8.4.5.1.cmml">,</mo><mi id="A3.p3.8.m8.2.2" xref="A3.p3.8.m8.2.2.cmml">L</mi><mo id="A3.p3.8.m8.4.5.2.3" xref="A3.p3.8.m8.4.5.1.cmml">,</mo><mi id="A3.p3.8.m8.3.3" xref="A3.p3.8.m8.3.3.cmml">L</mi><mo id="A3.p3.8.m8.4.5.2.4" xref="A3.p3.8.m8.4.5.1.cmml">,</mo><mi id="A3.p3.8.m8.4.4" xref="A3.p3.8.m8.4.4.cmml">L</mi><mo id="A3.p3.8.m8.4.5.2.5" stretchy="false" xref="A3.p3.8.m8.4.5.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.8.m8.4b"><vector id="A3.p3.8.m8.4.5.1.cmml" xref="A3.p3.8.m8.4.5.2"><ci id="A3.p3.8.m8.1.1.cmml" xref="A3.p3.8.m8.1.1">𝐻</ci><ci id="A3.p3.8.m8.2.2.cmml" xref="A3.p3.8.m8.2.2">𝐿</ci><ci id="A3.p3.8.m8.3.3.cmml" xref="A3.p3.8.m8.3.3">𝐿</ci><ci id="A3.p3.8.m8.4.4.cmml" xref="A3.p3.8.m8.4.4">𝐿</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.8.m8.4c">(H,L,L,L)</annotation><annotation encoding="application/x-llamapun" id="A3.p3.8.m8.4d">( italic_H , italic_L , italic_L , italic_L )</annotation></semantics></math>, would yield payoff 0 deterministically. But the agent may note that flipping the last <math alttext="L" class="ltx_Math" display="inline" id="A3.p3.9.m9.1"><semantics id="A3.p3.9.m9.1a"><mi id="A3.p3.9.m9.1.1" xref="A3.p3.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A3.p3.9.m9.1b"><ci id="A3.p3.9.m9.1.1.cmml" xref="A3.p3.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="A3.p3.9.m9.1d">italic_L</annotation></semantics></math> to <math alttext="H" class="ltx_Math" display="inline" id="A3.p3.10.m10.1"><semantics id="A3.p3.10.m10.1a"><mi id="A3.p3.10.m10.1.1" xref="A3.p3.10.m10.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A3.p3.10.m10.1b"><ci id="A3.p3.10.m10.1.1.cmml" xref="A3.p3.10.m10.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.10.m10.1c">H</annotation><annotation encoding="application/x-llamapun" id="A3.p3.10.m10.1d">italic_H</annotation></semantics></math> is “safe” in the sense that it is extremely unlikely that the other agent will differ in their reports on the first 3 tasks. So either the other agent reports <math alttext="r_{4}=L" class="ltx_Math" display="inline" id="A3.p3.11.m11.1"><semantics id="A3.p3.11.m11.1a"><mrow id="A3.p3.11.m11.1.1" xref="A3.p3.11.m11.1.1.cmml"><msub id="A3.p3.11.m11.1.1.2" xref="A3.p3.11.m11.1.1.2.cmml"><mi id="A3.p3.11.m11.1.1.2.2" xref="A3.p3.11.m11.1.1.2.2.cmml">r</mi><mn id="A3.p3.11.m11.1.1.2.3" xref="A3.p3.11.m11.1.1.2.3.cmml">4</mn></msub><mo id="A3.p3.11.m11.1.1.1" xref="A3.p3.11.m11.1.1.1.cmml">=</mo><mi id="A3.p3.11.m11.1.1.3" xref="A3.p3.11.m11.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.11.m11.1b"><apply id="A3.p3.11.m11.1.1.cmml" xref="A3.p3.11.m11.1.1"><eq id="A3.p3.11.m11.1.1.1.cmml" xref="A3.p3.11.m11.1.1.1"></eq><apply id="A3.p3.11.m11.1.1.2.cmml" xref="A3.p3.11.m11.1.1.2"><csymbol cd="ambiguous" id="A3.p3.11.m11.1.1.2.1.cmml" xref="A3.p3.11.m11.1.1.2">subscript</csymbol><ci id="A3.p3.11.m11.1.1.2.2.cmml" xref="A3.p3.11.m11.1.1.2.2">𝑟</ci><cn id="A3.p3.11.m11.1.1.2.3.cmml" type="integer" xref="A3.p3.11.m11.1.1.2.3">4</cn></apply><ci id="A3.p3.11.m11.1.1.3.cmml" xref="A3.p3.11.m11.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.11.m11.1c">r_{4}=L</annotation><annotation encoding="application/x-llamapun" id="A3.p3.11.m11.1d">italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_L</annotation></semantics></math>, in which case the payment is zero for both agents, or <math alttext="r_{4}=H" class="ltx_Math" display="inline" id="A3.p3.12.m12.1"><semantics id="A3.p3.12.m12.1a"><mrow id="A3.p3.12.m12.1.1" xref="A3.p3.12.m12.1.1.cmml"><msub id="A3.p3.12.m12.1.1.2" xref="A3.p3.12.m12.1.1.2.cmml"><mi id="A3.p3.12.m12.1.1.2.2" xref="A3.p3.12.m12.1.1.2.2.cmml">r</mi><mn id="A3.p3.12.m12.1.1.2.3" xref="A3.p3.12.m12.1.1.2.3.cmml">4</mn></msub><mo id="A3.p3.12.m12.1.1.1" xref="A3.p3.12.m12.1.1.1.cmml">=</mo><mi id="A3.p3.12.m12.1.1.3" xref="A3.p3.12.m12.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.12.m12.1b"><apply id="A3.p3.12.m12.1.1.cmml" xref="A3.p3.12.m12.1.1"><eq id="A3.p3.12.m12.1.1.1.cmml" xref="A3.p3.12.m12.1.1.1"></eq><apply id="A3.p3.12.m12.1.1.2.cmml" xref="A3.p3.12.m12.1.1.2"><csymbol cd="ambiguous" id="A3.p3.12.m12.1.1.2.1.cmml" xref="A3.p3.12.m12.1.1.2">subscript</csymbol><ci id="A3.p3.12.m12.1.1.2.2.cmml" xref="A3.p3.12.m12.1.1.2.2">𝑟</ci><cn id="A3.p3.12.m12.1.1.2.3.cmml" type="integer" xref="A3.p3.12.m12.1.1.2.3">4</cn></apply><ci id="A3.p3.12.m12.1.1.3.cmml" xref="A3.p3.12.m12.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.12.m12.1c">r_{4}=H</annotation><annotation encoding="application/x-llamapun" id="A3.p3.12.m12.1d">italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_H</annotation></semantics></math>, in which case both agents receive 1. As there is a reasonable chance of the latter case, about <math alttext="0.2819" class="ltx_Math" display="inline" id="A3.p3.13.m13.1"><semantics id="A3.p3.13.m13.1a"><mn id="A3.p3.13.m13.1.1" xref="A3.p3.13.m13.1.1.cmml">0.2819</mn><annotation-xml encoding="MathML-Content" id="A3.p3.13.m13.1b"><cn id="A3.p3.13.m13.1.1.cmml" type="float" xref="A3.p3.13.m13.1.1">0.2819</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.13.m13.1c">0.2819</annotation><annotation encoding="application/x-llamapun" id="A3.p3.13.m13.1d">0.2819</annotation></semantics></math>, that is roughly the expected score.</p> </div> <div class="ltx_para" id="A3.p4"> <p class="ltx_p" id="A3.p4.4">Thus, given real-valued signals <math alttext="(10,-10,-10,-1)" class="ltx_Math" display="inline" id="A3.p4.1.m1.4"><semantics id="A3.p4.1.m1.4a"><mrow id="A3.p4.1.m1.4.4.3" xref="A3.p4.1.m1.4.4.4.cmml"><mo id="A3.p4.1.m1.4.4.3.4" stretchy="false" xref="A3.p4.1.m1.4.4.4.cmml">(</mo><mn id="A3.p4.1.m1.1.1" xref="A3.p4.1.m1.1.1.cmml">10</mn><mo id="A3.p4.1.m1.4.4.3.5" xref="A3.p4.1.m1.4.4.4.cmml">,</mo><mrow id="A3.p4.1.m1.2.2.1.1" xref="A3.p4.1.m1.2.2.1.1.cmml"><mo id="A3.p4.1.m1.2.2.1.1a" xref="A3.p4.1.m1.2.2.1.1.cmml">−</mo><mn id="A3.p4.1.m1.2.2.1.1.2" xref="A3.p4.1.m1.2.2.1.1.2.cmml">10</mn></mrow><mo id="A3.p4.1.m1.4.4.3.6" xref="A3.p4.1.m1.4.4.4.cmml">,</mo><mrow id="A3.p4.1.m1.3.3.2.2" xref="A3.p4.1.m1.3.3.2.2.cmml"><mo id="A3.p4.1.m1.3.3.2.2a" xref="A3.p4.1.m1.3.3.2.2.cmml">−</mo><mn id="A3.p4.1.m1.3.3.2.2.2" xref="A3.p4.1.m1.3.3.2.2.2.cmml">10</mn></mrow><mo id="A3.p4.1.m1.4.4.3.7" xref="A3.p4.1.m1.4.4.4.cmml">,</mo><mrow id="A3.p4.1.m1.4.4.3.3" xref="A3.p4.1.m1.4.4.3.3.cmml"><mo id="A3.p4.1.m1.4.4.3.3a" xref="A3.p4.1.m1.4.4.3.3.cmml">−</mo><mn id="A3.p4.1.m1.4.4.3.3.2" xref="A3.p4.1.m1.4.4.3.3.2.cmml">1</mn></mrow><mo id="A3.p4.1.m1.4.4.3.8" stretchy="false" xref="A3.p4.1.m1.4.4.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.1.m1.4b"><vector id="A3.p4.1.m1.4.4.4.cmml" xref="A3.p4.1.m1.4.4.3"><cn id="A3.p4.1.m1.1.1.cmml" type="integer" xref="A3.p4.1.m1.1.1">10</cn><apply id="A3.p4.1.m1.2.2.1.1.cmml" xref="A3.p4.1.m1.2.2.1.1"><minus id="A3.p4.1.m1.2.2.1.1.1.cmml" xref="A3.p4.1.m1.2.2.1.1"></minus><cn id="A3.p4.1.m1.2.2.1.1.2.cmml" type="integer" xref="A3.p4.1.m1.2.2.1.1.2">10</cn></apply><apply id="A3.p4.1.m1.3.3.2.2.cmml" xref="A3.p4.1.m1.3.3.2.2"><minus id="A3.p4.1.m1.3.3.2.2.1.cmml" xref="A3.p4.1.m1.3.3.2.2"></minus><cn id="A3.p4.1.m1.3.3.2.2.2.cmml" type="integer" xref="A3.p4.1.m1.3.3.2.2.2">10</cn></apply><apply id="A3.p4.1.m1.4.4.3.3.cmml" xref="A3.p4.1.m1.4.4.3.3"><minus id="A3.p4.1.m1.4.4.3.3.1.cmml" xref="A3.p4.1.m1.4.4.3.3"></minus><cn id="A3.p4.1.m1.4.4.3.3.2.cmml" type="integer" xref="A3.p4.1.m1.4.4.3.3.2">1</cn></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.1.m1.4c">(10,-10,-10,-1)</annotation><annotation encoding="application/x-llamapun" id="A3.p4.1.m1.4d">( 10 , - 10 , - 10 , - 1 )</annotation></semantics></math>, the optimal report is <math alttext="(H,L,L,H)" class="ltx_Math" display="inline" id="A3.p4.2.m2.4"><semantics id="A3.p4.2.m2.4a"><mrow id="A3.p4.2.m2.4.5.2" xref="A3.p4.2.m2.4.5.1.cmml"><mo id="A3.p4.2.m2.4.5.2.1" stretchy="false" xref="A3.p4.2.m2.4.5.1.cmml">(</mo><mi id="A3.p4.2.m2.1.1" xref="A3.p4.2.m2.1.1.cmml">H</mi><mo id="A3.p4.2.m2.4.5.2.2" xref="A3.p4.2.m2.4.5.1.cmml">,</mo><mi id="A3.p4.2.m2.2.2" xref="A3.p4.2.m2.2.2.cmml">L</mi><mo id="A3.p4.2.m2.4.5.2.3" xref="A3.p4.2.m2.4.5.1.cmml">,</mo><mi id="A3.p4.2.m2.3.3" xref="A3.p4.2.m2.3.3.cmml">L</mi><mo id="A3.p4.2.m2.4.5.2.4" xref="A3.p4.2.m2.4.5.1.cmml">,</mo><mi id="A3.p4.2.m2.4.4" xref="A3.p4.2.m2.4.4.cmml">H</mi><mo id="A3.p4.2.m2.4.5.2.5" stretchy="false" xref="A3.p4.2.m2.4.5.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.2.m2.4b"><vector id="A3.p4.2.m2.4.5.1.cmml" xref="A3.p4.2.m2.4.5.2"><ci id="A3.p4.2.m2.1.1.cmml" xref="A3.p4.2.m2.1.1">𝐻</ci><ci id="A3.p4.2.m2.2.2.cmml" xref="A3.p4.2.m2.2.2">𝐿</ci><ci id="A3.p4.2.m2.3.3.cmml" xref="A3.p4.2.m2.3.3">𝐿</ci><ci id="A3.p4.2.m2.4.4.cmml" xref="A3.p4.2.m2.4.4">𝐻</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.2.m2.4c">(H,L,L,H)</annotation><annotation encoding="application/x-llamapun" id="A3.p4.2.m2.4d">( italic_H , italic_L , italic_L , italic_H )</annotation></semantics></math>, which is not “truthful” for the prescribed threshold <math alttext="\tau=0" class="ltx_Math" display="inline" id="A3.p4.3.m3.1"><semantics id="A3.p4.3.m3.1a"><mrow id="A3.p4.3.m3.1.1" xref="A3.p4.3.m3.1.1.cmml"><mi id="A3.p4.3.m3.1.1.2" xref="A3.p4.3.m3.1.1.2.cmml">τ</mi><mo id="A3.p4.3.m3.1.1.1" xref="A3.p4.3.m3.1.1.1.cmml">=</mo><mn id="A3.p4.3.m3.1.1.3" xref="A3.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.3.m3.1b"><apply id="A3.p4.3.m3.1.1.cmml" xref="A3.p4.3.m3.1.1"><eq id="A3.p4.3.m3.1.1.1.cmml" xref="A3.p4.3.m3.1.1.1"></eq><ci id="A3.p4.3.m3.1.1.2.cmml" xref="A3.p4.3.m3.1.1.2">𝜏</ci><cn id="A3.p4.3.m3.1.1.3.cmml" type="integer" xref="A3.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.3.m3.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A3.p4.3.m3.1d">italic_τ = 0</annotation></semantics></math>. Hence DMI is no longer truthful in this joint-task sense, even at the equilibrium threshold <math alttext="\tau=0" class="ltx_Math" display="inline" id="A3.p4.4.m4.1"><semantics id="A3.p4.4.m4.1a"><mrow id="A3.p4.4.m4.1.1" xref="A3.p4.4.m4.1.1.cmml"><mi id="A3.p4.4.m4.1.1.2" xref="A3.p4.4.m4.1.1.2.cmml">τ</mi><mo id="A3.p4.4.m4.1.1.1" xref="A3.p4.4.m4.1.1.1.cmml">=</mo><mn id="A3.p4.4.m4.1.1.3" xref="A3.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.4.m4.1b"><apply id="A3.p4.4.m4.1.1.cmml" xref="A3.p4.4.m4.1.1"><eq id="A3.p4.4.m4.1.1.1.cmml" xref="A3.p4.4.m4.1.1.1"></eq><ci id="A3.p4.4.m4.1.1.2.cmml" xref="A3.p4.4.m4.1.1.2">𝜏</ci><cn id="A3.p4.4.m4.1.1.3.cmml" type="integer" xref="A3.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.4.m4.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A3.p4.4.m4.1d">italic_τ = 0</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="A3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">C.1 </span>Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> </h3> <div class="ltx_para" id="A3.SS1.p1"> <p class="ltx_p" id="A3.SS1.p1.1">First, we prove that in the binary signal model, truthfulness is an equilibrium among all joint-task strotegies <math alttext="\sigma:\{H,L\}^{T}\to\Delta(\{H,L\}^{T})" class="ltx_Math" display="inline" id="A3.SS1.p1.1.m1.5"><semantics id="A3.SS1.p1.1.m1.5a"><mrow id="A3.SS1.p1.1.m1.5.5" xref="A3.SS1.p1.1.m1.5.5.cmml"><mi id="A3.SS1.p1.1.m1.5.5.3" xref="A3.SS1.p1.1.m1.5.5.3.cmml">σ</mi><mo id="A3.SS1.p1.1.m1.5.5.2" lspace="0.278em" rspace="0.278em" xref="A3.SS1.p1.1.m1.5.5.2.cmml">:</mo><mrow id="A3.SS1.p1.1.m1.5.5.1" xref="A3.SS1.p1.1.m1.5.5.1.cmml"><msup id="A3.SS1.p1.1.m1.5.5.1.3" xref="A3.SS1.p1.1.m1.5.5.1.3.cmml"><mrow id="A3.SS1.p1.1.m1.5.5.1.3.2.2" xref="A3.SS1.p1.1.m1.5.5.1.3.2.1.cmml"><mo id="A3.SS1.p1.1.m1.5.5.1.3.2.2.1" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.3.2.1.cmml">{</mo><mi id="A3.SS1.p1.1.m1.1.1" xref="A3.SS1.p1.1.m1.1.1.cmml">H</mi><mo id="A3.SS1.p1.1.m1.5.5.1.3.2.2.2" xref="A3.SS1.p1.1.m1.5.5.1.3.2.1.cmml">,</mo><mi id="A3.SS1.p1.1.m1.2.2" xref="A3.SS1.p1.1.m1.2.2.cmml">L</mi><mo id="A3.SS1.p1.1.m1.5.5.1.3.2.2.3" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.3.2.1.cmml">}</mo></mrow><mi id="A3.SS1.p1.1.m1.5.5.1.3.3" xref="A3.SS1.p1.1.m1.5.5.1.3.3.cmml">T</mi></msup><mo id="A3.SS1.p1.1.m1.5.5.1.2" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.2.cmml">→</mo><mrow id="A3.SS1.p1.1.m1.5.5.1.1" xref="A3.SS1.p1.1.m1.5.5.1.1.cmml"><mi id="A3.SS1.p1.1.m1.5.5.1.1.3" mathvariant="normal" xref="A3.SS1.p1.1.m1.5.5.1.1.3.cmml">Δ</mi><mo id="A3.SS1.p1.1.m1.5.5.1.1.2" xref="A3.SS1.p1.1.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="A3.SS1.p1.1.m1.5.5.1.1.1.1" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml"><mo id="A3.SS1.p1.1.m1.5.5.1.1.1.1.2" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml">(</mo><msup id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml"><mrow id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml"><mo id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2.1" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml">{</mo><mi id="A3.SS1.p1.1.m1.3.3" xref="A3.SS1.p1.1.m1.3.3.cmml">H</mi><mo id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2.2" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml">,</mo><mi id="A3.SS1.p1.1.m1.4.4" xref="A3.SS1.p1.1.m1.4.4.cmml">L</mi><mo id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2.3" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml">}</mo></mrow><mi id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.3" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.3.cmml">T</mi></msup><mo id="A3.SS1.p1.1.m1.5.5.1.1.1.1.3" stretchy="false" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p1.1.m1.5b"><apply id="A3.SS1.p1.1.m1.5.5.cmml" xref="A3.SS1.p1.1.m1.5.5"><ci id="A3.SS1.p1.1.m1.5.5.2.cmml" xref="A3.SS1.p1.1.m1.5.5.2">:</ci><ci id="A3.SS1.p1.1.m1.5.5.3.cmml" xref="A3.SS1.p1.1.m1.5.5.3">𝜎</ci><apply id="A3.SS1.p1.1.m1.5.5.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1"><ci id="A3.SS1.p1.1.m1.5.5.1.2.cmml" xref="A3.SS1.p1.1.m1.5.5.1.2">→</ci><apply id="A3.SS1.p1.1.m1.5.5.1.3.cmml" xref="A3.SS1.p1.1.m1.5.5.1.3"><csymbol cd="ambiguous" id="A3.SS1.p1.1.m1.5.5.1.3.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.3">superscript</csymbol><set id="A3.SS1.p1.1.m1.5.5.1.3.2.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.3.2.2"><ci id="A3.SS1.p1.1.m1.1.1.cmml" xref="A3.SS1.p1.1.m1.1.1">𝐻</ci><ci id="A3.SS1.p1.1.m1.2.2.cmml" xref="A3.SS1.p1.1.m1.2.2">𝐿</ci></set><ci id="A3.SS1.p1.1.m1.5.5.1.3.3.cmml" xref="A3.SS1.p1.1.m1.5.5.1.3.3">𝑇</ci></apply><apply id="A3.SS1.p1.1.m1.5.5.1.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1"><times id="A3.SS1.p1.1.m1.5.5.1.1.2.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.2"></times><ci id="A3.SS1.p1.1.m1.5.5.1.1.3.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.3">Δ</ci><apply id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1">superscript</csymbol><set id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.2.2"><ci id="A3.SS1.p1.1.m1.3.3.cmml" xref="A3.SS1.p1.1.m1.3.3">𝐻</ci><ci id="A3.SS1.p1.1.m1.4.4.cmml" xref="A3.SS1.p1.1.m1.4.4">𝐿</ci></set><ci id="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.3.cmml" xref="A3.SS1.p1.1.m1.5.5.1.1.1.1.1.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p1.1.m1.5c">\sigma:\{H,L\}^{T}\to\Delta(\{H,L\}^{T})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p1.1.m1.5d">italic_σ : { italic_H , italic_L } start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT → roman_Δ ( { italic_H , italic_L } start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT )</annotation></semantics></math>. Truthfulness remains a best response even when the other agent deviates from truthful, but crucially, still plays a <em class="ltx_emph ltx_font_italic" id="A3.SS1.p1.1.1">consistent</em> strategy, which here means strategies that (a) map each signal to a distribution over reports, and (b) use the same such map for every task.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmdefinition3.1.1.1">Definition 3</span></span><span class="ltx_text ltx_font_bold" id="Thmdefinition3.2.2">.</span> </h6> <div class="ltx_para" id="Thmdefinition3.p1"> <p class="ltx_p" id="Thmdefinition3.p1.3">A strategy <math alttext="\sigma:\{H,L\}^{T}\to\Delta(\{H,L\}^{T})" class="ltx_Math" display="inline" id="Thmdefinition3.p1.1.m1.5"><semantics id="Thmdefinition3.p1.1.m1.5a"><mrow id="Thmdefinition3.p1.1.m1.5.5" 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xref="Thmdefinition3.p1.1.m1.5.5.1.3.2.1.cmml">}</mo></mrow><mi id="Thmdefinition3.p1.1.m1.5.5.1.3.3" xref="Thmdefinition3.p1.1.m1.5.5.1.3.3.cmml">T</mi></msup><mo id="Thmdefinition3.p1.1.m1.5.5.1.2" stretchy="false" xref="Thmdefinition3.p1.1.m1.5.5.1.2.cmml">→</mo><mrow id="Thmdefinition3.p1.1.m1.5.5.1.1" xref="Thmdefinition3.p1.1.m1.5.5.1.1.cmml"><mi id="Thmdefinition3.p1.1.m1.5.5.1.1.3" mathvariant="normal" xref="Thmdefinition3.p1.1.m1.5.5.1.1.3.cmml">Δ</mi><mo id="Thmdefinition3.p1.1.m1.5.5.1.1.2" xref="Thmdefinition3.p1.1.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.cmml"><mo id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.2" stretchy="false" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.cmml">(</mo><msup id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.cmml"><mrow id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.2.2" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml"><mo 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xref="Thmdefinition3.p1.1.m1.5.5.2">:</ci><ci id="Thmdefinition3.p1.1.m1.5.5.3.cmml" xref="Thmdefinition3.p1.1.m1.5.5.3">𝜎</ci><apply id="Thmdefinition3.p1.1.m1.5.5.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1"><ci id="Thmdefinition3.p1.1.m1.5.5.1.2.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.2">→</ci><apply id="Thmdefinition3.p1.1.m1.5.5.1.3.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.3"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.5.5.1.3.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.3">superscript</csymbol><set id="Thmdefinition3.p1.1.m1.5.5.1.3.2.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.3.2.2"><ci id="Thmdefinition3.p1.1.m1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.1.1">𝐻</ci><ci id="Thmdefinition3.p1.1.m1.2.2.cmml" xref="Thmdefinition3.p1.1.m1.2.2">𝐿</ci></set><ci id="Thmdefinition3.p1.1.m1.5.5.1.3.3.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.3.3">𝑇</ci></apply><apply id="Thmdefinition3.p1.1.m1.5.5.1.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1"><times id="Thmdefinition3.p1.1.m1.5.5.1.1.2.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.2"></times><ci id="Thmdefinition3.p1.1.m1.5.5.1.1.3.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.3">Δ</ci><apply id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1">superscript</csymbol><set id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.2.1.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.2.2"><ci id="Thmdefinition3.p1.1.m1.3.3.cmml" xref="Thmdefinition3.p1.1.m1.3.3">𝐻</ci><ci id="Thmdefinition3.p1.1.m1.4.4.cmml" xref="Thmdefinition3.p1.1.m1.4.4">𝐿</ci></set><ci id="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.3.cmml" xref="Thmdefinition3.p1.1.m1.5.5.1.1.1.1.1.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.1.m1.5c">\sigma:\{H,L\}^{T}\to\Delta(\{H,L\}^{T})</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.1.m1.5d">italic_σ : { italic_H , italic_L } start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT → roman_Δ ( { italic_H , italic_L } start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT )</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="Thmdefinition3.p1.3.1">consistent</em> if <math alttext="\sigma(s_{1..T})(r_{1..T})=\prod_{t=1}^{T}\hat{\sigma}(s_{t})(r_{t})" class="ltx_math_unparsed" display="inline" id="Thmdefinition3.p1.2.m2.6"><semantics id="Thmdefinition3.p1.2.m2.6a"><mrow id="Thmdefinition3.p1.2.m2.6.6"><mrow id="Thmdefinition3.p1.2.m2.4.4.2"><mi id="Thmdefinition3.p1.2.m2.4.4.2.4">σ</mi><mo id="Thmdefinition3.p1.2.m2.4.4.2.3">⁢</mo><mrow id="Thmdefinition3.p1.2.m2.3.3.1.1.1"><mo id="Thmdefinition3.p1.2.m2.3.3.1.1.1.2" stretchy="false">(</mo><msub id="Thmdefinition3.p1.2.m2.3.3.1.1.1.1"><mi id="Thmdefinition3.p1.2.m2.3.3.1.1.1.1.2">s</mi><mrow id="Thmdefinition3.p1.2.m2.1.1.1"><mn id="Thmdefinition3.p1.2.m2.1.1.1.1">1</mn><mo id="Thmdefinition3.p1.2.m2.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="Thmdefinition3.p1.2.m2.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="Thmdefinition3.p1.2.m2.1.1.1.4">T</mi></mrow></msub><mo id="Thmdefinition3.p1.2.m2.3.3.1.1.1.3" stretchy="false">)</mo></mrow><mo id="Thmdefinition3.p1.2.m2.4.4.2.3a">⁢</mo><mrow id="Thmdefinition3.p1.2.m2.4.4.2.2.1"><mo id="Thmdefinition3.p1.2.m2.4.4.2.2.1.2" stretchy="false">(</mo><msub id="Thmdefinition3.p1.2.m2.4.4.2.2.1.1"><mi id="Thmdefinition3.p1.2.m2.4.4.2.2.1.1.2">r</mi><mrow id="Thmdefinition3.p1.2.m2.2.2.1"><mn id="Thmdefinition3.p1.2.m2.2.2.1.1">1</mn><mo id="Thmdefinition3.p1.2.m2.2.2.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="Thmdefinition3.p1.2.m2.2.2.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="Thmdefinition3.p1.2.m2.2.2.1.4">T</mi></mrow></msub><mo id="Thmdefinition3.p1.2.m2.4.4.2.2.1.3" stretchy="false">)</mo></mrow></mrow><mo id="Thmdefinition3.p1.2.m2.6.6.5" rspace="0.111em">=</mo><mrow id="Thmdefinition3.p1.2.m2.6.6.4"><msubsup id="Thmdefinition3.p1.2.m2.6.6.4.3"><mo id="Thmdefinition3.p1.2.m2.6.6.4.3.2.2">∏</mo><mrow id="Thmdefinition3.p1.2.m2.6.6.4.3.2.3"><mi id="Thmdefinition3.p1.2.m2.6.6.4.3.2.3.2">t</mi><mo id="Thmdefinition3.p1.2.m2.6.6.4.3.2.3.1">=</mo><mn id="Thmdefinition3.p1.2.m2.6.6.4.3.2.3.3">1</mn></mrow><mi id="Thmdefinition3.p1.2.m2.6.6.4.3.3">T</mi></msubsup><mrow id="Thmdefinition3.p1.2.m2.6.6.4.2"><mover accent="true" id="Thmdefinition3.p1.2.m2.6.6.4.2.4"><mi id="Thmdefinition3.p1.2.m2.6.6.4.2.4.2">σ</mi><mo id="Thmdefinition3.p1.2.m2.6.6.4.2.4.1">^</mo></mover><mo id="Thmdefinition3.p1.2.m2.6.6.4.2.3">⁢</mo><mrow id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1"><mo id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1.2" stretchy="false">(</mo><msub id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1.1"><mi id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1.1.2">s</mi><mi id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1.1.3">t</mi></msub><mo id="Thmdefinition3.p1.2.m2.5.5.3.1.1.1.3" stretchy="false">)</mo></mrow><mo id="Thmdefinition3.p1.2.m2.6.6.4.2.3a">⁢</mo><mrow id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1"><mo id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1.2" stretchy="false">(</mo><msub id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1.1"><mi id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1.1.2">r</mi><mi id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1.1.3">t</mi></msub><mo id="Thmdefinition3.p1.2.m2.6.6.4.2.2.1.3" stretchy="false">)</mo></mrow></mrow></mrow></mrow><annotation encoding="application/x-tex" id="Thmdefinition3.p1.2.m2.6b">\sigma(s_{1..T})(r_{1..T})=\prod_{t=1}^{T}\hat{\sigma}(s_{t})(r_{t})</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.2.m2.6c">italic_σ ( italic_s start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ) ( italic_r start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ) = ∏ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT over^ start_ARG italic_σ end_ARG ( italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ( italic_r start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> for some <math alttext="\hat{\sigma}:\{H,L\}\to\Delta(\{H,L\})" class="ltx_Math" display="inline" id="Thmdefinition3.p1.3.m3.5"><semantics id="Thmdefinition3.p1.3.m3.5a"><mrow id="Thmdefinition3.p1.3.m3.5.5" xref="Thmdefinition3.p1.3.m3.5.5.cmml"><mover accent="true" id="Thmdefinition3.p1.3.m3.5.5.3" xref="Thmdefinition3.p1.3.m3.5.5.3.cmml"><mi id="Thmdefinition3.p1.3.m3.5.5.3.2" xref="Thmdefinition3.p1.3.m3.5.5.3.2.cmml">σ</mi><mo id="Thmdefinition3.p1.3.m3.5.5.3.1" xref="Thmdefinition3.p1.3.m3.5.5.3.1.cmml">^</mo></mover><mo id="Thmdefinition3.p1.3.m3.5.5.2" lspace="0.278em" rspace="0.278em" xref="Thmdefinition3.p1.3.m3.5.5.2.cmml">:</mo><mrow id="Thmdefinition3.p1.3.m3.5.5.1" xref="Thmdefinition3.p1.3.m3.5.5.1.cmml"><mrow id="Thmdefinition3.p1.3.m3.5.5.1.3.2" xref="Thmdefinition3.p1.3.m3.5.5.1.3.1.cmml"><mo id="Thmdefinition3.p1.3.m3.5.5.1.3.2.1" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.3.1.cmml">{</mo><mi id="Thmdefinition3.p1.3.m3.1.1" xref="Thmdefinition3.p1.3.m3.1.1.cmml">H</mi><mo id="Thmdefinition3.p1.3.m3.5.5.1.3.2.2" xref="Thmdefinition3.p1.3.m3.5.5.1.3.1.cmml">,</mo><mi id="Thmdefinition3.p1.3.m3.2.2" xref="Thmdefinition3.p1.3.m3.2.2.cmml">L</mi><mo id="Thmdefinition3.p1.3.m3.5.5.1.3.2.3" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.3.1.cmml">}</mo></mrow><mo id="Thmdefinition3.p1.3.m3.5.5.1.2" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.2.cmml">→</mo><mrow id="Thmdefinition3.p1.3.m3.5.5.1.1" xref="Thmdefinition3.p1.3.m3.5.5.1.1.cmml"><mi id="Thmdefinition3.p1.3.m3.5.5.1.1.3" mathvariant="normal" xref="Thmdefinition3.p1.3.m3.5.5.1.1.3.cmml">Δ</mi><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.2" xref="Thmdefinition3.p1.3.m3.5.5.1.1.2.cmml">⁢</mo><mrow id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1" xref="Thmdefinition3.p1.3.m3.5.5.1.1.cmml"><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.2" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.1.cmml">(</mo><mrow id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.2" xref="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.1.cmml"><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.2.1" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.1.cmml">{</mo><mi id="Thmdefinition3.p1.3.m3.3.3" xref="Thmdefinition3.p1.3.m3.3.3.cmml">H</mi><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.2.2" xref="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.1.cmml">,</mo><mi id="Thmdefinition3.p1.3.m3.4.4" xref="Thmdefinition3.p1.3.m3.4.4.cmml">L</mi><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.2.3" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.1.cmml">}</mo></mrow><mo id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.3" stretchy="false" xref="Thmdefinition3.p1.3.m3.5.5.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.3.m3.5b"><apply id="Thmdefinition3.p1.3.m3.5.5.cmml" xref="Thmdefinition3.p1.3.m3.5.5"><ci id="Thmdefinition3.p1.3.m3.5.5.2.cmml" xref="Thmdefinition3.p1.3.m3.5.5.2">:</ci><apply id="Thmdefinition3.p1.3.m3.5.5.3.cmml" xref="Thmdefinition3.p1.3.m3.5.5.3"><ci id="Thmdefinition3.p1.3.m3.5.5.3.1.cmml" xref="Thmdefinition3.p1.3.m3.5.5.3.1">^</ci><ci id="Thmdefinition3.p1.3.m3.5.5.3.2.cmml" xref="Thmdefinition3.p1.3.m3.5.5.3.2">𝜎</ci></apply><apply id="Thmdefinition3.p1.3.m3.5.5.1.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1"><ci id="Thmdefinition3.p1.3.m3.5.5.1.2.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.2">→</ci><set id="Thmdefinition3.p1.3.m3.5.5.1.3.1.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.3.2"><ci id="Thmdefinition3.p1.3.m3.1.1.cmml" xref="Thmdefinition3.p1.3.m3.1.1">𝐻</ci><ci id="Thmdefinition3.p1.3.m3.2.2.cmml" xref="Thmdefinition3.p1.3.m3.2.2">𝐿</ci></set><apply id="Thmdefinition3.p1.3.m3.5.5.1.1.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.1"><times id="Thmdefinition3.p1.3.m3.5.5.1.1.2.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.1.2"></times><ci id="Thmdefinition3.p1.3.m3.5.5.1.1.3.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.1.3">Δ</ci><set id="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.1.cmml" xref="Thmdefinition3.p1.3.m3.5.5.1.1.1.1.1.2"><ci id="Thmdefinition3.p1.3.m3.3.3.cmml" xref="Thmdefinition3.p1.3.m3.3.3">𝐻</ci><ci id="Thmdefinition3.p1.3.m3.4.4.cmml" xref="Thmdefinition3.p1.3.m3.4.4">𝐿</ci></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.3.m3.5c">\hat{\sigma}:\{H,L\}\to\Delta(\{H,L\})</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.3.m3.5d">over^ start_ARG italic_σ end_ARG : { italic_H , italic_L } → roman_Δ ( { italic_H , italic_L } )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="A3.SS1.p2"> <p class="ltx_p" id="A3.SS1.p2.5">For example, the truthful strategy <math alttext="\sigma_{\text{true}}:s\mapsto\delta_{s}" class="ltx_Math" display="inline" id="A3.SS1.p2.1.m1.1"><semantics id="A3.SS1.p2.1.m1.1a"><mrow id="A3.SS1.p2.1.m1.1.1" xref="A3.SS1.p2.1.m1.1.1.cmml"><msub id="A3.SS1.p2.1.m1.1.1.2" xref="A3.SS1.p2.1.m1.1.1.2.cmml"><mi id="A3.SS1.p2.1.m1.1.1.2.2" xref="A3.SS1.p2.1.m1.1.1.2.2.cmml">σ</mi><mtext id="A3.SS1.p2.1.m1.1.1.2.3" xref="A3.SS1.p2.1.m1.1.1.2.3a.cmml">true</mtext></msub><mo id="A3.SS1.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="A3.SS1.p2.1.m1.1.1.1.cmml">:</mo><mrow id="A3.SS1.p2.1.m1.1.1.3" xref="A3.SS1.p2.1.m1.1.1.3.cmml"><mi id="A3.SS1.p2.1.m1.1.1.3.2" xref="A3.SS1.p2.1.m1.1.1.3.2.cmml">s</mi><mo id="A3.SS1.p2.1.m1.1.1.3.1" stretchy="false" xref="A3.SS1.p2.1.m1.1.1.3.1.cmml">↦</mo><msub id="A3.SS1.p2.1.m1.1.1.3.3" xref="A3.SS1.p2.1.m1.1.1.3.3.cmml"><mi id="A3.SS1.p2.1.m1.1.1.3.3.2" xref="A3.SS1.p2.1.m1.1.1.3.3.2.cmml">δ</mi><mi id="A3.SS1.p2.1.m1.1.1.3.3.3" xref="A3.SS1.p2.1.m1.1.1.3.3.3.cmml">s</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.1.m1.1b"><apply id="A3.SS1.p2.1.m1.1.1.cmml" xref="A3.SS1.p2.1.m1.1.1"><ci id="A3.SS1.p2.1.m1.1.1.1.cmml" xref="A3.SS1.p2.1.m1.1.1.1">:</ci><apply id="A3.SS1.p2.1.m1.1.1.2.cmml" xref="A3.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.p2.1.m1.1.1.2.1.cmml" xref="A3.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS1.p2.1.m1.1.1.2.2.cmml" xref="A3.SS1.p2.1.m1.1.1.2.2">𝜎</ci><ci id="A3.SS1.p2.1.m1.1.1.2.3a.cmml" xref="A3.SS1.p2.1.m1.1.1.2.3"><mtext id="A3.SS1.p2.1.m1.1.1.2.3.cmml" mathsize="70%" xref="A3.SS1.p2.1.m1.1.1.2.3">true</mtext></ci></apply><apply id="A3.SS1.p2.1.m1.1.1.3.cmml" xref="A3.SS1.p2.1.m1.1.1.3"><csymbol cd="latexml" id="A3.SS1.p2.1.m1.1.1.3.1.cmml" xref="A3.SS1.p2.1.m1.1.1.3.1">maps-to</csymbol><ci id="A3.SS1.p2.1.m1.1.1.3.2.cmml" xref="A3.SS1.p2.1.m1.1.1.3.2">𝑠</ci><apply id="A3.SS1.p2.1.m1.1.1.3.3.cmml" xref="A3.SS1.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="A3.SS1.p2.1.m1.1.1.3.3.1.cmml" xref="A3.SS1.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="A3.SS1.p2.1.m1.1.1.3.3.2.cmml" xref="A3.SS1.p2.1.m1.1.1.3.3.2">𝛿</ci><ci id="A3.SS1.p2.1.m1.1.1.3.3.3.cmml" xref="A3.SS1.p2.1.m1.1.1.3.3.3">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.1.m1.1c">\sigma_{\text{true}}:s\mapsto\delta_{s}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT true end_POSTSUBSCRIPT : italic_s ↦ italic_δ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\delta_{s}" class="ltx_Math" display="inline" id="A3.SS1.p2.2.m2.1"><semantics id="A3.SS1.p2.2.m2.1a"><msub id="A3.SS1.p2.2.m2.1.1" xref="A3.SS1.p2.2.m2.1.1.cmml"><mi id="A3.SS1.p2.2.m2.1.1.2" xref="A3.SS1.p2.2.m2.1.1.2.cmml">δ</mi><mi id="A3.SS1.p2.2.m2.1.1.3" xref="A3.SS1.p2.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.2.m2.1b"><apply id="A3.SS1.p2.2.m2.1.1.cmml" xref="A3.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS1.p2.2.m2.1.1.1.cmml" xref="A3.SS1.p2.2.m2.1.1">subscript</csymbol><ci id="A3.SS1.p2.2.m2.1.1.2.cmml" xref="A3.SS1.p2.2.m2.1.1.2">𝛿</ci><ci id="A3.SS1.p2.2.m2.1.1.3.cmml" xref="A3.SS1.p2.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.2.m2.1c">\delta_{s}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is the point distribution on <math alttext="s" class="ltx_Math" display="inline" id="A3.SS1.p2.3.m3.1"><semantics id="A3.SS1.p2.3.m3.1a"><mi id="A3.SS1.p2.3.m3.1.1" xref="A3.SS1.p2.3.m3.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.3.m3.1b"><ci id="A3.SS1.p2.3.m3.1.1.cmml" xref="A3.SS1.p2.3.m3.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.3.m3.1c">s</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.3.m3.1d">italic_s</annotation></semantics></math>, is a consistent strategy, with <math alttext="\hat{\sigma}:s\mapsto\delta_{s}" class="ltx_Math" display="inline" id="A3.SS1.p2.4.m4.1"><semantics id="A3.SS1.p2.4.m4.1a"><mrow id="A3.SS1.p2.4.m4.1.1" xref="A3.SS1.p2.4.m4.1.1.cmml"><mover accent="true" id="A3.SS1.p2.4.m4.1.1.2" xref="A3.SS1.p2.4.m4.1.1.2.cmml"><mi id="A3.SS1.p2.4.m4.1.1.2.2" xref="A3.SS1.p2.4.m4.1.1.2.2.cmml">σ</mi><mo id="A3.SS1.p2.4.m4.1.1.2.1" xref="A3.SS1.p2.4.m4.1.1.2.1.cmml">^</mo></mover><mo id="A3.SS1.p2.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="A3.SS1.p2.4.m4.1.1.1.cmml">:</mo><mrow id="A3.SS1.p2.4.m4.1.1.3" xref="A3.SS1.p2.4.m4.1.1.3.cmml"><mi id="A3.SS1.p2.4.m4.1.1.3.2" xref="A3.SS1.p2.4.m4.1.1.3.2.cmml">s</mi><mo id="A3.SS1.p2.4.m4.1.1.3.1" stretchy="false" xref="A3.SS1.p2.4.m4.1.1.3.1.cmml">↦</mo><msub id="A3.SS1.p2.4.m4.1.1.3.3" xref="A3.SS1.p2.4.m4.1.1.3.3.cmml"><mi id="A3.SS1.p2.4.m4.1.1.3.3.2" xref="A3.SS1.p2.4.m4.1.1.3.3.2.cmml">δ</mi><mi id="A3.SS1.p2.4.m4.1.1.3.3.3" xref="A3.SS1.p2.4.m4.1.1.3.3.3.cmml">s</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.4.m4.1b"><apply id="A3.SS1.p2.4.m4.1.1.cmml" xref="A3.SS1.p2.4.m4.1.1"><ci id="A3.SS1.p2.4.m4.1.1.1.cmml" xref="A3.SS1.p2.4.m4.1.1.1">:</ci><apply id="A3.SS1.p2.4.m4.1.1.2.cmml" xref="A3.SS1.p2.4.m4.1.1.2"><ci id="A3.SS1.p2.4.m4.1.1.2.1.cmml" xref="A3.SS1.p2.4.m4.1.1.2.1">^</ci><ci id="A3.SS1.p2.4.m4.1.1.2.2.cmml" xref="A3.SS1.p2.4.m4.1.1.2.2">𝜎</ci></apply><apply id="A3.SS1.p2.4.m4.1.1.3.cmml" xref="A3.SS1.p2.4.m4.1.1.3"><csymbol cd="latexml" id="A3.SS1.p2.4.m4.1.1.3.1.cmml" xref="A3.SS1.p2.4.m4.1.1.3.1">maps-to</csymbol><ci id="A3.SS1.p2.4.m4.1.1.3.2.cmml" xref="A3.SS1.p2.4.m4.1.1.3.2">𝑠</ci><apply id="A3.SS1.p2.4.m4.1.1.3.3.cmml" xref="A3.SS1.p2.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="A3.SS1.p2.4.m4.1.1.3.3.1.cmml" xref="A3.SS1.p2.4.m4.1.1.3.3">subscript</csymbol><ci id="A3.SS1.p2.4.m4.1.1.3.3.2.cmml" xref="A3.SS1.p2.4.m4.1.1.3.3.2">𝛿</ci><ci id="A3.SS1.p2.4.m4.1.1.3.3.3.cmml" xref="A3.SS1.p2.4.m4.1.1.3.3.3">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.4.m4.1c">\hat{\sigma}:s\mapsto\delta_{s}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.4.m4.1d">over^ start_ARG italic_σ end_ARG : italic_s ↦ italic_δ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="s\in\{H,L\}" class="ltx_Math" display="inline" id="A3.SS1.p2.5.m5.2"><semantics id="A3.SS1.p2.5.m5.2a"><mrow id="A3.SS1.p2.5.m5.2.3" xref="A3.SS1.p2.5.m5.2.3.cmml"><mi id="A3.SS1.p2.5.m5.2.3.2" xref="A3.SS1.p2.5.m5.2.3.2.cmml">s</mi><mo id="A3.SS1.p2.5.m5.2.3.1" xref="A3.SS1.p2.5.m5.2.3.1.cmml">∈</mo><mrow id="A3.SS1.p2.5.m5.2.3.3.2" xref="A3.SS1.p2.5.m5.2.3.3.1.cmml"><mo id="A3.SS1.p2.5.m5.2.3.3.2.1" stretchy="false" xref="A3.SS1.p2.5.m5.2.3.3.1.cmml">{</mo><mi id="A3.SS1.p2.5.m5.1.1" xref="A3.SS1.p2.5.m5.1.1.cmml">H</mi><mo id="A3.SS1.p2.5.m5.2.3.3.2.2" xref="A3.SS1.p2.5.m5.2.3.3.1.cmml">,</mo><mi id="A3.SS1.p2.5.m5.2.2" xref="A3.SS1.p2.5.m5.2.2.cmml">L</mi><mo id="A3.SS1.p2.5.m5.2.3.3.2.3" stretchy="false" xref="A3.SS1.p2.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.5.m5.2b"><apply id="A3.SS1.p2.5.m5.2.3.cmml" xref="A3.SS1.p2.5.m5.2.3"><in id="A3.SS1.p2.5.m5.2.3.1.cmml" xref="A3.SS1.p2.5.m5.2.3.1"></in><ci id="A3.SS1.p2.5.m5.2.3.2.cmml" xref="A3.SS1.p2.5.m5.2.3.2">𝑠</ci><set id="A3.SS1.p2.5.m5.2.3.3.1.cmml" xref="A3.SS1.p2.5.m5.2.3.3.2"><ci id="A3.SS1.p2.5.m5.1.1.cmml" xref="A3.SS1.p2.5.m5.1.1">𝐻</ci><ci id="A3.SS1.p2.5.m5.2.2.cmml" xref="A3.SS1.p2.5.m5.2.2">𝐿</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.5.m5.2c">s\in\{H,L\}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.5.m5.2d">italic_s ∈ { italic_H , italic_L }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS1.p3"> <p class="ltx_p" id="A3.SS1.p3.3">We show a stronger property: truthfulness maximizes the expected payoff of <math alttext="M_{\textrm{DMI}}" class="ltx_Math" display="inline" id="A3.SS1.p3.1.m1.1"><semantics id="A3.SS1.p3.1.m1.1a"><msub id="A3.SS1.p3.1.m1.1.1" xref="A3.SS1.p3.1.m1.1.1.cmml"><mi id="A3.SS1.p3.1.m1.1.1.2" xref="A3.SS1.p3.1.m1.1.1.2.cmml">M</mi><mtext id="A3.SS1.p3.1.m1.1.1.3" xref="A3.SS1.p3.1.m1.1.1.3a.cmml">DMI</mtext></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.p3.1.m1.1b"><apply id="A3.SS1.p3.1.m1.1.1.cmml" xref="A3.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A3.SS1.p3.1.m1.1.1.1.cmml" xref="A3.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="A3.SS1.p3.1.m1.1.1.2.cmml" xref="A3.SS1.p3.1.m1.1.1.2">𝑀</ci><ci id="A3.SS1.p3.1.m1.1.1.3a.cmml" xref="A3.SS1.p3.1.m1.1.1.3"><mtext id="A3.SS1.p3.1.m1.1.1.3.cmml" mathsize="70%" xref="A3.SS1.p3.1.m1.1.1.3">DMI</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p3.1.m1.1c">M_{\textrm{DMI}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p3.1.m1.1d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT</annotation></semantics></math> when all other agents play consistent strategies. As a corollary, truthfulness is a Bayes Nash equilibrium even over the larger space of strategies that allow one to choose all <math alttext="T" class="ltx_Math" display="inline" id="A3.SS1.p3.2.m2.1"><semantics id="A3.SS1.p3.2.m2.1a"><mi id="A3.SS1.p3.2.m2.1.1" xref="A3.SS1.p3.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.p3.2.m2.1b"><ci id="A3.SS1.p3.2.m2.1.1.cmml" xref="A3.SS1.p3.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p3.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p3.2.m2.1d">italic_T</annotation></semantics></math> reports simultaneously after looking at all <math alttext="T" class="ltx_Math" display="inline" id="A3.SS1.p3.3.m3.1"><semantics id="A3.SS1.p3.3.m3.1a"><mi id="A3.SS1.p3.3.m3.1.1" xref="A3.SS1.p3.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.p3.3.m3.1b"><ci id="A3.SS1.p3.3.m3.1.1.cmml" xref="A3.SS1.p3.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p3.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p3.3.m3.1d">italic_T</annotation></semantics></math> signals.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem9.1.1.1">Theorem 9</span></span><span class="ltx_text ltx_font_bold" id="Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="Thmtheorem9.p1"> <p class="ltx_p" id="Thmtheorem9.p1.4">In the binary signal model, suppose all agents <math alttext="j\neq i" class="ltx_Math" display="inline" id="Thmtheorem9.p1.1.m1.1"><semantics id="Thmtheorem9.p1.1.m1.1a"><mrow id="Thmtheorem9.p1.1.m1.1.1" xref="Thmtheorem9.p1.1.m1.1.1.cmml"><mi id="Thmtheorem9.p1.1.m1.1.1.2" xref="Thmtheorem9.p1.1.m1.1.1.2.cmml">j</mi><mo id="Thmtheorem9.p1.1.m1.1.1.1" xref="Thmtheorem9.p1.1.m1.1.1.1.cmml">≠</mo><mi id="Thmtheorem9.p1.1.m1.1.1.3" xref="Thmtheorem9.p1.1.m1.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem9.p1.1.m1.1b"><apply id="Thmtheorem9.p1.1.m1.1.1.cmml" xref="Thmtheorem9.p1.1.m1.1.1"><neq id="Thmtheorem9.p1.1.m1.1.1.1.cmml" xref="Thmtheorem9.p1.1.m1.1.1.1"></neq><ci id="Thmtheorem9.p1.1.m1.1.1.2.cmml" xref="Thmtheorem9.p1.1.m1.1.1.2">𝑗</ci><ci id="Thmtheorem9.p1.1.m1.1.1.3.cmml" xref="Thmtheorem9.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem9.p1.1.m1.1c">j\neq i</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem9.p1.1.m1.1d">italic_j ≠ italic_i</annotation></semantics></math> play consistent strategies. Then <math alttext="\sigma_{\text{true}}" class="ltx_Math" display="inline" id="Thmtheorem9.p1.2.m2.1"><semantics id="Thmtheorem9.p1.2.m2.1a"><msub id="Thmtheorem9.p1.2.m2.1.1" xref="Thmtheorem9.p1.2.m2.1.1.cmml"><mi id="Thmtheorem9.p1.2.m2.1.1.2" xref="Thmtheorem9.p1.2.m2.1.1.2.cmml">σ</mi><mtext id="Thmtheorem9.p1.2.m2.1.1.3" xref="Thmtheorem9.p1.2.m2.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem9.p1.2.m2.1b"><apply id="Thmtheorem9.p1.2.m2.1.1.cmml" xref="Thmtheorem9.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmtheorem9.p1.2.m2.1.1.1.cmml" xref="Thmtheorem9.p1.2.m2.1.1">subscript</csymbol><ci id="Thmtheorem9.p1.2.m2.1.1.2.cmml" xref="Thmtheorem9.p1.2.m2.1.1.2">𝜎</ci><ci id="Thmtheorem9.p1.2.m2.1.1.3a.cmml" xref="Thmtheorem9.p1.2.m2.1.1.3"><mtext id="Thmtheorem9.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="Thmtheorem9.p1.2.m2.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem9.p1.2.m2.1c">\sigma_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem9.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> maximizes agent <math alttext="i" class="ltx_Math" display="inline" id="Thmtheorem9.p1.3.m3.1"><semantics id="Thmtheorem9.p1.3.m3.1a"><mi id="Thmtheorem9.p1.3.m3.1.1" xref="Thmtheorem9.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmtheorem9.p1.3.m3.1b"><ci id="Thmtheorem9.p1.3.m3.1.1.cmml" xref="Thmtheorem9.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem9.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem9.p1.3.m3.1d">italic_i</annotation></semantics></math>’s expected payment under <math alttext="M_{\textrm{DMI}}" class="ltx_Math" display="inline" id="Thmtheorem9.p1.4.m4.1"><semantics id="Thmtheorem9.p1.4.m4.1a"><msub id="Thmtheorem9.p1.4.m4.1.1" xref="Thmtheorem9.p1.4.m4.1.1.cmml"><mi id="Thmtheorem9.p1.4.m4.1.1.2" xref="Thmtheorem9.p1.4.m4.1.1.2.cmml">M</mi><mtext id="Thmtheorem9.p1.4.m4.1.1.3" xref="Thmtheorem9.p1.4.m4.1.1.3a.cmml">DMI</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem9.p1.4.m4.1b"><apply id="Thmtheorem9.p1.4.m4.1.1.cmml" xref="Thmtheorem9.p1.4.m4.1.1"><csymbol cd="ambiguous" id="Thmtheorem9.p1.4.m4.1.1.1.cmml" xref="Thmtheorem9.p1.4.m4.1.1">subscript</csymbol><ci id="Thmtheorem9.p1.4.m4.1.1.2.cmml" xref="Thmtheorem9.p1.4.m4.1.1.2">𝑀</ci><ci id="Thmtheorem9.p1.4.m4.1.1.3a.cmml" xref="Thmtheorem9.p1.4.m4.1.1.3"><mtext id="Thmtheorem9.p1.4.m4.1.1.3.cmml" mathsize="70%" xref="Thmtheorem9.p1.4.m4.1.1.3">DMI</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem9.p1.4.m4.1c">M_{\textrm{DMI}}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem9.p1.4.m4.1d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="Thmcorollary6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmcorollary6.1.1.1">Corollary 6</span></span><span class="ltx_text ltx_font_bold" id="Thmcorollary6.2.2">.</span> </h6> <div class="ltx_para" id="Thmcorollary6.p1"> <p class="ltx_p" id="Thmcorollary6.p1.1">The truthful strategy <math alttext="\sigma_{\text{true}}" class="ltx_Math" display="inline" id="Thmcorollary6.p1.1.m1.1"><semantics id="Thmcorollary6.p1.1.m1.1a"><msub id="Thmcorollary6.p1.1.m1.1.1" xref="Thmcorollary6.p1.1.m1.1.1.cmml"><mi id="Thmcorollary6.p1.1.m1.1.1.2" xref="Thmcorollary6.p1.1.m1.1.1.2.cmml">σ</mi><mtext id="Thmcorollary6.p1.1.m1.1.1.3" xref="Thmcorollary6.p1.1.m1.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmcorollary6.p1.1.m1.1b"><apply id="Thmcorollary6.p1.1.m1.1.1.cmml" xref="Thmcorollary6.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmcorollary6.p1.1.m1.1.1.1.cmml" xref="Thmcorollary6.p1.1.m1.1.1">subscript</csymbol><ci id="Thmcorollary6.p1.1.m1.1.1.2.cmml" xref="Thmcorollary6.p1.1.m1.1.1.2">𝜎</ci><ci id="Thmcorollary6.p1.1.m1.1.1.3a.cmml" xref="Thmcorollary6.p1.1.m1.1.1.3"><mtext id="Thmcorollary6.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="Thmcorollary6.p1.1.m1.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmcorollary6.p1.1.m1.1c">\sigma_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="Thmcorollary6.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> is a Bayes Nash equilibrium of the DMI mechanism.</p> </div> </div> <div class="ltx_para" id="A3.SS1.p4"> <p class="ltx_p" id="A3.SS1.p4.1">The key observation needed for Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> is the following characterization of the possible expected values of each determinant term in the definition of <math alttext="M_{\textrm{DMI}}" class="ltx_Math" display="inline" id="A3.SS1.p4.1.m1.1"><semantics id="A3.SS1.p4.1.m1.1a"><msub id="A3.SS1.p4.1.m1.1.1" xref="A3.SS1.p4.1.m1.1.1.cmml"><mi id="A3.SS1.p4.1.m1.1.1.2" xref="A3.SS1.p4.1.m1.1.1.2.cmml">M</mi><mtext id="A3.SS1.p4.1.m1.1.1.3" xref="A3.SS1.p4.1.m1.1.1.3a.cmml">DMI</mtext></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.p4.1.m1.1b"><apply id="A3.SS1.p4.1.m1.1.1.cmml" xref="A3.SS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="A3.SS1.p4.1.m1.1.1.1.cmml" xref="A3.SS1.p4.1.m1.1.1">subscript</csymbol><ci id="A3.SS1.p4.1.m1.1.1.2.cmml" xref="A3.SS1.p4.1.m1.1.1.2">𝑀</ci><ci id="A3.SS1.p4.1.m1.1.1.3a.cmml" xref="A3.SS1.p4.1.m1.1.1.3"><mtext id="A3.SS1.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="A3.SS1.p4.1.m1.1.1.3">DMI</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p4.1.m1.1c">M_{\textrm{DMI}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p4.1.m1.1d">italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="Thmlemma1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmlemma1.1.1.1">Lemma 1</span></span><span class="ltx_text ltx_font_bold" id="Thmlemma1.2.2">.</span> </h6> <div class="ltx_para" id="Thmlemma1.p1"> <p class="ltx_p" id="Thmlemma1.p1.9">Suppose agent <math alttext="j" class="ltx_Math" display="inline" id="Thmlemma1.p1.1.m1.1"><semantics id="Thmlemma1.p1.1.m1.1a"><mi id="Thmlemma1.p1.1.m1.1.1" xref="Thmlemma1.p1.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.1.m1.1b"><ci id="Thmlemma1.p1.1.m1.1.1.cmml" xref="Thmlemma1.p1.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.1.m1.1d">italic_j</annotation></semantics></math> plays a consistent strategy. Let <math alttext="R^{\prime}_{t}" class="ltx_Math" display="inline" id="Thmlemma1.p1.2.m2.1"><semantics id="Thmlemma1.p1.2.m2.1a"><msubsup id="Thmlemma1.p1.2.m2.1.1" xref="Thmlemma1.p1.2.m2.1.1.cmml"><mi id="Thmlemma1.p1.2.m2.1.1.2.2" xref="Thmlemma1.p1.2.m2.1.1.2.2.cmml">R</mi><mi id="Thmlemma1.p1.2.m2.1.1.3" xref="Thmlemma1.p1.2.m2.1.1.3.cmml">t</mi><mo id="Thmlemma1.p1.2.m2.1.1.2.3" xref="Thmlemma1.p1.2.m2.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.2.m2.1b"><apply id="Thmlemma1.p1.2.m2.1.1.cmml" xref="Thmlemma1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.2.m2.1.1.1.cmml" xref="Thmlemma1.p1.2.m2.1.1">subscript</csymbol><apply id="Thmlemma1.p1.2.m2.1.1.2.cmml" xref="Thmlemma1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.2.m2.1.1.2.1.cmml" xref="Thmlemma1.p1.2.m2.1.1">superscript</csymbol><ci id="Thmlemma1.p1.2.m2.1.1.2.2.cmml" xref="Thmlemma1.p1.2.m2.1.1.2.2">𝑅</ci><ci id="Thmlemma1.p1.2.m2.1.1.2.3.cmml" xref="Thmlemma1.p1.2.m2.1.1.2.3">′</ci></apply><ci id="Thmlemma1.p1.2.m2.1.1.3.cmml" xref="Thmlemma1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.2.m2.1c">R^{\prime}_{t}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.2.m2.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> be the random variable on <math alttext="\{H,L\}" class="ltx_Math" display="inline" id="Thmlemma1.p1.3.m3.2"><semantics id="Thmlemma1.p1.3.m3.2a"><mrow id="Thmlemma1.p1.3.m3.2.3.2" xref="Thmlemma1.p1.3.m3.2.3.1.cmml"><mo id="Thmlemma1.p1.3.m3.2.3.2.1" stretchy="false" xref="Thmlemma1.p1.3.m3.2.3.1.cmml">{</mo><mi id="Thmlemma1.p1.3.m3.1.1" xref="Thmlemma1.p1.3.m3.1.1.cmml">H</mi><mo id="Thmlemma1.p1.3.m3.2.3.2.2" xref="Thmlemma1.p1.3.m3.2.3.1.cmml">,</mo><mi id="Thmlemma1.p1.3.m3.2.2" xref="Thmlemma1.p1.3.m3.2.2.cmml">L</mi><mo id="Thmlemma1.p1.3.m3.2.3.2.3" stretchy="false" xref="Thmlemma1.p1.3.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.3.m3.2b"><set id="Thmlemma1.p1.3.m3.2.3.1.cmml" xref="Thmlemma1.p1.3.m3.2.3.2"><ci id="Thmlemma1.p1.3.m3.1.1.cmml" xref="Thmlemma1.p1.3.m3.1.1">𝐻</ci><ci id="Thmlemma1.p1.3.m3.2.2.cmml" xref="Thmlemma1.p1.3.m3.2.2">𝐿</ci></set></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.3.m3.2c">\{H,L\}</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.3.m3.2d">{ italic_H , italic_L }</annotation></semantics></math> resulting agent <math alttext="j" class="ltx_Math" display="inline" id="Thmlemma1.p1.4.m4.1"><semantics id="Thmlemma1.p1.4.m4.1a"><mi id="Thmlemma1.p1.4.m4.1.1" xref="Thmlemma1.p1.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.4.m4.1b"><ci id="Thmlemma1.p1.4.m4.1.1.cmml" xref="Thmlemma1.p1.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.4.m4.1d">italic_j</annotation></semantics></math>’s strategy, i.e., <math alttext="R_{t}^{\prime}\sim\hat{\sigma}(S_{t}^{\prime})" class="ltx_Math" display="inline" id="Thmlemma1.p1.5.m5.1"><semantics id="Thmlemma1.p1.5.m5.1a"><mrow id="Thmlemma1.p1.5.m5.1.1" xref="Thmlemma1.p1.5.m5.1.1.cmml"><msubsup id="Thmlemma1.p1.5.m5.1.1.3" xref="Thmlemma1.p1.5.m5.1.1.3.cmml"><mi id="Thmlemma1.p1.5.m5.1.1.3.2.2" xref="Thmlemma1.p1.5.m5.1.1.3.2.2.cmml">R</mi><mi id="Thmlemma1.p1.5.m5.1.1.3.2.3" xref="Thmlemma1.p1.5.m5.1.1.3.2.3.cmml">t</mi><mo id="Thmlemma1.p1.5.m5.1.1.3.3" xref="Thmlemma1.p1.5.m5.1.1.3.3.cmml">′</mo></msubsup><mo id="Thmlemma1.p1.5.m5.1.1.2" xref="Thmlemma1.p1.5.m5.1.1.2.cmml">∼</mo><mrow id="Thmlemma1.p1.5.m5.1.1.1" xref="Thmlemma1.p1.5.m5.1.1.1.cmml"><mover accent="true" id="Thmlemma1.p1.5.m5.1.1.1.3" xref="Thmlemma1.p1.5.m5.1.1.1.3.cmml"><mi id="Thmlemma1.p1.5.m5.1.1.1.3.2" xref="Thmlemma1.p1.5.m5.1.1.1.3.2.cmml">σ</mi><mo id="Thmlemma1.p1.5.m5.1.1.1.3.1" xref="Thmlemma1.p1.5.m5.1.1.1.3.1.cmml">^</mo></mover><mo id="Thmlemma1.p1.5.m5.1.1.1.2" xref="Thmlemma1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="Thmlemma1.p1.5.m5.1.1.1.1.1" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="Thmlemma1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msubsup id="Thmlemma1.p1.5.m5.1.1.1.1.1.1" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.2" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.2.cmml">S</mi><mi id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.3" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.3.cmml">t</mi><mo id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.3" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="Thmlemma1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.5.m5.1b"><apply id="Thmlemma1.p1.5.m5.1.1.cmml" xref="Thmlemma1.p1.5.m5.1.1"><csymbol cd="latexml" id="Thmlemma1.p1.5.m5.1.1.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.2">similar-to</csymbol><apply id="Thmlemma1.p1.5.m5.1.1.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.5.m5.1.1.3.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.3">superscript</csymbol><apply id="Thmlemma1.p1.5.m5.1.1.3.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.5.m5.1.1.3.2.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.3">subscript</csymbol><ci id="Thmlemma1.p1.5.m5.1.1.3.2.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.3.2.2">𝑅</ci><ci id="Thmlemma1.p1.5.m5.1.1.3.2.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.3.2.3">𝑡</ci></apply><ci id="Thmlemma1.p1.5.m5.1.1.3.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.3.3">′</ci></apply><apply id="Thmlemma1.p1.5.m5.1.1.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.1"><times id="Thmlemma1.p1.5.m5.1.1.1.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.2"></times><apply id="Thmlemma1.p1.5.m5.1.1.1.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.3"><ci id="Thmlemma1.p1.5.m5.1.1.1.3.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.3.1">^</ci><ci id="Thmlemma1.p1.5.m5.1.1.1.3.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.3.2">𝜎</ci></apply><apply id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1">superscript</csymbol><apply id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.1.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.2.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.2">𝑆</ci><ci id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.2.3">𝑡</ci></apply><ci id="Thmlemma1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="Thmlemma1.p1.5.m5.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.5.m5.1c">R_{t}^{\prime}\sim\hat{\sigma}(S_{t}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.5.m5.1d">italic_R start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ over^ start_ARG italic_σ end_ARG ( italic_S start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> for each task <math alttext="t" class="ltx_Math" display="inline" id="Thmlemma1.p1.6.m6.1"><semantics id="Thmlemma1.p1.6.m6.1a"><mi id="Thmlemma1.p1.6.m6.1.1" xref="Thmlemma1.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.6.m6.1b"><ci id="Thmlemma1.p1.6.m6.1.1.cmml" xref="Thmlemma1.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.6.m6.1d">italic_t</annotation></semantics></math>. Let <math alttext="p(s_{t})=\Pr[R_{t}^{\prime}=H\mid S_{t}=s_{t}]" class="ltx_Math" display="inline" id="Thmlemma1.p1.7.m7.3"><semantics id="Thmlemma1.p1.7.m7.3a"><mrow id="Thmlemma1.p1.7.m7.3.3" xref="Thmlemma1.p1.7.m7.3.3.cmml"><mrow id="Thmlemma1.p1.7.m7.2.2.1" xref="Thmlemma1.p1.7.m7.2.2.1.cmml"><mi id="Thmlemma1.p1.7.m7.2.2.1.3" xref="Thmlemma1.p1.7.m7.2.2.1.3.cmml">p</mi><mo id="Thmlemma1.p1.7.m7.2.2.1.2" xref="Thmlemma1.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="Thmlemma1.p1.7.m7.2.2.1.1.1" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="Thmlemma1.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><msub id="Thmlemma1.p1.7.m7.2.2.1.1.1.1" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.2" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.2.cmml">s</mi><mi id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.3" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.3.cmml">t</mi></msub><mo id="Thmlemma1.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmlemma1.p1.7.m7.3.3.3" xref="Thmlemma1.p1.7.m7.3.3.3.cmml">=</mo><mrow id="Thmlemma1.p1.7.m7.3.3.2.1" xref="Thmlemma1.p1.7.m7.3.3.2.2.cmml"><mi id="Thmlemma1.p1.7.m7.1.1" xref="Thmlemma1.p1.7.m7.1.1.cmml">Pr</mi><mo id="Thmlemma1.p1.7.m7.3.3.2.1a" xref="Thmlemma1.p1.7.m7.3.3.2.2.cmml">⁡</mo><mrow id="Thmlemma1.p1.7.m7.3.3.2.1.1" xref="Thmlemma1.p1.7.m7.3.3.2.2.cmml"><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.2" stretchy="false" xref="Thmlemma1.p1.7.m7.3.3.2.2.cmml">[</mo><mrow id="Thmlemma1.p1.7.m7.3.3.2.1.1.1" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.cmml"><msubsup id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.cmml"><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.2" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.2.cmml">R</mi><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.3.cmml">t</mi><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.3.cmml">′</mo></msubsup><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.3.cmml">=</mo><mrow id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.cmml"><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.2" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.2.cmml">H</mi><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.1" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.1.cmml">∣</mo><msub id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.cmml"><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.2" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.2.cmml">S</mi><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.3.cmml">t</mi></msub></mrow><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.5" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.5.cmml">=</mo><msub id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.cmml"><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.2" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.2.cmml">s</mi><mi id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.3" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.3.cmml">t</mi></msub></mrow><mo id="Thmlemma1.p1.7.m7.3.3.2.1.1.3" stretchy="false" xref="Thmlemma1.p1.7.m7.3.3.2.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.7.m7.3b"><apply id="Thmlemma1.p1.7.m7.3.3.cmml" xref="Thmlemma1.p1.7.m7.3.3"><eq id="Thmlemma1.p1.7.m7.3.3.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.3"></eq><apply id="Thmlemma1.p1.7.m7.2.2.1.cmml" xref="Thmlemma1.p1.7.m7.2.2.1"><times id="Thmlemma1.p1.7.m7.2.2.1.2.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.2"></times><ci id="Thmlemma1.p1.7.m7.2.2.1.3.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.3">𝑝</ci><apply id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.1.1">subscript</csymbol><ci id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.2">𝑠</ci><ci id="Thmlemma1.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="Thmlemma1.p1.7.m7.2.2.1.1.1.1.3">𝑡</ci></apply></apply><apply id="Thmlemma1.p1.7.m7.3.3.2.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1"><ci id="Thmlemma1.p1.7.m7.1.1.cmml" xref="Thmlemma1.p1.7.m7.1.1">Pr</ci><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1"><and id="Thmlemma1.p1.7.m7.3.3.2.1.1.1a.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1"></and><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1b.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1"><eq id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.3"></eq><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2">superscript</csymbol><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2"><csymbol cd="ambiguous" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2">subscript</csymbol><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.2">𝑅</ci><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.2.3">𝑡</ci></apply><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.2.3">′</ci></apply><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4"><csymbol cd="latexml" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.1">conditional</csymbol><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.2">𝐻</ci><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3"><csymbol cd="ambiguous" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3">subscript</csymbol><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.2">𝑆</ci><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.3.3">𝑡</ci></apply></apply></apply><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1c.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1"><eq id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.5.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#Thmlemma1.p1.7.m7.3.3.2.1.1.1.4.cmml" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1d.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1"></share><apply id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6"><csymbol cd="ambiguous" id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.1.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6">subscript</csymbol><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.2.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.2">𝑠</ci><ci id="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.3.cmml" xref="Thmlemma1.p1.7.m7.3.3.2.1.1.1.6.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.7.m7.3c">p(s_{t})=\Pr[R_{t}^{\prime}=H\mid S_{t}=s_{t}]</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.7.m7.3d">italic_p ( italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) = roman_Pr [ italic_R start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_H ∣ italic_S start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ]</annotation></semantics></math>, which by assumption is independent of <math alttext="t" class="ltx_Math" display="inline" id="Thmlemma1.p1.8.m8.1"><semantics id="Thmlemma1.p1.8.m8.1a"><mi id="Thmlemma1.p1.8.m8.1.1" xref="Thmlemma1.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.8.m8.1b"><ci id="Thmlemma1.p1.8.m8.1.1.cmml" xref="Thmlemma1.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.8.m8.1d">italic_t</annotation></semantics></math>. Then for agent <math alttext="i" class="ltx_Math" display="inline" id="Thmlemma1.p1.9.m9.1"><semantics id="Thmlemma1.p1.9.m9.1a"><mi id="Thmlemma1.p1.9.m9.1.1" xref="Thmlemma1.p1.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="Thmlemma1.p1.9.m9.1b"><ci id="Thmlemma1.p1.9.m9.1.1.cmml" xref="Thmlemma1.p1.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmlemma1.p1.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="Thmlemma1.p1.9.m9.1d">italic_i</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx10"> <tbody id="A3.Ex17"><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\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]\;=\;\begin{cases% }0,&\text{if }r_{1}=r_{2}\text{ or }s_{1}=s_{2},\\ p(H)-p(L),&\text{if }(r_{1},r_{2})=(s_{1},s_{2}),\\ p(L)-p(H),&\text{if }(r_{1},r_{2})=(s_{2},s_{1}).\end{cases}" class="ltx_Math" display="inline" id="A3.Ex17.m1.7"><semantics id="A3.Ex17.m1.7a"><mrow id="A3.Ex17.m1.7.7" xref="A3.Ex17.m1.7.7.cmml"><mrow id="A3.Ex17.m1.7.7.1" xref="A3.Ex17.m1.7.7.1.cmml"><mo id="A3.Ex17.m1.7.7.1.2" movablelimits="false" rspace="0em" xref="A3.Ex17.m1.7.7.1.2.cmml">𝔼</mo><mrow id="A3.Ex17.m1.7.7.1.1.1" xref="A3.Ex17.m1.7.7.1.1.2.cmml"><mo id="A3.Ex17.m1.7.7.1.1.1.2" stretchy="false" xref="A3.Ex17.m1.7.7.1.1.2.1.cmml">[</mo><mrow id="A3.Ex17.m1.7.7.1.1.1.1" xref="A3.Ex17.m1.7.7.1.1.1.1.cmml"><mrow id="A3.Ex17.m1.7.7.1.1.1.1.4" xref="A3.Ex17.m1.7.7.1.1.1.1.4.cmml"><mo id="A3.Ex17.m1.7.7.1.1.1.1.4.1" lspace="0em" movablelimits="false" rspace="0.167em" xref="A3.Ex17.m1.7.7.1.1.1.1.4.1.cmml">det</mo><msub id="A3.Ex17.m1.7.7.1.1.1.1.4.2" xref="A3.Ex17.m1.7.7.1.1.1.1.4.2.cmml"><mi id="A3.Ex17.m1.7.7.1.1.1.1.4.2.2" xref="A3.Ex17.m1.7.7.1.1.1.1.4.2.2.cmml">M</mi><mn id="A3.Ex17.m1.7.7.1.1.1.1.4.2.3" xref="A3.Ex17.m1.7.7.1.1.1.1.4.2.3.cmml">12</mn></msub></mrow><mo id="A3.Ex17.m1.7.7.1.1.1.1.3" xref="A3.Ex17.m1.7.7.1.1.1.1.3.cmml">∣</mo><mrow id="A3.Ex17.m1.7.7.1.1.1.1.2.2" xref="A3.Ex17.m1.7.7.1.1.1.1.2.3.cmml"><msub id="A3.Ex17.m1.7.7.1.1.1.1.1.1.1" xref="A3.Ex17.m1.7.7.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex17.m1.7.7.1.1.1.1.1.1.1.2" xref="A3.Ex17.m1.7.7.1.1.1.1.1.1.1.2.cmml">S</mi><mn id="A3.Ex17.m1.7.7.1.1.1.1.1.1.1.3" xref="A3.Ex17.m1.7.7.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.Ex17.m1.7.7.1.1.1.1.2.2.3" xref="A3.Ex17.m1.7.7.1.1.1.1.2.3.cmml">,</mo><msub id="A3.Ex17.m1.7.7.1.1.1.1.2.2.2" xref="A3.Ex17.m1.7.7.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex17.m1.7.7.1.1.1.1.2.2.2.2" xref="A3.Ex17.m1.7.7.1.1.1.1.2.2.2.2.cmml">S</mi><mn id="A3.Ex17.m1.7.7.1.1.1.1.2.2.2.3" xref="A3.Ex17.m1.7.7.1.1.1.1.2.2.2.3.cmml">2</mn></msub></mrow></mrow><mo id="A3.Ex17.m1.7.7.1.1.1.3" rspace="0.280em" stretchy="false" xref="A3.Ex17.m1.7.7.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="A3.Ex17.m1.7.7.2" rspace="0.558em" xref="A3.Ex17.m1.7.7.2.cmml">=</mo><mrow id="A3.Ex17.m1.6.6a" xref="A3.Ex17.m1.7.7.3.1.cmml"><mo id="A3.Ex17.m1.6.6a.7" xref="A3.Ex17.m1.7.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" id="A3.Ex17.m1.6.6.6a" rowspacing="0pt" xref="A3.Ex17.m1.7.7.3.1.cmml"><mtr id="A3.Ex17.m1.6.6.6aa" xref="A3.Ex17.m1.7.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex17.m1.6.6.6ab" xref="A3.Ex17.m1.7.7.3.1.cmml"><mrow id="A3.Ex17.m1.1.1.1.1.1.1.3" xref="A3.Ex17.m1.7.7.3.1.cmml"><mn id="A3.Ex17.m1.1.1.1.1.1.1.1" xref="A3.Ex17.m1.1.1.1.1.1.1.1.cmml">0</mn><mo id="A3.Ex17.m1.1.1.1.1.1.1.3.1" xref="A3.Ex17.m1.7.7.3.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex17.m1.6.6.6ac" xref="A3.Ex17.m1.7.7.3.1.cmml"><mrow id="A3.Ex17.m1.2.2.2.2.2.1.1" xref="A3.Ex17.m1.2.2.2.2.2.1.1.1.cmml"><mrow id="A3.Ex17.m1.2.2.2.2.2.1.1.1" 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S_{1},S_{2}]\;=\;\begin{cases% }0,&\text{if }r_{1}=r_{2}\text{ or }s_{1}=s_{2},\\ p(H)-p(L),&\text{if }(r_{1},r_{2})=(s_{1},s_{2}),\\ p(L)-p(H),&\text{if }(r_{1},r_{2})=(s_{2},s_{1}).\end{cases}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex17.m1.7d">blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = { start_ROW start_CELL 0 , end_CELL start_CELL if italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT or italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_p ( italic_H ) - italic_p ( italic_L ) , end_CELL start_CELL if ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL italic_p ( italic_L ) - italic_p ( italic_H ) , end_CELL start_CELL if ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="Thmlemma1.p1.10">where as before</p> <table class="ltx_equation ltx_eqn_table" id="A3.Ex18"> <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="M_{12}\;=\;1_{r_{1}}\,1_{R_{1}^{\prime}}^{\top}\;+\;1_{r_{2}}\,1_{R_{2}^{% \prime}}^{\top}~{}." class="ltx_Math" display="block" id="A3.Ex18.m1.1"><semantics id="A3.Ex18.m1.1a"><mrow id="A3.Ex18.m1.1.1.1" 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cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.2">subscript</csymbol><cn id="A3.Ex18.m1.1.1.1.1.3.2.2.2.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.2.2.2">1</cn><apply id="A3.Ex18.m1.1.1.1.1.3.2.2.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.2.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.2.3">subscript</csymbol><ci id="A3.Ex18.m1.1.1.1.1.3.2.2.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.2.3.2">𝑟</ci><cn id="A3.Ex18.m1.1.1.1.1.3.2.2.3.3.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.2.2.3.3">1</cn></apply></apply><apply id="A3.Ex18.m1.1.1.1.1.3.2.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3">superscript</csymbol><apply id="A3.Ex18.m1.1.1.1.1.3.2.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.3.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3">subscript</csymbol><cn id="A3.Ex18.m1.1.1.1.1.3.2.3.2.2.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.2">1</cn><apply id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3">superscript</csymbol><apply id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3">subscript</csymbol><ci id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.2">𝑅</ci><cn id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.3.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.2.3">1</cn></apply><ci id="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.2.3.3">′</ci></apply></apply><csymbol cd="latexml" id="A3.Ex18.m1.1.1.1.1.3.2.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.2.3.3">top</csymbol></apply></apply><apply id="A3.Ex18.m1.1.1.1.1.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3"><times id="A3.Ex18.m1.1.1.1.1.3.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.1"></times><apply id="A3.Ex18.m1.1.1.1.1.3.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.2">subscript</csymbol><cn id="A3.Ex18.m1.1.1.1.1.3.3.2.2.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.3.2.2">1</cn><apply id="A3.Ex18.m1.1.1.1.1.3.3.2.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.2.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.2.3">subscript</csymbol><ci id="A3.Ex18.m1.1.1.1.1.3.3.2.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.2.3.2">𝑟</ci><cn id="A3.Ex18.m1.1.1.1.1.3.3.2.3.3.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.3.2.3.3">2</cn></apply></apply><apply id="A3.Ex18.m1.1.1.1.1.3.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A3.Ex18.m1.1.1.1.1.3.3.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3">subscript</csymbol><cn id="A3.Ex18.m1.1.1.1.1.3.3.3.2.2.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.2">1</cn><apply id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3">superscript</csymbol><apply id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3"><csymbol cd="ambiguous" id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.1.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3">subscript</csymbol><ci id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.2.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.2">𝑅</ci><cn id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.3.cmml" type="integer" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.2.3">2</cn></apply><ci id="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.2.3.3">′</ci></apply></apply><csymbol cd="latexml" id="A3.Ex18.m1.1.1.1.1.3.3.3.3.cmml" xref="A3.Ex18.m1.1.1.1.1.3.3.3.3">top</csymbol></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex18.m1.1c">M_{12}\;=\;1_{r_{1}}\,1_{R_{1}^{\prime}}^{\top}\;+\;1_{r_{2}}\,1_{R_{2}^{% \prime}}^{\top}~{}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex18.m1.1d">italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT = 1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + 1 start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A3.SS1.3"> <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.p1.4">If <math alttext="r_{1}=r_{2}" class="ltx_Math" display="inline" id="A3.SS1.1.p1.1.m1.1"><semantics id="A3.SS1.1.p1.1.m1.1a"><mrow id="A3.SS1.1.p1.1.m1.1.1" xref="A3.SS1.1.p1.1.m1.1.1.cmml"><msub id="A3.SS1.1.p1.1.m1.1.1.2" xref="A3.SS1.1.p1.1.m1.1.1.2.cmml"><mi id="A3.SS1.1.p1.1.m1.1.1.2.2" xref="A3.SS1.1.p1.1.m1.1.1.2.2.cmml">r</mi><mn id="A3.SS1.1.p1.1.m1.1.1.2.3" xref="A3.SS1.1.p1.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="A3.SS1.1.p1.1.m1.1.1.1" xref="A3.SS1.1.p1.1.m1.1.1.1.cmml">=</mo><msub id="A3.SS1.1.p1.1.m1.1.1.3" xref="A3.SS1.1.p1.1.m1.1.1.3.cmml"><mi id="A3.SS1.1.p1.1.m1.1.1.3.2" xref="A3.SS1.1.p1.1.m1.1.1.3.2.cmml">r</mi><mn id="A3.SS1.1.p1.1.m1.1.1.3.3" xref="A3.SS1.1.p1.1.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.1.m1.1b"><apply id="A3.SS1.1.p1.1.m1.1.1.cmml" xref="A3.SS1.1.p1.1.m1.1.1"><eq id="A3.SS1.1.p1.1.m1.1.1.1.cmml" xref="A3.SS1.1.p1.1.m1.1.1.1"></eq><apply id="A3.SS1.1.p1.1.m1.1.1.2.cmml" xref="A3.SS1.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.1.m1.1.1.2.1.cmml" xref="A3.SS1.1.p1.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS1.1.p1.1.m1.1.1.2.2.cmml" xref="A3.SS1.1.p1.1.m1.1.1.2.2">𝑟</ci><cn id="A3.SS1.1.p1.1.m1.1.1.2.3.cmml" type="integer" xref="A3.SS1.1.p1.1.m1.1.1.2.3">1</cn></apply><apply id="A3.SS1.1.p1.1.m1.1.1.3.cmml" xref="A3.SS1.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.1.p1.1.m1.1.1.3.1.cmml" xref="A3.SS1.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="A3.SS1.1.p1.1.m1.1.1.3.2.cmml" xref="A3.SS1.1.p1.1.m1.1.1.3.2">𝑟</ci><cn id="A3.SS1.1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="A3.SS1.1.p1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.1.m1.1c">r_{1}=r_{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.1.m1.1d">italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="M_{12}" class="ltx_Math" display="inline" id="A3.SS1.1.p1.2.m2.1"><semantics id="A3.SS1.1.p1.2.m2.1a"><msub id="A3.SS1.1.p1.2.m2.1.1" xref="A3.SS1.1.p1.2.m2.1.1.cmml"><mi id="A3.SS1.1.p1.2.m2.1.1.2" xref="A3.SS1.1.p1.2.m2.1.1.2.cmml">M</mi><mn id="A3.SS1.1.p1.2.m2.1.1.3" xref="A3.SS1.1.p1.2.m2.1.1.3.cmml">12</mn></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.2.m2.1b"><apply id="A3.SS1.1.p1.2.m2.1.1.cmml" xref="A3.SS1.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.2.m2.1.1.1.cmml" xref="A3.SS1.1.p1.2.m2.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.2.m2.1.1.2.cmml" xref="A3.SS1.1.p1.2.m2.1.1.2">𝑀</ci><cn id="A3.SS1.1.p1.2.m2.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.2.m2.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.2.m2.1c">M_{12}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.2.m2.1d">italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT</annotation></semantics></math> has rank one and <math alttext="\det M_{12}=0" 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.m3.1.1" xref="A3.SS1.1.p1.3.m3.1.1.cmml"><mrow id="A3.SS1.1.p1.3.m3.1.1.2" xref="A3.SS1.1.p1.3.m3.1.1.2.cmml"><mo id="A3.SS1.1.p1.3.m3.1.1.2.1" rspace="0.167em" xref="A3.SS1.1.p1.3.m3.1.1.2.1.cmml">det</mo><msub id="A3.SS1.1.p1.3.m3.1.1.2.2" xref="A3.SS1.1.p1.3.m3.1.1.2.2.cmml"><mi id="A3.SS1.1.p1.3.m3.1.1.2.2.2" xref="A3.SS1.1.p1.3.m3.1.1.2.2.2.cmml">M</mi><mn id="A3.SS1.1.p1.3.m3.1.1.2.2.3" xref="A3.SS1.1.p1.3.m3.1.1.2.2.3.cmml">12</mn></msub></mrow><mo id="A3.SS1.1.p1.3.m3.1.1.1" xref="A3.SS1.1.p1.3.m3.1.1.1.cmml">=</mo><mn id="A3.SS1.1.p1.3.m3.1.1.3" xref="A3.SS1.1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.3.m3.1b"><apply id="A3.SS1.1.p1.3.m3.1.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1"><eq id="A3.SS1.1.p1.3.m3.1.1.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.1"></eq><apply id="A3.SS1.1.p1.3.m3.1.1.2.cmml" xref="A3.SS1.1.p1.3.m3.1.1.2"><determinant id="A3.SS1.1.p1.3.m3.1.1.2.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.2.1"></determinant><apply id="A3.SS1.1.p1.3.m3.1.1.2.2.cmml" xref="A3.SS1.1.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.3.m3.1.1.2.2.1.cmml" xref="A3.SS1.1.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="A3.SS1.1.p1.3.m3.1.1.2.2.2.cmml" xref="A3.SS1.1.p1.3.m3.1.1.2.2.2">𝑀</ci><cn id="A3.SS1.1.p1.3.m3.1.1.2.2.3.cmml" type="integer" xref="A3.SS1.1.p1.3.m3.1.1.2.2.3">12</cn></apply></apply><cn id="A3.SS1.1.p1.3.m3.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.3.m3.1c">\det M_{12}=0</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.3.m3.1d">roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT = 0</annotation></semantics></math> deterministically. Suppose next <math alttext="(r_{1},r_{2})=(H,L)" class="ltx_Math" display="inline" id="A3.SS1.1.p1.4.m4.4"><semantics id="A3.SS1.1.p1.4.m4.4a"><mrow id="A3.SS1.1.p1.4.m4.4.4" xref="A3.SS1.1.p1.4.m4.4.4.cmml"><mrow id="A3.SS1.1.p1.4.m4.4.4.2.2" xref="A3.SS1.1.p1.4.m4.4.4.2.3.cmml"><mo id="A3.SS1.1.p1.4.m4.4.4.2.2.3" stretchy="false" xref="A3.SS1.1.p1.4.m4.4.4.2.3.cmml">(</mo><msub id="A3.SS1.1.p1.4.m4.3.3.1.1.1" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1.cmml"><mi id="A3.SS1.1.p1.4.m4.3.3.1.1.1.2" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1.2.cmml">r</mi><mn id="A3.SS1.1.p1.4.m4.3.3.1.1.1.3" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.1.p1.4.m4.4.4.2.2.4" xref="A3.SS1.1.p1.4.m4.4.4.2.3.cmml">,</mo><msub id="A3.SS1.1.p1.4.m4.4.4.2.2.2" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2.cmml"><mi id="A3.SS1.1.p1.4.m4.4.4.2.2.2.2" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2.2.cmml">r</mi><mn id="A3.SS1.1.p1.4.m4.4.4.2.2.2.3" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.1.p1.4.m4.4.4.2.2.5" stretchy="false" xref="A3.SS1.1.p1.4.m4.4.4.2.3.cmml">)</mo></mrow><mo id="A3.SS1.1.p1.4.m4.4.4.3" xref="A3.SS1.1.p1.4.m4.4.4.3.cmml">=</mo><mrow id="A3.SS1.1.p1.4.m4.4.4.4.2" xref="A3.SS1.1.p1.4.m4.4.4.4.1.cmml"><mo id="A3.SS1.1.p1.4.m4.4.4.4.2.1" stretchy="false" xref="A3.SS1.1.p1.4.m4.4.4.4.1.cmml">(</mo><mi id="A3.SS1.1.p1.4.m4.1.1" xref="A3.SS1.1.p1.4.m4.1.1.cmml">H</mi><mo id="A3.SS1.1.p1.4.m4.4.4.4.2.2" xref="A3.SS1.1.p1.4.m4.4.4.4.1.cmml">,</mo><mi id="A3.SS1.1.p1.4.m4.2.2" xref="A3.SS1.1.p1.4.m4.2.2.cmml">L</mi><mo id="A3.SS1.1.p1.4.m4.4.4.4.2.3" stretchy="false" xref="A3.SS1.1.p1.4.m4.4.4.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.4.m4.4b"><apply id="A3.SS1.1.p1.4.m4.4.4.cmml" xref="A3.SS1.1.p1.4.m4.4.4"><eq id="A3.SS1.1.p1.4.m4.4.4.3.cmml" xref="A3.SS1.1.p1.4.m4.4.4.3"></eq><interval closure="open" id="A3.SS1.1.p1.4.m4.4.4.2.3.cmml" xref="A3.SS1.1.p1.4.m4.4.4.2.2"><apply id="A3.SS1.1.p1.4.m4.3.3.1.1.1.cmml" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.4.m4.3.3.1.1.1.1.cmml" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.4.m4.3.3.1.1.1.2.cmml" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1.2">𝑟</ci><cn id="A3.SS1.1.p1.4.m4.3.3.1.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.4.m4.3.3.1.1.1.3">1</cn></apply><apply id="A3.SS1.1.p1.4.m4.4.4.2.2.2.cmml" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.4.m4.4.4.2.2.2.1.cmml" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2">subscript</csymbol><ci id="A3.SS1.1.p1.4.m4.4.4.2.2.2.2.cmml" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2.2">𝑟</ci><cn id="A3.SS1.1.p1.4.m4.4.4.2.2.2.3.cmml" type="integer" xref="A3.SS1.1.p1.4.m4.4.4.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.1.p1.4.m4.4.4.4.1.cmml" xref="A3.SS1.1.p1.4.m4.4.4.4.2"><ci id="A3.SS1.1.p1.4.m4.1.1.cmml" xref="A3.SS1.1.p1.4.m4.1.1">𝐻</ci><ci id="A3.SS1.1.p1.4.m4.2.2.cmml" xref="A3.SS1.1.p1.4.m4.2.2">𝐿</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.4.m4.4c">(r_{1},r_{2})=(H,L)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.4.m4.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_H , italic_L )</annotation></semantics></math>. We have</p> <table class="ltx_equation ltx_eqn_table" id="A3.Ex19"> <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="\det M_{12}\;=\;\begin{cases}+1,&\text{if }(R_{1}^{\prime},R_{2}^{\prime})=(H,% L),\\ -1,&\text{if }(R_{1}^{\prime},R_{2}^{\prime})=(L,H),\\ 0,&\text{otherwise}.\end{cases}" class="ltx_Math" display="block" id="A3.Ex19.m1.6"><semantics id="A3.Ex19.m1.6a"><mrow id="A3.Ex19.m1.6.7" xref="A3.Ex19.m1.6.7.cmml"><mrow id="A3.Ex19.m1.6.7.2" xref="A3.Ex19.m1.6.7.2.cmml"><mo id="A3.Ex19.m1.6.7.2.1" movablelimits="false" rspace="0.167em" xref="A3.Ex19.m1.6.7.2.1.cmml">det</mo><msub id="A3.Ex19.m1.6.7.2.2" xref="A3.Ex19.m1.6.7.2.2.cmml"><mi id="A3.Ex19.m1.6.7.2.2.2" xref="A3.Ex19.m1.6.7.2.2.2.cmml">M</mi><mn id="A3.Ex19.m1.6.7.2.2.3" xref="A3.Ex19.m1.6.7.2.2.3.cmml">12</mn></msub></mrow><mo id="A3.Ex19.m1.6.7.1" rspace="0.558em" xref="A3.Ex19.m1.6.7.1.cmml">=</mo><mrow id="A3.Ex19.m1.6.6" xref="A3.Ex19.m1.6.7.3.1.cmml"><mo id="A3.Ex19.m1.6.6.7" xref="A3.Ex19.m1.6.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="A3.Ex19.m1.6.6.6" rowspacing="0pt" xref="A3.Ex19.m1.6.7.3.1.cmml"><mtr id="A3.Ex19.m1.6.6.6a" xref="A3.Ex19.m1.6.7.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex19.m1.6.6.6b" xref="A3.Ex19.m1.6.7.3.1.cmml"><mrow id="A3.Ex19.m1.1.1.1.1.1.1.1" xref="A3.Ex19.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex19.m1.1.1.1.1.1.1.1.1" xref="A3.Ex19.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex19.m1.1.1.1.1.1.1.1.1a" xref="A3.Ex19.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="A3.Ex19.m1.1.1.1.1.1.1.1.1.2" xref="A3.Ex19.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="A3.Ex19.m1.1.1.1.1.1.1.1.2" xref="A3.Ex19.m1.1.1.1.1.1.1.1.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex19.m1.6.6.6c" xref="A3.Ex19.m1.6.7.3.1.cmml"><mrow id="A3.Ex19.m1.2.2.2.2.2.1.3" 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xref="A3.Ex19.m1.5.5.5.5.1.1.1">0</cn><ci id="A3.Ex19.m1.6.6.6.6.2.1.1a.cmml" xref="A3.Ex19.m1.6.6.6.6.2.1.3"><mtext id="A3.Ex19.m1.6.6.6.6.2.1.1.cmml" xref="A3.Ex19.m1.6.6.6.6.2.1.1">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex19.m1.6c">\det M_{12}\;=\;\begin{cases}+1,&\text{if }(R_{1}^{\prime},R_{2}^{\prime})=(H,% L),\\ -1,&\text{if }(R_{1}^{\prime},R_{2}^{\prime})=(L,H),\\ 0,&\text{otherwise}.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex19.m1.6d">roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT = { start_ROW start_CELL + 1 , end_CELL start_CELL if ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( italic_H , italic_L ) , end_CELL end_ROW start_ROW start_CELL - 1 , end_CELL start_CELL if ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( italic_L , italic_H ) , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.1.p1.5">As <math alttext="R_{1}^{\prime},R_{2}^{\prime}" class="ltx_Math" display="inline" id="A3.SS1.1.p1.5.m1.2"><semantics id="A3.SS1.1.p1.5.m1.2a"><mrow id="A3.SS1.1.p1.5.m1.2.2.2" xref="A3.SS1.1.p1.5.m1.2.2.3.cmml"><msubsup id="A3.SS1.1.p1.5.m1.1.1.1.1" xref="A3.SS1.1.p1.5.m1.1.1.1.1.cmml"><mi id="A3.SS1.1.p1.5.m1.1.1.1.1.2.2" xref="A3.SS1.1.p1.5.m1.1.1.1.1.2.2.cmml">R</mi><mn id="A3.SS1.1.p1.5.m1.1.1.1.1.2.3" xref="A3.SS1.1.p1.5.m1.1.1.1.1.2.3.cmml">1</mn><mo id="A3.SS1.1.p1.5.m1.1.1.1.1.3" xref="A3.SS1.1.p1.5.m1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="A3.SS1.1.p1.5.m1.2.2.2.3" xref="A3.SS1.1.p1.5.m1.2.2.3.cmml">,</mo><msubsup id="A3.SS1.1.p1.5.m1.2.2.2.2" xref="A3.SS1.1.p1.5.m1.2.2.2.2.cmml"><mi id="A3.SS1.1.p1.5.m1.2.2.2.2.2.2" xref="A3.SS1.1.p1.5.m1.2.2.2.2.2.2.cmml">R</mi><mn id="A3.SS1.1.p1.5.m1.2.2.2.2.2.3" xref="A3.SS1.1.p1.5.m1.2.2.2.2.2.3.cmml">2</mn><mo id="A3.SS1.1.p1.5.m1.2.2.2.2.3" xref="A3.SS1.1.p1.5.m1.2.2.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.5.m1.2b"><list id="A3.SS1.1.p1.5.m1.2.2.3.cmml" xref="A3.SS1.1.p1.5.m1.2.2.2"><apply id="A3.SS1.1.p1.5.m1.1.1.1.1.cmml" xref="A3.SS1.1.p1.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.5.m1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.5.m1.1.1.1.1">superscript</csymbol><apply id="A3.SS1.1.p1.5.m1.1.1.1.1.2.cmml" xref="A3.SS1.1.p1.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.5.m1.1.1.1.1.2.1.cmml" xref="A3.SS1.1.p1.5.m1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.5.m1.1.1.1.1.2.2.cmml" xref="A3.SS1.1.p1.5.m1.1.1.1.1.2.2">𝑅</ci><cn 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encoding="application/x-llamapun" id="A3.SS1.1.p1.5.m1.2d">italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are independent, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx11"> <tbody id="A3.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\Pr\bigl{[}(R_{1}^{\prime},R_{2}^{\prime})=(H,L)\mid S_{1},S_{2}% \bigr{]}" class="ltx_Math" display="inline" id="A3.Ex20.m1.4"><semantics id="A3.Ex20.m1.4a"><mrow id="A3.Ex20.m1.4.4.1" xref="A3.Ex20.m1.4.4.2.cmml"><mi id="A3.Ex20.m1.3.3" xref="A3.Ex20.m1.3.3.cmml">Pr</mi><mo id="A3.Ex20.m1.4.4.1a" xref="A3.Ex20.m1.4.4.2.cmml">⁡</mo><mrow id="A3.Ex20.m1.4.4.1.1" xref="A3.Ex20.m1.4.4.2.cmml"><mo 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encoding="application/x-tex" id="A3.Ex20.m1.4c">\displaystyle\Pr\bigl{[}(R_{1}^{\prime},R_{2}^{\prime})=(H,L)\mid S_{1},S_{2}% \bigr{]}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex20.m1.4d">roman_Pr [ ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( italic_H , italic_L ) ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=p(s_{1})\,\bigl{(}1-p(s_{2})\bigr{)}" class="ltx_Math" display="inline" id="A3.Ex20.m2.2"><semantics id="A3.Ex20.m2.2a"><mrow id="A3.Ex20.m2.2.2" xref="A3.Ex20.m2.2.2.cmml"><mi id="A3.Ex20.m2.2.2.4" xref="A3.Ex20.m2.2.2.4.cmml"></mi><mo id="A3.Ex20.m2.2.2.3" xref="A3.Ex20.m2.2.2.3.cmml">=</mo><mrow id="A3.Ex20.m2.2.2.2" xref="A3.Ex20.m2.2.2.2.cmml"><mi id="A3.Ex20.m2.2.2.2.4" xref="A3.Ex20.m2.2.2.2.4.cmml">p</mi><mo id="A3.Ex20.m2.2.2.2.3" xref="A3.Ex20.m2.2.2.2.3.cmml">⁢</mo><mrow id="A3.Ex20.m2.1.1.1.1.1" xref="A3.Ex20.m2.1.1.1.1.1.1.cmml"><mo id="A3.Ex20.m2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex20.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A3.Ex20.m2.1.1.1.1.1.1" xref="A3.Ex20.m2.1.1.1.1.1.1.cmml"><mi id="A3.Ex20.m2.1.1.1.1.1.1.2" xref="A3.Ex20.m2.1.1.1.1.1.1.2.cmml">s</mi><mn id="A3.Ex20.m2.1.1.1.1.1.1.3" xref="A3.Ex20.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.Ex20.m2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex20.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex20.m2.2.2.2.3a" lspace="0.170em" xref="A3.Ex20.m2.2.2.2.3.cmml">⁢</mo><mrow id="A3.Ex20.m2.2.2.2.2.1" xref="A3.Ex20.m2.2.2.2.2.1.1.cmml"><mo id="A3.Ex20.m2.2.2.2.2.1.2" maxsize="120%" minsize="120%" xref="A3.Ex20.m2.2.2.2.2.1.1.cmml">(</mo><mrow id="A3.Ex20.m2.2.2.2.2.1.1" xref="A3.Ex20.m2.2.2.2.2.1.1.cmml"><mn id="A3.Ex20.m2.2.2.2.2.1.1.3" xref="A3.Ex20.m2.2.2.2.2.1.1.3.cmml">1</mn><mo id="A3.Ex20.m2.2.2.2.2.1.1.2" xref="A3.Ex20.m2.2.2.2.2.1.1.2.cmml">−</mo><mrow id="A3.Ex20.m2.2.2.2.2.1.1.1" xref="A3.Ex20.m2.2.2.2.2.1.1.1.cmml"><mi id="A3.Ex20.m2.2.2.2.2.1.1.1.3" xref="A3.Ex20.m2.2.2.2.2.1.1.1.3.cmml">p</mi><mo id="A3.Ex20.m2.2.2.2.2.1.1.1.2" xref="A3.Ex20.m2.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.cmml"><mi id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.2" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.2.cmml">s</mi><mn id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.3" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex20.m2.2.2.2.2.1.3" maxsize="120%" minsize="120%" xref="A3.Ex20.m2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex20.m2.2b"><apply id="A3.Ex20.m2.2.2.cmml" xref="A3.Ex20.m2.2.2"><eq id="A3.Ex20.m2.2.2.3.cmml" xref="A3.Ex20.m2.2.2.3"></eq><csymbol cd="latexml" id="A3.Ex20.m2.2.2.4.cmml" xref="A3.Ex20.m2.2.2.4">absent</csymbol><apply id="A3.Ex20.m2.2.2.2.cmml" xref="A3.Ex20.m2.2.2.2"><times id="A3.Ex20.m2.2.2.2.3.cmml" xref="A3.Ex20.m2.2.2.2.3"></times><ci id="A3.Ex20.m2.2.2.2.4.cmml" xref="A3.Ex20.m2.2.2.2.4">𝑝</ci><apply id="A3.Ex20.m2.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex20.m2.1.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex20.m2.1.1.1.1.1.1.2.cmml" xref="A3.Ex20.m2.1.1.1.1.1.1.2">𝑠</ci><cn id="A3.Ex20.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex20.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="A3.Ex20.m2.2.2.2.2.1.1.cmml" xref="A3.Ex20.m2.2.2.2.2.1"><minus id="A3.Ex20.m2.2.2.2.2.1.1.2.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.2"></minus><cn id="A3.Ex20.m2.2.2.2.2.1.1.3.cmml" type="integer" xref="A3.Ex20.m2.2.2.2.2.1.1.3">1</cn><apply id="A3.Ex20.m2.2.2.2.2.1.1.1.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1"><times id="A3.Ex20.m2.2.2.2.2.1.1.1.2.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1.2"></times><ci id="A3.Ex20.m2.2.2.2.2.1.1.1.3.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1.3">𝑝</ci><apply id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.2">𝑠</ci><cn id="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex20.m2.2.2.2.2.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex20.m2.2c">\displaystyle=p(s_{1})\,\bigl{(}1-p(s_{2})\bigr{)}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex20.m2.2d">= italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( 1 - italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex21"><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\Pr\bigl{[}(R_{1}^{\prime},R_{2}^{\prime})=(L,H)\mid S_{1},S_{2}% \bigr{]}" class="ltx_Math" display="inline" id="A3.Ex21.m1.4"><semantics id="A3.Ex21.m1.4a"><mrow id="A3.Ex21.m1.4.4.1" xref="A3.Ex21.m1.4.4.2.cmml"><mi id="A3.Ex21.m1.3.3" xref="A3.Ex21.m1.3.3.cmml">Pr</mi><mo id="A3.Ex21.m1.4.4.1a" xref="A3.Ex21.m1.4.4.2.cmml">⁡</mo><mrow id="A3.Ex21.m1.4.4.1.1" xref="A3.Ex21.m1.4.4.2.cmml"><mo id="A3.Ex21.m1.4.4.1.1.2" maxsize="120%" minsize="120%" xref="A3.Ex21.m1.4.4.2.cmml">[</mo><mrow 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id="A3.Ex21.m1.4c">\displaystyle\Pr\bigl{[}(R_{1}^{\prime},R_{2}^{\prime})=(L,H)\mid S_{1},S_{2}% \bigr{]}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex21.m1.4d">roman_Pr [ ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ( italic_L , italic_H ) ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\bigl{(}1-p(s_{1})\bigr{)}\,p(s_{2})~{}." class="ltx_Math" display="inline" id="A3.Ex21.m2.1"><semantics id="A3.Ex21.m2.1a"><mrow id="A3.Ex21.m2.1.1.1" xref="A3.Ex21.m2.1.1.1.1.cmml"><mrow id="A3.Ex21.m2.1.1.1.1" xref="A3.Ex21.m2.1.1.1.1.cmml"><mi id="A3.Ex21.m2.1.1.1.1.4" xref="A3.Ex21.m2.1.1.1.1.4.cmml"></mi><mo id="A3.Ex21.m2.1.1.1.1.3" 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xref="A3.Ex21.m2.1.1.1.1.2.4">𝑝</ci><apply id="A3.Ex21.m2.1.1.1.1.2.2.1.1.cmml" xref="A3.Ex21.m2.1.1.1.1.2.2.1"><csymbol cd="ambiguous" id="A3.Ex21.m2.1.1.1.1.2.2.1.1.1.cmml" xref="A3.Ex21.m2.1.1.1.1.2.2.1">subscript</csymbol><ci id="A3.Ex21.m2.1.1.1.1.2.2.1.1.2.cmml" xref="A3.Ex21.m2.1.1.1.1.2.2.1.1.2">𝑠</ci><cn id="A3.Ex21.m2.1.1.1.1.2.2.1.1.3.cmml" type="integer" xref="A3.Ex21.m2.1.1.1.1.2.2.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex21.m2.1c">\displaystyle=\bigl{(}1-p(s_{1})\bigr{)}\,p(s_{2})~{}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex21.m2.1d">= ( 1 - italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.1.p1.8">We thus have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx12"> <tbody id="A3.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]" class="ltx_Math" display="inline" id="A3.Ex22.m1.1"><semantics id="A3.Ex22.m1.1a"><mrow id="A3.Ex22.m1.1.1" xref="A3.Ex22.m1.1.1.cmml"><mo id="A3.Ex22.m1.1.1.2" movablelimits="false" rspace="0em" xref="A3.Ex22.m1.1.1.2.cmml">𝔼</mo><mrow id="A3.Ex22.m1.1.1.1.1" xref="A3.Ex22.m1.1.1.1.2.cmml"><mo id="A3.Ex22.m1.1.1.1.1.2" stretchy="false" xref="A3.Ex22.m1.1.1.1.2.1.cmml">[</mo><mrow id="A3.Ex22.m1.1.1.1.1.1" xref="A3.Ex22.m1.1.1.1.1.1.cmml"><mrow id="A3.Ex22.m1.1.1.1.1.1.4" xref="A3.Ex22.m1.1.1.1.1.1.4.cmml"><mo id="A3.Ex22.m1.1.1.1.1.1.4.1" lspace="0em" movablelimits="false" rspace="0.167em" xref="A3.Ex22.m1.1.1.1.1.1.4.1.cmml">det</mo><msub id="A3.Ex22.m1.1.1.1.1.1.4.2" xref="A3.Ex22.m1.1.1.1.1.1.4.2.cmml"><mi id="A3.Ex22.m1.1.1.1.1.1.4.2.2" xref="A3.Ex22.m1.1.1.1.1.1.4.2.2.cmml">M</mi><mn id="A3.Ex22.m1.1.1.1.1.1.4.2.3" xref="A3.Ex22.m1.1.1.1.1.1.4.2.3.cmml">12</mn></msub></mrow><mo id="A3.Ex22.m1.1.1.1.1.1.3" xref="A3.Ex22.m1.1.1.1.1.1.3.cmml">∣</mo><mrow id="A3.Ex22.m1.1.1.1.1.1.2.2" xref="A3.Ex22.m1.1.1.1.1.1.2.3.cmml"><msub id="A3.Ex22.m1.1.1.1.1.1.1.1.1" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex22.m1.1.1.1.1.1.1.1.1.2" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1.2.cmml">S</mi><mn id="A3.Ex22.m1.1.1.1.1.1.1.1.1.3" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.Ex22.m1.1.1.1.1.1.2.2.3" xref="A3.Ex22.m1.1.1.1.1.1.2.3.cmml">,</mo><msub id="A3.Ex22.m1.1.1.1.1.1.2.2.2" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex22.m1.1.1.1.1.1.2.2.2.2" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2.2.cmml">S</mi><mn id="A3.Ex22.m1.1.1.1.1.1.2.2.2.3" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2.3.cmml">2</mn></msub></mrow></mrow><mo id="A3.Ex22.m1.1.1.1.1.3" stretchy="false" xref="A3.Ex22.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex22.m1.1b"><apply id="A3.Ex22.m1.1.1.cmml" xref="A3.Ex22.m1.1.1"><ci id="A3.Ex22.m1.1.1.2.cmml" xref="A3.Ex22.m1.1.1.2">𝔼</ci><apply id="A3.Ex22.m1.1.1.1.2.cmml" xref="A3.Ex22.m1.1.1.1.1"><csymbol cd="latexml" id="A3.Ex22.m1.1.1.1.2.1.cmml" xref="A3.Ex22.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="A3.Ex22.m1.1.1.1.1.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1"><csymbol cd="latexml" id="A3.Ex22.m1.1.1.1.1.1.3.cmml" xref="A3.Ex22.m1.1.1.1.1.1.3">conditional</csymbol><apply id="A3.Ex22.m1.1.1.1.1.1.4.cmml" xref="A3.Ex22.m1.1.1.1.1.1.4"><determinant id="A3.Ex22.m1.1.1.1.1.1.4.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1.4.1"></determinant><apply id="A3.Ex22.m1.1.1.1.1.1.4.2.cmml" xref="A3.Ex22.m1.1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="A3.Ex22.m1.1.1.1.1.1.4.2.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1.4.2">subscript</csymbol><ci id="A3.Ex22.m1.1.1.1.1.1.4.2.2.cmml" xref="A3.Ex22.m1.1.1.1.1.1.4.2.2">𝑀</ci><cn id="A3.Ex22.m1.1.1.1.1.1.4.2.3.cmml" type="integer" xref="A3.Ex22.m1.1.1.1.1.1.4.2.3">12</cn></apply></apply><list id="A3.Ex22.m1.1.1.1.1.1.2.3.cmml" xref="A3.Ex22.m1.1.1.1.1.1.2.2"><apply id="A3.Ex22.m1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex22.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.Ex22.m1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1.2">𝑆</ci><cn id="A3.Ex22.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex22.m1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="A3.Ex22.m1.1.1.1.1.1.2.2.2.cmml" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.Ex22.m1.1.1.1.1.1.2.2.2.1.cmml" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A3.Ex22.m1.1.1.1.1.1.2.2.2.2.cmml" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2.2">𝑆</ci><cn id="A3.Ex22.m1.1.1.1.1.1.2.2.2.3.cmml" type="integer" xref="A3.Ex22.m1.1.1.1.1.1.2.2.2.3">2</cn></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex22.m1.1c">\displaystyle\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]</annotation><annotation encoding="application/x-llamapun" id="A3.Ex22.m1.1d">blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=p(s_{1})\,\bigl{(}1-p(s_{2})\bigr{)}-\bigl{(}1-p(s_{1})\bigr{)}% \,p(s_{2})" class="ltx_Math" display="inline" id="A3.Ex22.m2.4"><semantics id="A3.Ex22.m2.4a"><mrow id="A3.Ex22.m2.4.4" xref="A3.Ex22.m2.4.4.cmml"><mi id="A3.Ex22.m2.4.4.6" xref="A3.Ex22.m2.4.4.6.cmml"></mi><mo id="A3.Ex22.m2.4.4.5" xref="A3.Ex22.m2.4.4.5.cmml">=</mo><mrow id="A3.Ex22.m2.4.4.4" xref="A3.Ex22.m2.4.4.4.cmml"><mrow 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type="integer" xref="A3.Ex22.m2.4.4.4.4.2.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex22.m2.4c">\displaystyle=p(s_{1})\,\bigl{(}1-p(s_{2})\bigr{)}-\bigl{(}1-p(s_{1})\bigr{)}% \,p(s_{2})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex22.m2.4d">= italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( 1 - italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) - ( 1 - italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=p(s_{1})-p(s_{2})~{}." class="ltx_Math" display="inline" id="A3.Ex23.m1.1"><semantics id="A3.Ex23.m1.1a"><mrow id="A3.Ex23.m1.1.1.1" xref="A3.Ex23.m1.1.1.1.1.cmml"><mrow id="A3.Ex23.m1.1.1.1.1" xref="A3.Ex23.m1.1.1.1.1.cmml"><mi id="A3.Ex23.m1.1.1.1.1.4" xref="A3.Ex23.m1.1.1.1.1.4.cmml"></mi><mo id="A3.Ex23.m1.1.1.1.1.3" xref="A3.Ex23.m1.1.1.1.1.3.cmml">=</mo><mrow id="A3.Ex23.m1.1.1.1.1.2" xref="A3.Ex23.m1.1.1.1.1.2.cmml"><mrow id="A3.Ex23.m1.1.1.1.1.1.1" xref="A3.Ex23.m1.1.1.1.1.1.1.cmml"><mi id="A3.Ex23.m1.1.1.1.1.1.1.3" xref="A3.Ex23.m1.1.1.1.1.1.1.3.cmml">p</mi><mo id="A3.Ex23.m1.1.1.1.1.1.1.2" xref="A3.Ex23.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex23.m1.1.1.1.1.1.1.1.1" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex23.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="A3.Ex23.m1.1.1.1.1.1.1.1.1.1" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.2.cmml">s</mi><mn id="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.Ex23.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex23.m1.1.1.1.1.2.3" xref="A3.Ex23.m1.1.1.1.1.2.3.cmml">−</mo><mrow id="A3.Ex23.m1.1.1.1.1.2.2" xref="A3.Ex23.m1.1.1.1.1.2.2.cmml"><mi id="A3.Ex23.m1.1.1.1.1.2.2.3" xref="A3.Ex23.m1.1.1.1.1.2.2.3.cmml">p</mi><mo id="A3.Ex23.m1.1.1.1.1.2.2.2" xref="A3.Ex23.m1.1.1.1.1.2.2.2.cmml">⁢</mo><mrow id="A3.Ex23.m1.1.1.1.1.2.2.1.1" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.cmml"><mo id="A3.Ex23.m1.1.1.1.1.2.2.1.1.2" stretchy="false" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.cmml">(</mo><msub id="A3.Ex23.m1.1.1.1.1.2.2.1.1.1" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.cmml"><mi id="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.2" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.2.cmml">s</mi><mn id="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.3" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.3.cmml">2</mn></msub><mo id="A3.Ex23.m1.1.1.1.1.2.2.1.1.3" rspace="0.052em" stretchy="false" xref="A3.Ex23.m1.1.1.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A3.Ex23.m1.1.1.1.2" xref="A3.Ex23.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex23.m1.1b"><apply id="A3.Ex23.m1.1.1.1.1.cmml" xref="A3.Ex23.m1.1.1.1"><eq id="A3.Ex23.m1.1.1.1.1.3.cmml" xref="A3.Ex23.m1.1.1.1.1.3"></eq><csymbol cd="latexml" id="A3.Ex23.m1.1.1.1.1.4.cmml" xref="A3.Ex23.m1.1.1.1.1.4">absent</csymbol><apply id="A3.Ex23.m1.1.1.1.1.2.cmml" xref="A3.Ex23.m1.1.1.1.1.2"><minus id="A3.Ex23.m1.1.1.1.1.2.3.cmml" xref="A3.Ex23.m1.1.1.1.1.2.3"></minus><apply id="A3.Ex23.m1.1.1.1.1.1.1.cmml" xref="A3.Ex23.m1.1.1.1.1.1.1"><times id="A3.Ex23.m1.1.1.1.1.1.1.2.cmml" xref="A3.Ex23.m1.1.1.1.1.1.1.2"></times><ci id="A3.Ex23.m1.1.1.1.1.1.1.3.cmml" xref="A3.Ex23.m1.1.1.1.1.1.1.3">𝑝</ci><apply id="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex23.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex23.m1.1.1.1.1.1.1.1.1.1.1.cmml" 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id="A3.Ex23.m1.1c">\displaystyle=p(s_{1})-p(s_{2})~{}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex23.m1.1d">= italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.1.p1.7">The same argument gives <math alttext="\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]=p(s_{2})-p(s_{1})" class="ltx_Math" display="inline" id="A3.SS1.1.p1.6.m1.3"><semantics id="A3.SS1.1.p1.6.m1.3a"><mrow id="A3.SS1.1.p1.6.m1.3.3" xref="A3.SS1.1.p1.6.m1.3.3.cmml"><mrow id="A3.SS1.1.p1.6.m1.1.1.1" xref="A3.SS1.1.p1.6.m1.1.1.1.cmml"><mo id="A3.SS1.1.p1.6.m1.1.1.1.2" rspace="0em" xref="A3.SS1.1.p1.6.m1.1.1.1.2.cmml">𝔼</mo><mrow id="A3.SS1.1.p1.6.m1.1.1.1.1.1" xref="A3.SS1.1.p1.6.m1.1.1.1.1.2.cmml"><mo id="A3.SS1.1.p1.6.m1.1.1.1.1.1.2" stretchy="false" xref="A3.SS1.1.p1.6.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.cmml"><mrow id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.cmml"><mo id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.1" lspace="0em" rspace="0.167em" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.1.cmml">det</mo><msub id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.cmml"><mi id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.2" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.2.cmml">M</mi><mn id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.3" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.3.cmml">12</mn></msub></mrow><mo id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.3" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.3.cmml">∣</mo><mrow id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.3.cmml"><msub id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.2" 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xref="A3.SS1.1.p1.6.m1.1.1.1.1.1"><csymbol cd="latexml" id="A3.SS1.1.p1.6.m1.1.1.1.1.2.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.3.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.3">conditional</csymbol><apply id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4"><determinant id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.1"></determinant><apply id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2">subscript</csymbol><ci id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.2.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.2">𝑀</ci><cn id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.3.cmml" type="integer" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.4.2.3">12</cn></apply></apply><list id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.3.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2"><apply id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.2">𝑆</ci><cn id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.1.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.2.cmml" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.2">𝑆</ci><cn id="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.3.cmml" type="integer" xref="A3.SS1.1.p1.6.m1.1.1.1.1.1.1.2.2.2.3">2</cn></apply></list></apply></apply></apply><apply id="A3.SS1.1.p1.6.m1.3.3.3.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3"><minus id="A3.SS1.1.p1.6.m1.3.3.3.3.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.3"></minus><apply id="A3.SS1.1.p1.6.m1.2.2.2.1.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1"><times id="A3.SS1.1.p1.6.m1.2.2.2.1.2.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1.2"></times><ci id="A3.SS1.1.p1.6.m1.2.2.2.1.3.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1.3">𝑝</ci><apply id="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.2.cmml" xref="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.2">𝑠</ci><cn id="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.6.m1.2.2.2.1.1.1.1.3">2</cn></apply></apply><apply id="A3.SS1.1.p1.6.m1.3.3.3.2.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2"><times id="A3.SS1.1.p1.6.m1.3.3.3.2.2.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2.2"></times><ci id="A3.SS1.1.p1.6.m1.3.3.3.2.3.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2.3">𝑝</ci><apply id="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.1.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.2.cmml" xref="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.2">𝑠</ci><cn id="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.6.m1.3.3.3.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.6.m1.3c">\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]=p(s_{2})-p(s_{1})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.6.m1.3d">blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> when <math alttext="(r_{1},r_{2})=(L,H)" class="ltx_Math" display="inline" id="A3.SS1.1.p1.7.m2.4"><semantics id="A3.SS1.1.p1.7.m2.4a"><mrow id="A3.SS1.1.p1.7.m2.4.4" xref="A3.SS1.1.p1.7.m2.4.4.cmml"><mrow id="A3.SS1.1.p1.7.m2.4.4.2.2" xref="A3.SS1.1.p1.7.m2.4.4.2.3.cmml"><mo id="A3.SS1.1.p1.7.m2.4.4.2.2.3" stretchy="false" xref="A3.SS1.1.p1.7.m2.4.4.2.3.cmml">(</mo><msub id="A3.SS1.1.p1.7.m2.3.3.1.1.1" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1.cmml"><mi id="A3.SS1.1.p1.7.m2.3.3.1.1.1.2" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1.2.cmml">r</mi><mn id="A3.SS1.1.p1.7.m2.3.3.1.1.1.3" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.1.p1.7.m2.4.4.2.2.4" xref="A3.SS1.1.p1.7.m2.4.4.2.3.cmml">,</mo><msub id="A3.SS1.1.p1.7.m2.4.4.2.2.2" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2.cmml"><mi id="A3.SS1.1.p1.7.m2.4.4.2.2.2.2" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2.2.cmml">r</mi><mn id="A3.SS1.1.p1.7.m2.4.4.2.2.2.3" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.1.p1.7.m2.4.4.2.2.5" stretchy="false" xref="A3.SS1.1.p1.7.m2.4.4.2.3.cmml">)</mo></mrow><mo id="A3.SS1.1.p1.7.m2.4.4.3" xref="A3.SS1.1.p1.7.m2.4.4.3.cmml">=</mo><mrow id="A3.SS1.1.p1.7.m2.4.4.4.2" xref="A3.SS1.1.p1.7.m2.4.4.4.1.cmml"><mo id="A3.SS1.1.p1.7.m2.4.4.4.2.1" stretchy="false" xref="A3.SS1.1.p1.7.m2.4.4.4.1.cmml">(</mo><mi id="A3.SS1.1.p1.7.m2.1.1" xref="A3.SS1.1.p1.7.m2.1.1.cmml">L</mi><mo id="A3.SS1.1.p1.7.m2.4.4.4.2.2" xref="A3.SS1.1.p1.7.m2.4.4.4.1.cmml">,</mo><mi id="A3.SS1.1.p1.7.m2.2.2" xref="A3.SS1.1.p1.7.m2.2.2.cmml">H</mi><mo id="A3.SS1.1.p1.7.m2.4.4.4.2.3" stretchy="false" xref="A3.SS1.1.p1.7.m2.4.4.4.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.1.p1.7.m2.4b"><apply id="A3.SS1.1.p1.7.m2.4.4.cmml" xref="A3.SS1.1.p1.7.m2.4.4"><eq id="A3.SS1.1.p1.7.m2.4.4.3.cmml" xref="A3.SS1.1.p1.7.m2.4.4.3"></eq><interval closure="open" id="A3.SS1.1.p1.7.m2.4.4.2.3.cmml" xref="A3.SS1.1.p1.7.m2.4.4.2.2"><apply id="A3.SS1.1.p1.7.m2.3.3.1.1.1.cmml" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.1.p1.7.m2.3.3.1.1.1.1.cmml" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1">subscript</csymbol><ci id="A3.SS1.1.p1.7.m2.3.3.1.1.1.2.cmml" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1.2">𝑟</ci><cn id="A3.SS1.1.p1.7.m2.3.3.1.1.1.3.cmml" type="integer" xref="A3.SS1.1.p1.7.m2.3.3.1.1.1.3">1</cn></apply><apply id="A3.SS1.1.p1.7.m2.4.4.2.2.2.cmml" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.1.p1.7.m2.4.4.2.2.2.1.cmml" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2">subscript</csymbol><ci id="A3.SS1.1.p1.7.m2.4.4.2.2.2.2.cmml" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2.2">𝑟</ci><cn id="A3.SS1.1.p1.7.m2.4.4.2.2.2.3.cmml" type="integer" xref="A3.SS1.1.p1.7.m2.4.4.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.1.p1.7.m2.4.4.4.1.cmml" xref="A3.SS1.1.p1.7.m2.4.4.4.2"><ci id="A3.SS1.1.p1.7.m2.1.1.cmml" xref="A3.SS1.1.p1.7.m2.1.1">𝐿</ci><ci id="A3.SS1.1.p1.7.m2.2.2.cmml" xref="A3.SS1.1.p1.7.m2.2.2">𝐻</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.1.p1.7.m2.4c">(r_{1},r_{2})=(L,H)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.1.p1.7.m2.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_L , italic_H )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS1.2.p2"> <p class="ltx_p" id="A3.SS1.2.p2.10">We have now established</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx13"> <tbody id="A3.Ex24"><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\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]\;=\;\begin{cases% }0,&\text{if }r_{1}=r_{2},\\ p(s_{1})-p(s_{2}),&\text{if }(r_{1},r_{2})=(H,L),\\ p(s_{2})-p(s_{1}),&\text{if }(r_{1},r_{2})=(L,H).\end{cases}" class="ltx_Math" display="inline" id="A3.Ex24.m1.7"><semantics id="A3.Ex24.m1.7a"><mrow id="A3.Ex24.m1.7.7" xref="A3.Ex24.m1.7.7.cmml"><mrow id="A3.Ex24.m1.7.7.1" xref="A3.Ex24.m1.7.7.1.cmml"><mo id="A3.Ex24.m1.7.7.1.2" movablelimits="false" rspace="0em" xref="A3.Ex24.m1.7.7.1.2.cmml">𝔼</mo><mrow id="A3.Ex24.m1.7.7.1.1.1" xref="A3.Ex24.m1.7.7.1.1.2.cmml"><mo id="A3.Ex24.m1.7.7.1.1.1.2" stretchy="false" xref="A3.Ex24.m1.7.7.1.1.2.1.cmml">[</mo><mrow id="A3.Ex24.m1.7.7.1.1.1.1" xref="A3.Ex24.m1.7.7.1.1.1.1.cmml"><mrow id="A3.Ex24.m1.7.7.1.1.1.1.4" xref="A3.Ex24.m1.7.7.1.1.1.1.4.cmml"><mo id="A3.Ex24.m1.7.7.1.1.1.1.4.1" lspace="0em" movablelimits="false" rspace="0.167em" xref="A3.Ex24.m1.7.7.1.1.1.1.4.1.cmml">det</mo><msub id="A3.Ex24.m1.7.7.1.1.1.1.4.2" xref="A3.Ex24.m1.7.7.1.1.1.1.4.2.cmml"><mi 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xref="A3.Ex24.m1.6.6.6.6.2.1.3.1.2.2.2.2.3">2</cn></apply></interval></apply><interval closure="open" id="A3.Ex24.m1.6.6.6.6.2.1.3.1.4.1.cmml" xref="A3.Ex24.m1.6.6.6.6.2.1.3.1.4.2"><ci id="A3.Ex24.m1.6.6.6.6.2.1.1.cmml" xref="A3.Ex24.m1.6.6.6.6.2.1.1">𝐿</ci><ci id="A3.Ex24.m1.6.6.6.6.2.1.2.cmml" xref="A3.Ex24.m1.6.6.6.6.2.1.2">𝐻</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex24.m1.7c">\displaystyle\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]\;=\;\begin{cases% }0,&\text{if }r_{1}=r_{2},\\ p(s_{1})-p(s_{2}),&\text{if }(r_{1},r_{2})=(H,L),\\ p(s_{2})-p(s_{1}),&\text{if }(r_{1},r_{2})=(L,H).\end{cases}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex24.m1.7d">blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = { start_ROW start_CELL 0 , end_CELL start_CELL if italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , end_CELL start_CELL if ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_H , italic_L ) , end_CELL end_ROW start_ROW start_CELL italic_p ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_p ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , end_CELL start_CELL if ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_L , italic_H ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.2.p2.9">To finish the proof, consider the cases <math alttext="s_{1}=s_{2}" class="ltx_Math" 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xref="A3.SS1.2.p2.3.m3.4.4.2.2.2.2">𝑠</ci><cn id="A3.SS1.2.p2.3.m3.4.4.2.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.3.m3.4.4.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.2.p2.3.m3.4.4.4.1.cmml" xref="A3.SS1.2.p2.3.m3.4.4.4.2"><ci id="A3.SS1.2.p2.3.m3.1.1.cmml" xref="A3.SS1.2.p2.3.m3.1.1">𝐿</ci><ci id="A3.SS1.2.p2.3.m3.2.2.cmml" xref="A3.SS1.2.p2.3.m3.2.2">𝐻</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.3.m3.4c">(s_{1},s_{2})=(L,H)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.3.m3.4d">( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_L , italic_H )</annotation></semantics></math>. From the above, we obtain 0 in the first case. In the second, we have <math alttext="p(H)-p(L)" class="ltx_Math" display="inline" id="A3.SS1.2.p2.4.m4.2"><semantics id="A3.SS1.2.p2.4.m4.2a"><mrow id="A3.SS1.2.p2.4.m4.2.3" xref="A3.SS1.2.p2.4.m4.2.3.cmml"><mrow id="A3.SS1.2.p2.4.m4.2.3.2" xref="A3.SS1.2.p2.4.m4.2.3.2.cmml"><mi id="A3.SS1.2.p2.4.m4.2.3.2.2" xref="A3.SS1.2.p2.4.m4.2.3.2.2.cmml">p</mi><mo id="A3.SS1.2.p2.4.m4.2.3.2.1" xref="A3.SS1.2.p2.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="A3.SS1.2.p2.4.m4.2.3.2.3.2" xref="A3.SS1.2.p2.4.m4.2.3.2.cmml"><mo id="A3.SS1.2.p2.4.m4.2.3.2.3.2.1" stretchy="false" xref="A3.SS1.2.p2.4.m4.2.3.2.cmml">(</mo><mi id="A3.SS1.2.p2.4.m4.1.1" xref="A3.SS1.2.p2.4.m4.1.1.cmml">H</mi><mo id="A3.SS1.2.p2.4.m4.2.3.2.3.2.2" stretchy="false" xref="A3.SS1.2.p2.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="A3.SS1.2.p2.4.m4.2.3.1" xref="A3.SS1.2.p2.4.m4.2.3.1.cmml">−</mo><mrow id="A3.SS1.2.p2.4.m4.2.3.3" xref="A3.SS1.2.p2.4.m4.2.3.3.cmml"><mi id="A3.SS1.2.p2.4.m4.2.3.3.2" xref="A3.SS1.2.p2.4.m4.2.3.3.2.cmml">p</mi><mo id="A3.SS1.2.p2.4.m4.2.3.3.1" xref="A3.SS1.2.p2.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="A3.SS1.2.p2.4.m4.2.3.3.3.2" xref="A3.SS1.2.p2.4.m4.2.3.3.cmml"><mo id="A3.SS1.2.p2.4.m4.2.3.3.3.2.1" stretchy="false" xref="A3.SS1.2.p2.4.m4.2.3.3.cmml">(</mo><mi id="A3.SS1.2.p2.4.m4.2.2" xref="A3.SS1.2.p2.4.m4.2.2.cmml">L</mi><mo id="A3.SS1.2.p2.4.m4.2.3.3.3.2.2" stretchy="false" xref="A3.SS1.2.p2.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.4.m4.2b"><apply id="A3.SS1.2.p2.4.m4.2.3.cmml" xref="A3.SS1.2.p2.4.m4.2.3"><minus id="A3.SS1.2.p2.4.m4.2.3.1.cmml" xref="A3.SS1.2.p2.4.m4.2.3.1"></minus><apply id="A3.SS1.2.p2.4.m4.2.3.2.cmml" xref="A3.SS1.2.p2.4.m4.2.3.2"><times id="A3.SS1.2.p2.4.m4.2.3.2.1.cmml" xref="A3.SS1.2.p2.4.m4.2.3.2.1"></times><ci id="A3.SS1.2.p2.4.m4.2.3.2.2.cmml" xref="A3.SS1.2.p2.4.m4.2.3.2.2">𝑝</ci><ci id="A3.SS1.2.p2.4.m4.1.1.cmml" xref="A3.SS1.2.p2.4.m4.1.1">𝐻</ci></apply><apply id="A3.SS1.2.p2.4.m4.2.3.3.cmml" xref="A3.SS1.2.p2.4.m4.2.3.3"><times id="A3.SS1.2.p2.4.m4.2.3.3.1.cmml" xref="A3.SS1.2.p2.4.m4.2.3.3.1"></times><ci id="A3.SS1.2.p2.4.m4.2.3.3.2.cmml" xref="A3.SS1.2.p2.4.m4.2.3.3.2">𝑝</ci><ci id="A3.SS1.2.p2.4.m4.2.2.cmml" xref="A3.SS1.2.p2.4.m4.2.2">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.4.m4.2c">p(H)-p(L)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.4.m4.2d">italic_p ( italic_H ) - italic_p ( italic_L )</annotation></semantics></math> if <math alttext="(r_{1},r_{2})=(s_{1},s_{2})" class="ltx_Math" display="inline" id="A3.SS1.2.p2.5.m5.4"><semantics id="A3.SS1.2.p2.5.m5.4a"><mrow id="A3.SS1.2.p2.5.m5.4.4" xref="A3.SS1.2.p2.5.m5.4.4.cmml"><mrow id="A3.SS1.2.p2.5.m5.2.2.2.2" xref="A3.SS1.2.p2.5.m5.2.2.2.3.cmml"><mo id="A3.SS1.2.p2.5.m5.2.2.2.2.3" stretchy="false" xref="A3.SS1.2.p2.5.m5.2.2.2.3.cmml">(</mo><msub id="A3.SS1.2.p2.5.m5.1.1.1.1.1" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1.cmml"><mi id="A3.SS1.2.p2.5.m5.1.1.1.1.1.2" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1.2.cmml">r</mi><mn id="A3.SS1.2.p2.5.m5.1.1.1.1.1.3" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.5.m5.2.2.2.2.4" xref="A3.SS1.2.p2.5.m5.2.2.2.3.cmml">,</mo><msub id="A3.SS1.2.p2.5.m5.2.2.2.2.2" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2.cmml"><mi id="A3.SS1.2.p2.5.m5.2.2.2.2.2.2" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2.2.cmml">r</mi><mn id="A3.SS1.2.p2.5.m5.2.2.2.2.2.3" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.5.m5.2.2.2.2.5" stretchy="false" xref="A3.SS1.2.p2.5.m5.2.2.2.3.cmml">)</mo></mrow><mo id="A3.SS1.2.p2.5.m5.4.4.5" xref="A3.SS1.2.p2.5.m5.4.4.5.cmml">=</mo><mrow id="A3.SS1.2.p2.5.m5.4.4.4.2" xref="A3.SS1.2.p2.5.m5.4.4.4.3.cmml"><mo id="A3.SS1.2.p2.5.m5.4.4.4.2.3" stretchy="false" xref="A3.SS1.2.p2.5.m5.4.4.4.3.cmml">(</mo><msub id="A3.SS1.2.p2.5.m5.3.3.3.1.1" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1.cmml"><mi id="A3.SS1.2.p2.5.m5.3.3.3.1.1.2" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1.2.cmml">s</mi><mn id="A3.SS1.2.p2.5.m5.3.3.3.1.1.3" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.5.m5.4.4.4.2.4" xref="A3.SS1.2.p2.5.m5.4.4.4.3.cmml">,</mo><msub id="A3.SS1.2.p2.5.m5.4.4.4.2.2" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2.cmml"><mi id="A3.SS1.2.p2.5.m5.4.4.4.2.2.2" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2.2.cmml">s</mi><mn id="A3.SS1.2.p2.5.m5.4.4.4.2.2.3" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.5.m5.4.4.4.2.5" stretchy="false" xref="A3.SS1.2.p2.5.m5.4.4.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.5.m5.4b"><apply id="A3.SS1.2.p2.5.m5.4.4.cmml" xref="A3.SS1.2.p2.5.m5.4.4"><eq id="A3.SS1.2.p2.5.m5.4.4.5.cmml" xref="A3.SS1.2.p2.5.m5.4.4.5"></eq><interval closure="open" id="A3.SS1.2.p2.5.m5.2.2.2.3.cmml" xref="A3.SS1.2.p2.5.m5.2.2.2.2"><apply id="A3.SS1.2.p2.5.m5.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.5.m5.1.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.5.m5.1.1.1.1.1.2.cmml" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1.2">𝑟</ci><cn id="A3.SS1.2.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.5.m5.1.1.1.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.5.m5.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.5.m5.2.2.2.2.2.1.cmml" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.5.m5.2.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2.2">𝑟</ci><cn id="A3.SS1.2.p2.5.m5.2.2.2.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.5.m5.2.2.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.2.p2.5.m5.4.4.4.3.cmml" xref="A3.SS1.2.p2.5.m5.4.4.4.2"><apply id="A3.SS1.2.p2.5.m5.3.3.3.1.1.cmml" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.5.m5.3.3.3.1.1.1.cmml" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.5.m5.3.3.3.1.1.2.cmml" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1.2">𝑠</ci><cn id="A3.SS1.2.p2.5.m5.3.3.3.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.5.m5.3.3.3.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.5.m5.4.4.4.2.2.cmml" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.5.m5.4.4.4.2.2.1.cmml" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.5.m5.4.4.4.2.2.2.cmml" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2.2">𝑠</ci><cn id="A3.SS1.2.p2.5.m5.4.4.4.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.5.m5.4.4.4.2.2.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.5.m5.4c">(r_{1},r_{2})=(s_{1},s_{2})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.5.m5.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and its negation when <math alttext="(r_{1},r_{2})=(s_{2},s_{1})" class="ltx_Math" display="inline" id="A3.SS1.2.p2.6.m6.4"><semantics id="A3.SS1.2.p2.6.m6.4a"><mrow id="A3.SS1.2.p2.6.m6.4.4" xref="A3.SS1.2.p2.6.m6.4.4.cmml"><mrow id="A3.SS1.2.p2.6.m6.2.2.2.2" xref="A3.SS1.2.p2.6.m6.2.2.2.3.cmml"><mo id="A3.SS1.2.p2.6.m6.2.2.2.2.3" stretchy="false" xref="A3.SS1.2.p2.6.m6.2.2.2.3.cmml">(</mo><msub id="A3.SS1.2.p2.6.m6.1.1.1.1.1" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1.cmml"><mi id="A3.SS1.2.p2.6.m6.1.1.1.1.1.2" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1.2.cmml">r</mi><mn id="A3.SS1.2.p2.6.m6.1.1.1.1.1.3" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.6.m6.2.2.2.2.4" xref="A3.SS1.2.p2.6.m6.2.2.2.3.cmml">,</mo><msub id="A3.SS1.2.p2.6.m6.2.2.2.2.2" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2.cmml"><mi id="A3.SS1.2.p2.6.m6.2.2.2.2.2.2" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2.2.cmml">r</mi><mn id="A3.SS1.2.p2.6.m6.2.2.2.2.2.3" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.6.m6.2.2.2.2.5" stretchy="false" xref="A3.SS1.2.p2.6.m6.2.2.2.3.cmml">)</mo></mrow><mo id="A3.SS1.2.p2.6.m6.4.4.5" xref="A3.SS1.2.p2.6.m6.4.4.5.cmml">=</mo><mrow id="A3.SS1.2.p2.6.m6.4.4.4.2" xref="A3.SS1.2.p2.6.m6.4.4.4.3.cmml"><mo id="A3.SS1.2.p2.6.m6.4.4.4.2.3" stretchy="false" xref="A3.SS1.2.p2.6.m6.4.4.4.3.cmml">(</mo><msub id="A3.SS1.2.p2.6.m6.3.3.3.1.1" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1.cmml"><mi id="A3.SS1.2.p2.6.m6.3.3.3.1.1.2" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1.2.cmml">s</mi><mn id="A3.SS1.2.p2.6.m6.3.3.3.1.1.3" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.6.m6.4.4.4.2.4" xref="A3.SS1.2.p2.6.m6.4.4.4.3.cmml">,</mo><msub id="A3.SS1.2.p2.6.m6.4.4.4.2.2" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2.cmml"><mi id="A3.SS1.2.p2.6.m6.4.4.4.2.2.2" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2.2.cmml">s</mi><mn id="A3.SS1.2.p2.6.m6.4.4.4.2.2.3" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.6.m6.4.4.4.2.5" stretchy="false" xref="A3.SS1.2.p2.6.m6.4.4.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.6.m6.4b"><apply id="A3.SS1.2.p2.6.m6.4.4.cmml" xref="A3.SS1.2.p2.6.m6.4.4"><eq id="A3.SS1.2.p2.6.m6.4.4.5.cmml" xref="A3.SS1.2.p2.6.m6.4.4.5"></eq><interval closure="open" id="A3.SS1.2.p2.6.m6.2.2.2.3.cmml" xref="A3.SS1.2.p2.6.m6.2.2.2.2"><apply id="A3.SS1.2.p2.6.m6.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.6.m6.1.1.1.1.1.2.cmml" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1.2">𝑟</ci><cn id="A3.SS1.2.p2.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.6.m6.1.1.1.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.6.m6.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.6.m6.2.2.2.2.2.1.cmml" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.6.m6.2.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2.2">𝑟</ci><cn id="A3.SS1.2.p2.6.m6.2.2.2.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.6.m6.2.2.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.2.p2.6.m6.4.4.4.3.cmml" xref="A3.SS1.2.p2.6.m6.4.4.4.2"><apply id="A3.SS1.2.p2.6.m6.3.3.3.1.1.cmml" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.6.m6.3.3.3.1.1.1.cmml" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.6.m6.3.3.3.1.1.2.cmml" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1.2">𝑠</ci><cn id="A3.SS1.2.p2.6.m6.3.3.3.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.6.m6.3.3.3.1.1.3">2</cn></apply><apply id="A3.SS1.2.p2.6.m6.4.4.4.2.2.cmml" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.6.m6.4.4.4.2.2.1.cmml" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.6.m6.4.4.4.2.2.2.cmml" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2.2">𝑠</ci><cn id="A3.SS1.2.p2.6.m6.4.4.4.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.6.m6.4.4.4.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.6.m6.4c">(r_{1},r_{2})=(s_{2},s_{1})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.6.m6.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. In the third, we again have <math alttext="p(H)-p(L)" class="ltx_Math" display="inline" id="A3.SS1.2.p2.7.m7.2"><semantics id="A3.SS1.2.p2.7.m7.2a"><mrow id="A3.SS1.2.p2.7.m7.2.3" xref="A3.SS1.2.p2.7.m7.2.3.cmml"><mrow id="A3.SS1.2.p2.7.m7.2.3.2" xref="A3.SS1.2.p2.7.m7.2.3.2.cmml"><mi id="A3.SS1.2.p2.7.m7.2.3.2.2" xref="A3.SS1.2.p2.7.m7.2.3.2.2.cmml">p</mi><mo id="A3.SS1.2.p2.7.m7.2.3.2.1" xref="A3.SS1.2.p2.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="A3.SS1.2.p2.7.m7.2.3.2.3.2" xref="A3.SS1.2.p2.7.m7.2.3.2.cmml"><mo id="A3.SS1.2.p2.7.m7.2.3.2.3.2.1" stretchy="false" xref="A3.SS1.2.p2.7.m7.2.3.2.cmml">(</mo><mi id="A3.SS1.2.p2.7.m7.1.1" xref="A3.SS1.2.p2.7.m7.1.1.cmml">H</mi><mo id="A3.SS1.2.p2.7.m7.2.3.2.3.2.2" stretchy="false" xref="A3.SS1.2.p2.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="A3.SS1.2.p2.7.m7.2.3.1" xref="A3.SS1.2.p2.7.m7.2.3.1.cmml">−</mo><mrow id="A3.SS1.2.p2.7.m7.2.3.3" xref="A3.SS1.2.p2.7.m7.2.3.3.cmml"><mi id="A3.SS1.2.p2.7.m7.2.3.3.2" xref="A3.SS1.2.p2.7.m7.2.3.3.2.cmml">p</mi><mo id="A3.SS1.2.p2.7.m7.2.3.3.1" xref="A3.SS1.2.p2.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="A3.SS1.2.p2.7.m7.2.3.3.3.2" xref="A3.SS1.2.p2.7.m7.2.3.3.cmml"><mo id="A3.SS1.2.p2.7.m7.2.3.3.3.2.1" stretchy="false" xref="A3.SS1.2.p2.7.m7.2.3.3.cmml">(</mo><mi id="A3.SS1.2.p2.7.m7.2.2" xref="A3.SS1.2.p2.7.m7.2.2.cmml">L</mi><mo id="A3.SS1.2.p2.7.m7.2.3.3.3.2.2" stretchy="false" xref="A3.SS1.2.p2.7.m7.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.7.m7.2b"><apply id="A3.SS1.2.p2.7.m7.2.3.cmml" xref="A3.SS1.2.p2.7.m7.2.3"><minus id="A3.SS1.2.p2.7.m7.2.3.1.cmml" xref="A3.SS1.2.p2.7.m7.2.3.1"></minus><apply id="A3.SS1.2.p2.7.m7.2.3.2.cmml" xref="A3.SS1.2.p2.7.m7.2.3.2"><times id="A3.SS1.2.p2.7.m7.2.3.2.1.cmml" xref="A3.SS1.2.p2.7.m7.2.3.2.1"></times><ci id="A3.SS1.2.p2.7.m7.2.3.2.2.cmml" xref="A3.SS1.2.p2.7.m7.2.3.2.2">𝑝</ci><ci id="A3.SS1.2.p2.7.m7.1.1.cmml" xref="A3.SS1.2.p2.7.m7.1.1">𝐻</ci></apply><apply id="A3.SS1.2.p2.7.m7.2.3.3.cmml" xref="A3.SS1.2.p2.7.m7.2.3.3"><times id="A3.SS1.2.p2.7.m7.2.3.3.1.cmml" xref="A3.SS1.2.p2.7.m7.2.3.3.1"></times><ci id="A3.SS1.2.p2.7.m7.2.3.3.2.cmml" xref="A3.SS1.2.p2.7.m7.2.3.3.2">𝑝</ci><ci id="A3.SS1.2.p2.7.m7.2.2.cmml" xref="A3.SS1.2.p2.7.m7.2.2">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.7.m7.2c">p(H)-p(L)</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.7.m7.2d">italic_p ( italic_H ) - italic_p ( italic_L )</annotation></semantics></math> if <math alttext="(r_{1},r_{2})=(s_{1},s_{2})" class="ltx_Math" display="inline" id="A3.SS1.2.p2.8.m8.4"><semantics id="A3.SS1.2.p2.8.m8.4a"><mrow id="A3.SS1.2.p2.8.m8.4.4" xref="A3.SS1.2.p2.8.m8.4.4.cmml"><mrow id="A3.SS1.2.p2.8.m8.2.2.2.2" xref="A3.SS1.2.p2.8.m8.2.2.2.3.cmml"><mo id="A3.SS1.2.p2.8.m8.2.2.2.2.3" stretchy="false" xref="A3.SS1.2.p2.8.m8.2.2.2.3.cmml">(</mo><msub id="A3.SS1.2.p2.8.m8.1.1.1.1.1" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1.cmml"><mi id="A3.SS1.2.p2.8.m8.1.1.1.1.1.2" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1.2.cmml">r</mi><mn id="A3.SS1.2.p2.8.m8.1.1.1.1.1.3" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.8.m8.2.2.2.2.4" xref="A3.SS1.2.p2.8.m8.2.2.2.3.cmml">,</mo><msub id="A3.SS1.2.p2.8.m8.2.2.2.2.2" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2.cmml"><mi id="A3.SS1.2.p2.8.m8.2.2.2.2.2.2" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2.2.cmml">r</mi><mn id="A3.SS1.2.p2.8.m8.2.2.2.2.2.3" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.8.m8.2.2.2.2.5" stretchy="false" xref="A3.SS1.2.p2.8.m8.2.2.2.3.cmml">)</mo></mrow><mo id="A3.SS1.2.p2.8.m8.4.4.5" xref="A3.SS1.2.p2.8.m8.4.4.5.cmml">=</mo><mrow id="A3.SS1.2.p2.8.m8.4.4.4.2" xref="A3.SS1.2.p2.8.m8.4.4.4.3.cmml"><mo id="A3.SS1.2.p2.8.m8.4.4.4.2.3" stretchy="false" xref="A3.SS1.2.p2.8.m8.4.4.4.3.cmml">(</mo><msub id="A3.SS1.2.p2.8.m8.3.3.3.1.1" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1.cmml"><mi id="A3.SS1.2.p2.8.m8.3.3.3.1.1.2" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1.2.cmml">s</mi><mn id="A3.SS1.2.p2.8.m8.3.3.3.1.1.3" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.8.m8.4.4.4.2.4" xref="A3.SS1.2.p2.8.m8.4.4.4.3.cmml">,</mo><msub id="A3.SS1.2.p2.8.m8.4.4.4.2.2" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2.cmml"><mi id="A3.SS1.2.p2.8.m8.4.4.4.2.2.2" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2.2.cmml">s</mi><mn id="A3.SS1.2.p2.8.m8.4.4.4.2.2.3" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.8.m8.4.4.4.2.5" stretchy="false" xref="A3.SS1.2.p2.8.m8.4.4.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.8.m8.4b"><apply id="A3.SS1.2.p2.8.m8.4.4.cmml" xref="A3.SS1.2.p2.8.m8.4.4"><eq id="A3.SS1.2.p2.8.m8.4.4.5.cmml" xref="A3.SS1.2.p2.8.m8.4.4.5"></eq><interval closure="open" id="A3.SS1.2.p2.8.m8.2.2.2.3.cmml" xref="A3.SS1.2.p2.8.m8.2.2.2.2"><apply id="A3.SS1.2.p2.8.m8.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.8.m8.1.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.8.m8.1.1.1.1.1.2.cmml" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1.2">𝑟</ci><cn id="A3.SS1.2.p2.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.8.m8.1.1.1.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.8.m8.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.8.m8.2.2.2.2.2.1.cmml" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.8.m8.2.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2.2">𝑟</ci><cn id="A3.SS1.2.p2.8.m8.2.2.2.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.8.m8.2.2.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.2.p2.8.m8.4.4.4.3.cmml" xref="A3.SS1.2.p2.8.m8.4.4.4.2"><apply id="A3.SS1.2.p2.8.m8.3.3.3.1.1.cmml" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.8.m8.3.3.3.1.1.1.cmml" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.8.m8.3.3.3.1.1.2.cmml" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1.2">𝑠</ci><cn id="A3.SS1.2.p2.8.m8.3.3.3.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.8.m8.3.3.3.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.8.m8.4.4.4.2.2.cmml" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.8.m8.4.4.4.2.2.1.cmml" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.8.m8.4.4.4.2.2.2.cmml" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2.2">𝑠</ci><cn id="A3.SS1.2.p2.8.m8.4.4.4.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.8.m8.4.4.4.2.2.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.8.m8.4c">(r_{1},r_{2})=(s_{1},s_{2})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.8.m8.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and its negation when <math alttext="(r_{1},r_{2})=(s_{2},s_{1})" class="ltx_Math" display="inline" id="A3.SS1.2.p2.9.m9.4"><semantics id="A3.SS1.2.p2.9.m9.4a"><mrow id="A3.SS1.2.p2.9.m9.4.4" xref="A3.SS1.2.p2.9.m9.4.4.cmml"><mrow id="A3.SS1.2.p2.9.m9.2.2.2.2" xref="A3.SS1.2.p2.9.m9.2.2.2.3.cmml"><mo id="A3.SS1.2.p2.9.m9.2.2.2.2.3" stretchy="false" xref="A3.SS1.2.p2.9.m9.2.2.2.3.cmml">(</mo><msub id="A3.SS1.2.p2.9.m9.1.1.1.1.1" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1.cmml"><mi id="A3.SS1.2.p2.9.m9.1.1.1.1.1.2" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1.2.cmml">r</mi><mn id="A3.SS1.2.p2.9.m9.1.1.1.1.1.3" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.9.m9.2.2.2.2.4" xref="A3.SS1.2.p2.9.m9.2.2.2.3.cmml">,</mo><msub id="A3.SS1.2.p2.9.m9.2.2.2.2.2" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2.cmml"><mi id="A3.SS1.2.p2.9.m9.2.2.2.2.2.2" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2.2.cmml">r</mi><mn id="A3.SS1.2.p2.9.m9.2.2.2.2.2.3" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.9.m9.2.2.2.2.5" stretchy="false" xref="A3.SS1.2.p2.9.m9.2.2.2.3.cmml">)</mo></mrow><mo id="A3.SS1.2.p2.9.m9.4.4.5" xref="A3.SS1.2.p2.9.m9.4.4.5.cmml">=</mo><mrow id="A3.SS1.2.p2.9.m9.4.4.4.2" xref="A3.SS1.2.p2.9.m9.4.4.4.3.cmml"><mo id="A3.SS1.2.p2.9.m9.4.4.4.2.3" stretchy="false" xref="A3.SS1.2.p2.9.m9.4.4.4.3.cmml">(</mo><msub id="A3.SS1.2.p2.9.m9.3.3.3.1.1" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1.cmml"><mi id="A3.SS1.2.p2.9.m9.3.3.3.1.1.2" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1.2.cmml">s</mi><mn id="A3.SS1.2.p2.9.m9.3.3.3.1.1.3" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1.3.cmml">2</mn></msub><mo id="A3.SS1.2.p2.9.m9.4.4.4.2.4" xref="A3.SS1.2.p2.9.m9.4.4.4.3.cmml">,</mo><msub id="A3.SS1.2.p2.9.m9.4.4.4.2.2" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2.cmml"><mi id="A3.SS1.2.p2.9.m9.4.4.4.2.2.2" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2.2.cmml">s</mi><mn id="A3.SS1.2.p2.9.m9.4.4.4.2.2.3" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2.3.cmml">1</mn></msub><mo id="A3.SS1.2.p2.9.m9.4.4.4.2.5" stretchy="false" xref="A3.SS1.2.p2.9.m9.4.4.4.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.2.p2.9.m9.4b"><apply id="A3.SS1.2.p2.9.m9.4.4.cmml" xref="A3.SS1.2.p2.9.m9.4.4"><eq id="A3.SS1.2.p2.9.m9.4.4.5.cmml" xref="A3.SS1.2.p2.9.m9.4.4.5"></eq><interval closure="open" id="A3.SS1.2.p2.9.m9.2.2.2.3.cmml" xref="A3.SS1.2.p2.9.m9.2.2.2.2"><apply id="A3.SS1.2.p2.9.m9.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.9.m9.1.1.1.1.1.1.cmml" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.9.m9.1.1.1.1.1.2.cmml" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1.2">𝑟</ci><cn id="A3.SS1.2.p2.9.m9.1.1.1.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.9.m9.1.1.1.1.1.3">1</cn></apply><apply id="A3.SS1.2.p2.9.m9.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.9.m9.2.2.2.2.2.1.cmml" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.9.m9.2.2.2.2.2.2.cmml" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2.2">𝑟</ci><cn id="A3.SS1.2.p2.9.m9.2.2.2.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.9.m9.2.2.2.2.2.3">2</cn></apply></interval><interval closure="open" id="A3.SS1.2.p2.9.m9.4.4.4.3.cmml" xref="A3.SS1.2.p2.9.m9.4.4.4.2"><apply id="A3.SS1.2.p2.9.m9.3.3.3.1.1.cmml" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1"><csymbol cd="ambiguous" id="A3.SS1.2.p2.9.m9.3.3.3.1.1.1.cmml" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1">subscript</csymbol><ci id="A3.SS1.2.p2.9.m9.3.3.3.1.1.2.cmml" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1.2">𝑠</ci><cn id="A3.SS1.2.p2.9.m9.3.3.3.1.1.3.cmml" type="integer" xref="A3.SS1.2.p2.9.m9.3.3.3.1.1.3">2</cn></apply><apply id="A3.SS1.2.p2.9.m9.4.4.4.2.2.cmml" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2"><csymbol cd="ambiguous" id="A3.SS1.2.p2.9.m9.4.4.4.2.2.1.cmml" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2">subscript</csymbol><ci id="A3.SS1.2.p2.9.m9.4.4.4.2.2.2.cmml" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2.2">𝑠</ci><cn id="A3.SS1.2.p2.9.m9.4.4.4.2.2.3.cmml" type="integer" xref="A3.SS1.2.p2.9.m9.4.4.4.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.2.p2.9.m9.4c">(r_{1},r_{2})=(s_{2},s_{1})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.2.p2.9.m9.4d">( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS1.3.p3"> <p class="ltx_p" id="A3.SS1.3.p3.1">∎</p> </div> </div> <div class="ltx_para" id="A3.SS1.p5"> <p class="ltx_p" id="A3.SS1.p5.4">Clearly Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmlemma1" title="Lemma 1. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> also applies to <math alttext="M_{34}" class="ltx_Math" display="inline" id="A3.SS1.p5.1.m1.1"><semantics id="A3.SS1.p5.1.m1.1a"><msub id="A3.SS1.p5.1.m1.1.1" xref="A3.SS1.p5.1.m1.1.1.cmml"><mi id="A3.SS1.p5.1.m1.1.1.2" xref="A3.SS1.p5.1.m1.1.1.2.cmml">M</mi><mn id="A3.SS1.p5.1.m1.1.1.3" xref="A3.SS1.p5.1.m1.1.1.3.cmml">34</mn></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.p5.1.m1.1b"><apply id="A3.SS1.p5.1.m1.1.1.cmml" xref="A3.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="A3.SS1.p5.1.m1.1.1.1.cmml" xref="A3.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="A3.SS1.p5.1.m1.1.1.2.cmml" xref="A3.SS1.p5.1.m1.1.1.2">𝑀</ci><cn id="A3.SS1.p5.1.m1.1.1.3.cmml" type="integer" xref="A3.SS1.p5.1.m1.1.1.3">34</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p5.1.m1.1c">M_{34}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p5.1.m1.1d">italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT</annotation></semantics></math>. And since <math alttext="\det M_{12}" class="ltx_Math" display="inline" id="A3.SS1.p5.2.m2.1"><semantics id="A3.SS1.p5.2.m2.1a"><mrow id="A3.SS1.p5.2.m2.1.1" xref="A3.SS1.p5.2.m2.1.1.cmml"><mo id="A3.SS1.p5.2.m2.1.1.1" rspace="0.167em" xref="A3.SS1.p5.2.m2.1.1.1.cmml">det</mo><msub id="A3.SS1.p5.2.m2.1.1.2" xref="A3.SS1.p5.2.m2.1.1.2.cmml"><mi id="A3.SS1.p5.2.m2.1.1.2.2" xref="A3.SS1.p5.2.m2.1.1.2.2.cmml">M</mi><mn id="A3.SS1.p5.2.m2.1.1.2.3" xref="A3.SS1.p5.2.m2.1.1.2.3.cmml">12</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p5.2.m2.1b"><apply id="A3.SS1.p5.2.m2.1.1.cmml" xref="A3.SS1.p5.2.m2.1.1"><determinant id="A3.SS1.p5.2.m2.1.1.1.cmml" xref="A3.SS1.p5.2.m2.1.1.1"></determinant><apply id="A3.SS1.p5.2.m2.1.1.2.cmml" xref="A3.SS1.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.p5.2.m2.1.1.2.1.cmml" xref="A3.SS1.p5.2.m2.1.1.2">subscript</csymbol><ci id="A3.SS1.p5.2.m2.1.1.2.2.cmml" xref="A3.SS1.p5.2.m2.1.1.2.2">𝑀</ci><cn id="A3.SS1.p5.2.m2.1.1.2.3.cmml" type="integer" xref="A3.SS1.p5.2.m2.1.1.2.3">12</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p5.2.m2.1c">\det M_{12}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p5.2.m2.1d">roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\det M_{34}" class="ltx_Math" display="inline" id="A3.SS1.p5.3.m3.1"><semantics id="A3.SS1.p5.3.m3.1a"><mrow id="A3.SS1.p5.3.m3.1.1" xref="A3.SS1.p5.3.m3.1.1.cmml"><mo id="A3.SS1.p5.3.m3.1.1.1" rspace="0.167em" xref="A3.SS1.p5.3.m3.1.1.1.cmml">det</mo><msub id="A3.SS1.p5.3.m3.1.1.2" xref="A3.SS1.p5.3.m3.1.1.2.cmml"><mi id="A3.SS1.p5.3.m3.1.1.2.2" xref="A3.SS1.p5.3.m3.1.1.2.2.cmml">M</mi><mn id="A3.SS1.p5.3.m3.1.1.2.3" xref="A3.SS1.p5.3.m3.1.1.2.3.cmml">34</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p5.3.m3.1b"><apply id="A3.SS1.p5.3.m3.1.1.cmml" xref="A3.SS1.p5.3.m3.1.1"><determinant id="A3.SS1.p5.3.m3.1.1.1.cmml" xref="A3.SS1.p5.3.m3.1.1.1"></determinant><apply id="A3.SS1.p5.3.m3.1.1.2.cmml" xref="A3.SS1.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.p5.3.m3.1.1.2.1.cmml" xref="A3.SS1.p5.3.m3.1.1.2">subscript</csymbol><ci id="A3.SS1.p5.3.m3.1.1.2.2.cmml" xref="A3.SS1.p5.3.m3.1.1.2.2">𝑀</ci><cn id="A3.SS1.p5.3.m3.1.1.2.3.cmml" type="integer" xref="A3.SS1.p5.3.m3.1.1.2.3">34</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p5.3.m3.1c">\det M_{34}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p5.3.m3.1d">roman_det italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT</annotation></semantics></math> are independent when conditioned on <math alttext="S_{1..T}" class="ltx_math_unparsed" display="inline" id="A3.SS1.p5.4.m4.1"><semantics id="A3.SS1.p5.4.m4.1a"><msub id="A3.SS1.p5.4.m4.1.2"><mi id="A3.SS1.p5.4.m4.1.2.2">S</mi><mrow id="A3.SS1.p5.4.m4.1.1.1"><mn id="A3.SS1.p5.4.m4.1.1.1.1">1</mn><mo id="A3.SS1.p5.4.m4.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.p5.4.m4.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.p5.4.m4.1.1.1.4">T</mi></mrow></msub><annotation encoding="application/x-tex" id="A3.SS1.p5.4.m4.1b">S_{1..T}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p5.4.m4.1c">italic_S start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a> follows.</p> </div> <div class="ltx_proof" id="A3.SS1.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem9" title="Theorem 9. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">9</span></a>.</h6> <div class="ltx_para" id="A3.SS1.4.p1"> <p class="ltx_p" id="A3.SS1.4.p1.3">Let <math alttext="s_{1..T}" class="ltx_math_unparsed" display="inline" id="A3.SS1.4.p1.1.m1.1"><semantics id="A3.SS1.4.p1.1.m1.1a"><msub id="A3.SS1.4.p1.1.m1.1.2"><mi id="A3.SS1.4.p1.1.m1.1.2.2">s</mi><mrow id="A3.SS1.4.p1.1.m1.1.1.1"><mn id="A3.SS1.4.p1.1.m1.1.1.1.1">1</mn><mo id="A3.SS1.4.p1.1.m1.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.4.p1.1.m1.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.4.p1.1.m1.1.1.1.4">T</mi></mrow></msub><annotation encoding="application/x-tex" id="A3.SS1.4.p1.1.m1.1b">s_{1..T}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.1.m1.1c">italic_s start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT</annotation></semantics></math> be the signal realization for agent <math alttext="i" class="ltx_Math" display="inline" id="A3.SS1.4.p1.2.m2.1"><semantics id="A3.SS1.4.p1.2.m2.1a"><mi id="A3.SS1.4.p1.2.m2.1.1" xref="A3.SS1.4.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.2.m2.1b"><ci id="A3.SS1.4.p1.2.m2.1.1.cmml" xref="A3.SS1.4.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.2.m2.1d">italic_i</annotation></semantics></math>, and consider any report <math alttext="r_{1..T}" class="ltx_math_unparsed" display="inline" id="A3.SS1.4.p1.3.m3.1"><semantics id="A3.SS1.4.p1.3.m3.1a"><msub id="A3.SS1.4.p1.3.m3.1.2"><mi id="A3.SS1.4.p1.3.m3.1.2.2">r</mi><mrow id="A3.SS1.4.p1.3.m3.1.1.1"><mn id="A3.SS1.4.p1.3.m3.1.1.1.1">1</mn><mo id="A3.SS1.4.p1.3.m3.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.4.p1.3.m3.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.4.p1.3.m3.1.1.1.4">T</mi></mrow></msub><annotation encoding="application/x-tex" id="A3.SS1.4.p1.3.m3.1b">r_{1..T}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.3.m3.1c">italic_r start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT</annotation></semantics></math>. We have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx14"> <tbody id="A3.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})% \mid S_{1..T}]" class="ltx_math_unparsed" display="inline" id="A3.Ex25.m1.4"><semantics id="A3.Ex25.m1.4a"><mrow id="A3.Ex25.m1.4.4"><mo id="A3.Ex25.m1.4.4.2" movablelimits="false" rspace="0em">𝔼</mo><mrow id="A3.Ex25.m1.4.4.1.1"><mo id="A3.Ex25.m1.4.4.1.1.2" stretchy="false">[</mo><mrow id="A3.Ex25.m1.4.4.1.1.1"><mrow id="A3.Ex25.m1.4.4.1.1.1.2"><msub id="A3.Ex25.m1.4.4.1.1.1.2.4"><mi id="A3.Ex25.m1.4.4.1.1.1.2.4.2">M</mi><mtext id="A3.Ex25.m1.4.4.1.1.1.2.4.3">DMI</mtext></msub><mo id="A3.Ex25.m1.4.4.1.1.1.2.3">⁢</mo><mrow id="A3.Ex25.m1.4.4.1.1.1.2.2.2"><mo id="A3.Ex25.m1.4.4.1.1.1.2.2.2.3" stretchy="false">(</mo><msub id="A3.Ex25.m1.4.4.1.1.1.1.1.1.1"><mi id="A3.Ex25.m1.4.4.1.1.1.1.1.1.1.2">r</mi><mrow id="A3.Ex25.m1.1.1.1"><mn id="A3.Ex25.m1.1.1.1.1">1</mn><mo id="A3.Ex25.m1.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex25.m1.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex25.m1.1.1.1.4">T</mi></mrow></msub><mo id="A3.Ex25.m1.4.4.1.1.1.2.2.2.4">,</mo><msubsup id="A3.Ex25.m1.4.4.1.1.1.2.2.2.2"><mi id="A3.Ex25.m1.4.4.1.1.1.2.2.2.2.2.2">R</mi><mrow id="A3.Ex25.m1.2.2.1"><mn id="A3.Ex25.m1.2.2.1.1">1</mn><mo id="A3.Ex25.m1.2.2.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex25.m1.2.2.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex25.m1.2.2.1.4">T</mi></mrow><mo id="A3.Ex25.m1.4.4.1.1.1.2.2.2.2.2.3">′</mo></msubsup><mo id="A3.Ex25.m1.4.4.1.1.1.2.2.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex25.m1.4.4.1.1.1.3">∣</mo><msub id="A3.Ex25.m1.4.4.1.1.1.4"><mi id="A3.Ex25.m1.4.4.1.1.1.4.2">S</mi><mrow id="A3.Ex25.m1.3.3.1"><mn id="A3.Ex25.m1.3.3.1.1">1</mn><mo id="A3.Ex25.m1.3.3.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex25.m1.3.3.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex25.m1.3.3.1.4">T</mi></mrow></msub></mrow><mo id="A3.Ex25.m1.4.4.1.1.3" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.Ex25.m1.4b">\displaystyle\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})% \mid S_{1..T}]</annotation><annotation encoding="application/x-llamapun" id="A3.Ex25.m1.4c">blackboard_E [ italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT , italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ) ∣ italic_S start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]\mathop{\mathbb{% E}}[\det M_{34}\mid S_{3},S_{4}]~{}." class="ltx_Math" display="inline" id="A3.Ex25.m2.1"><semantics id="A3.Ex25.m2.1a"><mrow id="A3.Ex25.m2.1.1.1" xref="A3.Ex25.m2.1.1.1.1.cmml"><mrow id="A3.Ex25.m2.1.1.1.1" xref="A3.Ex25.m2.1.1.1.1.cmml"><mi id="A3.Ex25.m2.1.1.1.1.4" xref="A3.Ex25.m2.1.1.1.1.4.cmml"></mi><mo id="A3.Ex25.m2.1.1.1.1.3" rspace="0.1389em" xref="A3.Ex25.m2.1.1.1.1.3.cmml">=</mo><mrow id="A3.Ex25.m2.1.1.1.1.2" xref="A3.Ex25.m2.1.1.1.1.2.cmml"><mo id="A3.Ex25.m2.1.1.1.1.2.3" lspace="0.1389em" movablelimits="false" rspace="0em" xref="A3.Ex25.m2.1.1.1.1.2.3.cmml">𝔼</mo><mrow id="A3.Ex25.m2.1.1.1.1.2.2" xref="A3.Ex25.m2.1.1.1.1.2.2.cmml"><mrow id="A3.Ex25.m2.1.1.1.1.1.1.1.1" xref="A3.Ex25.m2.1.1.1.1.1.1.1.2.cmml"><mo id="A3.Ex25.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex25.m2.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.cmml"><mo id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.1" lspace="0em" movablelimits="false" rspace="0.167em" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.1.cmml">det</mo><msub id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2.cmml"><mi id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2.2.cmml">M</mi><mn id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2.3" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.4.2.3.cmml">12</mn></msub></mrow><mo id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.3.cmml">∣</mo><mrow id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.3.cmml"><msub id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">S</mi><mn id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.3" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.3.cmml">,</mo><msub id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">S</mi><mn id="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2.3" xref="A3.Ex25.m2.1.1.1.1.1.1.1.1.1.2.2.2.3.cmml">2</mn></msub></mrow></mrow><mo id="A3.Ex25.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex25.m2.1.1.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="A3.Ex25.m2.1.1.1.1.2.2.3" lspace="0.167em" xref="A3.Ex25.m2.1.1.1.1.2.2.3.cmml">⁢</mo><mrow id="A3.Ex25.m2.1.1.1.1.2.2.2" xref="A3.Ex25.m2.1.1.1.1.2.2.2.cmml"><mo id="A3.Ex25.m2.1.1.1.1.2.2.2.2" movablelimits="false" rspace="0em" xref="A3.Ex25.m2.1.1.1.1.2.2.2.2.cmml">𝔼</mo><mrow id="A3.Ex25.m2.1.1.1.1.2.2.2.1.1" xref="A3.Ex25.m2.1.1.1.1.2.2.2.1.2.cmml"><mo id="A3.Ex25.m2.1.1.1.1.2.2.2.1.1.2" stretchy="false" xref="A3.Ex25.m2.1.1.1.1.2.2.2.1.2.1.cmml">[</mo><mrow 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S_{1},S_{2}]\mathop{\mathbb{% E}}[\det M_{34}\mid S_{3},S_{4}]~{}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex25.m2.1d">= blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex26"><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=\mathop{\mathbb{E}}[\det M_{12}\mid S_{1},S_{2}]\mathop{\mathbb{% E}}[\det M_{34}\mid S_{3},S_{4}]~{}." class="ltx_Math" display="inline" 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encoding="application/x-llamapun" id="A3.Ex26.m1.1d">= blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] blackboard_E [ roman_det italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT ∣ italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.4.p1.10">Applying Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmlemma1" title="Lemma 1. ‣ C.1 Proof of Theorem 9 ‣ Appendix C Omitted Proofs for DMI ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">1</span></a> to both <math alttext="M_{12}" class="ltx_Math" display="inline" id="A3.SS1.4.p1.4.m1.1"><semantics id="A3.SS1.4.p1.4.m1.1a"><msub id="A3.SS1.4.p1.4.m1.1.1" xref="A3.SS1.4.p1.4.m1.1.1.cmml"><mi id="A3.SS1.4.p1.4.m1.1.1.2" xref="A3.SS1.4.p1.4.m1.1.1.2.cmml">M</mi><mn id="A3.SS1.4.p1.4.m1.1.1.3" xref="A3.SS1.4.p1.4.m1.1.1.3.cmml">12</mn></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.4.m1.1b"><apply id="A3.SS1.4.p1.4.m1.1.1.cmml" xref="A3.SS1.4.p1.4.m1.1.1"><csymbol cd="ambiguous" id="A3.SS1.4.p1.4.m1.1.1.1.cmml" xref="A3.SS1.4.p1.4.m1.1.1">subscript</csymbol><ci id="A3.SS1.4.p1.4.m1.1.1.2.cmml" xref="A3.SS1.4.p1.4.m1.1.1.2">𝑀</ci><cn id="A3.SS1.4.p1.4.m1.1.1.3.cmml" type="integer" xref="A3.SS1.4.p1.4.m1.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.4.m1.1c">M_{12}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.4.m1.1d">italic_M start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="M_{34}" class="ltx_Math" display="inline" id="A3.SS1.4.p1.5.m2.1"><semantics id="A3.SS1.4.p1.5.m2.1a"><msub id="A3.SS1.4.p1.5.m2.1.1" xref="A3.SS1.4.p1.5.m2.1.1.cmml"><mi id="A3.SS1.4.p1.5.m2.1.1.2" xref="A3.SS1.4.p1.5.m2.1.1.2.cmml">M</mi><mn id="A3.SS1.4.p1.5.m2.1.1.3" xref="A3.SS1.4.p1.5.m2.1.1.3.cmml">34</mn></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.5.m2.1b"><apply id="A3.SS1.4.p1.5.m2.1.1.cmml" xref="A3.SS1.4.p1.5.m2.1.1"><csymbol cd="ambiguous" id="A3.SS1.4.p1.5.m2.1.1.1.cmml" xref="A3.SS1.4.p1.5.m2.1.1">subscript</csymbol><ci id="A3.SS1.4.p1.5.m2.1.1.2.cmml" xref="A3.SS1.4.p1.5.m2.1.1.2">𝑀</ci><cn id="A3.SS1.4.p1.5.m2.1.1.3.cmml" type="integer" xref="A3.SS1.4.p1.5.m2.1.1.3">34</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.5.m2.1c">M_{34}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.5.m2.1d">italic_M start_POSTSUBSCRIPT 34 end_POSTSUBSCRIPT</annotation></semantics></math>, observe that if <math alttext="s_{1}=s_{2}" class="ltx_Math" display="inline" id="A3.SS1.4.p1.6.m3.1"><semantics id="A3.SS1.4.p1.6.m3.1a"><mrow id="A3.SS1.4.p1.6.m3.1.1" xref="A3.SS1.4.p1.6.m3.1.1.cmml"><msub id="A3.SS1.4.p1.6.m3.1.1.2" xref="A3.SS1.4.p1.6.m3.1.1.2.cmml"><mi id="A3.SS1.4.p1.6.m3.1.1.2.2" xref="A3.SS1.4.p1.6.m3.1.1.2.2.cmml">s</mi><mn id="A3.SS1.4.p1.6.m3.1.1.2.3" xref="A3.SS1.4.p1.6.m3.1.1.2.3.cmml">1</mn></msub><mo id="A3.SS1.4.p1.6.m3.1.1.1" xref="A3.SS1.4.p1.6.m3.1.1.1.cmml">=</mo><msub id="A3.SS1.4.p1.6.m3.1.1.3" xref="A3.SS1.4.p1.6.m3.1.1.3.cmml"><mi id="A3.SS1.4.p1.6.m3.1.1.3.2" xref="A3.SS1.4.p1.6.m3.1.1.3.2.cmml">s</mi><mn id="A3.SS1.4.p1.6.m3.1.1.3.3" xref="A3.SS1.4.p1.6.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.6.m3.1b"><apply id="A3.SS1.4.p1.6.m3.1.1.cmml" xref="A3.SS1.4.p1.6.m3.1.1"><eq id="A3.SS1.4.p1.6.m3.1.1.1.cmml" xref="A3.SS1.4.p1.6.m3.1.1.1"></eq><apply id="A3.SS1.4.p1.6.m3.1.1.2.cmml" xref="A3.SS1.4.p1.6.m3.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.4.p1.6.m3.1.1.2.1.cmml" xref="A3.SS1.4.p1.6.m3.1.1.2">subscript</csymbol><ci id="A3.SS1.4.p1.6.m3.1.1.2.2.cmml" xref="A3.SS1.4.p1.6.m3.1.1.2.2">𝑠</ci><cn id="A3.SS1.4.p1.6.m3.1.1.2.3.cmml" type="integer" xref="A3.SS1.4.p1.6.m3.1.1.2.3">1</cn></apply><apply id="A3.SS1.4.p1.6.m3.1.1.3.cmml" xref="A3.SS1.4.p1.6.m3.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.4.p1.6.m3.1.1.3.1.cmml" xref="A3.SS1.4.p1.6.m3.1.1.3">subscript</csymbol><ci id="A3.SS1.4.p1.6.m3.1.1.3.2.cmml" xref="A3.SS1.4.p1.6.m3.1.1.3.2">𝑠</ci><cn id="A3.SS1.4.p1.6.m3.1.1.3.3.cmml" type="integer" xref="A3.SS1.4.p1.6.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.6.m3.1c">s_{1}=s_{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.6.m3.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="s_{3}=s_{4}" class="ltx_Math" display="inline" id="A3.SS1.4.p1.7.m4.1"><semantics id="A3.SS1.4.p1.7.m4.1a"><mrow id="A3.SS1.4.p1.7.m4.1.1" xref="A3.SS1.4.p1.7.m4.1.1.cmml"><msub id="A3.SS1.4.p1.7.m4.1.1.2" xref="A3.SS1.4.p1.7.m4.1.1.2.cmml"><mi id="A3.SS1.4.p1.7.m4.1.1.2.2" xref="A3.SS1.4.p1.7.m4.1.1.2.2.cmml">s</mi><mn id="A3.SS1.4.p1.7.m4.1.1.2.3" xref="A3.SS1.4.p1.7.m4.1.1.2.3.cmml">3</mn></msub><mo id="A3.SS1.4.p1.7.m4.1.1.1" xref="A3.SS1.4.p1.7.m4.1.1.1.cmml">=</mo><msub id="A3.SS1.4.p1.7.m4.1.1.3" xref="A3.SS1.4.p1.7.m4.1.1.3.cmml"><mi id="A3.SS1.4.p1.7.m4.1.1.3.2" xref="A3.SS1.4.p1.7.m4.1.1.3.2.cmml">s</mi><mn id="A3.SS1.4.p1.7.m4.1.1.3.3" xref="A3.SS1.4.p1.7.m4.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.7.m4.1b"><apply id="A3.SS1.4.p1.7.m4.1.1.cmml" xref="A3.SS1.4.p1.7.m4.1.1"><eq id="A3.SS1.4.p1.7.m4.1.1.1.cmml" xref="A3.SS1.4.p1.7.m4.1.1.1"></eq><apply id="A3.SS1.4.p1.7.m4.1.1.2.cmml" xref="A3.SS1.4.p1.7.m4.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.4.p1.7.m4.1.1.2.1.cmml" xref="A3.SS1.4.p1.7.m4.1.1.2">subscript</csymbol><ci id="A3.SS1.4.p1.7.m4.1.1.2.2.cmml" xref="A3.SS1.4.p1.7.m4.1.1.2.2">𝑠</ci><cn id="A3.SS1.4.p1.7.m4.1.1.2.3.cmml" type="integer" xref="A3.SS1.4.p1.7.m4.1.1.2.3">3</cn></apply><apply id="A3.SS1.4.p1.7.m4.1.1.3.cmml" xref="A3.SS1.4.p1.7.m4.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.4.p1.7.m4.1.1.3.1.cmml" xref="A3.SS1.4.p1.7.m4.1.1.3">subscript</csymbol><ci id="A3.SS1.4.p1.7.m4.1.1.3.2.cmml" xref="A3.SS1.4.p1.7.m4.1.1.3.2">𝑠</ci><cn id="A3.SS1.4.p1.7.m4.1.1.3.3.cmml" type="integer" xref="A3.SS1.4.p1.7.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.7.m4.1c">s_{3}=s_{4}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.7.m4.1d">italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> we have <math alttext="\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})\mid S_{1..T}]=0" class="ltx_math_unparsed" display="inline" id="A3.SS1.4.p1.8.m5.4"><semantics id="A3.SS1.4.p1.8.m5.4a"><mrow id="A3.SS1.4.p1.8.m5.4.4"><mrow id="A3.SS1.4.p1.8.m5.4.4.1"><mo id="A3.SS1.4.p1.8.m5.4.4.1.2" rspace="0em">𝔼</mo><mrow id="A3.SS1.4.p1.8.m5.4.4.1.1.1"><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.2" stretchy="false">[</mo><mrow id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1"><mrow id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2"><msub id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.4"><mi id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.4.2">M</mi><mtext id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.4.3">DMI</mtext></msub><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.3">⁢</mo><mrow id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2"><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.3" stretchy="false">(</mo><msub id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.1.1.1.1"><mi id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.1.1.1.1.2">r</mi><mrow id="A3.SS1.4.p1.8.m5.1.1.1"><mn id="A3.SS1.4.p1.8.m5.1.1.1.1">1</mn><mo id="A3.SS1.4.p1.8.m5.1.1.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.4.p1.8.m5.1.1.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.4.p1.8.m5.1.1.1.4">T</mi></mrow></msub><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.4">,</mo><msubsup id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.2"><mi id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.2.2.2">R</mi><mrow id="A3.SS1.4.p1.8.m5.2.2.1"><mn id="A3.SS1.4.p1.8.m5.2.2.1.1">1</mn><mo id="A3.SS1.4.p1.8.m5.2.2.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.4.p1.8.m5.2.2.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.4.p1.8.m5.2.2.1.4">T</mi></mrow><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.2.2.3">′</mo></msubsup><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.2.2.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.3">∣</mo><msub id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.4"><mi id="A3.SS1.4.p1.8.m5.4.4.1.1.1.1.4.2">S</mi><mrow id="A3.SS1.4.p1.8.m5.3.3.1"><mn id="A3.SS1.4.p1.8.m5.3.3.1.1">1</mn><mo id="A3.SS1.4.p1.8.m5.3.3.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.SS1.4.p1.8.m5.3.3.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.SS1.4.p1.8.m5.3.3.1.4">T</mi></mrow></msub></mrow><mo id="A3.SS1.4.p1.8.m5.4.4.1.1.1.3" stretchy="false">]</mo></mrow></mrow><mo id="A3.SS1.4.p1.8.m5.4.4.2">=</mo><mn id="A3.SS1.4.p1.8.m5.4.4.3">0</mn></mrow><annotation encoding="application/x-tex" id="A3.SS1.4.p1.8.m5.4b">\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})\mid S_{1..T}]=0</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.8.m5.4c">blackboard_E [ italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT , italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ) ∣ italic_S start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ] = 0</annotation></semantics></math> regardless of <math alttext="r" class="ltx_Math" display="inline" id="A3.SS1.4.p1.9.m6.1"><semantics id="A3.SS1.4.p1.9.m6.1a"><mi id="A3.SS1.4.p1.9.m6.1.1" xref="A3.SS1.4.p1.9.m6.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.9.m6.1b"><ci id="A3.SS1.4.p1.9.m6.1.1.cmml" xref="A3.SS1.4.p1.9.m6.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.9.m6.1c">r</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.9.m6.1d">italic_r</annotation></semantics></math>. In particular, <math alttext="\sigma_{\text{true}}" class="ltx_Math" display="inline" id="A3.SS1.4.p1.10.m7.1"><semantics id="A3.SS1.4.p1.10.m7.1a"><msub id="A3.SS1.4.p1.10.m7.1.1" xref="A3.SS1.4.p1.10.m7.1.1.cmml"><mi id="A3.SS1.4.p1.10.m7.1.1.2" xref="A3.SS1.4.p1.10.m7.1.1.2.cmml">σ</mi><mtext id="A3.SS1.4.p1.10.m7.1.1.3" xref="A3.SS1.4.p1.10.m7.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.4.p1.10.m7.1b"><apply id="A3.SS1.4.p1.10.m7.1.1.cmml" xref="A3.SS1.4.p1.10.m7.1.1"><csymbol cd="ambiguous" id="A3.SS1.4.p1.10.m7.1.1.1.cmml" xref="A3.SS1.4.p1.10.m7.1.1">subscript</csymbol><ci id="A3.SS1.4.p1.10.m7.1.1.2.cmml" xref="A3.SS1.4.p1.10.m7.1.1.2">𝜎</ci><ci id="A3.SS1.4.p1.10.m7.1.1.3a.cmml" xref="A3.SS1.4.p1.10.m7.1.1.3"><mtext id="A3.SS1.4.p1.10.m7.1.1.3.cmml" mathsize="70%" xref="A3.SS1.4.p1.10.m7.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.4.p1.10.m7.1c">\sigma_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.4.p1.10.m7.1d">italic_σ start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> maximizes the expected payoff.</p> </div> <div class="ltx_para" id="A3.SS1.5.p2"> <p class="ltx_p" id="A3.SS1.5.p2.6">Now suppose <math alttext="s_{1}\neq s_{2}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.1.m1.1"><semantics id="A3.SS1.5.p2.1.m1.1a"><mrow id="A3.SS1.5.p2.1.m1.1.1" xref="A3.SS1.5.p2.1.m1.1.1.cmml"><msub id="A3.SS1.5.p2.1.m1.1.1.2" xref="A3.SS1.5.p2.1.m1.1.1.2.cmml"><mi id="A3.SS1.5.p2.1.m1.1.1.2.2" xref="A3.SS1.5.p2.1.m1.1.1.2.2.cmml">s</mi><mn id="A3.SS1.5.p2.1.m1.1.1.2.3" xref="A3.SS1.5.p2.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="A3.SS1.5.p2.1.m1.1.1.1" xref="A3.SS1.5.p2.1.m1.1.1.1.cmml">≠</mo><msub id="A3.SS1.5.p2.1.m1.1.1.3" xref="A3.SS1.5.p2.1.m1.1.1.3.cmml"><mi id="A3.SS1.5.p2.1.m1.1.1.3.2" xref="A3.SS1.5.p2.1.m1.1.1.3.2.cmml">s</mi><mn id="A3.SS1.5.p2.1.m1.1.1.3.3" xref="A3.SS1.5.p2.1.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.1.m1.1b"><apply id="A3.SS1.5.p2.1.m1.1.1.cmml" xref="A3.SS1.5.p2.1.m1.1.1"><neq id="A3.SS1.5.p2.1.m1.1.1.1.cmml" xref="A3.SS1.5.p2.1.m1.1.1.1"></neq><apply id="A3.SS1.5.p2.1.m1.1.1.2.cmml" xref="A3.SS1.5.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.5.p2.1.m1.1.1.2.1.cmml" xref="A3.SS1.5.p2.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS1.5.p2.1.m1.1.1.2.2.cmml" xref="A3.SS1.5.p2.1.m1.1.1.2.2">𝑠</ci><cn id="A3.SS1.5.p2.1.m1.1.1.2.3.cmml" type="integer" xref="A3.SS1.5.p2.1.m1.1.1.2.3">1</cn></apply><apply id="A3.SS1.5.p2.1.m1.1.1.3.cmml" xref="A3.SS1.5.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.5.p2.1.m1.1.1.3.1.cmml" xref="A3.SS1.5.p2.1.m1.1.1.3">subscript</csymbol><ci id="A3.SS1.5.p2.1.m1.1.1.3.2.cmml" xref="A3.SS1.5.p2.1.m1.1.1.3.2">𝑠</ci><cn id="A3.SS1.5.p2.1.m1.1.1.3.3.cmml" type="integer" xref="A3.SS1.5.p2.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.1.m1.1c">s_{1}\neq s_{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.1.m1.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="s_{3}\neq s_{4}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.2.m2.1"><semantics id="A3.SS1.5.p2.2.m2.1a"><mrow id="A3.SS1.5.p2.2.m2.1.1" xref="A3.SS1.5.p2.2.m2.1.1.cmml"><msub id="A3.SS1.5.p2.2.m2.1.1.2" xref="A3.SS1.5.p2.2.m2.1.1.2.cmml"><mi id="A3.SS1.5.p2.2.m2.1.1.2.2" xref="A3.SS1.5.p2.2.m2.1.1.2.2.cmml">s</mi><mn id="A3.SS1.5.p2.2.m2.1.1.2.3" xref="A3.SS1.5.p2.2.m2.1.1.2.3.cmml">3</mn></msub><mo id="A3.SS1.5.p2.2.m2.1.1.1" xref="A3.SS1.5.p2.2.m2.1.1.1.cmml">≠</mo><msub id="A3.SS1.5.p2.2.m2.1.1.3" xref="A3.SS1.5.p2.2.m2.1.1.3.cmml"><mi id="A3.SS1.5.p2.2.m2.1.1.3.2" xref="A3.SS1.5.p2.2.m2.1.1.3.2.cmml">s</mi><mn id="A3.SS1.5.p2.2.m2.1.1.3.3" xref="A3.SS1.5.p2.2.m2.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.2.m2.1b"><apply id="A3.SS1.5.p2.2.m2.1.1.cmml" xref="A3.SS1.5.p2.2.m2.1.1"><neq id="A3.SS1.5.p2.2.m2.1.1.1.cmml" xref="A3.SS1.5.p2.2.m2.1.1.1"></neq><apply id="A3.SS1.5.p2.2.m2.1.1.2.cmml" xref="A3.SS1.5.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.5.p2.2.m2.1.1.2.1.cmml" xref="A3.SS1.5.p2.2.m2.1.1.2">subscript</csymbol><ci id="A3.SS1.5.p2.2.m2.1.1.2.2.cmml" xref="A3.SS1.5.p2.2.m2.1.1.2.2">𝑠</ci><cn id="A3.SS1.5.p2.2.m2.1.1.2.3.cmml" type="integer" xref="A3.SS1.5.p2.2.m2.1.1.2.3">3</cn></apply><apply id="A3.SS1.5.p2.2.m2.1.1.3.cmml" xref="A3.SS1.5.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.5.p2.2.m2.1.1.3.1.cmml" xref="A3.SS1.5.p2.2.m2.1.1.3">subscript</csymbol><ci id="A3.SS1.5.p2.2.m2.1.1.3.2.cmml" xref="A3.SS1.5.p2.2.m2.1.1.3.2">𝑠</ci><cn id="A3.SS1.5.p2.2.m2.1.1.3.3.cmml" type="integer" xref="A3.SS1.5.p2.2.m2.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.2.m2.1c">s_{3}\neq s_{4}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.2.m2.1d">italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≠ italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>. Again we have a zero expectation if <math alttext="r_{1}=r_{2}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.3.m3.1"><semantics id="A3.SS1.5.p2.3.m3.1a"><mrow id="A3.SS1.5.p2.3.m3.1.1" xref="A3.SS1.5.p2.3.m3.1.1.cmml"><msub id="A3.SS1.5.p2.3.m3.1.1.2" xref="A3.SS1.5.p2.3.m3.1.1.2.cmml"><mi id="A3.SS1.5.p2.3.m3.1.1.2.2" xref="A3.SS1.5.p2.3.m3.1.1.2.2.cmml">r</mi><mn id="A3.SS1.5.p2.3.m3.1.1.2.3" xref="A3.SS1.5.p2.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="A3.SS1.5.p2.3.m3.1.1.1" xref="A3.SS1.5.p2.3.m3.1.1.1.cmml">=</mo><msub id="A3.SS1.5.p2.3.m3.1.1.3" xref="A3.SS1.5.p2.3.m3.1.1.3.cmml"><mi id="A3.SS1.5.p2.3.m3.1.1.3.2" xref="A3.SS1.5.p2.3.m3.1.1.3.2.cmml">r</mi><mn id="A3.SS1.5.p2.3.m3.1.1.3.3" xref="A3.SS1.5.p2.3.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.3.m3.1b"><apply id="A3.SS1.5.p2.3.m3.1.1.cmml" xref="A3.SS1.5.p2.3.m3.1.1"><eq id="A3.SS1.5.p2.3.m3.1.1.1.cmml" xref="A3.SS1.5.p2.3.m3.1.1.1"></eq><apply id="A3.SS1.5.p2.3.m3.1.1.2.cmml" xref="A3.SS1.5.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.5.p2.3.m3.1.1.2.1.cmml" xref="A3.SS1.5.p2.3.m3.1.1.2">subscript</csymbol><ci id="A3.SS1.5.p2.3.m3.1.1.2.2.cmml" xref="A3.SS1.5.p2.3.m3.1.1.2.2">𝑟</ci><cn id="A3.SS1.5.p2.3.m3.1.1.2.3.cmml" type="integer" xref="A3.SS1.5.p2.3.m3.1.1.2.3">1</cn></apply><apply id="A3.SS1.5.p2.3.m3.1.1.3.cmml" xref="A3.SS1.5.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.5.p2.3.m3.1.1.3.1.cmml" xref="A3.SS1.5.p2.3.m3.1.1.3">subscript</csymbol><ci id="A3.SS1.5.p2.3.m3.1.1.3.2.cmml" xref="A3.SS1.5.p2.3.m3.1.1.3.2">𝑟</ci><cn id="A3.SS1.5.p2.3.m3.1.1.3.3.cmml" type="integer" xref="A3.SS1.5.p2.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.3.m3.1c">r_{1}=r_{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.3.m3.1d">italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="r_{3}=r_{4}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.4.m4.1"><semantics id="A3.SS1.5.p2.4.m4.1a"><mrow id="A3.SS1.5.p2.4.m4.1.1" xref="A3.SS1.5.p2.4.m4.1.1.cmml"><msub id="A3.SS1.5.p2.4.m4.1.1.2" xref="A3.SS1.5.p2.4.m4.1.1.2.cmml"><mi id="A3.SS1.5.p2.4.m4.1.1.2.2" xref="A3.SS1.5.p2.4.m4.1.1.2.2.cmml">r</mi><mn id="A3.SS1.5.p2.4.m4.1.1.2.3" xref="A3.SS1.5.p2.4.m4.1.1.2.3.cmml">3</mn></msub><mo id="A3.SS1.5.p2.4.m4.1.1.1" xref="A3.SS1.5.p2.4.m4.1.1.1.cmml">=</mo><msub id="A3.SS1.5.p2.4.m4.1.1.3" xref="A3.SS1.5.p2.4.m4.1.1.3.cmml"><mi id="A3.SS1.5.p2.4.m4.1.1.3.2" xref="A3.SS1.5.p2.4.m4.1.1.3.2.cmml">r</mi><mn id="A3.SS1.5.p2.4.m4.1.1.3.3" xref="A3.SS1.5.p2.4.m4.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.4.m4.1b"><apply id="A3.SS1.5.p2.4.m4.1.1.cmml" xref="A3.SS1.5.p2.4.m4.1.1"><eq id="A3.SS1.5.p2.4.m4.1.1.1.cmml" xref="A3.SS1.5.p2.4.m4.1.1.1"></eq><apply id="A3.SS1.5.p2.4.m4.1.1.2.cmml" xref="A3.SS1.5.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="A3.SS1.5.p2.4.m4.1.1.2.1.cmml" xref="A3.SS1.5.p2.4.m4.1.1.2">subscript</csymbol><ci id="A3.SS1.5.p2.4.m4.1.1.2.2.cmml" xref="A3.SS1.5.p2.4.m4.1.1.2.2">𝑟</ci><cn id="A3.SS1.5.p2.4.m4.1.1.2.3.cmml" type="integer" xref="A3.SS1.5.p2.4.m4.1.1.2.3">3</cn></apply><apply id="A3.SS1.5.p2.4.m4.1.1.3.cmml" xref="A3.SS1.5.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.SS1.5.p2.4.m4.1.1.3.1.cmml" xref="A3.SS1.5.p2.4.m4.1.1.3">subscript</csymbol><ci id="A3.SS1.5.p2.4.m4.1.1.3.2.cmml" xref="A3.SS1.5.p2.4.m4.1.1.3.2">𝑟</ci><cn id="A3.SS1.5.p2.4.m4.1.1.3.3.cmml" type="integer" xref="A3.SS1.5.p2.4.m4.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.4.m4.1c">r_{3}=r_{4}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.4.m4.1d">italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>. Otherwise, for each matrix, we have two cases: <math alttext="r" class="ltx_Math" display="inline" id="A3.SS1.5.p2.5.m5.1"><semantics id="A3.SS1.5.p2.5.m5.1a"><mi id="A3.SS1.5.p2.5.m5.1.1" xref="A3.SS1.5.p2.5.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.5.m5.1b"><ci id="A3.SS1.5.p2.5.m5.1.1.cmml" xref="A3.SS1.5.p2.5.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.5.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.5.m5.1d">italic_r</annotation></semantics></math> matches <math alttext="s" class="ltx_Math" display="inline" id="A3.SS1.5.p2.6.m6.1"><semantics id="A3.SS1.5.p2.6.m6.1a"><mi id="A3.SS1.5.p2.6.m6.1.1" xref="A3.SS1.5.p2.6.m6.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.6.m6.1b"><ci id="A3.SS1.5.p2.6.m6.1.1.cmml" xref="A3.SS1.5.p2.6.m6.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.6.m6.1c">s</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.6.m6.1d">italic_s</annotation></semantics></math> or not. In particular,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx15"> <tbody id="A3.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})% \mid S_{1..T}]=\begin{cases}(p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3}% ,r_{4})=(s_{3},s_{4})\\ -(p(H)-p(L))^{2}&(r_{1},r_{2})\neq(s_{1},s_{2}),(r_{3},r_{4})=(s_{3},s_{4})\\ -(p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3},r_{4})\neq(s_{3},s_{4})\\ (p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3},r_{4})=(s_{3},s_{4})\end{cases}" class="ltx_math_unparsed" display="inline" id="A3.Ex27.m1.12"><semantics id="A3.Ex27.m1.12a"><mrow id="A3.Ex27.m1.12.12"><mrow id="A3.Ex27.m1.12.12.1"><mo id="A3.Ex27.m1.12.12.1.2" movablelimits="false" rspace="0em">𝔼</mo><mrow id="A3.Ex27.m1.12.12.1.1.1"><mo id="A3.Ex27.m1.12.12.1.1.1.2" stretchy="false">[</mo><mrow id="A3.Ex27.m1.12.12.1.1.1.1"><mrow id="A3.Ex27.m1.12.12.1.1.1.1.2"><msub id="A3.Ex27.m1.12.12.1.1.1.1.2.4"><mi id="A3.Ex27.m1.12.12.1.1.1.1.2.4.2">M</mi><mtext id="A3.Ex27.m1.12.12.1.1.1.1.2.4.3">DMI</mtext></msub><mo id="A3.Ex27.m1.12.12.1.1.1.1.2.3">⁢</mo><mrow id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2"><mo id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.12.12.1.1.1.1.1.1.1.1"><mi id="A3.Ex27.m1.12.12.1.1.1.1.1.1.1.1.2">r</mi><mrow id="A3.Ex27.m1.9.9.1"><mn id="A3.Ex27.m1.9.9.1.1">1</mn><mo id="A3.Ex27.m1.9.9.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex27.m1.9.9.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex27.m1.9.9.1.4">T</mi></mrow></msub><mo id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.4">,</mo><msubsup id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.2"><mi id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.2.2.2">R</mi><mrow id="A3.Ex27.m1.10.10.1"><mn id="A3.Ex27.m1.10.10.1.1">1</mn><mo id="A3.Ex27.m1.10.10.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex27.m1.10.10.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex27.m1.10.10.1.4">T</mi></mrow><mo id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.2.2.3">′</mo></msubsup><mo id="A3.Ex27.m1.12.12.1.1.1.1.2.2.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.12.12.1.1.1.1.3">∣</mo><msub id="A3.Ex27.m1.12.12.1.1.1.1.4"><mi id="A3.Ex27.m1.12.12.1.1.1.1.4.2">S</mi><mrow id="A3.Ex27.m1.11.11.1"><mn id="A3.Ex27.m1.11.11.1.1">1</mn><mo id="A3.Ex27.m1.11.11.1.2" lspace="0em" rspace="0.0835em">.</mo><mo id="A3.Ex27.m1.11.11.1.3" lspace="0.0835em" rspace="0.167em">.</mo><mi id="A3.Ex27.m1.11.11.1.4">T</mi></mrow></msub></mrow><mo id="A3.Ex27.m1.12.12.1.1.1.3" stretchy="false">]</mo></mrow></mrow><mo id="A3.Ex27.m1.12.12.2">=</mo><mrow id="A3.Ex27.m1.8.8a"><mo id="A3.Ex27.m1.8.8a.9">{</mo><mtable columnspacing="5pt" id="A3.Ex27.m1.8.8.8a" rowspacing="0pt"><mtr id="A3.Ex27.m1.8.8.8aa"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ab"><msup id="A3.Ex27.m1.1.1.1.1.1.1"><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1"><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.2" stretchy="false">(</mo><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1"><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2"><mi id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2.2">p</mi><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2.1">⁢</mo><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2.3.2"><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.1.1.1.1.1.1.1">H</mi><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.2.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.1">−</mo><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3"><mi id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3.2">p</mi><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3.1">⁢</mo><mrow id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3.3.2"><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.1.1.1.1.1.1.2">L</mi><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A3.Ex27.m1.1.1.1.1.1.1.3.1.3" stretchy="false">)</mo></mrow><mn id="A3.Ex27.m1.1.1.1.1.1.1.5">2</mn></msup></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ac"><mrow id="A3.Ex27.m1.2.2.2.2.2.1.2"><mrow id="A3.Ex27.m1.2.2.2.2.2.1.1.1"><mrow id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2"><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.1.1.1.1.1"><mi id="A3.Ex27.m1.2.2.2.2.2.1.1.1.1.1.1.2">r</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.1.1.1.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.4">,</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.2"><mi id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.2.2">r</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.5">=</mo><mrow id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2"><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.1.1.3.1.1"><mi id="A3.Ex27.m1.2.2.2.2.2.1.1.1.3.1.1.2">s</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.1.1.3.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.4">,</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.2"><mi id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.2.2">s</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.1.1.4.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.3">,</mo><mrow id="A3.Ex27.m1.2.2.2.2.2.1.2.2"><mrow id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2"><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.2.2.1.1.1"><mi id="A3.Ex27.m1.2.2.2.2.2.1.2.2.1.1.1.2">r</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.2.2.1.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.4">,</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.2"><mi id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.2.2">r</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.5">=</mo><mrow id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2"><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.2.2.3.1.1"><mi id="A3.Ex27.m1.2.2.2.2.2.1.2.2.3.1.1.2">s</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.2.2.3.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.4">,</mo><msub id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.2"><mi id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.2.2">s</mi><mn id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.2.2.2.2.2.1.2.2.4.2.5" stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="A3.Ex27.m1.8.8.8ad"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ae"><mrow id="A3.Ex27.m1.3.3.3.3.1.1"><mo id="A3.Ex27.m1.3.3.3.3.1.1a">−</mo><msup id="A3.Ex27.m1.3.3.3.3.1.1.3"><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1"><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.2" stretchy="false">(</mo><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1"><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2"><mi id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2.2">p</mi><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2.1">⁢</mo><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2.3.2"><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.3.3.3.3.1.1.1">H</mi><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.2.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.1">−</mo><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3"><mi id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3.2">p</mi><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3.1">⁢</mo><mrow id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3.3.2"><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.3.3.3.3.1.1.2">L</mi><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A3.Ex27.m1.3.3.3.3.1.1.3.1.1.3" stretchy="false">)</mo></mrow><mn id="A3.Ex27.m1.3.3.3.3.1.1.3.3">2</mn></msup></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8af"><mrow id="A3.Ex27.m1.4.4.4.4.2.1.2"><mrow id="A3.Ex27.m1.4.4.4.4.2.1.1.1"><mrow id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2"><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.1.1.1.1.1"><mi id="A3.Ex27.m1.4.4.4.4.2.1.1.1.1.1.1.2">r</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.1.1.1.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.4">,</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.2"><mi id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.2.2">r</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.5">≠</mo><mrow id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2"><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.1.1.3.1.1"><mi id="A3.Ex27.m1.4.4.4.4.2.1.1.1.3.1.1.2">s</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.1.1.3.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.4">,</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.2"><mi id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.2.2">s</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.1.1.4.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.3">,</mo><mrow id="A3.Ex27.m1.4.4.4.4.2.1.2.2"><mrow id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2"><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.2.2.1.1.1"><mi id="A3.Ex27.m1.4.4.4.4.2.1.2.2.1.1.1.2">r</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.2.2.1.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.4">,</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.2"><mi id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.2.2">r</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.5">=</mo><mrow id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2"><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.2.2.3.1.1"><mi id="A3.Ex27.m1.4.4.4.4.2.1.2.2.3.1.1.2">s</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.2.2.3.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.4">,</mo><msub id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.2"><mi id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.2.2">s</mi><mn id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.4.4.4.4.2.1.2.2.4.2.5" stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="A3.Ex27.m1.8.8.8ag"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ah"><mrow id="A3.Ex27.m1.5.5.5.5.1.1"><mo id="A3.Ex27.m1.5.5.5.5.1.1a">−</mo><msup id="A3.Ex27.m1.5.5.5.5.1.1.3"><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1"><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.2" stretchy="false">(</mo><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1"><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2"><mi id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2.2">p</mi><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2.1">⁢</mo><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2.3.2"><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.5.5.5.5.1.1.1">H</mi><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.2.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.1">−</mo><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3"><mi id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3.2">p</mi><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3.1">⁢</mo><mrow id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3.3.2"><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.5.5.5.5.1.1.2">L</mi><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A3.Ex27.m1.5.5.5.5.1.1.3.1.1.3" stretchy="false">)</mo></mrow><mn id="A3.Ex27.m1.5.5.5.5.1.1.3.3">2</mn></msup></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ai"><mrow id="A3.Ex27.m1.6.6.6.6.2.1.2"><mrow id="A3.Ex27.m1.6.6.6.6.2.1.1.1"><mrow id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2"><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.6.6.6.6.2.1.1.1.1.1.1"><mi id="A3.Ex27.m1.6.6.6.6.2.1.1.1.1.1.1.2">r</mi><mn id="A3.Ex27.m1.6.6.6.6.2.1.1.1.1.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.4">,</mo><msub id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.2"><mi id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.2.2">r</mi><mn id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.5">=</mo><mrow id="A3.Ex27.m1.6.6.6.6.2.1.1.1.4.2"><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.6.6.6.6.2.1.1.1.3.1.1"><mi id="A3.Ex27.m1.6.6.6.6.2.1.1.1.3.1.1.2">s</mi><mn id="A3.Ex27.m1.6.6.6.6.2.1.1.1.3.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.6.6.6.6.2.1.1.1.4.2.4">,</mo><msub id="A3.Ex27.m1.6.6.6.6.2.1.1.1.4.2.2"><mi 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id="A3.Ex27.m1.6.6.6.6.2.1.2.2.3.1.1.2">s</mi><mn id="A3.Ex27.m1.6.6.6.6.2.1.2.2.3.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.6.6.6.6.2.1.2.2.4.2.4">,</mo><msub id="A3.Ex27.m1.6.6.6.6.2.1.2.2.4.2.2"><mi id="A3.Ex27.m1.6.6.6.6.2.1.2.2.4.2.2.2">s</mi><mn id="A3.Ex27.m1.6.6.6.6.2.1.2.2.4.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.6.6.6.6.2.1.2.2.4.2.5" stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="A3.Ex27.m1.8.8.8aj"><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8ak"><msup id="A3.Ex27.m1.7.7.7.7.1.1"><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1"><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.2" stretchy="false">(</mo><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1"><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2"><mi id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2.2">p</mi><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2.1">⁢</mo><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2.3.2"><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.7.7.7.7.1.1.1">H</mi><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.2.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.1">−</mo><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3"><mi id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3.2">p</mi><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3.1">⁢</mo><mrow id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3.3.2"><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3.3.2.1" stretchy="false">(</mo><mi id="A3.Ex27.m1.7.7.7.7.1.1.2">L</mi><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A3.Ex27.m1.7.7.7.7.1.1.3.1.3" stretchy="false">)</mo></mrow><mn id="A3.Ex27.m1.7.7.7.7.1.1.5">2</mn></msup></mtd><mtd class="ltx_align_left" columnalign="left" id="A3.Ex27.m1.8.8.8al"><mrow id="A3.Ex27.m1.8.8.8.8.2.1.2"><mrow id="A3.Ex27.m1.8.8.8.8.2.1.1.1"><mrow id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2"><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.1.1.1.1.1"><mi id="A3.Ex27.m1.8.8.8.8.2.1.1.1.1.1.1.2">r</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.1.1.1.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.4">,</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.2"><mi id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.2.2">r</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.5">=</mo><mrow id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2"><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.1.1.3.1.1"><mi id="A3.Ex27.m1.8.8.8.8.2.1.1.1.3.1.1.2">s</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.1.1.3.1.1.3">1</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.4">,</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.2"><mi id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.2.2">s</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.2.3">2</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.1.1.4.2.5" stretchy="false">)</mo></mrow></mrow><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.3">,</mo><mrow id="A3.Ex27.m1.8.8.8.8.2.1.2.2"><mrow id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2"><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.2.2.1.1.1"><mi id="A3.Ex27.m1.8.8.8.8.2.1.2.2.1.1.1.2">r</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.2.2.1.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.4">,</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.2"><mi id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.2.2">r</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.2.2.5" stretchy="false">)</mo></mrow><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.5">=</mo><mrow id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2"><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.3" stretchy="false">(</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.2.2.3.1.1"><mi id="A3.Ex27.m1.8.8.8.8.2.1.2.2.3.1.1.2">s</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.2.2.3.1.1.3">3</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.4">,</mo><msub id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.2"><mi id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.2.2">s</mi><mn id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.2.3">4</mn></msub><mo id="A3.Ex27.m1.8.8.8.8.2.1.2.2.4.2.5" stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex" id="A3.Ex27.m1.12b">\displaystyle\mathop{\mathbb{E}}[M_{\textrm{DMI}}(r_{1..T},R^{\prime}_{1..T})% \mid S_{1..T}]=\begin{cases}(p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3}% ,r_{4})=(s_{3},s_{4})\\ -(p(H)-p(L))^{2}&(r_{1},r_{2})\neq(s_{1},s_{2}),(r_{3},r_{4})=(s_{3},s_{4})\\ -(p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3},r_{4})\neq(s_{3},s_{4})\\ (p(H)-p(L))^{2}&(r_{1},r_{2})=(s_{1},s_{2}),(r_{3},r_{4})=(s_{3},s_{4})\end{cases}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex27.m1.12c">blackboard_E [ italic_M start_POSTSUBSCRIPT DMI end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT , italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ) ∣ italic_S start_POSTSUBSCRIPT 1 . . italic_T end_POSTSUBSCRIPT ] = { start_ROW start_CELL ( italic_p ( italic_H ) - italic_p ( italic_L ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - ( italic_p ( italic_H ) - italic_p ( italic_L ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≠ ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - ( italic_p ( italic_H ) - italic_p ( italic_L ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ≠ ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL ( italic_p ( italic_H ) - italic_p ( italic_L ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) = ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.5.p2.10">In summary, the highest expected payoff in this case is <math alttext="(p(H)-p(L))^{2}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.7.m1.3"><semantics id="A3.SS1.5.p2.7.m1.3a"><msup id="A3.SS1.5.p2.7.m1.3.3" xref="A3.SS1.5.p2.7.m1.3.3.cmml"><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.cmml"><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.2" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.cmml">(</mo><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1.1" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.cmml"><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.cmml"><mi id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.2" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.2.cmml">p</mi><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.1" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.3.2" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.cmml"><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.3.2.1" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.cmml">(</mo><mi id="A3.SS1.5.p2.7.m1.1.1" xref="A3.SS1.5.p2.7.m1.1.1.cmml">H</mi><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.3.2.2" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.1" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.1.cmml">−</mo><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.cmml"><mi id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.2" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.2.cmml">p</mi><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.1" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.3.2" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.cmml"><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.3.2.1" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.cmml">(</mo><mi id="A3.SS1.5.p2.7.m1.2.2" xref="A3.SS1.5.p2.7.m1.2.2.cmml">L</mi><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.3.2.2" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A3.SS1.5.p2.7.m1.3.3.1.1.3" stretchy="false" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.cmml">)</mo></mrow><mn id="A3.SS1.5.p2.7.m1.3.3.3" xref="A3.SS1.5.p2.7.m1.3.3.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.7.m1.3b"><apply id="A3.SS1.5.p2.7.m1.3.3.cmml" xref="A3.SS1.5.p2.7.m1.3.3"><csymbol cd="ambiguous" id="A3.SS1.5.p2.7.m1.3.3.2.cmml" xref="A3.SS1.5.p2.7.m1.3.3">superscript</csymbol><apply id="A3.SS1.5.p2.7.m1.3.3.1.1.1.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1"><minus id="A3.SS1.5.p2.7.m1.3.3.1.1.1.1.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.1"></minus><apply id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2"><times id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.1.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.1"></times><ci id="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.2.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.2.2">𝑝</ci><ci id="A3.SS1.5.p2.7.m1.1.1.cmml" xref="A3.SS1.5.p2.7.m1.1.1">𝐻</ci></apply><apply id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3"><times id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.1.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.1"></times><ci id="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.2.cmml" xref="A3.SS1.5.p2.7.m1.3.3.1.1.1.3.2">𝑝</ci><ci id="A3.SS1.5.p2.7.m1.2.2.cmml" xref="A3.SS1.5.p2.7.m1.2.2">𝐿</ci></apply></apply><cn id="A3.SS1.5.p2.7.m1.3.3.3.cmml" type="integer" xref="A3.SS1.5.p2.7.m1.3.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.7.m1.3c">(p(H)-p(L))^{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.7.m1.3d">( italic_p ( italic_H ) - italic_p ( italic_L ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, which is achieved by <math alttext="\sigma_{\text{true}}" class="ltx_Math" display="inline" id="A3.SS1.5.p2.8.m2.1"><semantics id="A3.SS1.5.p2.8.m2.1a"><msub id="A3.SS1.5.p2.8.m2.1.1" xref="A3.SS1.5.p2.8.m2.1.1.cmml"><mi id="A3.SS1.5.p2.8.m2.1.1.2" xref="A3.SS1.5.p2.8.m2.1.1.2.cmml">σ</mi><mtext id="A3.SS1.5.p2.8.m2.1.1.3" xref="A3.SS1.5.p2.8.m2.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.8.m2.1b"><apply id="A3.SS1.5.p2.8.m2.1.1.cmml" xref="A3.SS1.5.p2.8.m2.1.1"><csymbol cd="ambiguous" id="A3.SS1.5.p2.8.m2.1.1.1.cmml" xref="A3.SS1.5.p2.8.m2.1.1">subscript</csymbol><ci id="A3.SS1.5.p2.8.m2.1.1.2.cmml" xref="A3.SS1.5.p2.8.m2.1.1.2">𝜎</ci><ci id="A3.SS1.5.p2.8.m2.1.1.3a.cmml" xref="A3.SS1.5.p2.8.m2.1.1.3"><mtext id="A3.SS1.5.p2.8.m2.1.1.3.cmml" mathsize="70%" xref="A3.SS1.5.p2.8.m2.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.8.m2.1c">\sigma_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.8.m2.1d">italic_σ start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> and the permutation strategy which swaps <math alttext="H" class="ltx_Math" display="inline" id="A3.SS1.5.p2.9.m3.1"><semantics id="A3.SS1.5.p2.9.m3.1a"><mi id="A3.SS1.5.p2.9.m3.1.1" xref="A3.SS1.5.p2.9.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.9.m3.1b"><ci id="A3.SS1.5.p2.9.m3.1.1.cmml" xref="A3.SS1.5.p2.9.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.9.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.9.m3.1d">italic_H</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="A3.SS1.5.p2.10.m4.1"><semantics id="A3.SS1.5.p2.10.m4.1a"><mi id="A3.SS1.5.p2.10.m4.1.1" xref="A3.SS1.5.p2.10.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A3.SS1.5.p2.10.m4.1b"><ci id="A3.SS1.5.p2.10.m4.1.1.cmml" xref="A3.SS1.5.p2.10.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.5.p2.10.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.5.p2.10.m4.1d">italic_L</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix D </span>Omitted Proofs for RBTS</h2> <section class="ltx_subsection" id="A4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.1 </span>Gaussian Model</h3> <div class="ltx_para" id="A4.SS1.p1"> <p class="ltx_p" id="A4.SS1.p1.2">We first prove the form of the functions <math alttext="\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.p1.1.m1.2"><semantics id="A4.SS1.p1.1.m1.2a"><mrow id="A4.SS1.p1.1.m1.2.3" xref="A4.SS1.p1.1.m1.2.3.cmml"><mover accent="true" id="A4.SS1.p1.1.m1.2.3.2" xref="A4.SS1.p1.1.m1.2.3.2.cmml"><mi id="A4.SS1.p1.1.m1.2.3.2.2" xref="A4.SS1.p1.1.m1.2.3.2.2.cmml">P</mi><mo id="A4.SS1.p1.1.m1.2.3.2.1" xref="A4.SS1.p1.1.m1.2.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.p1.1.m1.2.3.1" xref="A4.SS1.p1.1.m1.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.p1.1.m1.2.3.3.2" xref="A4.SS1.p1.1.m1.2.3.3.1.cmml"><mo id="A4.SS1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="A4.SS1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="A4.SS1.p1.1.m1.1.1" xref="A4.SS1.p1.1.m1.1.1.cmml">τ</mi><mo id="A4.SS1.p1.1.m1.2.3.3.2.2" xref="A4.SS1.p1.1.m1.2.3.3.1.cmml">;</mo><mi id="A4.SS1.p1.1.m1.2.2" xref="A4.SS1.p1.1.m1.2.2.cmml">x</mi><mo id="A4.SS1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="A4.SS1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.p1.1.m1.2b"><apply id="A4.SS1.p1.1.m1.2.3.cmml" xref="A4.SS1.p1.1.m1.2.3"><times id="A4.SS1.p1.1.m1.2.3.1.cmml" xref="A4.SS1.p1.1.m1.2.3.1"></times><apply id="A4.SS1.p1.1.m1.2.3.2.cmml" xref="A4.SS1.p1.1.m1.2.3.2"><ci id="A4.SS1.p1.1.m1.2.3.2.1.cmml" xref="A4.SS1.p1.1.m1.2.3.2.1">¯</ci><ci id="A4.SS1.p1.1.m1.2.3.2.2.cmml" xref="A4.SS1.p1.1.m1.2.3.2.2">𝑃</ci></apply><list id="A4.SS1.p1.1.m1.2.3.3.1.cmml" xref="A4.SS1.p1.1.m1.2.3.3.2"><ci id="A4.SS1.p1.1.m1.1.1.cmml" xref="A4.SS1.p1.1.m1.1.1">𝜏</ci><ci id="A4.SS1.p1.1.m1.2.2.cmml" xref="A4.SS1.p1.1.m1.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p1.1.m1.2c">\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p1.1.m1.2d">over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math> and <math alttext="Q(x)" class="ltx_Math" display="inline" id="A4.SS1.p1.2.m2.1"><semantics id="A4.SS1.p1.2.m2.1a"><mrow id="A4.SS1.p1.2.m2.1.2" xref="A4.SS1.p1.2.m2.1.2.cmml"><mi id="A4.SS1.p1.2.m2.1.2.2" xref="A4.SS1.p1.2.m2.1.2.2.cmml">Q</mi><mo id="A4.SS1.p1.2.m2.1.2.1" xref="A4.SS1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="A4.SS1.p1.2.m2.1.2.3.2" xref="A4.SS1.p1.2.m2.1.2.cmml"><mo id="A4.SS1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="A4.SS1.p1.2.m2.1.2.cmml">(</mo><mi id="A4.SS1.p1.2.m2.1.1" xref="A4.SS1.p1.2.m2.1.1.cmml">x</mi><mo id="A4.SS1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="A4.SS1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.p1.2.m2.1b"><apply id="A4.SS1.p1.2.m2.1.2.cmml" xref="A4.SS1.p1.2.m2.1.2"><times id="A4.SS1.p1.2.m2.1.2.1.cmml" xref="A4.SS1.p1.2.m2.1.2.1"></times><ci id="A4.SS1.p1.2.m2.1.2.2.cmml" xref="A4.SS1.p1.2.m2.1.2.2">𝑄</ci><ci id="A4.SS1.p1.2.m2.1.1.cmml" xref="A4.SS1.p1.2.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p1.2.m2.1c">Q(x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p1.2.m2.1d">italic_Q ( italic_x )</annotation></semantics></math> for the Gaussian model.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition11.1.1.1">Proposition 11</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition11.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition11.p1"> <p class="ltx_p" id="Thmproposition11.p1.1">In the Gaussian model under the RBTS mechanism,</p> <table class="ltx_equation ltx_eqn_table" id="A4.E20"> <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="\bar{P}(\tau;x)=\Phi\left(\frac{\tau-\rho^{2}x}{\sqrt{b^{2}(1+\rho^{2})(1+\rho% )}}\right)," class="ltx_Math" display="block" id="A4.E20.m1.5"><semantics id="A4.E20.m1.5a"><mrow id="A4.E20.m1.5.5.1" 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xref="A4.E21.m1.3.3.3.2.2.2.1.1.3.2">𝜌</ci><cn id="A4.E21.m1.3.3.3.2.2.2.1.1.3.3.cmml" type="integer" xref="A4.E21.m1.3.3.3.2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.E21.m1.5c">Q(x)=\Phi\left(\frac{(1-\rho^{2})\tau}{b\sqrt{(1+\rho)(1+\rho^{2})}}\right).</annotation><annotation encoding="application/x-llamapun" id="A4.E21.m1.5d">italic_Q ( italic_x ) = roman_Φ ( divide start_ARG ( 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_τ end_ARG start_ARG italic_b square-root start_ARG ( 1 + italic_ρ ) ( 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) 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">(21)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A4.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A4.SS1.1.p1"> <p class="ltx_p" id="A4.SS1.1.p1.1">Note that for a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="A4.SS1.1.p1.1.m1.1"><semantics id="A4.SS1.1.p1.1.m1.1a"><mi id="A4.SS1.1.p1.1.m1.1.1" xref="A4.SS1.1.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.1.p1.1.m1.1b"><ci id="A4.SS1.1.p1.1.m1.1.1.cmml" xref="A4.SS1.1.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.1.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.1.p1.1.m1.1d">italic_τ</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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rspace="0.167em">𝔼</mo><mrow id="A4.Ex28.m1.2.2.2"><msup id="A4.Ex28.m1.2.2.2.4"><mi id="A4.Ex28.m1.2.2.2.4.2">X</mi><mo id="A4.Ex28.m1.2.2.2.4.3">′</mo></msup><mo id="A4.Ex28.m1.2.2.2.3">∼</mo><mrow id="A4.Ex28.m1.2.2.2.2"><mi id="A4.Ex28.m1.2.2.2.2.4">N</mi><mo id="A4.Ex28.m1.2.2.2.2.3">⁢</mo><mrow id="A4.Ex28.m1.2.2.2.2.2.2"><mo id="A4.Ex28.m1.2.2.2.2.2.2.3" stretchy="false">(</mo><mrow id="A4.Ex28.m1.1.1.1.1.1.1.1"><mi id="A4.Ex28.m1.1.1.1.1.1.1.1.2">ρ</mi><mo id="A4.Ex28.m1.1.1.1.1.1.1.1.1">⁢</mo><mi id="A4.Ex28.m1.1.1.1.1.1.1.1.3">x</mi></mrow><mo id="A4.Ex28.m1.2.2.2.2.2.2.4">,</mo><mrow id="A4.Ex28.m1.2.2.2.2.2.2.2"><msup id="A4.Ex28.m1.2.2.2.2.2.2.2.3"><mi id="A4.Ex28.m1.2.2.2.2.2.2.2.3.2">b</mi><mn id="A4.Ex28.m1.2.2.2.2.2.2.2.3.3">2</mn></msup><mo id="A4.Ex28.m1.2.2.2.2.2.2.2.2">⁢</mo><mrow id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1"><mo id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.2" stretchy="false">(</mo><mrow id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.1"><mn id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.1.2">1</mn><mo id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.1.1">+</mo><mi id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.1.3">ρ</mi></mrow><mo id="A4.Ex28.m1.2.2.2.2.2.2.2.1.1.3" stretchy="false">)</mo></mrow></mrow><mo id="A4.Ex28.m1.2.2.2.2.2.2.5" stretchy="false">)</mo></mrow></mrow></mrow></munder><mrow id="A4.Ex28.m1.7.7.1.1.6.2"><mi id="A4.Ex28.m1.7.7.1.1.6.2.2" mathvariant="normal">Φ</mi><mo id="A4.Ex28.m1.7.7.1.1.6.2.1">⁢</mo><mrow id="A4.Ex28.m1.7.7.1.1.6.2.3.2"><mo id="A4.Ex28.m1.7.7.1.1.6.2.3.2.1">(</mo><mfrac id="A4.Ex28.m1.6.6"><mrow id="A4.Ex28.m1.6.6.2"><mi id="A4.Ex28.m1.6.6.2.2">τ</mi><mo id="A4.Ex28.m1.6.6.2.1">−</mo><mrow id="A4.Ex28.m1.6.6.2.3"><mi id="A4.Ex28.m1.6.6.2.3.2">ρ</mi><mo id="A4.Ex28.m1.6.6.2.3.1">⁢</mo><msup id="A4.Ex28.m1.6.6.2.3.3"><mi id="A4.Ex28.m1.6.6.2.3.3.2">x</mi><mo id="A4.Ex28.m1.6.6.2.3.3.3">′</mo></msup></mrow></mrow><mrow id="A4.Ex28.m1.6.6.3"><mi id="A4.Ex28.m1.6.6.3.2">b</mi><mo id="A4.Ex28.m1.6.6.3.1">⁢</mo><msqrt id="A4.Ex28.m1.6.6.3.3"><mrow id="A4.Ex28.m1.6.6.3.3.2"><mn id="A4.Ex28.m1.6.6.3.3.2.2">1</mn><mo id="A4.Ex28.m1.6.6.3.3.2.1">+</mo><mi id="A4.Ex28.m1.6.6.3.3.2.3">ρ</mi></mrow></msqrt></mrow></mfrac><mo id="A4.Ex28.m1.7.7.1.1.6.2.3.2.2">)</mo></mrow></mrow></mrow></mrow><mo id="A4.Ex28.m1.7.7.1.2">,</mo></mrow><annotation encoding="application/x-tex" id="A4.Ex28.m1.7b">\bar{P}(\tau;x)=\mathop{\mathbb{E}}_{x^{\prime}\sim\Pr[\cdot\mid x]}\Pr[X^{% \prime\prime}\leq\tau\mid X^{\prime}=x^{\prime}]=\mathop{\mathbb{E}}_{X^{% \prime}\sim N(\rho x,b^{2}(1+\rho))}\Phi\left(\frac{\tau-\rho x^{\prime}}{b% \sqrt{1+\rho}}\right),</annotation><annotation encoding="application/x-llamapun" id="A4.Ex28.m1.7c">over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x ) = blackboard_E start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ roman_Pr [ ⋅ ∣ italic_x ] end_POSTSUBSCRIPT roman_Pr [ italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≤ italic_τ ∣ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] = blackboard_E start_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∼ italic_N ( italic_ρ italic_x , italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ρ ) ) end_POSTSUBSCRIPT roman_Φ ( divide start_ARG italic_τ - italic_ρ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ 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="A4.SS1.1.p1.3">and since <math alttext="X^{\prime}=\rho x+b\sqrt{1+\rho}X" class="ltx_Math" display="inline" id="A4.SS1.1.p1.2.m1.1"><semantics id="A4.SS1.1.p1.2.m1.1a"><mrow id="A4.SS1.1.p1.2.m1.1.1" xref="A4.SS1.1.p1.2.m1.1.1.cmml"><msup id="A4.SS1.1.p1.2.m1.1.1.2" xref="A4.SS1.1.p1.2.m1.1.1.2.cmml"><mi id="A4.SS1.1.p1.2.m1.1.1.2.2" xref="A4.SS1.1.p1.2.m1.1.1.2.2.cmml">X</mi><mo id="A4.SS1.1.p1.2.m1.1.1.2.3" xref="A4.SS1.1.p1.2.m1.1.1.2.3.cmml">′</mo></msup><mo id="A4.SS1.1.p1.2.m1.1.1.1" xref="A4.SS1.1.p1.2.m1.1.1.1.cmml">=</mo><mrow id="A4.SS1.1.p1.2.m1.1.1.3" xref="A4.SS1.1.p1.2.m1.1.1.3.cmml"><mrow id="A4.SS1.1.p1.2.m1.1.1.3.2" xref="A4.SS1.1.p1.2.m1.1.1.3.2.cmml"><mi id="A4.SS1.1.p1.2.m1.1.1.3.2.2" xref="A4.SS1.1.p1.2.m1.1.1.3.2.2.cmml">ρ</mi><mo id="A4.SS1.1.p1.2.m1.1.1.3.2.1" xref="A4.SS1.1.p1.2.m1.1.1.3.2.1.cmml">⁢</mo><mi id="A4.SS1.1.p1.2.m1.1.1.3.2.3" xref="A4.SS1.1.p1.2.m1.1.1.3.2.3.cmml">x</mi></mrow><mo id="A4.SS1.1.p1.2.m1.1.1.3.1" xref="A4.SS1.1.p1.2.m1.1.1.3.1.cmml">+</mo><mrow id="A4.SS1.1.p1.2.m1.1.1.3.3" xref="A4.SS1.1.p1.2.m1.1.1.3.3.cmml"><mi id="A4.SS1.1.p1.2.m1.1.1.3.3.2" xref="A4.SS1.1.p1.2.m1.1.1.3.3.2.cmml">b</mi><mo id="A4.SS1.1.p1.2.m1.1.1.3.3.1" xref="A4.SS1.1.p1.2.m1.1.1.3.3.1.cmml">⁢</mo><msqrt id="A4.SS1.1.p1.2.m1.1.1.3.3.3" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.cmml"><mrow id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.cmml"><mn id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.2" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.2.cmml">1</mn><mo id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.1" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.1.cmml">+</mo><mi id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.3" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.3.cmml">ρ</mi></mrow></msqrt><mo id="A4.SS1.1.p1.2.m1.1.1.3.3.1a" xref="A4.SS1.1.p1.2.m1.1.1.3.3.1.cmml">⁢</mo><mi id="A4.SS1.1.p1.2.m1.1.1.3.3.4" xref="A4.SS1.1.p1.2.m1.1.1.3.3.4.cmml">X</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.1.p1.2.m1.1b"><apply id="A4.SS1.1.p1.2.m1.1.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1"><eq id="A4.SS1.1.p1.2.m1.1.1.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.1"></eq><apply id="A4.SS1.1.p1.2.m1.1.1.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.1.p1.2.m1.1.1.2.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.2">superscript</csymbol><ci id="A4.SS1.1.p1.2.m1.1.1.2.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.2.2">𝑋</ci><ci id="A4.SS1.1.p1.2.m1.1.1.2.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.2.3">′</ci></apply><apply id="A4.SS1.1.p1.2.m1.1.1.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3"><plus id="A4.SS1.1.p1.2.m1.1.1.3.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.1"></plus><apply id="A4.SS1.1.p1.2.m1.1.1.3.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.2"><times id="A4.SS1.1.p1.2.m1.1.1.3.2.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.2.1"></times><ci id="A4.SS1.1.p1.2.m1.1.1.3.2.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.2.2">𝜌</ci><ci id="A4.SS1.1.p1.2.m1.1.1.3.2.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.2.3">𝑥</ci></apply><apply id="A4.SS1.1.p1.2.m1.1.1.3.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3"><times id="A4.SS1.1.p1.2.m1.1.1.3.3.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.1"></times><ci id="A4.SS1.1.p1.2.m1.1.1.3.3.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.2">𝑏</ci><apply id="A4.SS1.1.p1.2.m1.1.1.3.3.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3"><root id="A4.SS1.1.p1.2.m1.1.1.3.3.3a.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3"></root><apply id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2"><plus id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.1.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.1"></plus><cn id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.2.cmml" type="integer" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.2">1</cn><ci id="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.3.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.3.2.3">𝜌</ci></apply></apply><ci id="A4.SS1.1.p1.2.m1.1.1.3.3.4.cmml" xref="A4.SS1.1.p1.2.m1.1.1.3.3.4">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.1.p1.2.m1.1c">X^{\prime}=\rho x+b\sqrt{1+\rho}X</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.1.p1.2.m1.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_ρ italic_x + italic_b square-root start_ARG 1 + italic_ρ end_ARG italic_X</annotation></semantics></math> for <math alttext="X\sim N(0,1)," class="ltx_Math" display="inline" id="A4.SS1.1.p1.3.m2.3"><semantics id="A4.SS1.1.p1.3.m2.3a"><mrow id="A4.SS1.1.p1.3.m2.3.3.1" xref="A4.SS1.1.p1.3.m2.3.3.1.1.cmml"><mrow id="A4.SS1.1.p1.3.m2.3.3.1.1" xref="A4.SS1.1.p1.3.m2.3.3.1.1.cmml"><mi id="A4.SS1.1.p1.3.m2.3.3.1.1.2" xref="A4.SS1.1.p1.3.m2.3.3.1.1.2.cmml">X</mi><mo id="A4.SS1.1.p1.3.m2.3.3.1.1.1" xref="A4.SS1.1.p1.3.m2.3.3.1.1.1.cmml">∼</mo><mrow id="A4.SS1.1.p1.3.m2.3.3.1.1.3" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.cmml"><mi id="A4.SS1.1.p1.3.m2.3.3.1.1.3.2" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.2.cmml">N</mi><mo id="A4.SS1.1.p1.3.m2.3.3.1.1.3.1" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.1.cmml">⁢</mo><mrow id="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.2" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.1.cmml"><mo id="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.2.1" stretchy="false" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.1.cmml">(</mo><mn id="A4.SS1.1.p1.3.m2.1.1" xref="A4.SS1.1.p1.3.m2.1.1.cmml">0</mn><mo id="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.2.2" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.1.cmml">,</mo><mn id="A4.SS1.1.p1.3.m2.2.2" xref="A4.SS1.1.p1.3.m2.2.2.cmml">1</mn><mo id="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.2.3" stretchy="false" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.SS1.1.p1.3.m2.3.3.1.2" xref="A4.SS1.1.p1.3.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.1.p1.3.m2.3b"><apply id="A4.SS1.1.p1.3.m2.3.3.1.1.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1"><csymbol cd="latexml" id="A4.SS1.1.p1.3.m2.3.3.1.1.1.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.1">similar-to</csymbol><ci id="A4.SS1.1.p1.3.m2.3.3.1.1.2.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.2">𝑋</ci><apply id="A4.SS1.1.p1.3.m2.3.3.1.1.3.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3"><times id="A4.SS1.1.p1.3.m2.3.3.1.1.3.1.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.1"></times><ci id="A4.SS1.1.p1.3.m2.3.3.1.1.3.2.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.2">𝑁</ci><interval closure="open" id="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.1.cmml" xref="A4.SS1.1.p1.3.m2.3.3.1.1.3.3.2"><cn id="A4.SS1.1.p1.3.m2.1.1.cmml" type="integer" xref="A4.SS1.1.p1.3.m2.1.1">0</cn><cn id="A4.SS1.1.p1.3.m2.2.2.cmml" type="integer" xref="A4.SS1.1.p1.3.m2.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.1.p1.3.m2.3c">X\sim N(0,1),</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.1.p1.3.m2.3d">italic_X ∼ italic_N ( 0 , 1 ) ,</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx16"> <tbody id="A4.Ex29"><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\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.Ex29.m1.2"><semantics id="A4.Ex29.m1.2a"><mrow id="A4.Ex29.m1.2.3" xref="A4.Ex29.m1.2.3.cmml"><mover accent="true" id="A4.Ex29.m1.2.3.2" xref="A4.Ex29.m1.2.3.2.cmml"><mi id="A4.Ex29.m1.2.3.2.2" xref="A4.Ex29.m1.2.3.2.2.cmml">P</mi><mo id="A4.Ex29.m1.2.3.2.1" xref="A4.Ex29.m1.2.3.2.1.cmml">¯</mo></mover><mo id="A4.Ex29.m1.2.3.1" xref="A4.Ex29.m1.2.3.1.cmml">⁢</mo><mrow id="A4.Ex29.m1.2.3.3.2" xref="A4.Ex29.m1.2.3.3.1.cmml"><mo id="A4.Ex29.m1.2.3.3.2.1" stretchy="false" xref="A4.Ex29.m1.2.3.3.1.cmml">(</mo><mi id="A4.Ex29.m1.1.1" xref="A4.Ex29.m1.1.1.cmml">τ</mi><mo id="A4.Ex29.m1.2.3.3.2.2" xref="A4.Ex29.m1.2.3.3.1.cmml">;</mo><mi id="A4.Ex29.m1.2.2" xref="A4.Ex29.m1.2.2.cmml">x</mi><mo id="A4.Ex29.m1.2.3.3.2.3" stretchy="false" xref="A4.Ex29.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex29.m1.2b"><apply id="A4.Ex29.m1.2.3.cmml" xref="A4.Ex29.m1.2.3"><times id="A4.Ex29.m1.2.3.1.cmml" xref="A4.Ex29.m1.2.3.1"></times><apply id="A4.Ex29.m1.2.3.2.cmml" xref="A4.Ex29.m1.2.3.2"><ci id="A4.Ex29.m1.2.3.2.1.cmml" xref="A4.Ex29.m1.2.3.2.1">¯</ci><ci id="A4.Ex29.m1.2.3.2.2.cmml" xref="A4.Ex29.m1.2.3.2.2">𝑃</ci></apply><list id="A4.Ex29.m1.2.3.3.1.cmml" xref="A4.Ex29.m1.2.3.3.2"><ci id="A4.Ex29.m1.1.1.cmml" 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encoding="application/x-tex" id="A4.Ex29.m2.1c">\displaystyle=\mathop{\mathbb{E}}_{X}\Phi\left(\frac{\tau-\rho(\rho x+b\sqrt{1% +\rho}X)}{b\sqrt{1+\rho}}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex29.m2.1d">= blackboard_E start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT roman_Φ ( divide start_ARG italic_τ - italic_ρ ( italic_ρ italic_x + italic_b square-root start_ARG 1 + italic_ρ end_ARG italic_X ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Pr\left[Y\leq\frac{\tau-\rho(\rho x+b\sqrt{1+\rho}X)}{b\sqrt{1+% \rho}}\right]" class="ltx_Math" display="inline" 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xref="A4.Ex30.m1.1.1.3.cmml"><mi id="A4.Ex30.m1.1.1.3.2" xref="A4.Ex30.m1.1.1.3.2.cmml">b</mi><mo id="A4.Ex30.m1.1.1.3.1" xref="A4.Ex30.m1.1.1.3.1.cmml">⁢</mo><msqrt id="A4.Ex30.m1.1.1.3.3" xref="A4.Ex30.m1.1.1.3.3.cmml"><mrow id="A4.Ex30.m1.1.1.3.3.2" xref="A4.Ex30.m1.1.1.3.3.2.cmml"><mn id="A4.Ex30.m1.1.1.3.3.2.2" xref="A4.Ex30.m1.1.1.3.3.2.2.cmml">1</mn><mo id="A4.Ex30.m1.1.1.3.3.2.1" xref="A4.Ex30.m1.1.1.3.3.2.1.cmml">+</mo><mi id="A4.Ex30.m1.1.1.3.3.2.3" xref="A4.Ex30.m1.1.1.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mstyle></mrow><mo id="A4.Ex30.m1.3.3.1.1.1.3" xref="A4.Ex30.m1.3.3.1.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex30.m1.3b"><apply id="A4.Ex30.m1.3.3.cmml" xref="A4.Ex30.m1.3.3"><eq id="A4.Ex30.m1.3.3.2.cmml" xref="A4.Ex30.m1.3.3.2"></eq><csymbol cd="latexml" id="A4.Ex30.m1.3.3.3.cmml" xref="A4.Ex30.m1.3.3.3">absent</csymbol><apply id="A4.Ex30.m1.3.3.1.2.cmml" xref="A4.Ex30.m1.3.3.1.1"><ci id="A4.Ex30.m1.2.2.cmml" 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encoding="application/x-tex" id="A4.Ex30.m1.3c">\displaystyle=\Pr\left[Y\leq\frac{\tau-\rho(\rho x+b\sqrt{1+\rho}X)}{b\sqrt{1+% \rho}}\right]</annotation><annotation encoding="application/x-llamapun" id="A4.Ex30.m1.3d">= roman_Pr [ italic_Y ≤ divide start_ARG italic_τ - italic_ρ ( italic_ρ italic_x + italic_b square-root start_ARG 1 + italic_ρ end_ARG italic_X ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Pr\left[Y+\rho X\leq\frac{\tau-\rho^{2}x}{b\sqrt{1+\rho}}\right]," class="ltx_Math" display="inline" id="A4.Ex31.m1.2"><semantics id="A4.Ex31.m1.2a"><mrow id="A4.Ex31.m1.2.2.1" xref="A4.Ex31.m1.2.2.1.1.cmml"><mrow id="A4.Ex31.m1.2.2.1.1" xref="A4.Ex31.m1.2.2.1.1.cmml"><mi id="A4.Ex31.m1.2.2.1.1.3" xref="A4.Ex31.m1.2.2.1.1.3.cmml"></mi><mo id="A4.Ex31.m1.2.2.1.1.2" xref="A4.Ex31.m1.2.2.1.1.2.cmml">=</mo><mrow id="A4.Ex31.m1.2.2.1.1.1.1" xref="A4.Ex31.m1.2.2.1.1.1.2.cmml"><mi id="A4.Ex31.m1.1.1" xref="A4.Ex31.m1.1.1.cmml">Pr</mi><mo id="A4.Ex31.m1.2.2.1.1.1.1a" xref="A4.Ex31.m1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1" xref="A4.Ex31.m1.2.2.1.1.1.2.cmml"><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.2" xref="A4.Ex31.m1.2.2.1.1.1.2.cmml">[</mo><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.cmml"><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.2.cmml">Y</mi><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.1.cmml">+</mo><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.cmml"><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.2.cmml">ρ</mi><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.2.3.3.cmml">X</mi></mrow></mrow><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.1.cmml">≤</mo><mstyle displaystyle="true" id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.cmml"><mfrac id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3a" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.cmml"><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.cmml"><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.2.cmml">τ</mi><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.1.cmml">−</mo><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.cmml"><msup id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.cmml"><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.2.cmml">ρ</mi><mn id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.3.cmml">2</mn></msup><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.3.cmml">x</mi></mrow></mrow><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.2.cmml">b</mi><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.1.cmml">⁢</mo><msqrt id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.cmml"><mrow id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.cmml"><mn id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.2" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.2.cmml">1</mn><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.1" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.1.cmml">+</mo><mi id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.3" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mstyle></mrow><mo id="A4.Ex31.m1.2.2.1.1.1.1.1.3" xref="A4.Ex31.m1.2.2.1.1.1.2.cmml">]</mo></mrow></mrow></mrow><mo id="A4.Ex31.m1.2.2.1.2" xref="A4.Ex31.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex31.m1.2b"><apply id="A4.Ex31.m1.2.2.1.1.cmml" xref="A4.Ex31.m1.2.2.1"><eq id="A4.Ex31.m1.2.2.1.1.2.cmml" xref="A4.Ex31.m1.2.2.1.1.2"></eq><csymbol cd="latexml" id="A4.Ex31.m1.2.2.1.1.3.cmml" xref="A4.Ex31.m1.2.2.1.1.3">absent</csymbol><apply id="A4.Ex31.m1.2.2.1.1.1.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1"><ci id="A4.Ex31.m1.1.1.cmml" xref="A4.Ex31.m1.1.1">Pr</ci><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.cmml" 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xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2"><minus id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.1.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.1"></minus><ci id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.2">𝜏</ci><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3"><times id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.1.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.1"></times><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.1.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2">superscript</csymbol><ci id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.2">𝜌</ci><cn id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.3.cmml" type="integer" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.2.3">2</cn></apply><ci id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.3.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.2.3.3">𝑥</ci></apply></apply><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3"><times id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.1"></times><ci id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.2">𝑏</ci><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3"><root id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3a.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3"></root><apply id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2"><plus id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.1.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.1"></plus><cn id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.2.cmml" type="integer" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.2">1</cn><ci id="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.3.cmml" xref="A4.Ex31.m1.2.2.1.1.1.1.1.1.3.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex31.m1.2c">\displaystyle=\Pr\left[Y+\rho X\leq\frac{\tau-\rho^{2}x}{b\sqrt{1+\rho}}\right],</annotation><annotation encoding="application/x-llamapun" id="A4.Ex31.m1.2d">= roman_Pr [ italic_Y + italic_ρ italic_X ≤ divide start_ARG italic_τ - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.2.p2"> <p class="ltx_p" id="A4.SS1.2.p2.2">where <math alttext="Y\sim N(0,1)" class="ltx_Math" display="inline" id="A4.SS1.2.p2.1.m1.2"><semantics id="A4.SS1.2.p2.1.m1.2a"><mrow id="A4.SS1.2.p2.1.m1.2.3" xref="A4.SS1.2.p2.1.m1.2.3.cmml"><mi id="A4.SS1.2.p2.1.m1.2.3.2" xref="A4.SS1.2.p2.1.m1.2.3.2.cmml">Y</mi><mo id="A4.SS1.2.p2.1.m1.2.3.1" xref="A4.SS1.2.p2.1.m1.2.3.1.cmml">∼</mo><mrow id="A4.SS1.2.p2.1.m1.2.3.3" xref="A4.SS1.2.p2.1.m1.2.3.3.cmml"><mi id="A4.SS1.2.p2.1.m1.2.3.3.2" xref="A4.SS1.2.p2.1.m1.2.3.3.2.cmml">N</mi><mo id="A4.SS1.2.p2.1.m1.2.3.3.1" xref="A4.SS1.2.p2.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.2.p2.1.m1.2.3.3.3.2" xref="A4.SS1.2.p2.1.m1.2.3.3.3.1.cmml"><mo id="A4.SS1.2.p2.1.m1.2.3.3.3.2.1" stretchy="false" xref="A4.SS1.2.p2.1.m1.2.3.3.3.1.cmml">(</mo><mn id="A4.SS1.2.p2.1.m1.1.1" xref="A4.SS1.2.p2.1.m1.1.1.cmml">0</mn><mo id="A4.SS1.2.p2.1.m1.2.3.3.3.2.2" xref="A4.SS1.2.p2.1.m1.2.3.3.3.1.cmml">,</mo><mn id="A4.SS1.2.p2.1.m1.2.2" xref="A4.SS1.2.p2.1.m1.2.2.cmml">1</mn><mo id="A4.SS1.2.p2.1.m1.2.3.3.3.2.3" stretchy="false" xref="A4.SS1.2.p2.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.1.m1.2b"><apply id="A4.SS1.2.p2.1.m1.2.3.cmml" xref="A4.SS1.2.p2.1.m1.2.3"><csymbol cd="latexml" id="A4.SS1.2.p2.1.m1.2.3.1.cmml" xref="A4.SS1.2.p2.1.m1.2.3.1">similar-to</csymbol><ci id="A4.SS1.2.p2.1.m1.2.3.2.cmml" xref="A4.SS1.2.p2.1.m1.2.3.2">𝑌</ci><apply id="A4.SS1.2.p2.1.m1.2.3.3.cmml" xref="A4.SS1.2.p2.1.m1.2.3.3"><times id="A4.SS1.2.p2.1.m1.2.3.3.1.cmml" xref="A4.SS1.2.p2.1.m1.2.3.3.1"></times><ci id="A4.SS1.2.p2.1.m1.2.3.3.2.cmml" xref="A4.SS1.2.p2.1.m1.2.3.3.2">𝑁</ci><interval closure="open" id="A4.SS1.2.p2.1.m1.2.3.3.3.1.cmml" xref="A4.SS1.2.p2.1.m1.2.3.3.3.2"><cn id="A4.SS1.2.p2.1.m1.1.1.cmml" type="integer" xref="A4.SS1.2.p2.1.m1.1.1">0</cn><cn id="A4.SS1.2.p2.1.m1.2.2.cmml" type="integer" xref="A4.SS1.2.p2.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.1.m1.2c">Y\sim N(0,1)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.1.m1.2d">italic_Y ∼ italic_N ( 0 , 1 )</annotation></semantics></math>. It follows that <math alttext="Y+\rho X\sim N(0,1+\rho^{2})" class="ltx_Math" display="inline" id="A4.SS1.2.p2.2.m2.2"><semantics id="A4.SS1.2.p2.2.m2.2a"><mrow id="A4.SS1.2.p2.2.m2.2.2" xref="A4.SS1.2.p2.2.m2.2.2.cmml"><mrow id="A4.SS1.2.p2.2.m2.2.2.3" xref="A4.SS1.2.p2.2.m2.2.2.3.cmml"><mi id="A4.SS1.2.p2.2.m2.2.2.3.2" xref="A4.SS1.2.p2.2.m2.2.2.3.2.cmml">Y</mi><mo id="A4.SS1.2.p2.2.m2.2.2.3.1" xref="A4.SS1.2.p2.2.m2.2.2.3.1.cmml">+</mo><mrow id="A4.SS1.2.p2.2.m2.2.2.3.3" xref="A4.SS1.2.p2.2.m2.2.2.3.3.cmml"><mi id="A4.SS1.2.p2.2.m2.2.2.3.3.2" xref="A4.SS1.2.p2.2.m2.2.2.3.3.2.cmml">ρ</mi><mo id="A4.SS1.2.p2.2.m2.2.2.3.3.1" xref="A4.SS1.2.p2.2.m2.2.2.3.3.1.cmml">⁢</mo><mi id="A4.SS1.2.p2.2.m2.2.2.3.3.3" xref="A4.SS1.2.p2.2.m2.2.2.3.3.3.cmml">X</mi></mrow></mrow><mo id="A4.SS1.2.p2.2.m2.2.2.2" xref="A4.SS1.2.p2.2.m2.2.2.2.cmml">∼</mo><mrow id="A4.SS1.2.p2.2.m2.2.2.1" xref="A4.SS1.2.p2.2.m2.2.2.1.cmml"><mi id="A4.SS1.2.p2.2.m2.2.2.1.3" xref="A4.SS1.2.p2.2.m2.2.2.1.3.cmml">N</mi><mo id="A4.SS1.2.p2.2.m2.2.2.1.2" xref="A4.SS1.2.p2.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="A4.SS1.2.p2.2.m2.2.2.1.1.1" xref="A4.SS1.2.p2.2.m2.2.2.1.1.2.cmml"><mo id="A4.SS1.2.p2.2.m2.2.2.1.1.1.2" stretchy="false" xref="A4.SS1.2.p2.2.m2.2.2.1.1.2.cmml">(</mo><mn id="A4.SS1.2.p2.2.m2.1.1" xref="A4.SS1.2.p2.2.m2.1.1.cmml">0</mn><mo id="A4.SS1.2.p2.2.m2.2.2.1.1.1.3" xref="A4.SS1.2.p2.2.m2.2.2.1.1.2.cmml">,</mo><mrow id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.cmml"><mn id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.2" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.2.cmml">1</mn><mo id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.1" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.1.cmml">+</mo><msup id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.cmml"><mi id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.2" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.2.cmml">ρ</mi><mn id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.3" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="A4.SS1.2.p2.2.m2.2.2.1.1.1.4" stretchy="false" xref="A4.SS1.2.p2.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.2.m2.2b"><apply id="A4.SS1.2.p2.2.m2.2.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2"><csymbol cd="latexml" id="A4.SS1.2.p2.2.m2.2.2.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.2">similar-to</csymbol><apply id="A4.SS1.2.p2.2.m2.2.2.3.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3"><plus id="A4.SS1.2.p2.2.m2.2.2.3.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.1"></plus><ci id="A4.SS1.2.p2.2.m2.2.2.3.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.2">𝑌</ci><apply id="A4.SS1.2.p2.2.m2.2.2.3.3.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.3"><times id="A4.SS1.2.p2.2.m2.2.2.3.3.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.3.1"></times><ci id="A4.SS1.2.p2.2.m2.2.2.3.3.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.3.2">𝜌</ci><ci id="A4.SS1.2.p2.2.m2.2.2.3.3.3.cmml" xref="A4.SS1.2.p2.2.m2.2.2.3.3.3">𝑋</ci></apply></apply><apply id="A4.SS1.2.p2.2.m2.2.2.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1"><times id="A4.SS1.2.p2.2.m2.2.2.1.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.2"></times><ci id="A4.SS1.2.p2.2.m2.2.2.1.3.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.3">𝑁</ci><interval closure="open" id="A4.SS1.2.p2.2.m2.2.2.1.1.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1"><cn id="A4.SS1.2.p2.2.m2.1.1.cmml" type="integer" xref="A4.SS1.2.p2.2.m2.1.1">0</cn><apply id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1"><plus id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.1"></plus><cn id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.2">1</cn><apply id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.1.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3">superscript</csymbol><ci id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.2.cmml" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.2">𝜌</ci><cn id="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="A4.SS1.2.p2.2.m2.2.2.1.1.1.1.3.3">2</cn></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.2.m2.2c">Y+\rho X\sim N(0,1+\rho^{2})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.2.m2.2d">italic_Y + italic_ρ italic_X ∼ italic_N ( 0 , 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, so</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex32"> <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="\bar{P}(\tau;x)=\Phi\left(\frac{\tau-\rho^{2}x}{b\sqrt{(1+\rho^{2})(1+\rho)}}% \right)." class="ltx_Math" display="block" id="A4.Ex32.m1.5"><semantics id="A4.Ex32.m1.5a"><mrow id="A4.Ex32.m1.5.5.1" xref="A4.Ex32.m1.5.5.1.1.cmml"><mrow id="A4.Ex32.m1.5.5.1.1" xref="A4.Ex32.m1.5.5.1.1.cmml"><mrow id="A4.Ex32.m1.5.5.1.1.2" xref="A4.Ex32.m1.5.5.1.1.2.cmml"><mover accent="true" id="A4.Ex32.m1.5.5.1.1.2.2" xref="A4.Ex32.m1.5.5.1.1.2.2.cmml"><mi id="A4.Ex32.m1.5.5.1.1.2.2.2" xref="A4.Ex32.m1.5.5.1.1.2.2.2.cmml">P</mi><mo id="A4.Ex32.m1.5.5.1.1.2.2.1" xref="A4.Ex32.m1.5.5.1.1.2.2.1.cmml">¯</mo></mover><mo id="A4.Ex32.m1.5.5.1.1.2.1" xref="A4.Ex32.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex32.m1.5.5.1.1.2.3.2" xref="A4.Ex32.m1.5.5.1.1.2.3.1.cmml"><mo id="A4.Ex32.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="A4.Ex32.m1.5.5.1.1.2.3.1.cmml">(</mo><mi id="A4.Ex32.m1.3.3" xref="A4.Ex32.m1.3.3.cmml">τ</mi><mo id="A4.Ex32.m1.5.5.1.1.2.3.2.2" xref="A4.Ex32.m1.5.5.1.1.2.3.1.cmml">;</mo><mi id="A4.Ex32.m1.4.4" xref="A4.Ex32.m1.4.4.cmml">x</mi><mo id="A4.Ex32.m1.5.5.1.1.2.3.2.3" stretchy="false" xref="A4.Ex32.m1.5.5.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex32.m1.5.5.1.1.1" xref="A4.Ex32.m1.5.5.1.1.1.cmml">=</mo><mrow id="A4.Ex32.m1.5.5.1.1.3" xref="A4.Ex32.m1.5.5.1.1.3.cmml"><mi id="A4.Ex32.m1.5.5.1.1.3.2" mathvariant="normal" xref="A4.Ex32.m1.5.5.1.1.3.2.cmml">Φ</mi><mo id="A4.Ex32.m1.5.5.1.1.3.1" xref="A4.Ex32.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex32.m1.5.5.1.1.3.3.2" xref="A4.Ex32.m1.2.2.cmml"><mo id="A4.Ex32.m1.5.5.1.1.3.3.2.1" xref="A4.Ex32.m1.2.2.cmml">(</mo><mfrac id="A4.Ex32.m1.2.2" xref="A4.Ex32.m1.2.2.cmml"><mrow id="A4.Ex32.m1.2.2.4" xref="A4.Ex32.m1.2.2.4.cmml"><mi id="A4.Ex32.m1.2.2.4.2" xref="A4.Ex32.m1.2.2.4.2.cmml">τ</mi><mo id="A4.Ex32.m1.2.2.4.1" xref="A4.Ex32.m1.2.2.4.1.cmml">−</mo><mrow id="A4.Ex32.m1.2.2.4.3" xref="A4.Ex32.m1.2.2.4.3.cmml"><msup id="A4.Ex32.m1.2.2.4.3.2" xref="A4.Ex32.m1.2.2.4.3.2.cmml"><mi id="A4.Ex32.m1.2.2.4.3.2.2" xref="A4.Ex32.m1.2.2.4.3.2.2.cmml">ρ</mi><mn id="A4.Ex32.m1.2.2.4.3.2.3" xref="A4.Ex32.m1.2.2.4.3.2.3.cmml">2</mn></msup><mo id="A4.Ex32.m1.2.2.4.3.1" xref="A4.Ex32.m1.2.2.4.3.1.cmml">⁢</mo><mi id="A4.Ex32.m1.2.2.4.3.3" xref="A4.Ex32.m1.2.2.4.3.3.cmml">x</mi></mrow></mrow><mrow id="A4.Ex32.m1.2.2.2" xref="A4.Ex32.m1.2.2.2.cmml"><mi id="A4.Ex32.m1.2.2.2.4" xref="A4.Ex32.m1.2.2.2.4.cmml">b</mi><mo id="A4.Ex32.m1.2.2.2.3" xref="A4.Ex32.m1.2.2.2.3.cmml">⁢</mo><msqrt id="A4.Ex32.m1.2.2.2.2" xref="A4.Ex32.m1.2.2.2.2.cmml"><mrow id="A4.Ex32.m1.2.2.2.2.2" xref="A4.Ex32.m1.2.2.2.2.2.cmml"><mrow id="A4.Ex32.m1.1.1.1.1.1.1.1" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex32.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex32.m1.1.1.1.1.1.1.1.1" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex32.m1.1.1.1.1.1.1.1.1.2" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex32.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><msup id="A4.Ex32.m1.1.1.1.1.1.1.1.1.3" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex32.m1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.3.2.cmml">ρ</mi><mn id="A4.Ex32.m1.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="A4.Ex32.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex32.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex32.m1.2.2.2.2.2.3" xref="A4.Ex32.m1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex32.m1.2.2.2.2.2.2.1" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.cmml"><mo id="A4.Ex32.m1.2.2.2.2.2.2.1.2" stretchy="false" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A4.Ex32.m1.2.2.2.2.2.2.1.1" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.cmml"><mn id="A4.Ex32.m1.2.2.2.2.2.2.1.1.2" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.2.cmml">1</mn><mo id="A4.Ex32.m1.2.2.2.2.2.2.1.1.1" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.1.cmml">+</mo><mi id="A4.Ex32.m1.2.2.2.2.2.2.1.1.3" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.3.cmml">ρ</mi></mrow><mo id="A4.Ex32.m1.2.2.2.2.2.2.1.3" stretchy="false" xref="A4.Ex32.m1.2.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msqrt></mrow></mfrac><mo id="A4.Ex32.m1.5.5.1.1.3.3.2.2" xref="A4.Ex32.m1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex32.m1.5.5.1.2" lspace="0em" xref="A4.Ex32.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" 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end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.2.p2.3">The definition of <math alttext="Q(x)=\bar{P}(x;x)" class="ltx_Math" display="inline" id="A4.SS1.2.p2.3.m1.3"><semantics id="A4.SS1.2.p2.3.m1.3a"><mrow id="A4.SS1.2.p2.3.m1.3.4" xref="A4.SS1.2.p2.3.m1.3.4.cmml"><mrow id="A4.SS1.2.p2.3.m1.3.4.2" xref="A4.SS1.2.p2.3.m1.3.4.2.cmml"><mi id="A4.SS1.2.p2.3.m1.3.4.2.2" xref="A4.SS1.2.p2.3.m1.3.4.2.2.cmml">Q</mi><mo id="A4.SS1.2.p2.3.m1.3.4.2.1" xref="A4.SS1.2.p2.3.m1.3.4.2.1.cmml">⁢</mo><mrow id="A4.SS1.2.p2.3.m1.3.4.2.3.2" xref="A4.SS1.2.p2.3.m1.3.4.2.cmml"><mo id="A4.SS1.2.p2.3.m1.3.4.2.3.2.1" stretchy="false" xref="A4.SS1.2.p2.3.m1.3.4.2.cmml">(</mo><mi id="A4.SS1.2.p2.3.m1.1.1" xref="A4.SS1.2.p2.3.m1.1.1.cmml">x</mi><mo id="A4.SS1.2.p2.3.m1.3.4.2.3.2.2" stretchy="false" xref="A4.SS1.2.p2.3.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.2.p2.3.m1.3.4.1" xref="A4.SS1.2.p2.3.m1.3.4.1.cmml">=</mo><mrow id="A4.SS1.2.p2.3.m1.3.4.3" xref="A4.SS1.2.p2.3.m1.3.4.3.cmml"><mover accent="true" id="A4.SS1.2.p2.3.m1.3.4.3.2" xref="A4.SS1.2.p2.3.m1.3.4.3.2.cmml"><mi id="A4.SS1.2.p2.3.m1.3.4.3.2.2" xref="A4.SS1.2.p2.3.m1.3.4.3.2.2.cmml">P</mi><mo id="A4.SS1.2.p2.3.m1.3.4.3.2.1" xref="A4.SS1.2.p2.3.m1.3.4.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.2.p2.3.m1.3.4.3.1" xref="A4.SS1.2.p2.3.m1.3.4.3.1.cmml">⁢</mo><mrow id="A4.SS1.2.p2.3.m1.3.4.3.3.2" xref="A4.SS1.2.p2.3.m1.3.4.3.3.1.cmml"><mo id="A4.SS1.2.p2.3.m1.3.4.3.3.2.1" stretchy="false" xref="A4.SS1.2.p2.3.m1.3.4.3.3.1.cmml">(</mo><mi id="A4.SS1.2.p2.3.m1.2.2" xref="A4.SS1.2.p2.3.m1.2.2.cmml">x</mi><mo id="A4.SS1.2.p2.3.m1.3.4.3.3.2.2" xref="A4.SS1.2.p2.3.m1.3.4.3.3.1.cmml">;</mo><mi id="A4.SS1.2.p2.3.m1.3.3" xref="A4.SS1.2.p2.3.m1.3.3.cmml">x</mi><mo id="A4.SS1.2.p2.3.m1.3.4.3.3.2.3" stretchy="false" xref="A4.SS1.2.p2.3.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.3.m1.3b"><apply id="A4.SS1.2.p2.3.m1.3.4.cmml" xref="A4.SS1.2.p2.3.m1.3.4"><eq id="A4.SS1.2.p2.3.m1.3.4.1.cmml" xref="A4.SS1.2.p2.3.m1.3.4.1"></eq><apply id="A4.SS1.2.p2.3.m1.3.4.2.cmml" xref="A4.SS1.2.p2.3.m1.3.4.2"><times id="A4.SS1.2.p2.3.m1.3.4.2.1.cmml" xref="A4.SS1.2.p2.3.m1.3.4.2.1"></times><ci id="A4.SS1.2.p2.3.m1.3.4.2.2.cmml" xref="A4.SS1.2.p2.3.m1.3.4.2.2">𝑄</ci><ci id="A4.SS1.2.p2.3.m1.1.1.cmml" xref="A4.SS1.2.p2.3.m1.1.1">𝑥</ci></apply><apply id="A4.SS1.2.p2.3.m1.3.4.3.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3"><times id="A4.SS1.2.p2.3.m1.3.4.3.1.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3.1"></times><apply id="A4.SS1.2.p2.3.m1.3.4.3.2.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3.2"><ci id="A4.SS1.2.p2.3.m1.3.4.3.2.1.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3.2.1">¯</ci><ci id="A4.SS1.2.p2.3.m1.3.4.3.2.2.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3.2.2">𝑃</ci></apply><list id="A4.SS1.2.p2.3.m1.3.4.3.3.1.cmml" xref="A4.SS1.2.p2.3.m1.3.4.3.3.2"><ci id="A4.SS1.2.p2.3.m1.2.2.cmml" xref="A4.SS1.2.p2.3.m1.2.2">𝑥</ci><ci id="A4.SS1.2.p2.3.m1.3.3.cmml" xref="A4.SS1.2.p2.3.m1.3.3">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.3.m1.3c">Q(x)=\bar{P}(x;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.3.m1.3d">italic_Q ( italic_x ) = over¯ start_ARG italic_P end_ARG ( italic_x ; italic_x )</annotation></semantics></math> follows. ∎</p> </div> </div> <div class="ltx_para" id="A4.SS1.p2"> <p class="ltx_p" id="A4.SS1.p2.1">Now we have the tools to prove the characterization of equilibria in the Gaussian model.</p> </div> <div class="ltx_proof" id="A4.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmcorollary5" title="Corollary 5. ‣ 5.2 Gaussian Model ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">5</span></a>.</h6> <div class="ltx_para" id="A4.SS1.3.p1"> <p class="ltx_p" id="A4.SS1.3.p1.7">Existence of equilibria at <math alttext="\pm\infty" class="ltx_Math" display="inline" id="A4.SS1.3.p1.1.m1.1"><semantics id="A4.SS1.3.p1.1.m1.1a"><mrow id="A4.SS1.3.p1.1.m1.1.1" xref="A4.SS1.3.p1.1.m1.1.1.cmml"><mo id="A4.SS1.3.p1.1.m1.1.1a" xref="A4.SS1.3.p1.1.m1.1.1.cmml">±</mo><mi id="A4.SS1.3.p1.1.m1.1.1.2" mathvariant="normal" xref="A4.SS1.3.p1.1.m1.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.1.m1.1b"><apply id="A4.SS1.3.p1.1.m1.1.1.cmml" xref="A4.SS1.3.p1.1.m1.1.1"><csymbol cd="latexml" id="A4.SS1.3.p1.1.m1.1.1.1.cmml" xref="A4.SS1.3.p1.1.m1.1.1">plus-or-minus</csymbol><infinity id="A4.SS1.3.p1.1.m1.1.1.2.cmml" xref="A4.SS1.3.p1.1.m1.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.1.m1.1c">\pm\infty</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.1.m1.1d">± ∞</annotation></semantics></math> follow as a corollary from <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition3" title="Proposition 3. ‣ Equilibrium results. ‣ 3.1 Equilibrium Characterization ‣ 3 Dasgupta-Ghosh ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">3</span></a>. Next, note all non-infinite equilibria must satisfy <math alttext="G(x)=Q(x)" class="ltx_Math" display="inline" id="A4.SS1.3.p1.2.m2.2"><semantics id="A4.SS1.3.p1.2.m2.2a"><mrow id="A4.SS1.3.p1.2.m2.2.3" xref="A4.SS1.3.p1.2.m2.2.3.cmml"><mrow id="A4.SS1.3.p1.2.m2.2.3.2" xref="A4.SS1.3.p1.2.m2.2.3.2.cmml"><mi id="A4.SS1.3.p1.2.m2.2.3.2.2" xref="A4.SS1.3.p1.2.m2.2.3.2.2.cmml">G</mi><mo id="A4.SS1.3.p1.2.m2.2.3.2.1" xref="A4.SS1.3.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.2.m2.2.3.2.3.2" xref="A4.SS1.3.p1.2.m2.2.3.2.cmml"><mo id="A4.SS1.3.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="A4.SS1.3.p1.2.m2.2.3.2.cmml">(</mo><mi id="A4.SS1.3.p1.2.m2.1.1" xref="A4.SS1.3.p1.2.m2.1.1.cmml">x</mi><mo id="A4.SS1.3.p1.2.m2.2.3.2.3.2.2" stretchy="false" xref="A4.SS1.3.p1.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.3.p1.2.m2.2.3.1" xref="A4.SS1.3.p1.2.m2.2.3.1.cmml">=</mo><mrow id="A4.SS1.3.p1.2.m2.2.3.3" xref="A4.SS1.3.p1.2.m2.2.3.3.cmml"><mi id="A4.SS1.3.p1.2.m2.2.3.3.2" xref="A4.SS1.3.p1.2.m2.2.3.3.2.cmml">Q</mi><mo id="A4.SS1.3.p1.2.m2.2.3.3.1" xref="A4.SS1.3.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.2.m2.2.3.3.3.2" xref="A4.SS1.3.p1.2.m2.2.3.3.cmml"><mo id="A4.SS1.3.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="A4.SS1.3.p1.2.m2.2.3.3.cmml">(</mo><mi id="A4.SS1.3.p1.2.m2.2.2" xref="A4.SS1.3.p1.2.m2.2.2.cmml">x</mi><mo id="A4.SS1.3.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="A4.SS1.3.p1.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.2.m2.2b"><apply id="A4.SS1.3.p1.2.m2.2.3.cmml" xref="A4.SS1.3.p1.2.m2.2.3"><eq id="A4.SS1.3.p1.2.m2.2.3.1.cmml" xref="A4.SS1.3.p1.2.m2.2.3.1"></eq><apply id="A4.SS1.3.p1.2.m2.2.3.2.cmml" xref="A4.SS1.3.p1.2.m2.2.3.2"><times id="A4.SS1.3.p1.2.m2.2.3.2.1.cmml" xref="A4.SS1.3.p1.2.m2.2.3.2.1"></times><ci id="A4.SS1.3.p1.2.m2.2.3.2.2.cmml" xref="A4.SS1.3.p1.2.m2.2.3.2.2">𝐺</ci><ci id="A4.SS1.3.p1.2.m2.1.1.cmml" xref="A4.SS1.3.p1.2.m2.1.1">𝑥</ci></apply><apply id="A4.SS1.3.p1.2.m2.2.3.3.cmml" xref="A4.SS1.3.p1.2.m2.2.3.3"><times id="A4.SS1.3.p1.2.m2.2.3.3.1.cmml" xref="A4.SS1.3.p1.2.m2.2.3.3.1"></times><ci id="A4.SS1.3.p1.2.m2.2.3.3.2.cmml" xref="A4.SS1.3.p1.2.m2.2.3.3.2">𝑄</ci><ci id="A4.SS1.3.p1.2.m2.2.2.cmml" xref="A4.SS1.3.p1.2.m2.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.2.m2.2c">G(x)=Q(x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.2.m2.2d">italic_G ( italic_x ) = italic_Q ( italic_x )</annotation></semantics></math> by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem6" title="Theorem 6. ‣ 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>. But <math alttext="G(x)" class="ltx_Math" display="inline" id="A4.SS1.3.p1.3.m3.1"><semantics id="A4.SS1.3.p1.3.m3.1a"><mrow id="A4.SS1.3.p1.3.m3.1.2" xref="A4.SS1.3.p1.3.m3.1.2.cmml"><mi id="A4.SS1.3.p1.3.m3.1.2.2" xref="A4.SS1.3.p1.3.m3.1.2.2.cmml">G</mi><mo id="A4.SS1.3.p1.3.m3.1.2.1" xref="A4.SS1.3.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.3.m3.1.2.3.2" xref="A4.SS1.3.p1.3.m3.1.2.cmml"><mo id="A4.SS1.3.p1.3.m3.1.2.3.2.1" stretchy="false" xref="A4.SS1.3.p1.3.m3.1.2.cmml">(</mo><mi id="A4.SS1.3.p1.3.m3.1.1" xref="A4.SS1.3.p1.3.m3.1.1.cmml">x</mi><mo id="A4.SS1.3.p1.3.m3.1.2.3.2.2" stretchy="false" xref="A4.SS1.3.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.3.m3.1b"><apply id="A4.SS1.3.p1.3.m3.1.2.cmml" xref="A4.SS1.3.p1.3.m3.1.2"><times id="A4.SS1.3.p1.3.m3.1.2.1.cmml" xref="A4.SS1.3.p1.3.m3.1.2.1"></times><ci id="A4.SS1.3.p1.3.m3.1.2.2.cmml" xref="A4.SS1.3.p1.3.m3.1.2.2">𝐺</ci><ci id="A4.SS1.3.p1.3.m3.1.1.cmml" xref="A4.SS1.3.p1.3.m3.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.3.m3.1c">G(x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.3.m3.1d">italic_G ( italic_x )</annotation></semantics></math> and <math alttext="Q(x)" class="ltx_Math" display="inline" id="A4.SS1.3.p1.4.m4.1"><semantics id="A4.SS1.3.p1.4.m4.1a"><mrow id="A4.SS1.3.p1.4.m4.1.2" xref="A4.SS1.3.p1.4.m4.1.2.cmml"><mi id="A4.SS1.3.p1.4.m4.1.2.2" xref="A4.SS1.3.p1.4.m4.1.2.2.cmml">Q</mi><mo id="A4.SS1.3.p1.4.m4.1.2.1" xref="A4.SS1.3.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.4.m4.1.2.3.2" xref="A4.SS1.3.p1.4.m4.1.2.cmml"><mo id="A4.SS1.3.p1.4.m4.1.2.3.2.1" stretchy="false" xref="A4.SS1.3.p1.4.m4.1.2.cmml">(</mo><mi id="A4.SS1.3.p1.4.m4.1.1" xref="A4.SS1.3.p1.4.m4.1.1.cmml">x</mi><mo id="A4.SS1.3.p1.4.m4.1.2.3.2.2" stretchy="false" xref="A4.SS1.3.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" 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id="A4.SS1.3.p1.5.m5.1.1.3.1" xref="A4.SS1.3.p1.5.m5.1.1.3.1.cmml">⁢</mo><msqrt id="A4.SS1.3.p1.5.m5.1.1.3.3" xref="A4.SS1.3.p1.5.m5.1.1.3.3.cmml"><mrow id="A4.SS1.3.p1.5.m5.1.1.3.3.2" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.cmml"><mn id="A4.SS1.3.p1.5.m5.1.1.3.3.2.2" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.2.cmml">1</mn><mo id="A4.SS1.3.p1.5.m5.1.1.3.3.2.1" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.1.cmml">+</mo><mi id="A4.SS1.3.p1.5.m5.1.1.3.3.2.3" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.5.m5.1b"><apply id="A4.SS1.3.p1.5.m5.1.2.cmml" xref="A4.SS1.3.p1.5.m5.1.2"><eq id="A4.SS1.3.p1.5.m5.1.2.1.cmml" xref="A4.SS1.3.p1.5.m5.1.2.1"></eq><apply id="A4.SS1.3.p1.5.m5.1.2.2.cmml" xref="A4.SS1.3.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="A4.SS1.3.p1.5.m5.1.2.2.1.cmml" xref="A4.SS1.3.p1.5.m5.1.2.2">subscript</csymbol><ci id="A4.SS1.3.p1.5.m5.1.2.2.2.cmml" xref="A4.SS1.3.p1.5.m5.1.2.2.2">𝑐</ci><ci 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id="A4.SS1.3.p1.5.m5.1.1.3.3.2.cmml" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2"><plus id="A4.SS1.3.p1.5.m5.1.1.3.3.2.1.cmml" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.1"></plus><cn id="A4.SS1.3.p1.5.m5.1.1.3.3.2.2.cmml" type="integer" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.2">1</cn><ci id="A4.SS1.3.p1.5.m5.1.1.3.3.2.3.cmml" xref="A4.SS1.3.p1.5.m5.1.1.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.5.m5.1c">c_{G}=\frac{(1-\rho)}{b\sqrt{1+\rho}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.5.m5.1d">italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT = divide start_ARG ( 1 - italic_ρ ) end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG</annotation></semantics></math> and <math alttext="c_{Q}=\frac{(1-\rho^{2})}{b\sqrt{(1+\rho)(1+\rho^{2})}}" class="ltx_Math" display="inline" id="A4.SS1.3.p1.6.m6.3"><semantics id="A4.SS1.3.p1.6.m6.3a"><mrow id="A4.SS1.3.p1.6.m6.3.4" 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id="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.3" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="A4.SS1.3.p1.6.m6.1.1.1.1.3" stretchy="false" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A4.SS1.3.p1.6.m6.3.3.3" xref="A4.SS1.3.p1.6.m6.3.3.3.cmml"><mi id="A4.SS1.3.p1.6.m6.3.3.3.4" xref="A4.SS1.3.p1.6.m6.3.3.3.4.cmml">b</mi><mo id="A4.SS1.3.p1.6.m6.3.3.3.3" xref="A4.SS1.3.p1.6.m6.3.3.3.3.cmml">⁢</mo><msqrt id="A4.SS1.3.p1.6.m6.3.3.3.2" xref="A4.SS1.3.p1.6.m6.3.3.3.2.cmml"><mrow id="A4.SS1.3.p1.6.m6.3.3.3.2.2" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.cmml"><mrow id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.cmml"><mo id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.2" stretchy="false" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.cmml"><mn id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.2" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.1" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.3" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.3.cmml">ρ</mi></mrow><mo id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.3" stretchy="false" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.SS1.3.p1.6.m6.3.3.3.2.2.3" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.3.cmml">⁢</mo><mrow id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.cmml"><mo id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.2" stretchy="false" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.cmml">(</mo><mrow id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.cmml"><mn id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.2" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.2.cmml">1</mn><mo id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.1" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.1.cmml">+</mo><msup id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.cmml"><mi id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.2" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.2.cmml">ρ</mi><mn id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.3" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.3.cmml">2</mn></msup></mrow><mo id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.3" stretchy="false" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.cmml">)</mo></mrow></mrow></msqrt></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.6.m6.3b"><apply id="A4.SS1.3.p1.6.m6.3.4.cmml" xref="A4.SS1.3.p1.6.m6.3.4"><eq id="A4.SS1.3.p1.6.m6.3.4.1.cmml" xref="A4.SS1.3.p1.6.m6.3.4.1"></eq><apply id="A4.SS1.3.p1.6.m6.3.4.2.cmml" xref="A4.SS1.3.p1.6.m6.3.4.2"><csymbol cd="ambiguous" id="A4.SS1.3.p1.6.m6.3.4.2.1.cmml" xref="A4.SS1.3.p1.6.m6.3.4.2">subscript</csymbol><ci id="A4.SS1.3.p1.6.m6.3.4.2.2.cmml" xref="A4.SS1.3.p1.6.m6.3.4.2.2">𝑐</ci><ci id="A4.SS1.3.p1.6.m6.3.4.2.3.cmml" xref="A4.SS1.3.p1.6.m6.3.4.2.3">𝑄</ci></apply><apply id="A4.SS1.3.p1.6.m6.3.3.cmml" xref="A4.SS1.3.p1.6.m6.3.3"><divide id="A4.SS1.3.p1.6.m6.3.3.4.cmml" xref="A4.SS1.3.p1.6.m6.3.3"></divide><apply id="A4.SS1.3.p1.6.m6.1.1.1.1.1.cmml" xref="A4.SS1.3.p1.6.m6.1.1.1.1"><minus id="A4.SS1.3.p1.6.m6.1.1.1.1.1.1.cmml" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.1"></minus><cn id="A4.SS1.3.p1.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.2">1</cn><apply id="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.cmml" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.1.cmml" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.3">superscript</csymbol><ci id="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.2.cmml" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.2">𝜌</ci><cn id="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="A4.SS1.3.p1.6.m6.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="A4.SS1.3.p1.6.m6.3.3.3.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3"><times id="A4.SS1.3.p1.6.m6.3.3.3.3.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.3"></times><ci id="A4.SS1.3.p1.6.m6.3.3.3.4.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.4">𝑏</ci><apply id="A4.SS1.3.p1.6.m6.3.3.3.2.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2"><root id="A4.SS1.3.p1.6.m6.3.3.3.2a.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2"></root><apply id="A4.SS1.3.p1.6.m6.3.3.3.2.2.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2"><times id="A4.SS1.3.p1.6.m6.3.3.3.2.2.3.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.3"></times><apply id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.cmml" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1"><plus id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.1.cmml" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.1"></plus><cn id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.2.cmml" type="integer" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.2">1</cn><ci id="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.3.cmml" xref="A4.SS1.3.p1.6.m6.2.2.2.1.1.1.1.1.3">𝜌</ci></apply><apply id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1"><plus id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.1.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.1"></plus><cn id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.2.cmml" type="integer" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.2">1</cn><apply id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.1.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3">superscript</csymbol><ci id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.2.cmml" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.2">𝜌</ci><cn id="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.3.cmml" type="integer" xref="A4.SS1.3.p1.6.m6.3.3.3.2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.6.m6.3c">c_{Q}=\frac{(1-\rho^{2})}{b\sqrt{(1+\rho)(1+\rho^{2})}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.6.m6.3d">italic_c start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = divide start_ARG ( 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_b square-root start_ARG ( 1 + italic_ρ ) ( 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG end_ARG</annotation></semantics></math>. Since <math alttext="\rho\in(0,1)" class="ltx_Math" display="inline" id="A4.SS1.3.p1.7.m7.2"><semantics id="A4.SS1.3.p1.7.m7.2a"><mrow id="A4.SS1.3.p1.7.m7.2.3" xref="A4.SS1.3.p1.7.m7.2.3.cmml"><mi id="A4.SS1.3.p1.7.m7.2.3.2" xref="A4.SS1.3.p1.7.m7.2.3.2.cmml">ρ</mi><mo id="A4.SS1.3.p1.7.m7.2.3.1" xref="A4.SS1.3.p1.7.m7.2.3.1.cmml">∈</mo><mrow id="A4.SS1.3.p1.7.m7.2.3.3.2" xref="A4.SS1.3.p1.7.m7.2.3.3.1.cmml"><mo id="A4.SS1.3.p1.7.m7.2.3.3.2.1" stretchy="false" xref="A4.SS1.3.p1.7.m7.2.3.3.1.cmml">(</mo><mn id="A4.SS1.3.p1.7.m7.1.1" xref="A4.SS1.3.p1.7.m7.1.1.cmml">0</mn><mo id="A4.SS1.3.p1.7.m7.2.3.3.2.2" xref="A4.SS1.3.p1.7.m7.2.3.3.1.cmml">,</mo><mn id="A4.SS1.3.p1.7.m7.2.2" xref="A4.SS1.3.p1.7.m7.2.2.cmml">1</mn><mo id="A4.SS1.3.p1.7.m7.2.3.3.2.3" stretchy="false" xref="A4.SS1.3.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.7.m7.2b"><apply id="A4.SS1.3.p1.7.m7.2.3.cmml" xref="A4.SS1.3.p1.7.m7.2.3"><in id="A4.SS1.3.p1.7.m7.2.3.1.cmml" xref="A4.SS1.3.p1.7.m7.2.3.1"></in><ci id="A4.SS1.3.p1.7.m7.2.3.2.cmml" xref="A4.SS1.3.p1.7.m7.2.3.2">𝜌</ci><interval closure="open" id="A4.SS1.3.p1.7.m7.2.3.3.1.cmml" xref="A4.SS1.3.p1.7.m7.2.3.3.2"><cn id="A4.SS1.3.p1.7.m7.1.1.cmml" type="integer" xref="A4.SS1.3.p1.7.m7.1.1">0</cn><cn id="A4.SS1.3.p1.7.m7.2.2.cmml" type="integer" xref="A4.SS1.3.p1.7.m7.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.7.m7.2c">\rho\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.7.m7.2d">italic_ρ ∈ ( 0 , 1 )</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\left(\frac{c_{Q}}{c_{G}}\right)^{2}=\frac{(1-\rho^{2})^{2}}{(1-\rho)^{2}(1+% \rho^{2})}=\frac{(1+\rho)^{2}}{1+\rho^{2}}>1." class="ltx_Math" display="block" id="A4.Ex33.m1.6"><semantics id="A4.Ex33.m1.6a"><mrow id="A4.Ex33.m1.6.6.1" xref="A4.Ex33.m1.6.6.1.1.cmml"><mrow id="A4.Ex33.m1.6.6.1.1" xref="A4.Ex33.m1.6.6.1.1.cmml"><msup id="A4.Ex33.m1.6.6.1.1.2" xref="A4.Ex33.m1.6.6.1.1.2.cmml"><mrow id="A4.Ex33.m1.6.6.1.1.2.2.2" xref="A4.Ex33.m1.5.5.cmml"><mo id="A4.Ex33.m1.6.6.1.1.2.2.2.1" xref="A4.Ex33.m1.5.5.cmml">(</mo><mfrac id="A4.Ex33.m1.5.5" xref="A4.Ex33.m1.5.5.cmml"><msub id="A4.Ex33.m1.5.5.2" xref="A4.Ex33.m1.5.5.2.cmml"><mi id="A4.Ex33.m1.5.5.2.2" xref="A4.Ex33.m1.5.5.2.2.cmml">c</mi><mi id="A4.Ex33.m1.5.5.2.3" xref="A4.Ex33.m1.5.5.2.3.cmml">Q</mi></msub><msub id="A4.Ex33.m1.5.5.3" xref="A4.Ex33.m1.5.5.3.cmml"><mi id="A4.Ex33.m1.5.5.3.2" xref="A4.Ex33.m1.5.5.3.2.cmml">c</mi><mi id="A4.Ex33.m1.5.5.3.3" xref="A4.Ex33.m1.5.5.3.3.cmml">G</mi></msub></mfrac><mo id="A4.Ex33.m1.6.6.1.1.2.2.2.2" xref="A4.Ex33.m1.5.5.cmml">)</mo></mrow><mn 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\rho^{2})}=\frac{(1+\rho)^{2}}{1+\rho^{2}}>1.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex33.m1.6d">( divide start_ARG italic_c start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG ( 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG = divide start_ARG ( 1 + italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG > 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.3.p1.10">Thus <math alttext="\tau=0" class="ltx_Math" display="inline" id="A4.SS1.3.p1.8.m1.1"><semantics id="A4.SS1.3.p1.8.m1.1a"><mrow id="A4.SS1.3.p1.8.m1.1.1" xref="A4.SS1.3.p1.8.m1.1.1.cmml"><mi id="A4.SS1.3.p1.8.m1.1.1.2" xref="A4.SS1.3.p1.8.m1.1.1.2.cmml">τ</mi><mo id="A4.SS1.3.p1.8.m1.1.1.1" xref="A4.SS1.3.p1.8.m1.1.1.1.cmml">=</mo><mn id="A4.SS1.3.p1.8.m1.1.1.3" xref="A4.SS1.3.p1.8.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.8.m1.1b"><apply id="A4.SS1.3.p1.8.m1.1.1.cmml" xref="A4.SS1.3.p1.8.m1.1.1"><eq id="A4.SS1.3.p1.8.m1.1.1.1.cmml" xref="A4.SS1.3.p1.8.m1.1.1.1"></eq><ci id="A4.SS1.3.p1.8.m1.1.1.2.cmml" xref="A4.SS1.3.p1.8.m1.1.1.2">𝜏</ci><cn id="A4.SS1.3.p1.8.m1.1.1.3.cmml" type="integer" xref="A4.SS1.3.p1.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.8.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.8.m1.1d">italic_τ = 0</annotation></semantics></math> is the unique point at which <math alttext="G(\tau)=Q(\tau)" class="ltx_Math" display="inline" id="A4.SS1.3.p1.9.m2.2"><semantics id="A4.SS1.3.p1.9.m2.2a"><mrow id="A4.SS1.3.p1.9.m2.2.3" xref="A4.SS1.3.p1.9.m2.2.3.cmml"><mrow id="A4.SS1.3.p1.9.m2.2.3.2" xref="A4.SS1.3.p1.9.m2.2.3.2.cmml"><mi id="A4.SS1.3.p1.9.m2.2.3.2.2" xref="A4.SS1.3.p1.9.m2.2.3.2.2.cmml">G</mi><mo id="A4.SS1.3.p1.9.m2.2.3.2.1" xref="A4.SS1.3.p1.9.m2.2.3.2.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.9.m2.2.3.2.3.2" xref="A4.SS1.3.p1.9.m2.2.3.2.cmml"><mo id="A4.SS1.3.p1.9.m2.2.3.2.3.2.1" stretchy="false" xref="A4.SS1.3.p1.9.m2.2.3.2.cmml">(</mo><mi id="A4.SS1.3.p1.9.m2.1.1" xref="A4.SS1.3.p1.9.m2.1.1.cmml">τ</mi><mo id="A4.SS1.3.p1.9.m2.2.3.2.3.2.2" stretchy="false" xref="A4.SS1.3.p1.9.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.3.p1.9.m2.2.3.1" xref="A4.SS1.3.p1.9.m2.2.3.1.cmml">=</mo><mrow id="A4.SS1.3.p1.9.m2.2.3.3" xref="A4.SS1.3.p1.9.m2.2.3.3.cmml"><mi id="A4.SS1.3.p1.9.m2.2.3.3.2" xref="A4.SS1.3.p1.9.m2.2.3.3.2.cmml">Q</mi><mo id="A4.SS1.3.p1.9.m2.2.3.3.1" xref="A4.SS1.3.p1.9.m2.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.3.p1.9.m2.2.3.3.3.2" xref="A4.SS1.3.p1.9.m2.2.3.3.cmml"><mo id="A4.SS1.3.p1.9.m2.2.3.3.3.2.1" stretchy="false" xref="A4.SS1.3.p1.9.m2.2.3.3.cmml">(</mo><mi id="A4.SS1.3.p1.9.m2.2.2" xref="A4.SS1.3.p1.9.m2.2.2.cmml">τ</mi><mo id="A4.SS1.3.p1.9.m2.2.3.3.3.2.2" stretchy="false" xref="A4.SS1.3.p1.9.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.9.m2.2b"><apply id="A4.SS1.3.p1.9.m2.2.3.cmml" xref="A4.SS1.3.p1.9.m2.2.3"><eq id="A4.SS1.3.p1.9.m2.2.3.1.cmml" xref="A4.SS1.3.p1.9.m2.2.3.1"></eq><apply id="A4.SS1.3.p1.9.m2.2.3.2.cmml" xref="A4.SS1.3.p1.9.m2.2.3.2"><times id="A4.SS1.3.p1.9.m2.2.3.2.1.cmml" xref="A4.SS1.3.p1.9.m2.2.3.2.1"></times><ci id="A4.SS1.3.p1.9.m2.2.3.2.2.cmml" xref="A4.SS1.3.p1.9.m2.2.3.2.2">𝐺</ci><ci id="A4.SS1.3.p1.9.m2.1.1.cmml" xref="A4.SS1.3.p1.9.m2.1.1">𝜏</ci></apply><apply id="A4.SS1.3.p1.9.m2.2.3.3.cmml" xref="A4.SS1.3.p1.9.m2.2.3.3"><times id="A4.SS1.3.p1.9.m2.2.3.3.1.cmml" xref="A4.SS1.3.p1.9.m2.2.3.3.1"></times><ci id="A4.SS1.3.p1.9.m2.2.3.3.2.cmml" xref="A4.SS1.3.p1.9.m2.2.3.3.2">𝑄</ci><ci id="A4.SS1.3.p1.9.m2.2.2.cmml" xref="A4.SS1.3.p1.9.m2.2.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.9.m2.2c">G(\tau)=Q(\tau)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.9.m2.2d">italic_G ( italic_τ ) = italic_Q ( italic_τ )</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S2.F2" title="Figure 2 ‣ 2.3 Dynamics ‣ 2 Output Agreement ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">2</span></a> for an example with <math alttext="a=b=1" class="ltx_Math" display="inline" id="A4.SS1.3.p1.10.m3.1"><semantics id="A4.SS1.3.p1.10.m3.1a"><mrow id="A4.SS1.3.p1.10.m3.1.1" xref="A4.SS1.3.p1.10.m3.1.1.cmml"><mi id="A4.SS1.3.p1.10.m3.1.1.2" xref="A4.SS1.3.p1.10.m3.1.1.2.cmml">a</mi><mo id="A4.SS1.3.p1.10.m3.1.1.3" xref="A4.SS1.3.p1.10.m3.1.1.3.cmml">=</mo><mi id="A4.SS1.3.p1.10.m3.1.1.4" xref="A4.SS1.3.p1.10.m3.1.1.4.cmml">b</mi><mo id="A4.SS1.3.p1.10.m3.1.1.5" xref="A4.SS1.3.p1.10.m3.1.1.5.cmml">=</mo><mn id="A4.SS1.3.p1.10.m3.1.1.6" xref="A4.SS1.3.p1.10.m3.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p1.10.m3.1b"><apply id="A4.SS1.3.p1.10.m3.1.1.cmml" xref="A4.SS1.3.p1.10.m3.1.1"><and id="A4.SS1.3.p1.10.m3.1.1a.cmml" xref="A4.SS1.3.p1.10.m3.1.1"></and><apply id="A4.SS1.3.p1.10.m3.1.1b.cmml" xref="A4.SS1.3.p1.10.m3.1.1"><eq id="A4.SS1.3.p1.10.m3.1.1.3.cmml" xref="A4.SS1.3.p1.10.m3.1.1.3"></eq><ci id="A4.SS1.3.p1.10.m3.1.1.2.cmml" xref="A4.SS1.3.p1.10.m3.1.1.2">𝑎</ci><ci id="A4.SS1.3.p1.10.m3.1.1.4.cmml" xref="A4.SS1.3.p1.10.m3.1.1.4">𝑏</ci></apply><apply id="A4.SS1.3.p1.10.m3.1.1c.cmml" xref="A4.SS1.3.p1.10.m3.1.1"><eq id="A4.SS1.3.p1.10.m3.1.1.5.cmml" xref="A4.SS1.3.p1.10.m3.1.1.5"></eq><share href="https://arxiv.org/html/2503.16280v1#A4.SS1.3.p1.10.m3.1.1.4.cmml" id="A4.SS1.3.p1.10.m3.1.1d.cmml" xref="A4.SS1.3.p1.10.m3.1.1"></share><cn id="A4.SS1.3.p1.10.m3.1.1.6.cmml" type="integer" xref="A4.SS1.3.p1.10.m3.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p1.10.m3.1c">a=b=1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p1.10.m3.1d">italic_a = italic_b = 1</annotation></semantics></math>), so there are no other potential non-infinite equilibria.</p> </div> <div class="ltx_para" id="A4.SS1.4.p2"> <p class="ltx_p" id="A4.SS1.4.p2.10">Now we prove <math alttext="\tau=0" class="ltx_Math" display="inline" id="A4.SS1.4.p2.1.m1.1"><semantics id="A4.SS1.4.p2.1.m1.1a"><mrow id="A4.SS1.4.p2.1.m1.1.1" xref="A4.SS1.4.p2.1.m1.1.1.cmml"><mi id="A4.SS1.4.p2.1.m1.1.1.2" xref="A4.SS1.4.p2.1.m1.1.1.2.cmml">τ</mi><mo id="A4.SS1.4.p2.1.m1.1.1.1" xref="A4.SS1.4.p2.1.m1.1.1.1.cmml">=</mo><mn id="A4.SS1.4.p2.1.m1.1.1.3" xref="A4.SS1.4.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.1.m1.1b"><apply id="A4.SS1.4.p2.1.m1.1.1.cmml" xref="A4.SS1.4.p2.1.m1.1.1"><eq id="A4.SS1.4.p2.1.m1.1.1.1.cmml" xref="A4.SS1.4.p2.1.m1.1.1.1"></eq><ci id="A4.SS1.4.p2.1.m1.1.1.2.cmml" xref="A4.SS1.4.p2.1.m1.1.1.2">𝜏</ci><cn id="A4.SS1.4.p2.1.m1.1.1.3.cmml" type="integer" xref="A4.SS1.4.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.1.m1.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.1.m1.1d">italic_τ = 0</annotation></semantics></math> actually is an equilibrium using the sufficiency condition in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem6" title="Theorem 6. ‣ 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a>. Note that when <math alttext="\tau=0" class="ltx_Math" display="inline" id="A4.SS1.4.p2.2.m2.1"><semantics id="A4.SS1.4.p2.2.m2.1a"><mrow id="A4.SS1.4.p2.2.m2.1.1" xref="A4.SS1.4.p2.2.m2.1.1.cmml"><mi id="A4.SS1.4.p2.2.m2.1.1.2" xref="A4.SS1.4.p2.2.m2.1.1.2.cmml">τ</mi><mo id="A4.SS1.4.p2.2.m2.1.1.1" xref="A4.SS1.4.p2.2.m2.1.1.1.cmml">=</mo><mn id="A4.SS1.4.p2.2.m2.1.1.3" xref="A4.SS1.4.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.2.m2.1b"><apply id="A4.SS1.4.p2.2.m2.1.1.cmml" xref="A4.SS1.4.p2.2.m2.1.1"><eq id="A4.SS1.4.p2.2.m2.1.1.1.cmml" xref="A4.SS1.4.p2.2.m2.1.1.1"></eq><ci id="A4.SS1.4.p2.2.m2.1.1.2.cmml" xref="A4.SS1.4.p2.2.m2.1.1.2">𝜏</ci><cn id="A4.SS1.4.p2.2.m2.1.1.3.cmml" type="integer" xref="A4.SS1.4.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.2.m2.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.2.m2.1d">italic_τ = 0</annotation></semantics></math>, <math alttext="P(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.4.p2.3.m3.2"><semantics id="A4.SS1.4.p2.3.m3.2a"><mrow id="A4.SS1.4.p2.3.m3.2.3" xref="A4.SS1.4.p2.3.m3.2.3.cmml"><mi id="A4.SS1.4.p2.3.m3.2.3.2" xref="A4.SS1.4.p2.3.m3.2.3.2.cmml">P</mi><mo id="A4.SS1.4.p2.3.m3.2.3.1" xref="A4.SS1.4.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.3.m3.2.3.3.2" xref="A4.SS1.4.p2.3.m3.2.3.3.1.cmml"><mo id="A4.SS1.4.p2.3.m3.2.3.3.2.1" stretchy="false" xref="A4.SS1.4.p2.3.m3.2.3.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.3.m3.1.1" xref="A4.SS1.4.p2.3.m3.1.1.cmml">τ</mi><mo id="A4.SS1.4.p2.3.m3.2.3.3.2.2" xref="A4.SS1.4.p2.3.m3.2.3.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.3.m3.2.2" xref="A4.SS1.4.p2.3.m3.2.2.cmml">x</mi><mo id="A4.SS1.4.p2.3.m3.2.3.3.2.3" stretchy="false" xref="A4.SS1.4.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.3.m3.2b"><apply id="A4.SS1.4.p2.3.m3.2.3.cmml" xref="A4.SS1.4.p2.3.m3.2.3"><times id="A4.SS1.4.p2.3.m3.2.3.1.cmml" xref="A4.SS1.4.p2.3.m3.2.3.1"></times><ci id="A4.SS1.4.p2.3.m3.2.3.2.cmml" xref="A4.SS1.4.p2.3.m3.2.3.2">𝑃</ci><list id="A4.SS1.4.p2.3.m3.2.3.3.1.cmml" xref="A4.SS1.4.p2.3.m3.2.3.3.2"><ci id="A4.SS1.4.p2.3.m3.1.1.cmml" xref="A4.SS1.4.p2.3.m3.1.1">𝜏</ci><ci id="A4.SS1.4.p2.3.m3.2.2.cmml" xref="A4.SS1.4.p2.3.m3.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.3.m3.2c">P(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.3.m3.2d">italic_P ( italic_τ ; italic_x )</annotation></semantics></math> and <math alttext="\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.4.p2.4.m4.2"><semantics id="A4.SS1.4.p2.4.m4.2a"><mrow id="A4.SS1.4.p2.4.m4.2.3" xref="A4.SS1.4.p2.4.m4.2.3.cmml"><mover accent="true" id="A4.SS1.4.p2.4.m4.2.3.2" xref="A4.SS1.4.p2.4.m4.2.3.2.cmml"><mi id="A4.SS1.4.p2.4.m4.2.3.2.2" xref="A4.SS1.4.p2.4.m4.2.3.2.2.cmml">P</mi><mo id="A4.SS1.4.p2.4.m4.2.3.2.1" xref="A4.SS1.4.p2.4.m4.2.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.4.p2.4.m4.2.3.1" xref="A4.SS1.4.p2.4.m4.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.4.m4.2.3.3.2" xref="A4.SS1.4.p2.4.m4.2.3.3.1.cmml"><mo id="A4.SS1.4.p2.4.m4.2.3.3.2.1" stretchy="false" xref="A4.SS1.4.p2.4.m4.2.3.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.4.m4.1.1" xref="A4.SS1.4.p2.4.m4.1.1.cmml">τ</mi><mo id="A4.SS1.4.p2.4.m4.2.3.3.2.2" xref="A4.SS1.4.p2.4.m4.2.3.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.4.m4.2.2" xref="A4.SS1.4.p2.4.m4.2.2.cmml">x</mi><mo id="A4.SS1.4.p2.4.m4.2.3.3.2.3" stretchy="false" xref="A4.SS1.4.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.4.m4.2b"><apply id="A4.SS1.4.p2.4.m4.2.3.cmml" xref="A4.SS1.4.p2.4.m4.2.3"><times id="A4.SS1.4.p2.4.m4.2.3.1.cmml" xref="A4.SS1.4.p2.4.m4.2.3.1"></times><apply id="A4.SS1.4.p2.4.m4.2.3.2.cmml" xref="A4.SS1.4.p2.4.m4.2.3.2"><ci id="A4.SS1.4.p2.4.m4.2.3.2.1.cmml" xref="A4.SS1.4.p2.4.m4.2.3.2.1">¯</ci><ci id="A4.SS1.4.p2.4.m4.2.3.2.2.cmml" xref="A4.SS1.4.p2.4.m4.2.3.2.2">𝑃</ci></apply><list id="A4.SS1.4.p2.4.m4.2.3.3.1.cmml" xref="A4.SS1.4.p2.4.m4.2.3.3.2"><ci id="A4.SS1.4.p2.4.m4.1.1.cmml" xref="A4.SS1.4.p2.4.m4.1.1">𝜏</ci><ci id="A4.SS1.4.p2.4.m4.2.2.cmml" xref="A4.SS1.4.p2.4.m4.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.4.m4.2c">\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.4.m4.2d">over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math> are both Normal CDFs over <math alttext="x" class="ltx_Math" display="inline" id="A4.SS1.4.p2.5.m5.1"><semantics id="A4.SS1.4.p2.5.m5.1a"><mi id="A4.SS1.4.p2.5.m5.1.1" xref="A4.SS1.4.p2.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.5.m5.1b"><ci id="A4.SS1.4.p2.5.m5.1.1.cmml" xref="A4.SS1.4.p2.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.5.m5.1d">italic_x</annotation></semantics></math> with negative (and distinct) coefficients. One can thus easily check that if <math alttext="x\leq\tau" class="ltx_Math" display="inline" id="A4.SS1.4.p2.6.m6.1"><semantics id="A4.SS1.4.p2.6.m6.1a"><mrow id="A4.SS1.4.p2.6.m6.1.1" xref="A4.SS1.4.p2.6.m6.1.1.cmml"><mi id="A4.SS1.4.p2.6.m6.1.1.2" xref="A4.SS1.4.p2.6.m6.1.1.2.cmml">x</mi><mo id="A4.SS1.4.p2.6.m6.1.1.1" xref="A4.SS1.4.p2.6.m6.1.1.1.cmml">≤</mo><mi id="A4.SS1.4.p2.6.m6.1.1.3" xref="A4.SS1.4.p2.6.m6.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.6.m6.1b"><apply id="A4.SS1.4.p2.6.m6.1.1.cmml" xref="A4.SS1.4.p2.6.m6.1.1"><leq id="A4.SS1.4.p2.6.m6.1.1.1.cmml" xref="A4.SS1.4.p2.6.m6.1.1.1"></leq><ci id="A4.SS1.4.p2.6.m6.1.1.2.cmml" xref="A4.SS1.4.p2.6.m6.1.1.2">𝑥</ci><ci id="A4.SS1.4.p2.6.m6.1.1.3.cmml" xref="A4.SS1.4.p2.6.m6.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.6.m6.1c">x\leq\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.6.m6.1d">italic_x ≤ italic_τ</annotation></semantics></math>, <math alttext="P(\tau;x)\geq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.4.p2.7.m7.4"><semantics id="A4.SS1.4.p2.7.m7.4a"><mrow id="A4.SS1.4.p2.7.m7.4.5" xref="A4.SS1.4.p2.7.m7.4.5.cmml"><mrow id="A4.SS1.4.p2.7.m7.4.5.2" xref="A4.SS1.4.p2.7.m7.4.5.2.cmml"><mi id="A4.SS1.4.p2.7.m7.4.5.2.2" xref="A4.SS1.4.p2.7.m7.4.5.2.2.cmml">P</mi><mo id="A4.SS1.4.p2.7.m7.4.5.2.1" xref="A4.SS1.4.p2.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.7.m7.4.5.2.3.2" xref="A4.SS1.4.p2.7.m7.4.5.2.3.1.cmml"><mo id="A4.SS1.4.p2.7.m7.4.5.2.3.2.1" stretchy="false" xref="A4.SS1.4.p2.7.m7.4.5.2.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.7.m7.1.1" xref="A4.SS1.4.p2.7.m7.1.1.cmml">τ</mi><mo id="A4.SS1.4.p2.7.m7.4.5.2.3.2.2" xref="A4.SS1.4.p2.7.m7.4.5.2.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.7.m7.2.2" xref="A4.SS1.4.p2.7.m7.2.2.cmml">x</mi><mo id="A4.SS1.4.p2.7.m7.4.5.2.3.2.3" stretchy="false" xref="A4.SS1.4.p2.7.m7.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.4.p2.7.m7.4.5.1" xref="A4.SS1.4.p2.7.m7.4.5.1.cmml">≥</mo><mrow id="A4.SS1.4.p2.7.m7.4.5.3" xref="A4.SS1.4.p2.7.m7.4.5.3.cmml"><mover accent="true" id="A4.SS1.4.p2.7.m7.4.5.3.2" xref="A4.SS1.4.p2.7.m7.4.5.3.2.cmml"><mi id="A4.SS1.4.p2.7.m7.4.5.3.2.2" xref="A4.SS1.4.p2.7.m7.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.4.p2.7.m7.4.5.3.2.1" xref="A4.SS1.4.p2.7.m7.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.4.p2.7.m7.4.5.3.1" xref="A4.SS1.4.p2.7.m7.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.7.m7.4.5.3.3.2" xref="A4.SS1.4.p2.7.m7.4.5.3.3.1.cmml"><mo id="A4.SS1.4.p2.7.m7.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.4.p2.7.m7.4.5.3.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.7.m7.3.3" xref="A4.SS1.4.p2.7.m7.3.3.cmml">τ</mi><mo id="A4.SS1.4.p2.7.m7.4.5.3.3.2.2" xref="A4.SS1.4.p2.7.m7.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.7.m7.4.4" xref="A4.SS1.4.p2.7.m7.4.4.cmml">x</mi><mo id="A4.SS1.4.p2.7.m7.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.4.p2.7.m7.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.7.m7.4b"><apply id="A4.SS1.4.p2.7.m7.4.5.cmml" xref="A4.SS1.4.p2.7.m7.4.5"><geq id="A4.SS1.4.p2.7.m7.4.5.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.1"></geq><apply id="A4.SS1.4.p2.7.m7.4.5.2.cmml" xref="A4.SS1.4.p2.7.m7.4.5.2"><times id="A4.SS1.4.p2.7.m7.4.5.2.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.2.1"></times><ci id="A4.SS1.4.p2.7.m7.4.5.2.2.cmml" xref="A4.SS1.4.p2.7.m7.4.5.2.2">𝑃</ci><list id="A4.SS1.4.p2.7.m7.4.5.2.3.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.2.3.2"><ci id="A4.SS1.4.p2.7.m7.1.1.cmml" xref="A4.SS1.4.p2.7.m7.1.1">𝜏</ci><ci id="A4.SS1.4.p2.7.m7.2.2.cmml" xref="A4.SS1.4.p2.7.m7.2.2">𝑥</ci></list></apply><apply id="A4.SS1.4.p2.7.m7.4.5.3.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3"><times id="A4.SS1.4.p2.7.m7.4.5.3.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3.1"></times><apply id="A4.SS1.4.p2.7.m7.4.5.3.2.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3.2"><ci id="A4.SS1.4.p2.7.m7.4.5.3.2.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3.2.1">¯</ci><ci id="A4.SS1.4.p2.7.m7.4.5.3.2.2.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.4.p2.7.m7.4.5.3.3.1.cmml" xref="A4.SS1.4.p2.7.m7.4.5.3.3.2"><ci id="A4.SS1.4.p2.7.m7.3.3.cmml" xref="A4.SS1.4.p2.7.m7.3.3">𝜏</ci><ci id="A4.SS1.4.p2.7.m7.4.4.cmml" xref="A4.SS1.4.p2.7.m7.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.7.m7.4c">P(\tau;x)\geq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.7.m7.4d">italic_P ( italic_τ ; italic_x ) ≥ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>. Moreover, if <math alttext="x>\tau" class="ltx_Math" display="inline" id="A4.SS1.4.p2.8.m8.1"><semantics id="A4.SS1.4.p2.8.m8.1a"><mrow id="A4.SS1.4.p2.8.m8.1.1" xref="A4.SS1.4.p2.8.m8.1.1.cmml"><mi id="A4.SS1.4.p2.8.m8.1.1.2" xref="A4.SS1.4.p2.8.m8.1.1.2.cmml">x</mi><mo id="A4.SS1.4.p2.8.m8.1.1.1" xref="A4.SS1.4.p2.8.m8.1.1.1.cmml">></mo><mi id="A4.SS1.4.p2.8.m8.1.1.3" xref="A4.SS1.4.p2.8.m8.1.1.3.cmml">τ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.8.m8.1b"><apply id="A4.SS1.4.p2.8.m8.1.1.cmml" xref="A4.SS1.4.p2.8.m8.1.1"><gt id="A4.SS1.4.p2.8.m8.1.1.1.cmml" xref="A4.SS1.4.p2.8.m8.1.1.1"></gt><ci id="A4.SS1.4.p2.8.m8.1.1.2.cmml" xref="A4.SS1.4.p2.8.m8.1.1.2">𝑥</ci><ci id="A4.SS1.4.p2.8.m8.1.1.3.cmml" xref="A4.SS1.4.p2.8.m8.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.8.m8.1c">x>\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.8.m8.1d">italic_x > italic_τ</annotation></semantics></math>, <math alttext="P(\tau;x)\leq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.4.p2.9.m9.4"><semantics id="A4.SS1.4.p2.9.m9.4a"><mrow id="A4.SS1.4.p2.9.m9.4.5" xref="A4.SS1.4.p2.9.m9.4.5.cmml"><mrow id="A4.SS1.4.p2.9.m9.4.5.2" xref="A4.SS1.4.p2.9.m9.4.5.2.cmml"><mi id="A4.SS1.4.p2.9.m9.4.5.2.2" xref="A4.SS1.4.p2.9.m9.4.5.2.2.cmml">P</mi><mo id="A4.SS1.4.p2.9.m9.4.5.2.1" xref="A4.SS1.4.p2.9.m9.4.5.2.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.9.m9.4.5.2.3.2" xref="A4.SS1.4.p2.9.m9.4.5.2.3.1.cmml"><mo id="A4.SS1.4.p2.9.m9.4.5.2.3.2.1" stretchy="false" xref="A4.SS1.4.p2.9.m9.4.5.2.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.9.m9.1.1" xref="A4.SS1.4.p2.9.m9.1.1.cmml">τ</mi><mo id="A4.SS1.4.p2.9.m9.4.5.2.3.2.2" xref="A4.SS1.4.p2.9.m9.4.5.2.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.9.m9.2.2" xref="A4.SS1.4.p2.9.m9.2.2.cmml">x</mi><mo id="A4.SS1.4.p2.9.m9.4.5.2.3.2.3" stretchy="false" xref="A4.SS1.4.p2.9.m9.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.4.p2.9.m9.4.5.1" xref="A4.SS1.4.p2.9.m9.4.5.1.cmml">≤</mo><mrow id="A4.SS1.4.p2.9.m9.4.5.3" xref="A4.SS1.4.p2.9.m9.4.5.3.cmml"><mover accent="true" id="A4.SS1.4.p2.9.m9.4.5.3.2" xref="A4.SS1.4.p2.9.m9.4.5.3.2.cmml"><mi id="A4.SS1.4.p2.9.m9.4.5.3.2.2" xref="A4.SS1.4.p2.9.m9.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.4.p2.9.m9.4.5.3.2.1" xref="A4.SS1.4.p2.9.m9.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.4.p2.9.m9.4.5.3.1" xref="A4.SS1.4.p2.9.m9.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.4.p2.9.m9.4.5.3.3.2" xref="A4.SS1.4.p2.9.m9.4.5.3.3.1.cmml"><mo id="A4.SS1.4.p2.9.m9.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.4.p2.9.m9.4.5.3.3.1.cmml">(</mo><mi id="A4.SS1.4.p2.9.m9.3.3" xref="A4.SS1.4.p2.9.m9.3.3.cmml">τ</mi><mo id="A4.SS1.4.p2.9.m9.4.5.3.3.2.2" xref="A4.SS1.4.p2.9.m9.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.4.p2.9.m9.4.4" xref="A4.SS1.4.p2.9.m9.4.4.cmml">x</mi><mo id="A4.SS1.4.p2.9.m9.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.4.p2.9.m9.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.9.m9.4b"><apply id="A4.SS1.4.p2.9.m9.4.5.cmml" xref="A4.SS1.4.p2.9.m9.4.5"><leq id="A4.SS1.4.p2.9.m9.4.5.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.1"></leq><apply id="A4.SS1.4.p2.9.m9.4.5.2.cmml" xref="A4.SS1.4.p2.9.m9.4.5.2"><times id="A4.SS1.4.p2.9.m9.4.5.2.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.2.1"></times><ci id="A4.SS1.4.p2.9.m9.4.5.2.2.cmml" xref="A4.SS1.4.p2.9.m9.4.5.2.2">𝑃</ci><list id="A4.SS1.4.p2.9.m9.4.5.2.3.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.2.3.2"><ci id="A4.SS1.4.p2.9.m9.1.1.cmml" xref="A4.SS1.4.p2.9.m9.1.1">𝜏</ci><ci id="A4.SS1.4.p2.9.m9.2.2.cmml" xref="A4.SS1.4.p2.9.m9.2.2">𝑥</ci></list></apply><apply id="A4.SS1.4.p2.9.m9.4.5.3.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3"><times id="A4.SS1.4.p2.9.m9.4.5.3.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3.1"></times><apply id="A4.SS1.4.p2.9.m9.4.5.3.2.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3.2"><ci id="A4.SS1.4.p2.9.m9.4.5.3.2.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3.2.1">¯</ci><ci id="A4.SS1.4.p2.9.m9.4.5.3.2.2.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.4.p2.9.m9.4.5.3.3.1.cmml" xref="A4.SS1.4.p2.9.m9.4.5.3.3.2"><ci id="A4.SS1.4.p2.9.m9.3.3.cmml" xref="A4.SS1.4.p2.9.m9.3.3">𝜏</ci><ci id="A4.SS1.4.p2.9.m9.4.4.cmml" xref="A4.SS1.4.p2.9.m9.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.9.m9.4c">P(\tau;x)\leq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.9.m9.4d">italic_P ( italic_τ ; italic_x ) ≤ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>. It follows by the sufficiency condition in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmtheorem6" title="Theorem 6. ‣ 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">6</span></a> that <math alttext="\tau=0" class="ltx_Math" display="inline" id="A4.SS1.4.p2.10.m10.1"><semantics id="A4.SS1.4.p2.10.m10.1a"><mrow id="A4.SS1.4.p2.10.m10.1.1" xref="A4.SS1.4.p2.10.m10.1.1.cmml"><mi id="A4.SS1.4.p2.10.m10.1.1.2" xref="A4.SS1.4.p2.10.m10.1.1.2.cmml">τ</mi><mo id="A4.SS1.4.p2.10.m10.1.1.1" xref="A4.SS1.4.p2.10.m10.1.1.1.cmml">=</mo><mn id="A4.SS1.4.p2.10.m10.1.1.3" xref="A4.SS1.4.p2.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p2.10.m10.1b"><apply id="A4.SS1.4.p2.10.m10.1.1.cmml" xref="A4.SS1.4.p2.10.m10.1.1"><eq id="A4.SS1.4.p2.10.m10.1.1.1.cmml" xref="A4.SS1.4.p2.10.m10.1.1.1"></eq><ci id="A4.SS1.4.p2.10.m10.1.1.2.cmml" xref="A4.SS1.4.p2.10.m10.1.1.2">𝜏</ci><cn id="A4.SS1.4.p2.10.m10.1.1.3.cmml" type="integer" xref="A4.SS1.4.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p2.10.m10.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p2.10.m10.1d">italic_τ = 0</annotation></semantics></math> is an equilibrium. ∎</p> </div> </div> <div class="ltx_para" id="A4.SS1.p3"> <p class="ltx_p" id="A4.SS1.p3.1">Next, we include a proof of the form of the best response in RBTS under the Gaussian model.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition12.1.1.1">Proposition 12</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition12.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition12.p1"> <p class="ltx_p" id="Thmproposition12.p1.1">When other agents are playing according to some threshold <math alttext="\tau" class="ltx_Math" display="inline" id="Thmproposition12.p1.1.m1.1"><semantics id="Thmproposition12.p1.1.m1.1a"><mi id="Thmproposition12.p1.1.m1.1.1" xref="Thmproposition12.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="Thmproposition12.p1.1.m1.1b"><ci id="Thmproposition12.p1.1.m1.1.1.cmml" xref="Thmproposition12.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition12.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="Thmproposition12.p1.1.m1.1d">italic_τ</annotation></semantics></math>, an agent will best respond with threshold strategy</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex34"> <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 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id="A4.Ex34.m1.1.1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="A4.Ex34.m1.1.1.1.1.1.1.2.2.3.3">2</cn></apply></apply></apply><ci id="A4.Ex34.m1.1.1.1.1.1.1.3.cmml" xref="A4.Ex34.m1.1.1.1.1.1.1.3">𝜌</ci></apply></apply></apply><ci id="A4.Ex34.m1.2.2.1.1.3.3.cmml" xref="A4.Ex34.m1.2.2.1.1.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex34.m1.2c">\hat{\tau}=\left(\frac{\sqrt{1+\rho^{2}}-1}{\rho(\sqrt{1+\rho^{2}}-\rho)}% \right)\tau.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex34.m1.2d">over^ start_ARG italic_τ end_ARG = ( divide start_ARG square-root start_ARG 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 end_ARG start_ARG italic_ρ ( square-root start_ARG 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - italic_ρ ) end_ARG ) italic_τ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A4.SS1.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A4.SS1.5.p1"> <p class="ltx_p" id="A4.SS1.5.p1.5">Let <math alttext="\tau" class="ltx_Math" display="inline" id="A4.SS1.5.p1.1.m1.1"><semantics id="A4.SS1.5.p1.1.m1.1a"><mi id="A4.SS1.5.p1.1.m1.1.1" xref="A4.SS1.5.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.1.m1.1b"><ci id="A4.SS1.5.p1.1.m1.1.1.cmml" xref="A4.SS1.5.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.1.m1.1d">italic_τ</annotation></semantics></math> be the current threshold strategy that all other agents are following. By equations (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E18" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">18</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#S5.E19" title="In 5.1 Equilibrium Characterization ‣ 5 Robust Bayesian Truth Serum ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">19</span></a>), an agent who receives signals <math alttext="x" class="ltx_Math" display="inline" id="A4.SS1.5.p1.2.m2.1"><semantics id="A4.SS1.5.p1.2.m2.1a"><mi id="A4.SS1.5.p1.2.m2.1.1" xref="A4.SS1.5.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.2.m2.1b"><ci id="A4.SS1.5.p1.2.m2.1.1.cmml" xref="A4.SS1.5.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.2.m2.1d">italic_x</annotation></semantics></math> will best respond with <math alttext="H" class="ltx_Math" display="inline" id="A4.SS1.5.p1.3.m3.1"><semantics id="A4.SS1.5.p1.3.m3.1a"><mi id="A4.SS1.5.p1.3.m3.1.1" xref="A4.SS1.5.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.3.m3.1b"><ci id="A4.SS1.5.p1.3.m3.1.1.cmml" xref="A4.SS1.5.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.3.m3.1d">italic_H</annotation></semantics></math> if <math alttext="P(\tau;x)\leq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.5.p1.4.m4.4"><semantics id="A4.SS1.5.p1.4.m4.4a"><mrow id="A4.SS1.5.p1.4.m4.4.5" xref="A4.SS1.5.p1.4.m4.4.5.cmml"><mrow id="A4.SS1.5.p1.4.m4.4.5.2" xref="A4.SS1.5.p1.4.m4.4.5.2.cmml"><mi id="A4.SS1.5.p1.4.m4.4.5.2.2" xref="A4.SS1.5.p1.4.m4.4.5.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.4.m4.4.5.2.1" xref="A4.SS1.5.p1.4.m4.4.5.2.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.4.m4.4.5.2.3.2" xref="A4.SS1.5.p1.4.m4.4.5.2.3.1.cmml"><mo id="A4.SS1.5.p1.4.m4.4.5.2.3.2.1" stretchy="false" xref="A4.SS1.5.p1.4.m4.4.5.2.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.4.m4.1.1" xref="A4.SS1.5.p1.4.m4.1.1.cmml">τ</mi><mo id="A4.SS1.5.p1.4.m4.4.5.2.3.2.2" xref="A4.SS1.5.p1.4.m4.4.5.2.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.4.m4.2.2" xref="A4.SS1.5.p1.4.m4.2.2.cmml">x</mi><mo id="A4.SS1.5.p1.4.m4.4.5.2.3.2.3" stretchy="false" xref="A4.SS1.5.p1.4.m4.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.5.p1.4.m4.4.5.1" xref="A4.SS1.5.p1.4.m4.4.5.1.cmml">≤</mo><mrow id="A4.SS1.5.p1.4.m4.4.5.3" xref="A4.SS1.5.p1.4.m4.4.5.3.cmml"><mover accent="true" id="A4.SS1.5.p1.4.m4.4.5.3.2" xref="A4.SS1.5.p1.4.m4.4.5.3.2.cmml"><mi id="A4.SS1.5.p1.4.m4.4.5.3.2.2" xref="A4.SS1.5.p1.4.m4.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.4.m4.4.5.3.2.1" xref="A4.SS1.5.p1.4.m4.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.5.p1.4.m4.4.5.3.1" xref="A4.SS1.5.p1.4.m4.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.4.m4.4.5.3.3.2" xref="A4.SS1.5.p1.4.m4.4.5.3.3.1.cmml"><mo id="A4.SS1.5.p1.4.m4.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.5.p1.4.m4.4.5.3.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.4.m4.3.3" xref="A4.SS1.5.p1.4.m4.3.3.cmml">τ</mi><mo id="A4.SS1.5.p1.4.m4.4.5.3.3.2.2" xref="A4.SS1.5.p1.4.m4.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.4.m4.4.4" xref="A4.SS1.5.p1.4.m4.4.4.cmml">x</mi><mo id="A4.SS1.5.p1.4.m4.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.5.p1.4.m4.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.4.m4.4b"><apply id="A4.SS1.5.p1.4.m4.4.5.cmml" xref="A4.SS1.5.p1.4.m4.4.5"><leq id="A4.SS1.5.p1.4.m4.4.5.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.1"></leq><apply id="A4.SS1.5.p1.4.m4.4.5.2.cmml" xref="A4.SS1.5.p1.4.m4.4.5.2"><times id="A4.SS1.5.p1.4.m4.4.5.2.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.2.1"></times><ci id="A4.SS1.5.p1.4.m4.4.5.2.2.cmml" xref="A4.SS1.5.p1.4.m4.4.5.2.2">𝑃</ci><list id="A4.SS1.5.p1.4.m4.4.5.2.3.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.2.3.2"><ci id="A4.SS1.5.p1.4.m4.1.1.cmml" xref="A4.SS1.5.p1.4.m4.1.1">𝜏</ci><ci id="A4.SS1.5.p1.4.m4.2.2.cmml" xref="A4.SS1.5.p1.4.m4.2.2">𝑥</ci></list></apply><apply id="A4.SS1.5.p1.4.m4.4.5.3.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3"><times id="A4.SS1.5.p1.4.m4.4.5.3.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3.1"></times><apply id="A4.SS1.5.p1.4.m4.4.5.3.2.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3.2"><ci id="A4.SS1.5.p1.4.m4.4.5.3.2.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3.2.1">¯</ci><ci id="A4.SS1.5.p1.4.m4.4.5.3.2.2.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.5.p1.4.m4.4.5.3.3.1.cmml" xref="A4.SS1.5.p1.4.m4.4.5.3.3.2"><ci id="A4.SS1.5.p1.4.m4.3.3.cmml" xref="A4.SS1.5.p1.4.m4.3.3">𝜏</ci><ci id="A4.SS1.5.p1.4.m4.4.4.cmml" xref="A4.SS1.5.p1.4.m4.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.4.m4.4c">P(\tau;x)\leq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.4.m4.4d">italic_P ( italic_τ ; italic_x ) ≤ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math>, and <math alttext="L" class="ltx_Math" display="inline" id="A4.SS1.5.p1.5.m5.1"><semantics id="A4.SS1.5.p1.5.m5.1a"><mi id="A4.SS1.5.p1.5.m5.1.1" xref="A4.SS1.5.p1.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.5.m5.1b"><ci id="A4.SS1.5.p1.5.m5.1.1.cmml" xref="A4.SS1.5.p1.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.5.m5.1d">italic_L</annotation></semantics></math> otherwise. In the Gaussian model, note that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A5.EGx17"> <tbody id="A4.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P(0;x)" class="ltx_Math" display="inline" id="A4.Ex35.m1.2"><semantics id="A4.Ex35.m1.2a"><mrow id="A4.Ex35.m1.2.3" xref="A4.Ex35.m1.2.3.cmml"><mi id="A4.Ex35.m1.2.3.2" xref="A4.Ex35.m1.2.3.2.cmml">P</mi><mo id="A4.Ex35.m1.2.3.1" xref="A4.Ex35.m1.2.3.1.cmml">⁢</mo><mrow id="A4.Ex35.m1.2.3.3.2" xref="A4.Ex35.m1.2.3.3.1.cmml"><mo id="A4.Ex35.m1.2.3.3.2.1" stretchy="false" xref="A4.Ex35.m1.2.3.3.1.cmml">(</mo><mn id="A4.Ex35.m1.1.1" xref="A4.Ex35.m1.1.1.cmml">0</mn><mo id="A4.Ex35.m1.2.3.3.2.2" xref="A4.Ex35.m1.2.3.3.1.cmml">;</mo><mi id="A4.Ex35.m1.2.2" xref="A4.Ex35.m1.2.2.cmml">x</mi><mo id="A4.Ex35.m1.2.3.3.2.3" stretchy="false" xref="A4.Ex35.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex35.m1.2b"><apply id="A4.Ex35.m1.2.3.cmml" xref="A4.Ex35.m1.2.3"><times id="A4.Ex35.m1.2.3.1.cmml" xref="A4.Ex35.m1.2.3.1"></times><ci id="A4.Ex35.m1.2.3.2.cmml" xref="A4.Ex35.m1.2.3.2">𝑃</ci><list id="A4.Ex35.m1.2.3.3.1.cmml" xref="A4.Ex35.m1.2.3.3.2"><cn id="A4.Ex35.m1.1.1.cmml" type="integer" xref="A4.Ex35.m1.1.1">0</cn><ci id="A4.Ex35.m1.2.2.cmml" xref="A4.Ex35.m1.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex35.m1.2c">\displaystyle P(0;x)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex35.m1.2d">italic_P ( 0 ; italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Phi\left(\frac{-\rho x}{b\sqrt{1+\rho}}\right)," class="ltx_Math" display="inline" id="A4.Ex35.m2.2"><semantics id="A4.Ex35.m2.2a"><mrow id="A4.Ex35.m2.2.2.1" xref="A4.Ex35.m2.2.2.1.1.cmml"><mrow id="A4.Ex35.m2.2.2.1.1" xref="A4.Ex35.m2.2.2.1.1.cmml"><mi id="A4.Ex35.m2.2.2.1.1.2" xref="A4.Ex35.m2.2.2.1.1.2.cmml"></mi><mo id="A4.Ex35.m2.2.2.1.1.1" xref="A4.Ex35.m2.2.2.1.1.1.cmml">=</mo><mrow id="A4.Ex35.m2.2.2.1.1.3" xref="A4.Ex35.m2.2.2.1.1.3.cmml"><mi id="A4.Ex35.m2.2.2.1.1.3.2" mathvariant="normal" xref="A4.Ex35.m2.2.2.1.1.3.2.cmml">Φ</mi><mo id="A4.Ex35.m2.2.2.1.1.3.1" xref="A4.Ex35.m2.2.2.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex35.m2.2.2.1.1.3.3.2" xref="A4.Ex35.m2.1.1.cmml"><mo id="A4.Ex35.m2.2.2.1.1.3.3.2.1" xref="A4.Ex35.m2.1.1.cmml">(</mo><mstyle displaystyle="true" id="A4.Ex35.m2.1.1" xref="A4.Ex35.m2.1.1.cmml"><mfrac id="A4.Ex35.m2.1.1a" xref="A4.Ex35.m2.1.1.cmml"><mrow id="A4.Ex35.m2.1.1.2" xref="A4.Ex35.m2.1.1.2.cmml"><mo id="A4.Ex35.m2.1.1.2a" xref="A4.Ex35.m2.1.1.2.cmml">−</mo><mrow id="A4.Ex35.m2.1.1.2.2" xref="A4.Ex35.m2.1.1.2.2.cmml"><mi id="A4.Ex35.m2.1.1.2.2.2" xref="A4.Ex35.m2.1.1.2.2.2.cmml">ρ</mi><mo id="A4.Ex35.m2.1.1.2.2.1" xref="A4.Ex35.m2.1.1.2.2.1.cmml">⁢</mo><mi id="A4.Ex35.m2.1.1.2.2.3" xref="A4.Ex35.m2.1.1.2.2.3.cmml">x</mi></mrow></mrow><mrow id="A4.Ex35.m2.1.1.3" xref="A4.Ex35.m2.1.1.3.cmml"><mi id="A4.Ex35.m2.1.1.3.2" xref="A4.Ex35.m2.1.1.3.2.cmml">b</mi><mo id="A4.Ex35.m2.1.1.3.1" xref="A4.Ex35.m2.1.1.3.1.cmml">⁢</mo><msqrt id="A4.Ex35.m2.1.1.3.3" xref="A4.Ex35.m2.1.1.3.3.cmml"><mrow id="A4.Ex35.m2.1.1.3.3.2" xref="A4.Ex35.m2.1.1.3.3.2.cmml"><mn id="A4.Ex35.m2.1.1.3.3.2.2" xref="A4.Ex35.m2.1.1.3.3.2.2.cmml">1</mn><mo id="A4.Ex35.m2.1.1.3.3.2.1" xref="A4.Ex35.m2.1.1.3.3.2.1.cmml">+</mo><mi id="A4.Ex35.m2.1.1.3.3.2.3" xref="A4.Ex35.m2.1.1.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac></mstyle><mo id="A4.Ex35.m2.2.2.1.1.3.3.2.2" xref="A4.Ex35.m2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex35.m2.2.2.1.2" xref="A4.Ex35.m2.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex35.m2.2b"><apply id="A4.Ex35.m2.2.2.1.1.cmml" xref="A4.Ex35.m2.2.2.1"><eq id="A4.Ex35.m2.2.2.1.1.1.cmml" xref="A4.Ex35.m2.2.2.1.1.1"></eq><csymbol cd="latexml" id="A4.Ex35.m2.2.2.1.1.2.cmml" xref="A4.Ex35.m2.2.2.1.1.2">absent</csymbol><apply id="A4.Ex35.m2.2.2.1.1.3.cmml" xref="A4.Ex35.m2.2.2.1.1.3"><times id="A4.Ex35.m2.2.2.1.1.3.1.cmml" xref="A4.Ex35.m2.2.2.1.1.3.1"></times><ci id="A4.Ex35.m2.2.2.1.1.3.2.cmml" xref="A4.Ex35.m2.2.2.1.1.3.2">Φ</ci><apply id="A4.Ex35.m2.1.1.cmml" xref="A4.Ex35.m2.2.2.1.1.3.3.2"><divide id="A4.Ex35.m2.1.1.1.cmml" xref="A4.Ex35.m2.2.2.1.1.3.3.2"></divide><apply id="A4.Ex35.m2.1.1.2.cmml" xref="A4.Ex35.m2.1.1.2"><minus id="A4.Ex35.m2.1.1.2.1.cmml" xref="A4.Ex35.m2.1.1.2"></minus><apply id="A4.Ex35.m2.1.1.2.2.cmml" xref="A4.Ex35.m2.1.1.2.2"><times id="A4.Ex35.m2.1.1.2.2.1.cmml" xref="A4.Ex35.m2.1.1.2.2.1"></times><ci id="A4.Ex35.m2.1.1.2.2.2.cmml" xref="A4.Ex35.m2.1.1.2.2.2">𝜌</ci><ci id="A4.Ex35.m2.1.1.2.2.3.cmml" xref="A4.Ex35.m2.1.1.2.2.3">𝑥</ci></apply></apply><apply id="A4.Ex35.m2.1.1.3.cmml" xref="A4.Ex35.m2.1.1.3"><times id="A4.Ex35.m2.1.1.3.1.cmml" xref="A4.Ex35.m2.1.1.3.1"></times><ci id="A4.Ex35.m2.1.1.3.2.cmml" xref="A4.Ex35.m2.1.1.3.2">𝑏</ci><apply id="A4.Ex35.m2.1.1.3.3.cmml" xref="A4.Ex35.m2.1.1.3.3"><root id="A4.Ex35.m2.1.1.3.3a.cmml" xref="A4.Ex35.m2.1.1.3.3"></root><apply id="A4.Ex35.m2.1.1.3.3.2.cmml" xref="A4.Ex35.m2.1.1.3.3.2"><plus id="A4.Ex35.m2.1.1.3.3.2.1.cmml" xref="A4.Ex35.m2.1.1.3.3.2.1"></plus><cn id="A4.Ex35.m2.1.1.3.3.2.2.cmml" type="integer" xref="A4.Ex35.m2.1.1.3.3.2.2">1</cn><ci id="A4.Ex35.m2.1.1.3.3.2.3.cmml" xref="A4.Ex35.m2.1.1.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex35.m2.2c">\displaystyle=\Phi\left(\frac{-\rho x}{b\sqrt{1+\rho}}\right),</annotation><annotation encoding="application/x-llamapun" id="A4.Ex35.m2.2d">= roman_Φ ( divide start_ARG - italic_ρ italic_x end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ end_ARG end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\bar{P}(0;x)" class="ltx_Math" display="inline" id="A4.Ex36.m1.2"><semantics id="A4.Ex36.m1.2a"><mrow id="A4.Ex36.m1.2.3" xref="A4.Ex36.m1.2.3.cmml"><mover accent="true" id="A4.Ex36.m1.2.3.2" xref="A4.Ex36.m1.2.3.2.cmml"><mi id="A4.Ex36.m1.2.3.2.2" xref="A4.Ex36.m1.2.3.2.2.cmml">P</mi><mo id="A4.Ex36.m1.2.3.2.1" xref="A4.Ex36.m1.2.3.2.1.cmml">¯</mo></mover><mo id="A4.Ex36.m1.2.3.1" xref="A4.Ex36.m1.2.3.1.cmml">⁢</mo><mrow id="A4.Ex36.m1.2.3.3.2" xref="A4.Ex36.m1.2.3.3.1.cmml"><mo id="A4.Ex36.m1.2.3.3.2.1" stretchy="false" xref="A4.Ex36.m1.2.3.3.1.cmml">(</mo><mn id="A4.Ex36.m1.1.1" xref="A4.Ex36.m1.1.1.cmml">0</mn><mo id="A4.Ex36.m1.2.3.3.2.2" xref="A4.Ex36.m1.2.3.3.1.cmml">;</mo><mi id="A4.Ex36.m1.2.2" xref="A4.Ex36.m1.2.2.cmml">x</mi><mo id="A4.Ex36.m1.2.3.3.2.3" stretchy="false" xref="A4.Ex36.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex36.m1.2b"><apply id="A4.Ex36.m1.2.3.cmml" xref="A4.Ex36.m1.2.3"><times id="A4.Ex36.m1.2.3.1.cmml" xref="A4.Ex36.m1.2.3.1"></times><apply id="A4.Ex36.m1.2.3.2.cmml" xref="A4.Ex36.m1.2.3.2"><ci id="A4.Ex36.m1.2.3.2.1.cmml" xref="A4.Ex36.m1.2.3.2.1">¯</ci><ci id="A4.Ex36.m1.2.3.2.2.cmml" xref="A4.Ex36.m1.2.3.2.2">𝑃</ci></apply><list id="A4.Ex36.m1.2.3.3.1.cmml" xref="A4.Ex36.m1.2.3.3.2"><cn id="A4.Ex36.m1.1.1.cmml" type="integer" xref="A4.Ex36.m1.1.1">0</cn><ci id="A4.Ex36.m1.2.2.cmml" xref="A4.Ex36.m1.2.2">𝑥</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex36.m1.2c">\displaystyle\bar{P}(0;x)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex36.m1.2d">over¯ start_ARG italic_P end_ARG ( 0 ; italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Phi\left(\frac{-\rho^{2}}{b\sqrt{(1+\rho^{2})(1+\rho)}}\right)." class="ltx_Math" display="inline" id="A4.Ex36.m2.3"><semantics id="A4.Ex36.m2.3a"><mrow id="A4.Ex36.m2.3.3.1" xref="A4.Ex36.m2.3.3.1.1.cmml"><mrow id="A4.Ex36.m2.3.3.1.1" xref="A4.Ex36.m2.3.3.1.1.cmml"><mi id="A4.Ex36.m2.3.3.1.1.2" xref="A4.Ex36.m2.3.3.1.1.2.cmml"></mi><mo id="A4.Ex36.m2.3.3.1.1.1" xref="A4.Ex36.m2.3.3.1.1.1.cmml">=</mo><mrow id="A4.Ex36.m2.3.3.1.1.3" xref="A4.Ex36.m2.3.3.1.1.3.cmml"><mi id="A4.Ex36.m2.3.3.1.1.3.2" mathvariant="normal" xref="A4.Ex36.m2.3.3.1.1.3.2.cmml">Φ</mi><mo id="A4.Ex36.m2.3.3.1.1.3.1" xref="A4.Ex36.m2.3.3.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex36.m2.3.3.1.1.3.3.2" xref="A4.Ex36.m2.2.2.cmml"><mo id="A4.Ex36.m2.3.3.1.1.3.3.2.1" xref="A4.Ex36.m2.2.2.cmml">(</mo><mstyle displaystyle="true" id="A4.Ex36.m2.2.2" xref="A4.Ex36.m2.2.2.cmml"><mfrac id="A4.Ex36.m2.2.2a" 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id="A4.Ex36.m2.2.2.2.2.2.2.1.1.3.cmml" xref="A4.Ex36.m2.2.2.2.2.2.2.1.1.3">𝜌</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex36.m2.3c">\displaystyle=\Phi\left(\frac{-\rho^{2}}{b\sqrt{(1+\rho^{2})(1+\rho)}}\right).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex36.m2.3d">= roman_Φ ( divide start_ARG - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_b square-root start_ARG ( 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( 1 + italic_ρ ) 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="A4.SS1.5.p1.12">Thus <math alttext="P(0;x)=\bar{P}(0;x)" class="ltx_Math" display="inline" id="A4.SS1.5.p1.6.m1.4"><semantics id="A4.SS1.5.p1.6.m1.4a"><mrow id="A4.SS1.5.p1.6.m1.4.5" xref="A4.SS1.5.p1.6.m1.4.5.cmml"><mrow id="A4.SS1.5.p1.6.m1.4.5.2" 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xref="A4.SS1.5.p1.6.m1.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.6.m1.4.5.3.2.1" xref="A4.SS1.5.p1.6.m1.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.5.p1.6.m1.4.5.3.1" xref="A4.SS1.5.p1.6.m1.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.6.m1.4.5.3.3.2" xref="A4.SS1.5.p1.6.m1.4.5.3.3.1.cmml"><mo id="A4.SS1.5.p1.6.m1.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.5.p1.6.m1.4.5.3.3.1.cmml">(</mo><mn id="A4.SS1.5.p1.6.m1.3.3" xref="A4.SS1.5.p1.6.m1.3.3.cmml">0</mn><mo id="A4.SS1.5.p1.6.m1.4.5.3.3.2.2" xref="A4.SS1.5.p1.6.m1.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.6.m1.4.4" xref="A4.SS1.5.p1.6.m1.4.4.cmml">x</mi><mo id="A4.SS1.5.p1.6.m1.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.5.p1.6.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.6.m1.4b"><apply id="A4.SS1.5.p1.6.m1.4.5.cmml" xref="A4.SS1.5.p1.6.m1.4.5"><eq id="A4.SS1.5.p1.6.m1.4.5.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.1"></eq><apply id="A4.SS1.5.p1.6.m1.4.5.2.cmml" xref="A4.SS1.5.p1.6.m1.4.5.2"><times id="A4.SS1.5.p1.6.m1.4.5.2.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.2.1"></times><ci id="A4.SS1.5.p1.6.m1.4.5.2.2.cmml" xref="A4.SS1.5.p1.6.m1.4.5.2.2">𝑃</ci><list id="A4.SS1.5.p1.6.m1.4.5.2.3.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.2.3.2"><cn id="A4.SS1.5.p1.6.m1.1.1.cmml" type="integer" xref="A4.SS1.5.p1.6.m1.1.1">0</cn><ci id="A4.SS1.5.p1.6.m1.2.2.cmml" xref="A4.SS1.5.p1.6.m1.2.2">𝑥</ci></list></apply><apply id="A4.SS1.5.p1.6.m1.4.5.3.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3"><times id="A4.SS1.5.p1.6.m1.4.5.3.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3.1"></times><apply id="A4.SS1.5.p1.6.m1.4.5.3.2.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3.2"><ci id="A4.SS1.5.p1.6.m1.4.5.3.2.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3.2.1">¯</ci><ci id="A4.SS1.5.p1.6.m1.4.5.3.2.2.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.5.p1.6.m1.4.5.3.3.1.cmml" xref="A4.SS1.5.p1.6.m1.4.5.3.3.2"><cn id="A4.SS1.5.p1.6.m1.3.3.cmml" type="integer" xref="A4.SS1.5.p1.6.m1.3.3">0</cn><ci id="A4.SS1.5.p1.6.m1.4.4.cmml" xref="A4.SS1.5.p1.6.m1.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.6.m1.4c">P(0;x)=\bar{P}(0;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.6.m1.4d">italic_P ( 0 ; italic_x ) = over¯ start_ARG italic_P end_ARG ( 0 ; italic_x )</annotation></semantics></math> at a unique point which can be verified with simple algebra to be <math alttext="x=\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.5.p1.7.m2.1"><semantics id="A4.SS1.5.p1.7.m2.1a"><mrow id="A4.SS1.5.p1.7.m2.1.1" xref="A4.SS1.5.p1.7.m2.1.1.cmml"><mi id="A4.SS1.5.p1.7.m2.1.1.2" xref="A4.SS1.5.p1.7.m2.1.1.2.cmml">x</mi><mo id="A4.SS1.5.p1.7.m2.1.1.1" xref="A4.SS1.5.p1.7.m2.1.1.1.cmml">=</mo><mover accent="true" id="A4.SS1.5.p1.7.m2.1.1.3" xref="A4.SS1.5.p1.7.m2.1.1.3.cmml"><mi id="A4.SS1.5.p1.7.m2.1.1.3.2" xref="A4.SS1.5.p1.7.m2.1.1.3.2.cmml">τ</mi><mo id="A4.SS1.5.p1.7.m2.1.1.3.1" xref="A4.SS1.5.p1.7.m2.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.7.m2.1b"><apply id="A4.SS1.5.p1.7.m2.1.1.cmml" xref="A4.SS1.5.p1.7.m2.1.1"><eq id="A4.SS1.5.p1.7.m2.1.1.1.cmml" xref="A4.SS1.5.p1.7.m2.1.1.1"></eq><ci id="A4.SS1.5.p1.7.m2.1.1.2.cmml" xref="A4.SS1.5.p1.7.m2.1.1.2">𝑥</ci><apply id="A4.SS1.5.p1.7.m2.1.1.3.cmml" xref="A4.SS1.5.p1.7.m2.1.1.3"><ci id="A4.SS1.5.p1.7.m2.1.1.3.1.cmml" xref="A4.SS1.5.p1.7.m2.1.1.3.1">^</ci><ci id="A4.SS1.5.p1.7.m2.1.1.3.2.cmml" xref="A4.SS1.5.p1.7.m2.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.7.m2.1c">x=\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.7.m2.1d">italic_x = over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. One can check for the specified value of <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.5.p1.8.m3.1"><semantics id="A4.SS1.5.p1.8.m3.1a"><mover accent="true" id="A4.SS1.5.p1.8.m3.1.1" xref="A4.SS1.5.p1.8.m3.1.1.cmml"><mi id="A4.SS1.5.p1.8.m3.1.1.2" xref="A4.SS1.5.p1.8.m3.1.1.2.cmml">τ</mi><mo id="A4.SS1.5.p1.8.m3.1.1.1" xref="A4.SS1.5.p1.8.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.8.m3.1b"><apply id="A4.SS1.5.p1.8.m3.1.1.cmml" xref="A4.SS1.5.p1.8.m3.1.1"><ci id="A4.SS1.5.p1.8.m3.1.1.1.cmml" xref="A4.SS1.5.p1.8.m3.1.1.1">^</ci><ci id="A4.SS1.5.p1.8.m3.1.1.2.cmml" xref="A4.SS1.5.p1.8.m3.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.8.m3.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.8.m3.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math>, <math alttext="P(\tau;x)\geq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.5.p1.9.m4.4"><semantics id="A4.SS1.5.p1.9.m4.4a"><mrow id="A4.SS1.5.p1.9.m4.4.5" xref="A4.SS1.5.p1.9.m4.4.5.cmml"><mrow id="A4.SS1.5.p1.9.m4.4.5.2" xref="A4.SS1.5.p1.9.m4.4.5.2.cmml"><mi id="A4.SS1.5.p1.9.m4.4.5.2.2" xref="A4.SS1.5.p1.9.m4.4.5.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.9.m4.4.5.2.1" xref="A4.SS1.5.p1.9.m4.4.5.2.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.9.m4.4.5.2.3.2" xref="A4.SS1.5.p1.9.m4.4.5.2.3.1.cmml"><mo id="A4.SS1.5.p1.9.m4.4.5.2.3.2.1" stretchy="false" xref="A4.SS1.5.p1.9.m4.4.5.2.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.9.m4.1.1" xref="A4.SS1.5.p1.9.m4.1.1.cmml">τ</mi><mo id="A4.SS1.5.p1.9.m4.4.5.2.3.2.2" xref="A4.SS1.5.p1.9.m4.4.5.2.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.9.m4.2.2" xref="A4.SS1.5.p1.9.m4.2.2.cmml">x</mi><mo id="A4.SS1.5.p1.9.m4.4.5.2.3.2.3" stretchy="false" xref="A4.SS1.5.p1.9.m4.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.5.p1.9.m4.4.5.1" xref="A4.SS1.5.p1.9.m4.4.5.1.cmml">≥</mo><mrow id="A4.SS1.5.p1.9.m4.4.5.3" xref="A4.SS1.5.p1.9.m4.4.5.3.cmml"><mover accent="true" id="A4.SS1.5.p1.9.m4.4.5.3.2" xref="A4.SS1.5.p1.9.m4.4.5.3.2.cmml"><mi id="A4.SS1.5.p1.9.m4.4.5.3.2.2" xref="A4.SS1.5.p1.9.m4.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.9.m4.4.5.3.2.1" xref="A4.SS1.5.p1.9.m4.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.5.p1.9.m4.4.5.3.1" xref="A4.SS1.5.p1.9.m4.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.9.m4.4.5.3.3.2" xref="A4.SS1.5.p1.9.m4.4.5.3.3.1.cmml"><mo id="A4.SS1.5.p1.9.m4.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.5.p1.9.m4.4.5.3.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.9.m4.3.3" xref="A4.SS1.5.p1.9.m4.3.3.cmml">τ</mi><mo id="A4.SS1.5.p1.9.m4.4.5.3.3.2.2" xref="A4.SS1.5.p1.9.m4.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.9.m4.4.4" xref="A4.SS1.5.p1.9.m4.4.4.cmml">x</mi><mo id="A4.SS1.5.p1.9.m4.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.5.p1.9.m4.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.9.m4.4b"><apply id="A4.SS1.5.p1.9.m4.4.5.cmml" xref="A4.SS1.5.p1.9.m4.4.5"><geq id="A4.SS1.5.p1.9.m4.4.5.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.1"></geq><apply id="A4.SS1.5.p1.9.m4.4.5.2.cmml" xref="A4.SS1.5.p1.9.m4.4.5.2"><times id="A4.SS1.5.p1.9.m4.4.5.2.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.2.1"></times><ci id="A4.SS1.5.p1.9.m4.4.5.2.2.cmml" xref="A4.SS1.5.p1.9.m4.4.5.2.2">𝑃</ci><list id="A4.SS1.5.p1.9.m4.4.5.2.3.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.2.3.2"><ci id="A4.SS1.5.p1.9.m4.1.1.cmml" xref="A4.SS1.5.p1.9.m4.1.1">𝜏</ci><ci id="A4.SS1.5.p1.9.m4.2.2.cmml" xref="A4.SS1.5.p1.9.m4.2.2">𝑥</ci></list></apply><apply id="A4.SS1.5.p1.9.m4.4.5.3.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3"><times id="A4.SS1.5.p1.9.m4.4.5.3.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3.1"></times><apply id="A4.SS1.5.p1.9.m4.4.5.3.2.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3.2"><ci id="A4.SS1.5.p1.9.m4.4.5.3.2.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3.2.1">¯</ci><ci id="A4.SS1.5.p1.9.m4.4.5.3.2.2.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.5.p1.9.m4.4.5.3.3.1.cmml" xref="A4.SS1.5.p1.9.m4.4.5.3.3.2"><ci id="A4.SS1.5.p1.9.m4.3.3.cmml" xref="A4.SS1.5.p1.9.m4.3.3">𝜏</ci><ci id="A4.SS1.5.p1.9.m4.4.4.cmml" xref="A4.SS1.5.p1.9.m4.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.9.m4.4c">P(\tau;x)\geq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.9.m4.4d">italic_P ( italic_τ ; italic_x ) ≥ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math> for <math alttext="x\leq\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.5.p1.10.m5.1"><semantics id="A4.SS1.5.p1.10.m5.1a"><mrow id="A4.SS1.5.p1.10.m5.1.1" xref="A4.SS1.5.p1.10.m5.1.1.cmml"><mi id="A4.SS1.5.p1.10.m5.1.1.2" xref="A4.SS1.5.p1.10.m5.1.1.2.cmml">x</mi><mo id="A4.SS1.5.p1.10.m5.1.1.1" xref="A4.SS1.5.p1.10.m5.1.1.1.cmml">≤</mo><mover accent="true" id="A4.SS1.5.p1.10.m5.1.1.3" xref="A4.SS1.5.p1.10.m5.1.1.3.cmml"><mi id="A4.SS1.5.p1.10.m5.1.1.3.2" xref="A4.SS1.5.p1.10.m5.1.1.3.2.cmml">τ</mi><mo id="A4.SS1.5.p1.10.m5.1.1.3.1" xref="A4.SS1.5.p1.10.m5.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.10.m5.1b"><apply id="A4.SS1.5.p1.10.m5.1.1.cmml" xref="A4.SS1.5.p1.10.m5.1.1"><leq id="A4.SS1.5.p1.10.m5.1.1.1.cmml" xref="A4.SS1.5.p1.10.m5.1.1.1"></leq><ci id="A4.SS1.5.p1.10.m5.1.1.2.cmml" xref="A4.SS1.5.p1.10.m5.1.1.2">𝑥</ci><apply id="A4.SS1.5.p1.10.m5.1.1.3.cmml" xref="A4.SS1.5.p1.10.m5.1.1.3"><ci id="A4.SS1.5.p1.10.m5.1.1.3.1.cmml" xref="A4.SS1.5.p1.10.m5.1.1.3.1">^</ci><ci id="A4.SS1.5.p1.10.m5.1.1.3.2.cmml" xref="A4.SS1.5.p1.10.m5.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.10.m5.1c">x\leq\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.10.m5.1d">italic_x ≤ over^ start_ARG italic_τ end_ARG</annotation></semantics></math> and <math alttext="P(\tau;x)\leq\bar{P}(\tau;x)" class="ltx_Math" display="inline" id="A4.SS1.5.p1.11.m6.4"><semantics id="A4.SS1.5.p1.11.m6.4a"><mrow id="A4.SS1.5.p1.11.m6.4.5" xref="A4.SS1.5.p1.11.m6.4.5.cmml"><mrow id="A4.SS1.5.p1.11.m6.4.5.2" xref="A4.SS1.5.p1.11.m6.4.5.2.cmml"><mi id="A4.SS1.5.p1.11.m6.4.5.2.2" xref="A4.SS1.5.p1.11.m6.4.5.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.11.m6.4.5.2.1" xref="A4.SS1.5.p1.11.m6.4.5.2.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.11.m6.4.5.2.3.2" xref="A4.SS1.5.p1.11.m6.4.5.2.3.1.cmml"><mo id="A4.SS1.5.p1.11.m6.4.5.2.3.2.1" stretchy="false" xref="A4.SS1.5.p1.11.m6.4.5.2.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.11.m6.1.1" xref="A4.SS1.5.p1.11.m6.1.1.cmml">τ</mi><mo id="A4.SS1.5.p1.11.m6.4.5.2.3.2.2" xref="A4.SS1.5.p1.11.m6.4.5.2.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.11.m6.2.2" xref="A4.SS1.5.p1.11.m6.2.2.cmml">x</mi><mo id="A4.SS1.5.p1.11.m6.4.5.2.3.2.3" stretchy="false" xref="A4.SS1.5.p1.11.m6.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.5.p1.11.m6.4.5.1" xref="A4.SS1.5.p1.11.m6.4.5.1.cmml">≤</mo><mrow id="A4.SS1.5.p1.11.m6.4.5.3" xref="A4.SS1.5.p1.11.m6.4.5.3.cmml"><mover accent="true" id="A4.SS1.5.p1.11.m6.4.5.3.2" xref="A4.SS1.5.p1.11.m6.4.5.3.2.cmml"><mi id="A4.SS1.5.p1.11.m6.4.5.3.2.2" xref="A4.SS1.5.p1.11.m6.4.5.3.2.2.cmml">P</mi><mo id="A4.SS1.5.p1.11.m6.4.5.3.2.1" xref="A4.SS1.5.p1.11.m6.4.5.3.2.1.cmml">¯</mo></mover><mo id="A4.SS1.5.p1.11.m6.4.5.3.1" xref="A4.SS1.5.p1.11.m6.4.5.3.1.cmml">⁢</mo><mrow id="A4.SS1.5.p1.11.m6.4.5.3.3.2" xref="A4.SS1.5.p1.11.m6.4.5.3.3.1.cmml"><mo id="A4.SS1.5.p1.11.m6.4.5.3.3.2.1" stretchy="false" xref="A4.SS1.5.p1.11.m6.4.5.3.3.1.cmml">(</mo><mi id="A4.SS1.5.p1.11.m6.3.3" xref="A4.SS1.5.p1.11.m6.3.3.cmml">τ</mi><mo id="A4.SS1.5.p1.11.m6.4.5.3.3.2.2" xref="A4.SS1.5.p1.11.m6.4.5.3.3.1.cmml">;</mo><mi id="A4.SS1.5.p1.11.m6.4.4" xref="A4.SS1.5.p1.11.m6.4.4.cmml">x</mi><mo id="A4.SS1.5.p1.11.m6.4.5.3.3.2.3" stretchy="false" xref="A4.SS1.5.p1.11.m6.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" 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xref="A4.SS1.5.p1.11.m6.4.5.3.2.2">𝑃</ci></apply><list id="A4.SS1.5.p1.11.m6.4.5.3.3.1.cmml" xref="A4.SS1.5.p1.11.m6.4.5.3.3.2"><ci id="A4.SS1.5.p1.11.m6.3.3.cmml" xref="A4.SS1.5.p1.11.m6.3.3">𝜏</ci><ci id="A4.SS1.5.p1.11.m6.4.4.cmml" xref="A4.SS1.5.p1.11.m6.4.4">𝑥</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.11.m6.4c">P(\tau;x)\leq\bar{P}(\tau;x)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.11.m6.4d">italic_P ( italic_τ ; italic_x ) ≤ over¯ start_ARG italic_P end_ARG ( italic_τ ; italic_x )</annotation></semantics></math> for <math alttext="x\geq\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.5.p1.12.m7.1"><semantics id="A4.SS1.5.p1.12.m7.1a"><mrow id="A4.SS1.5.p1.12.m7.1.1" xref="A4.SS1.5.p1.12.m7.1.1.cmml"><mi id="A4.SS1.5.p1.12.m7.1.1.2" xref="A4.SS1.5.p1.12.m7.1.1.2.cmml">x</mi><mo id="A4.SS1.5.p1.12.m7.1.1.1" xref="A4.SS1.5.p1.12.m7.1.1.1.cmml">≥</mo><mover accent="true" id="A4.SS1.5.p1.12.m7.1.1.3" xref="A4.SS1.5.p1.12.m7.1.1.3.cmml"><mi id="A4.SS1.5.p1.12.m7.1.1.3.2" xref="A4.SS1.5.p1.12.m7.1.1.3.2.cmml">τ</mi><mo id="A4.SS1.5.p1.12.m7.1.1.3.1" xref="A4.SS1.5.p1.12.m7.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p1.12.m7.1b"><apply id="A4.SS1.5.p1.12.m7.1.1.cmml" xref="A4.SS1.5.p1.12.m7.1.1"><geq id="A4.SS1.5.p1.12.m7.1.1.1.cmml" xref="A4.SS1.5.p1.12.m7.1.1.1"></geq><ci id="A4.SS1.5.p1.12.m7.1.1.2.cmml" xref="A4.SS1.5.p1.12.m7.1.1.2">𝑥</ci><apply id="A4.SS1.5.p1.12.m7.1.1.3.cmml" xref="A4.SS1.5.p1.12.m7.1.1.3"><ci id="A4.SS1.5.p1.12.m7.1.1.3.1.cmml" xref="A4.SS1.5.p1.12.m7.1.1.3.1">^</ci><ci id="A4.SS1.5.p1.12.m7.1.1.3.2.cmml" xref="A4.SS1.5.p1.12.m7.1.1.3.2">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p1.12.m7.1c">x\geq\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p1.12.m7.1d">italic_x ≥ over^ start_ARG italic_τ end_ARG</annotation></semantics></math>. The result follows. ∎</p> </div> </div> <div class="ltx_para" id="A4.SS1.p4"> <p class="ltx_p" id="A4.SS1.p4.1">Now we formally compare convergence of the DG and RBTS mechanisms under the Gaussian model.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="Thmproposition13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmproposition13.1.1.1">Proposition 13</span></span><span class="ltx_text ltx_font_bold" id="Thmproposition13.2.2">.</span> </h6> <div class="ltx_para" id="Thmproposition13.p1"> <p class="ltx_p" id="Thmproposition13.p1.2">In the Gaussian model, for any initial threshold <math alttext="\tau(0)\neq 0" class="ltx_Math" display="inline" id="Thmproposition13.p1.1.m1.1"><semantics id="Thmproposition13.p1.1.m1.1a"><mrow id="Thmproposition13.p1.1.m1.1.2" xref="Thmproposition13.p1.1.m1.1.2.cmml"><mrow id="Thmproposition13.p1.1.m1.1.2.2" xref="Thmproposition13.p1.1.m1.1.2.2.cmml"><mi id="Thmproposition13.p1.1.m1.1.2.2.2" xref="Thmproposition13.p1.1.m1.1.2.2.2.cmml">τ</mi><mo id="Thmproposition13.p1.1.m1.1.2.2.1" xref="Thmproposition13.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="Thmproposition13.p1.1.m1.1.2.2.3.2" xref="Thmproposition13.p1.1.m1.1.2.2.cmml"><mo id="Thmproposition13.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="Thmproposition13.p1.1.m1.1.2.2.cmml">(</mo><mn id="Thmproposition13.p1.1.m1.1.1" xref="Thmproposition13.p1.1.m1.1.1.cmml">0</mn><mo id="Thmproposition13.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="Thmproposition13.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproposition13.p1.1.m1.1.2.1" xref="Thmproposition13.p1.1.m1.1.2.1.cmml">≠</mo><mn id="Thmproposition13.p1.1.m1.1.2.3" xref="Thmproposition13.p1.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition13.p1.1.m1.1b"><apply id="Thmproposition13.p1.1.m1.1.2.cmml" xref="Thmproposition13.p1.1.m1.1.2"><neq id="Thmproposition13.p1.1.m1.1.2.1.cmml" xref="Thmproposition13.p1.1.m1.1.2.1"></neq><apply id="Thmproposition13.p1.1.m1.1.2.2.cmml" xref="Thmproposition13.p1.1.m1.1.2.2"><times id="Thmproposition13.p1.1.m1.1.2.2.1.cmml" xref="Thmproposition13.p1.1.m1.1.2.2.1"></times><ci id="Thmproposition13.p1.1.m1.1.2.2.2.cmml" xref="Thmproposition13.p1.1.m1.1.2.2.2">𝜏</ci><cn id="Thmproposition13.p1.1.m1.1.1.cmml" type="integer" xref="Thmproposition13.p1.1.m1.1.1">0</cn></apply><cn id="Thmproposition13.p1.1.m1.1.2.3.cmml" type="integer" xref="Thmproposition13.p1.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition13.p1.1.m1.1c">\tau(0)\neq 0</annotation><annotation encoding="application/x-llamapun" id="Thmproposition13.p1.1.m1.1d">italic_τ ( 0 ) ≠ 0</annotation></semantics></math>, convergence to the stable equilibrium <math alttext="\tau=0" class="ltx_Math" display="inline" id="Thmproposition13.p1.2.m2.1"><semantics id="Thmproposition13.p1.2.m2.1a"><mrow id="Thmproposition13.p1.2.m2.1.1" xref="Thmproposition13.p1.2.m2.1.1.cmml"><mi id="Thmproposition13.p1.2.m2.1.1.2" xref="Thmproposition13.p1.2.m2.1.1.2.cmml">τ</mi><mo id="Thmproposition13.p1.2.m2.1.1.1" xref="Thmproposition13.p1.2.m2.1.1.1.cmml">=</mo><mn id="Thmproposition13.p1.2.m2.1.1.3" xref="Thmproposition13.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition13.p1.2.m2.1b"><apply id="Thmproposition13.p1.2.m2.1.1.cmml" xref="Thmproposition13.p1.2.m2.1.1"><eq id="Thmproposition13.p1.2.m2.1.1.1.cmml" xref="Thmproposition13.p1.2.m2.1.1.1"></eq><ci id="Thmproposition13.p1.2.m2.1.1.2.cmml" xref="Thmproposition13.p1.2.m2.1.1.2">𝜏</ci><cn id="Thmproposition13.p1.2.m2.1.1.3.cmml" type="integer" xref="Thmproposition13.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition13.p1.2.m2.1c">\tau=0</annotation><annotation encoding="application/x-llamapun" id="Thmproposition13.p1.2.m2.1d">italic_τ = 0</annotation></semantics></math> is strictly slower in the RBTS mechanism than in the DG mechanism.</p> </div> </div> <div class="ltx_proof" id="A4.SS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A4.SS1.6.p1"> <p class="ltx_p" id="A4.SS1.6.p1.3">By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.16280v1#Thmproposition12" title="Proposition 12. ‣ D.1 Gaussian Model ‣ Appendix D Omitted Proofs for RBTS ‣ Binary-Report Peer Prediction for Real-Valued Signal Spaces"><span class="ltx_text ltx_ref_tag">12</span></a>, in the RBTS mechanism a best response <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.6.p1.1.m1.1"><semantics id="A4.SS1.6.p1.1.m1.1a"><mover accent="true" id="A4.SS1.6.p1.1.m1.1.1" xref="A4.SS1.6.p1.1.m1.1.1.cmml"><mi id="A4.SS1.6.p1.1.m1.1.1.2" xref="A4.SS1.6.p1.1.m1.1.1.2.cmml">τ</mi><mo id="A4.SS1.6.p1.1.m1.1.1.1" xref="A4.SS1.6.p1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.1.m1.1b"><apply id="A4.SS1.6.p1.1.m1.1.1.cmml" xref="A4.SS1.6.p1.1.m1.1.1"><ci id="A4.SS1.6.p1.1.m1.1.1.1.cmml" xref="A4.SS1.6.p1.1.m1.1.1.1">^</ci><ci id="A4.SS1.6.p1.1.m1.1.1.2.cmml" xref="A4.SS1.6.p1.1.m1.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.1.m1.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.1.m1.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> to all other agents playing according to threshold <math alttext="\tau" class="ltx_Math" display="inline" id="A4.SS1.6.p1.2.m2.1"><semantics id="A4.SS1.6.p1.2.m2.1a"><mi id="A4.SS1.6.p1.2.m2.1.1" xref="A4.SS1.6.p1.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.2.m2.1b"><ci id="A4.SS1.6.p1.2.m2.1.1.cmml" xref="A4.SS1.6.p1.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.2.m2.1d">italic_τ</annotation></semantics></math> is <math alttext="\hat{\tau}=m_{\textrm{RBTS}}(\rho)\tau" class="ltx_Math" display="inline" id="A4.SS1.6.p1.3.m3.1"><semantics id="A4.SS1.6.p1.3.m3.1a"><mrow id="A4.SS1.6.p1.3.m3.1.2" xref="A4.SS1.6.p1.3.m3.1.2.cmml"><mover accent="true" id="A4.SS1.6.p1.3.m3.1.2.2" xref="A4.SS1.6.p1.3.m3.1.2.2.cmml"><mi id="A4.SS1.6.p1.3.m3.1.2.2.2" xref="A4.SS1.6.p1.3.m3.1.2.2.2.cmml">τ</mi><mo id="A4.SS1.6.p1.3.m3.1.2.2.1" xref="A4.SS1.6.p1.3.m3.1.2.2.1.cmml">^</mo></mover><mo id="A4.SS1.6.p1.3.m3.1.2.1" xref="A4.SS1.6.p1.3.m3.1.2.1.cmml">=</mo><mrow id="A4.SS1.6.p1.3.m3.1.2.3" xref="A4.SS1.6.p1.3.m3.1.2.3.cmml"><msub id="A4.SS1.6.p1.3.m3.1.2.3.2" xref="A4.SS1.6.p1.3.m3.1.2.3.2.cmml"><mi id="A4.SS1.6.p1.3.m3.1.2.3.2.2" xref="A4.SS1.6.p1.3.m3.1.2.3.2.2.cmml">m</mi><mtext id="A4.SS1.6.p1.3.m3.1.2.3.2.3" xref="A4.SS1.6.p1.3.m3.1.2.3.2.3a.cmml">RBTS</mtext></msub><mo id="A4.SS1.6.p1.3.m3.1.2.3.1" xref="A4.SS1.6.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.6.p1.3.m3.1.2.3.3.2" xref="A4.SS1.6.p1.3.m3.1.2.3.cmml"><mo id="A4.SS1.6.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="A4.SS1.6.p1.3.m3.1.2.3.cmml">(</mo><mi id="A4.SS1.6.p1.3.m3.1.1" xref="A4.SS1.6.p1.3.m3.1.1.cmml">ρ</mi><mo id="A4.SS1.6.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="A4.SS1.6.p1.3.m3.1.2.3.cmml">)</mo></mrow><mo id="A4.SS1.6.p1.3.m3.1.2.3.1a" xref="A4.SS1.6.p1.3.m3.1.2.3.1.cmml">⁢</mo><mi id="A4.SS1.6.p1.3.m3.1.2.3.4" xref="A4.SS1.6.p1.3.m3.1.2.3.4.cmml">τ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.3.m3.1b"><apply id="A4.SS1.6.p1.3.m3.1.2.cmml" xref="A4.SS1.6.p1.3.m3.1.2"><eq id="A4.SS1.6.p1.3.m3.1.2.1.cmml" xref="A4.SS1.6.p1.3.m3.1.2.1"></eq><apply id="A4.SS1.6.p1.3.m3.1.2.2.cmml" xref="A4.SS1.6.p1.3.m3.1.2.2"><ci id="A4.SS1.6.p1.3.m3.1.2.2.1.cmml" xref="A4.SS1.6.p1.3.m3.1.2.2.1">^</ci><ci id="A4.SS1.6.p1.3.m3.1.2.2.2.cmml" xref="A4.SS1.6.p1.3.m3.1.2.2.2">𝜏</ci></apply><apply id="A4.SS1.6.p1.3.m3.1.2.3.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3"><times id="A4.SS1.6.p1.3.m3.1.2.3.1.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.1"></times><apply id="A4.SS1.6.p1.3.m3.1.2.3.2.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.6.p1.3.m3.1.2.3.2.1.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.2">subscript</csymbol><ci id="A4.SS1.6.p1.3.m3.1.2.3.2.2.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.2.2">𝑚</ci><ci id="A4.SS1.6.p1.3.m3.1.2.3.2.3a.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.2.3"><mtext id="A4.SS1.6.p1.3.m3.1.2.3.2.3.cmml" mathsize="70%" xref="A4.SS1.6.p1.3.m3.1.2.3.2.3">RBTS</mtext></ci></apply><ci id="A4.SS1.6.p1.3.m3.1.1.cmml" xref="A4.SS1.6.p1.3.m3.1.1">𝜌</ci><ci id="A4.SS1.6.p1.3.m3.1.2.3.4.cmml" xref="A4.SS1.6.p1.3.m3.1.2.3.4">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.3.m3.1c">\hat{\tau}=m_{\textrm{RBTS}}(\rho)\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.3.m3.1d">over^ start_ARG italic_τ end_ARG = italic_m start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT ( italic_ρ ) italic_τ</annotation></semantics></math> for</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex37"> <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="m_{\textrm{RBTS}}(\rho)=\frac{\sqrt{1+\rho^{2}}-1}{\rho(\sqrt{1+\rho^{2}}-\rho% )}." class="ltx_Math" display="block" id="A4.Ex37.m1.3"><semantics id="A4.Ex37.m1.3a"><mrow id="A4.Ex37.m1.3.3.1" xref="A4.Ex37.m1.3.3.1.1.cmml"><mrow id="A4.Ex37.m1.3.3.1.1" xref="A4.Ex37.m1.3.3.1.1.cmml"><mrow id="A4.Ex37.m1.3.3.1.1.2" xref="A4.Ex37.m1.3.3.1.1.2.cmml"><msub id="A4.Ex37.m1.3.3.1.1.2.2" xref="A4.Ex37.m1.3.3.1.1.2.2.cmml"><mi id="A4.Ex37.m1.3.3.1.1.2.2.2" xref="A4.Ex37.m1.3.3.1.1.2.2.2.cmml">m</mi><mtext id="A4.Ex37.m1.3.3.1.1.2.2.3" xref="A4.Ex37.m1.3.3.1.1.2.2.3a.cmml">RBTS</mtext></msub><mo id="A4.Ex37.m1.3.3.1.1.2.1" 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id="A4.Ex37.m1.3c">m_{\textrm{RBTS}}(\rho)=\frac{\sqrt{1+\rho^{2}}-1}{\rho(\sqrt{1+\rho^{2}}-\rho% )}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex37.m1.3d">italic_m start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT ( italic_ρ ) = divide start_ARG square-root start_ARG 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 end_ARG start_ARG italic_ρ ( square-root start_ARG 1 + italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - italic_ρ ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.6.p1.5">Meanwhile, the best response in DG to a fixed threshold <math alttext="\tau" class="ltx_Math" display="inline" id="A4.SS1.6.p1.4.m1.1"><semantics id="A4.SS1.6.p1.4.m1.1a"><mi id="A4.SS1.6.p1.4.m1.1.1" xref="A4.SS1.6.p1.4.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.4.m1.1b"><ci id="A4.SS1.6.p1.4.m1.1.1.cmml" xref="A4.SS1.6.p1.4.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.4.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.4.m1.1d">italic_τ</annotation></semantics></math> satisfies <math alttext="P(\tau;\hat{\tau})=F(\tau)" class="ltx_Math" display="inline" id="A4.SS1.6.p1.5.m2.3"><semantics id="A4.SS1.6.p1.5.m2.3a"><mrow id="A4.SS1.6.p1.5.m2.3.4" xref="A4.SS1.6.p1.5.m2.3.4.cmml"><mrow id="A4.SS1.6.p1.5.m2.3.4.2" xref="A4.SS1.6.p1.5.m2.3.4.2.cmml"><mi id="A4.SS1.6.p1.5.m2.3.4.2.2" xref="A4.SS1.6.p1.5.m2.3.4.2.2.cmml">P</mi><mo id="A4.SS1.6.p1.5.m2.3.4.2.1" xref="A4.SS1.6.p1.5.m2.3.4.2.1.cmml">⁢</mo><mrow id="A4.SS1.6.p1.5.m2.3.4.2.3.2" xref="A4.SS1.6.p1.5.m2.3.4.2.3.1.cmml"><mo id="A4.SS1.6.p1.5.m2.3.4.2.3.2.1" stretchy="false" xref="A4.SS1.6.p1.5.m2.3.4.2.3.1.cmml">(</mo><mi id="A4.SS1.6.p1.5.m2.1.1" xref="A4.SS1.6.p1.5.m2.1.1.cmml">τ</mi><mo id="A4.SS1.6.p1.5.m2.3.4.2.3.2.2" xref="A4.SS1.6.p1.5.m2.3.4.2.3.1.cmml">;</mo><mover accent="true" id="A4.SS1.6.p1.5.m2.2.2" xref="A4.SS1.6.p1.5.m2.2.2.cmml"><mi id="A4.SS1.6.p1.5.m2.2.2.2" xref="A4.SS1.6.p1.5.m2.2.2.2.cmml">τ</mi><mo id="A4.SS1.6.p1.5.m2.2.2.1" xref="A4.SS1.6.p1.5.m2.2.2.1.cmml">^</mo></mover><mo id="A4.SS1.6.p1.5.m2.3.4.2.3.2.3" stretchy="false" xref="A4.SS1.6.p1.5.m2.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.6.p1.5.m2.3.4.1" xref="A4.SS1.6.p1.5.m2.3.4.1.cmml">=</mo><mrow id="A4.SS1.6.p1.5.m2.3.4.3" xref="A4.SS1.6.p1.5.m2.3.4.3.cmml"><mi id="A4.SS1.6.p1.5.m2.3.4.3.2" xref="A4.SS1.6.p1.5.m2.3.4.3.2.cmml">F</mi><mo id="A4.SS1.6.p1.5.m2.3.4.3.1" xref="A4.SS1.6.p1.5.m2.3.4.3.1.cmml">⁢</mo><mrow id="A4.SS1.6.p1.5.m2.3.4.3.3.2" xref="A4.SS1.6.p1.5.m2.3.4.3.cmml"><mo id="A4.SS1.6.p1.5.m2.3.4.3.3.2.1" stretchy="false" xref="A4.SS1.6.p1.5.m2.3.4.3.cmml">(</mo><mi id="A4.SS1.6.p1.5.m2.3.3" xref="A4.SS1.6.p1.5.m2.3.3.cmml">τ</mi><mo id="A4.SS1.6.p1.5.m2.3.4.3.3.2.2" stretchy="false" xref="A4.SS1.6.p1.5.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.5.m2.3b"><apply id="A4.SS1.6.p1.5.m2.3.4.cmml" xref="A4.SS1.6.p1.5.m2.3.4"><eq id="A4.SS1.6.p1.5.m2.3.4.1.cmml" xref="A4.SS1.6.p1.5.m2.3.4.1"></eq><apply id="A4.SS1.6.p1.5.m2.3.4.2.cmml" xref="A4.SS1.6.p1.5.m2.3.4.2"><times id="A4.SS1.6.p1.5.m2.3.4.2.1.cmml" xref="A4.SS1.6.p1.5.m2.3.4.2.1"></times><ci id="A4.SS1.6.p1.5.m2.3.4.2.2.cmml" xref="A4.SS1.6.p1.5.m2.3.4.2.2">𝑃</ci><list id="A4.SS1.6.p1.5.m2.3.4.2.3.1.cmml" xref="A4.SS1.6.p1.5.m2.3.4.2.3.2"><ci id="A4.SS1.6.p1.5.m2.1.1.cmml" xref="A4.SS1.6.p1.5.m2.1.1">𝜏</ci><apply id="A4.SS1.6.p1.5.m2.2.2.cmml" xref="A4.SS1.6.p1.5.m2.2.2"><ci id="A4.SS1.6.p1.5.m2.2.2.1.cmml" xref="A4.SS1.6.p1.5.m2.2.2.1">^</ci><ci id="A4.SS1.6.p1.5.m2.2.2.2.cmml" xref="A4.SS1.6.p1.5.m2.2.2.2">𝜏</ci></apply></list></apply><apply id="A4.SS1.6.p1.5.m2.3.4.3.cmml" xref="A4.SS1.6.p1.5.m2.3.4.3"><times id="A4.SS1.6.p1.5.m2.3.4.3.1.cmml" xref="A4.SS1.6.p1.5.m2.3.4.3.1"></times><ci id="A4.SS1.6.p1.5.m2.3.4.3.2.cmml" xref="A4.SS1.6.p1.5.m2.3.4.3.2">𝐹</ci><ci id="A4.SS1.6.p1.5.m2.3.3.cmml" xref="A4.SS1.6.p1.5.m2.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.5.m2.3c">P(\tau;\hat{\tau})=F(\tau)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.5.m2.3d">italic_P ( italic_τ ; over^ start_ARG italic_τ end_ARG ) = italic_F ( italic_τ )</annotation></semantics></math>, or</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex38"> <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="\Phi\left(\frac{\sqrt{\rho}\tau}{a}\right)=\Phi\left(\frac{\tau-\rho\hat{\tau}% }{b\sqrt{1+\rho}}\right)." class="ltx_Math" display="block" id="A4.Ex38.m1.3"><semantics id="A4.Ex38.m1.3a"><mrow id="A4.Ex38.m1.3.3.1" xref="A4.Ex38.m1.3.3.1.1.cmml"><mrow id="A4.Ex38.m1.3.3.1.1" xref="A4.Ex38.m1.3.3.1.1.cmml"><mrow id="A4.Ex38.m1.3.3.1.1.2" xref="A4.Ex38.m1.3.3.1.1.2.cmml"><mi id="A4.Ex38.m1.3.3.1.1.2.2" mathvariant="normal" xref="A4.Ex38.m1.3.3.1.1.2.2.cmml">Φ</mi><mo id="A4.Ex38.m1.3.3.1.1.2.1" xref="A4.Ex38.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex38.m1.3.3.1.1.2.3.2" xref="A4.Ex38.m1.1.1.cmml"><mo id="A4.Ex38.m1.3.3.1.1.2.3.2.1" xref="A4.Ex38.m1.1.1.cmml">(</mo><mfrac id="A4.Ex38.m1.1.1" xref="A4.Ex38.m1.1.1.cmml"><mrow id="A4.Ex38.m1.1.1.2" xref="A4.Ex38.m1.1.1.2.cmml"><msqrt id="A4.Ex38.m1.1.1.2.2" xref="A4.Ex38.m1.1.1.2.2.cmml"><mi id="A4.Ex38.m1.1.1.2.2.2" xref="A4.Ex38.m1.1.1.2.2.2.cmml">ρ</mi></msqrt><mo id="A4.Ex38.m1.1.1.2.1" xref="A4.Ex38.m1.1.1.2.1.cmml">⁢</mo><mi id="A4.Ex38.m1.1.1.2.3" xref="A4.Ex38.m1.1.1.2.3.cmml">τ</mi></mrow><mi id="A4.Ex38.m1.1.1.3" xref="A4.Ex38.m1.1.1.3.cmml">a</mi></mfrac><mo id="A4.Ex38.m1.3.3.1.1.2.3.2.2" xref="A4.Ex38.m1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex38.m1.3.3.1.1.1" xref="A4.Ex38.m1.3.3.1.1.1.cmml">=</mo><mrow id="A4.Ex38.m1.3.3.1.1.3" xref="A4.Ex38.m1.3.3.1.1.3.cmml"><mi id="A4.Ex38.m1.3.3.1.1.3.2" mathvariant="normal" xref="A4.Ex38.m1.3.3.1.1.3.2.cmml">Φ</mi><mo id="A4.Ex38.m1.3.3.1.1.3.1" xref="A4.Ex38.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex38.m1.3.3.1.1.3.3.2" xref="A4.Ex38.m1.2.2.cmml"><mo id="A4.Ex38.m1.3.3.1.1.3.3.2.1" xref="A4.Ex38.m1.2.2.cmml">(</mo><mfrac id="A4.Ex38.m1.2.2" xref="A4.Ex38.m1.2.2.cmml"><mrow id="A4.Ex38.m1.2.2.2" xref="A4.Ex38.m1.2.2.2.cmml"><mi id="A4.Ex38.m1.2.2.2.2" xref="A4.Ex38.m1.2.2.2.2.cmml">τ</mi><mo id="A4.Ex38.m1.2.2.2.1" xref="A4.Ex38.m1.2.2.2.1.cmml">−</mo><mrow id="A4.Ex38.m1.2.2.2.3" xref="A4.Ex38.m1.2.2.2.3.cmml"><mi id="A4.Ex38.m1.2.2.2.3.2" xref="A4.Ex38.m1.2.2.2.3.2.cmml">ρ</mi><mo id="A4.Ex38.m1.2.2.2.3.1" xref="A4.Ex38.m1.2.2.2.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex38.m1.2.2.2.3.3" xref="A4.Ex38.m1.2.2.2.3.3.cmml"><mi id="A4.Ex38.m1.2.2.2.3.3.2" xref="A4.Ex38.m1.2.2.2.3.3.2.cmml">τ</mi><mo id="A4.Ex38.m1.2.2.2.3.3.1" xref="A4.Ex38.m1.2.2.2.3.3.1.cmml">^</mo></mover></mrow></mrow><mrow id="A4.Ex38.m1.2.2.3" xref="A4.Ex38.m1.2.2.3.cmml"><mi id="A4.Ex38.m1.2.2.3.2" xref="A4.Ex38.m1.2.2.3.2.cmml">b</mi><mo id="A4.Ex38.m1.2.2.3.1" xref="A4.Ex38.m1.2.2.3.1.cmml">⁢</mo><msqrt id="A4.Ex38.m1.2.2.3.3" xref="A4.Ex38.m1.2.2.3.3.cmml"><mrow id="A4.Ex38.m1.2.2.3.3.2" xref="A4.Ex38.m1.2.2.3.3.2.cmml"><mn id="A4.Ex38.m1.2.2.3.3.2.2" xref="A4.Ex38.m1.2.2.3.3.2.2.cmml">1</mn><mo id="A4.Ex38.m1.2.2.3.3.2.1" xref="A4.Ex38.m1.2.2.3.3.2.1.cmml">+</mo><mi id="A4.Ex38.m1.2.2.3.3.2.3" xref="A4.Ex38.m1.2.2.3.3.2.3.cmml">ρ</mi></mrow></msqrt></mrow></mfrac><mo id="A4.Ex38.m1.3.3.1.1.3.3.2.2" xref="A4.Ex38.m1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex38.m1.3.3.1.2" lspace="0em" xref="A4.Ex38.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex38.m1.3b"><apply id="A4.Ex38.m1.3.3.1.1.cmml" xref="A4.Ex38.m1.3.3.1"><eq id="A4.Ex38.m1.3.3.1.1.1.cmml" xref="A4.Ex38.m1.3.3.1.1.1"></eq><apply id="A4.Ex38.m1.3.3.1.1.2.cmml" xref="A4.Ex38.m1.3.3.1.1.2"><times id="A4.Ex38.m1.3.3.1.1.2.1.cmml" xref="A4.Ex38.m1.3.3.1.1.2.1"></times><ci id="A4.Ex38.m1.3.3.1.1.2.2.cmml" xref="A4.Ex38.m1.3.3.1.1.2.2">Φ</ci><apply id="A4.Ex38.m1.1.1.cmml" xref="A4.Ex38.m1.3.3.1.1.2.3.2"><divide id="A4.Ex38.m1.1.1.1.cmml" xref="A4.Ex38.m1.3.3.1.1.2.3.2"></divide><apply id="A4.Ex38.m1.1.1.2.cmml" xref="A4.Ex38.m1.1.1.2"><times id="A4.Ex38.m1.1.1.2.1.cmml" xref="A4.Ex38.m1.1.1.2.1"></times><apply id="A4.Ex38.m1.1.1.2.2.cmml" xref="A4.Ex38.m1.1.1.2.2"><root id="A4.Ex38.m1.1.1.2.2a.cmml" xref="A4.Ex38.m1.1.1.2.2"></root><ci id="A4.Ex38.m1.1.1.2.2.2.cmml" xref="A4.Ex38.m1.1.1.2.2.2">𝜌</ci></apply><ci id="A4.Ex38.m1.1.1.2.3.cmml" xref="A4.Ex38.m1.1.1.2.3">𝜏</ci></apply><ci id="A4.Ex38.m1.1.1.3.cmml" xref="A4.Ex38.m1.1.1.3">𝑎</ci></apply></apply><apply id="A4.Ex38.m1.3.3.1.1.3.cmml" xref="A4.Ex38.m1.3.3.1.1.3"><times id="A4.Ex38.m1.3.3.1.1.3.1.cmml" xref="A4.Ex38.m1.3.3.1.1.3.1"></times><ci id="A4.Ex38.m1.3.3.1.1.3.2.cmml" xref="A4.Ex38.m1.3.3.1.1.3.2">Φ</ci><apply id="A4.Ex38.m1.2.2.cmml" xref="A4.Ex38.m1.3.3.1.1.3.3.2"><divide id="A4.Ex38.m1.2.2.1.cmml" xref="A4.Ex38.m1.3.3.1.1.3.3.2"></divide><apply id="A4.Ex38.m1.2.2.2.cmml" xref="A4.Ex38.m1.2.2.2"><minus id="A4.Ex38.m1.2.2.2.1.cmml" xref="A4.Ex38.m1.2.2.2.1"></minus><ci id="A4.Ex38.m1.2.2.2.2.cmml" xref="A4.Ex38.m1.2.2.2.2">𝜏</ci><apply id="A4.Ex38.m1.2.2.2.3.cmml" xref="A4.Ex38.m1.2.2.2.3"><times id="A4.Ex38.m1.2.2.2.3.1.cmml" xref="A4.Ex38.m1.2.2.2.3.1"></times><ci id="A4.Ex38.m1.2.2.2.3.2.cmml" xref="A4.Ex38.m1.2.2.2.3.2">𝜌</ci><apply id="A4.Ex38.m1.2.2.2.3.3.cmml" xref="A4.Ex38.m1.2.2.2.3.3"><ci id="A4.Ex38.m1.2.2.2.3.3.1.cmml" xref="A4.Ex38.m1.2.2.2.3.3.1">^</ci><ci id="A4.Ex38.m1.2.2.2.3.3.2.cmml" xref="A4.Ex38.m1.2.2.2.3.3.2">𝜏</ci></apply></apply></apply><apply id="A4.Ex38.m1.2.2.3.cmml" xref="A4.Ex38.m1.2.2.3"><times id="A4.Ex38.m1.2.2.3.1.cmml" xref="A4.Ex38.m1.2.2.3.1"></times><ci id="A4.Ex38.m1.2.2.3.2.cmml" xref="A4.Ex38.m1.2.2.3.2">𝑏</ci><apply id="A4.Ex38.m1.2.2.3.3.cmml" xref="A4.Ex38.m1.2.2.3.3"><root id="A4.Ex38.m1.2.2.3.3a.cmml" xref="A4.Ex38.m1.2.2.3.3"></root><apply id="A4.Ex38.m1.2.2.3.3.2.cmml" xref="A4.Ex38.m1.2.2.3.3.2"><plus id="A4.Ex38.m1.2.2.3.3.2.1.cmml" xref="A4.Ex38.m1.2.2.3.3.2.1"></plus><cn id="A4.Ex38.m1.2.2.3.3.2.2.cmml" type="integer" xref="A4.Ex38.m1.2.2.3.3.2.2">1</cn><ci id="A4.Ex38.m1.2.2.3.3.2.3.cmml" xref="A4.Ex38.m1.2.2.3.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex38.m1.3c">\Phi\left(\frac{\sqrt{\rho}\tau}{a}\right)=\Phi\left(\frac{\tau-\rho\hat{\tau}% }{b\sqrt{1+\rho}}\right).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex38.m1.3d">roman_Φ ( divide start_ARG square-root start_ARG italic_ρ end_ARG italic_τ end_ARG start_ARG italic_a end_ARG ) = roman_Φ ( divide start_ARG italic_τ - italic_ρ over^ start_ARG italic_τ end_ARG end_ARG start_ARG italic_b square-root start_ARG 1 + italic_ρ 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="A4.SS1.6.p1.7">The solution to this equality is easily reached by setting the arguments to <math alttext="\Phi" class="ltx_Math" display="inline" id="A4.SS1.6.p1.6.m1.1"><semantics id="A4.SS1.6.p1.6.m1.1a"><mi id="A4.SS1.6.p1.6.m1.1.1" mathvariant="normal" xref="A4.SS1.6.p1.6.m1.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.6.m1.1b"><ci id="A4.SS1.6.p1.6.m1.1.1.cmml" xref="A4.SS1.6.p1.6.m1.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.6.m1.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.6.m1.1d">roman_Φ</annotation></semantics></math> equal to each other. We end up with <math alttext="\hat{\tau}=m_{\textrm{DG}}(\rho)\tau" class="ltx_Math" display="inline" id="A4.SS1.6.p1.7.m2.1"><semantics id="A4.SS1.6.p1.7.m2.1a"><mrow id="A4.SS1.6.p1.7.m2.1.2" xref="A4.SS1.6.p1.7.m2.1.2.cmml"><mover accent="true" id="A4.SS1.6.p1.7.m2.1.2.2" xref="A4.SS1.6.p1.7.m2.1.2.2.cmml"><mi id="A4.SS1.6.p1.7.m2.1.2.2.2" xref="A4.SS1.6.p1.7.m2.1.2.2.2.cmml">τ</mi><mo id="A4.SS1.6.p1.7.m2.1.2.2.1" xref="A4.SS1.6.p1.7.m2.1.2.2.1.cmml">^</mo></mover><mo id="A4.SS1.6.p1.7.m2.1.2.1" xref="A4.SS1.6.p1.7.m2.1.2.1.cmml">=</mo><mrow id="A4.SS1.6.p1.7.m2.1.2.3" xref="A4.SS1.6.p1.7.m2.1.2.3.cmml"><msub id="A4.SS1.6.p1.7.m2.1.2.3.2" xref="A4.SS1.6.p1.7.m2.1.2.3.2.cmml"><mi id="A4.SS1.6.p1.7.m2.1.2.3.2.2" xref="A4.SS1.6.p1.7.m2.1.2.3.2.2.cmml">m</mi><mtext id="A4.SS1.6.p1.7.m2.1.2.3.2.3" xref="A4.SS1.6.p1.7.m2.1.2.3.2.3a.cmml">DG</mtext></msub><mo id="A4.SS1.6.p1.7.m2.1.2.3.1" xref="A4.SS1.6.p1.7.m2.1.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.6.p1.7.m2.1.2.3.3.2" xref="A4.SS1.6.p1.7.m2.1.2.3.cmml"><mo id="A4.SS1.6.p1.7.m2.1.2.3.3.2.1" stretchy="false" xref="A4.SS1.6.p1.7.m2.1.2.3.cmml">(</mo><mi id="A4.SS1.6.p1.7.m2.1.1" xref="A4.SS1.6.p1.7.m2.1.1.cmml">ρ</mi><mo id="A4.SS1.6.p1.7.m2.1.2.3.3.2.2" stretchy="false" xref="A4.SS1.6.p1.7.m2.1.2.3.cmml">)</mo></mrow><mo id="A4.SS1.6.p1.7.m2.1.2.3.1a" xref="A4.SS1.6.p1.7.m2.1.2.3.1.cmml">⁢</mo><mi id="A4.SS1.6.p1.7.m2.1.2.3.4" xref="A4.SS1.6.p1.7.m2.1.2.3.4.cmml">τ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.7.m2.1b"><apply id="A4.SS1.6.p1.7.m2.1.2.cmml" xref="A4.SS1.6.p1.7.m2.1.2"><eq id="A4.SS1.6.p1.7.m2.1.2.1.cmml" xref="A4.SS1.6.p1.7.m2.1.2.1"></eq><apply id="A4.SS1.6.p1.7.m2.1.2.2.cmml" xref="A4.SS1.6.p1.7.m2.1.2.2"><ci id="A4.SS1.6.p1.7.m2.1.2.2.1.cmml" xref="A4.SS1.6.p1.7.m2.1.2.2.1">^</ci><ci id="A4.SS1.6.p1.7.m2.1.2.2.2.cmml" xref="A4.SS1.6.p1.7.m2.1.2.2.2">𝜏</ci></apply><apply id="A4.SS1.6.p1.7.m2.1.2.3.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3"><times id="A4.SS1.6.p1.7.m2.1.2.3.1.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.1"></times><apply id="A4.SS1.6.p1.7.m2.1.2.3.2.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.6.p1.7.m2.1.2.3.2.1.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.2">subscript</csymbol><ci id="A4.SS1.6.p1.7.m2.1.2.3.2.2.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.2.2">𝑚</ci><ci id="A4.SS1.6.p1.7.m2.1.2.3.2.3a.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.2.3"><mtext id="A4.SS1.6.p1.7.m2.1.2.3.2.3.cmml" mathsize="70%" xref="A4.SS1.6.p1.7.m2.1.2.3.2.3">DG</mtext></ci></apply><ci id="A4.SS1.6.p1.7.m2.1.1.cmml" xref="A4.SS1.6.p1.7.m2.1.1">𝜌</ci><ci id="A4.SS1.6.p1.7.m2.1.2.3.4.cmml" xref="A4.SS1.6.p1.7.m2.1.2.3.4">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.7.m2.1c">\hat{\tau}=m_{\textrm{DG}}(\rho)\tau</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.7.m2.1d">over^ start_ARG italic_τ end_ARG = italic_m start_POSTSUBSCRIPT DG end_POSTSUBSCRIPT ( italic_ρ ) italic_τ</annotation></semantics></math> for</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="m_{\textrm{DG}}(\rho)=\frac{a-b\sqrt{p(1+p)}}{a\rho}." class="ltx_Math" display="block" id="A4.Ex39.m1.3"><semantics id="A4.Ex39.m1.3a"><mrow id="A4.Ex39.m1.3.3.1" xref="A4.Ex39.m1.3.3.1.1.cmml"><mrow id="A4.Ex39.m1.3.3.1.1" xref="A4.Ex39.m1.3.3.1.1.cmml"><mrow id="A4.Ex39.m1.3.3.1.1.2" xref="A4.Ex39.m1.3.3.1.1.2.cmml"><msub id="A4.Ex39.m1.3.3.1.1.2.2" xref="A4.Ex39.m1.3.3.1.1.2.2.cmml"><mi id="A4.Ex39.m1.3.3.1.1.2.2.2" xref="A4.Ex39.m1.3.3.1.1.2.2.2.cmml">m</mi><mtext id="A4.Ex39.m1.3.3.1.1.2.2.3" xref="A4.Ex39.m1.3.3.1.1.2.2.3a.cmml">DG</mtext></msub><mo id="A4.Ex39.m1.3.3.1.1.2.1" xref="A4.Ex39.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex39.m1.3.3.1.1.2.3.2" xref="A4.Ex39.m1.3.3.1.1.2.cmml"><mo id="A4.Ex39.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="A4.Ex39.m1.3.3.1.1.2.cmml">(</mo><mi id="A4.Ex39.m1.2.2" xref="A4.Ex39.m1.2.2.cmml">ρ</mi><mo id="A4.Ex39.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="A4.Ex39.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.Ex39.m1.3.3.1.1.1" xref="A4.Ex39.m1.3.3.1.1.1.cmml">=</mo><mfrac id="A4.Ex39.m1.1.1" xref="A4.Ex39.m1.1.1.cmml"><mrow id="A4.Ex39.m1.1.1.1" xref="A4.Ex39.m1.1.1.1.cmml"><mi id="A4.Ex39.m1.1.1.1.3" xref="A4.Ex39.m1.1.1.1.3.cmml">a</mi><mo id="A4.Ex39.m1.1.1.1.2" xref="A4.Ex39.m1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex39.m1.1.1.1.4" xref="A4.Ex39.m1.1.1.1.4.cmml"><mi id="A4.Ex39.m1.1.1.1.4.2" xref="A4.Ex39.m1.1.1.1.4.2.cmml">b</mi><mo id="A4.Ex39.m1.1.1.1.4.1" xref="A4.Ex39.m1.1.1.1.4.1.cmml">⁢</mo><msqrt id="A4.Ex39.m1.1.1.1.1" xref="A4.Ex39.m1.1.1.1.1.cmml"><mrow id="A4.Ex39.m1.1.1.1.1.1" xref="A4.Ex39.m1.1.1.1.1.1.cmml"><mi id="A4.Ex39.m1.1.1.1.1.1.3" xref="A4.Ex39.m1.1.1.1.1.1.3.cmml">p</mi><mo id="A4.Ex39.m1.1.1.1.1.1.2" xref="A4.Ex39.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex39.m1.1.1.1.1.1.1.1" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex39.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex39.m1.1.1.1.1.1.1.1.1" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.cmml"><mn id="A4.Ex39.m1.1.1.1.1.1.1.1.1.2" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A4.Ex39.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex39.m1.1.1.1.1.1.1.1.1.3" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A4.Ex39.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msqrt></mrow></mrow><mrow id="A4.Ex39.m1.1.1.3" xref="A4.Ex39.m1.1.1.3.cmml"><mi id="A4.Ex39.m1.1.1.3.2" xref="A4.Ex39.m1.1.1.3.2.cmml">a</mi><mo id="A4.Ex39.m1.1.1.3.1" xref="A4.Ex39.m1.1.1.3.1.cmml">⁢</mo><mi id="A4.Ex39.m1.1.1.3.3" xref="A4.Ex39.m1.1.1.3.3.cmml">ρ</mi></mrow></mfrac></mrow><mo id="A4.Ex39.m1.3.3.1.2" lspace="0em" xref="A4.Ex39.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex39.m1.3b"><apply id="A4.Ex39.m1.3.3.1.1.cmml" xref="A4.Ex39.m1.3.3.1"><eq id="A4.Ex39.m1.3.3.1.1.1.cmml" xref="A4.Ex39.m1.3.3.1.1.1"></eq><apply id="A4.Ex39.m1.3.3.1.1.2.cmml" xref="A4.Ex39.m1.3.3.1.1.2"><times id="A4.Ex39.m1.3.3.1.1.2.1.cmml" xref="A4.Ex39.m1.3.3.1.1.2.1"></times><apply id="A4.Ex39.m1.3.3.1.1.2.2.cmml" xref="A4.Ex39.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex39.m1.3.3.1.1.2.2.1.cmml" xref="A4.Ex39.m1.3.3.1.1.2.2">subscript</csymbol><ci id="A4.Ex39.m1.3.3.1.1.2.2.2.cmml" xref="A4.Ex39.m1.3.3.1.1.2.2.2">𝑚</ci><ci id="A4.Ex39.m1.3.3.1.1.2.2.3a.cmml" xref="A4.Ex39.m1.3.3.1.1.2.2.3"><mtext id="A4.Ex39.m1.3.3.1.1.2.2.3.cmml" mathsize="70%" xref="A4.Ex39.m1.3.3.1.1.2.2.3">DG</mtext></ci></apply><ci id="A4.Ex39.m1.2.2.cmml" xref="A4.Ex39.m1.2.2">𝜌</ci></apply><apply id="A4.Ex39.m1.1.1.cmml" xref="A4.Ex39.m1.1.1"><divide id="A4.Ex39.m1.1.1.2.cmml" xref="A4.Ex39.m1.1.1"></divide><apply id="A4.Ex39.m1.1.1.1.cmml" xref="A4.Ex39.m1.1.1.1"><minus id="A4.Ex39.m1.1.1.1.2.cmml" xref="A4.Ex39.m1.1.1.1.2"></minus><ci id="A4.Ex39.m1.1.1.1.3.cmml" xref="A4.Ex39.m1.1.1.1.3">𝑎</ci><apply id="A4.Ex39.m1.1.1.1.4.cmml" xref="A4.Ex39.m1.1.1.1.4"><times id="A4.Ex39.m1.1.1.1.4.1.cmml" xref="A4.Ex39.m1.1.1.1.4.1"></times><ci id="A4.Ex39.m1.1.1.1.4.2.cmml" xref="A4.Ex39.m1.1.1.1.4.2">𝑏</ci><apply id="A4.Ex39.m1.1.1.1.1.cmml" xref="A4.Ex39.m1.1.1.1.1"><root id="A4.Ex39.m1.1.1.1.1a.cmml" xref="A4.Ex39.m1.1.1.1.1"></root><apply id="A4.Ex39.m1.1.1.1.1.1.cmml" xref="A4.Ex39.m1.1.1.1.1.1"><times id="A4.Ex39.m1.1.1.1.1.1.2.cmml" xref="A4.Ex39.m1.1.1.1.1.1.2"></times><ci id="A4.Ex39.m1.1.1.1.1.1.3.cmml" xref="A4.Ex39.m1.1.1.1.1.1.3">𝑝</ci><apply id="A4.Ex39.m1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex39.m1.1.1.1.1.1.1.1"><plus id="A4.Ex39.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.1"></plus><cn id="A4.Ex39.m1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.2">1</cn><ci id="A4.Ex39.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex39.m1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></apply><apply id="A4.Ex39.m1.1.1.3.cmml" xref="A4.Ex39.m1.1.1.3"><times id="A4.Ex39.m1.1.1.3.1.cmml" xref="A4.Ex39.m1.1.1.3.1"></times><ci id="A4.Ex39.m1.1.1.3.2.cmml" xref="A4.Ex39.m1.1.1.3.2">𝑎</ci><ci id="A4.Ex39.m1.1.1.3.3.cmml" xref="A4.Ex39.m1.1.1.3.3">𝜌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex39.m1.3c">m_{\textrm{DG}}(\rho)=\frac{a-b\sqrt{p(1+p)}}{a\rho}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex39.m1.3d">italic_m start_POSTSUBSCRIPT DG end_POSTSUBSCRIPT ( italic_ρ ) = divide start_ARG italic_a - italic_b square-root start_ARG italic_p ( 1 + italic_p ) end_ARG end_ARG start_ARG italic_a italic_ρ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.6.p1.10">Now, one can check that <math alttext="\frac{m_{\textrm{DG}}(\rho)}{m_{\textrm{RBTS}}(\rho)}>1" class="ltx_Math" display="inline" id="A4.SS1.6.p1.8.m1.2"><semantics id="A4.SS1.6.p1.8.m1.2a"><mrow id="A4.SS1.6.p1.8.m1.2.3" xref="A4.SS1.6.p1.8.m1.2.3.cmml"><mfrac id="A4.SS1.6.p1.8.m1.2.2" xref="A4.SS1.6.p1.8.m1.2.2.cmml"><mrow id="A4.SS1.6.p1.8.m1.1.1.1" xref="A4.SS1.6.p1.8.m1.1.1.1.cmml"><msub id="A4.SS1.6.p1.8.m1.1.1.1.3" xref="A4.SS1.6.p1.8.m1.1.1.1.3.cmml"><mi id="A4.SS1.6.p1.8.m1.1.1.1.3.2" xref="A4.SS1.6.p1.8.m1.1.1.1.3.2.cmml">m</mi><mtext id="A4.SS1.6.p1.8.m1.1.1.1.3.3" xref="A4.SS1.6.p1.8.m1.1.1.1.3.3a.cmml">DG</mtext></msub><mo id="A4.SS1.6.p1.8.m1.1.1.1.2" xref="A4.SS1.6.p1.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.6.p1.8.m1.1.1.1.4.2" xref="A4.SS1.6.p1.8.m1.1.1.1.cmml"><mo id="A4.SS1.6.p1.8.m1.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.6.p1.8.m1.1.1.1.cmml">(</mo><mi id="A4.SS1.6.p1.8.m1.1.1.1.1" xref="A4.SS1.6.p1.8.m1.1.1.1.1.cmml">ρ</mi><mo id="A4.SS1.6.p1.8.m1.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.6.p1.8.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="A4.SS1.6.p1.8.m1.2.2.2" xref="A4.SS1.6.p1.8.m1.2.2.2.cmml"><msub id="A4.SS1.6.p1.8.m1.2.2.2.3" xref="A4.SS1.6.p1.8.m1.2.2.2.3.cmml"><mi id="A4.SS1.6.p1.8.m1.2.2.2.3.2" xref="A4.SS1.6.p1.8.m1.2.2.2.3.2.cmml">m</mi><mtext id="A4.SS1.6.p1.8.m1.2.2.2.3.3" xref="A4.SS1.6.p1.8.m1.2.2.2.3.3a.cmml">RBTS</mtext></msub><mo id="A4.SS1.6.p1.8.m1.2.2.2.2" xref="A4.SS1.6.p1.8.m1.2.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.6.p1.8.m1.2.2.2.4.2" xref="A4.SS1.6.p1.8.m1.2.2.2.cmml"><mo id="A4.SS1.6.p1.8.m1.2.2.2.4.2.1" stretchy="false" xref="A4.SS1.6.p1.8.m1.2.2.2.cmml">(</mo><mi id="A4.SS1.6.p1.8.m1.2.2.2.1" xref="A4.SS1.6.p1.8.m1.2.2.2.1.cmml">ρ</mi><mo id="A4.SS1.6.p1.8.m1.2.2.2.4.2.2" stretchy="false" xref="A4.SS1.6.p1.8.m1.2.2.2.cmml">)</mo></mrow></mrow></mfrac><mo id="A4.SS1.6.p1.8.m1.2.3.1" xref="A4.SS1.6.p1.8.m1.2.3.1.cmml">></mo><mn id="A4.SS1.6.p1.8.m1.2.3.2" 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encoding="application/x-tex" id="A4.SS1.6.p1.8.m1.2c">\frac{m_{\textrm{DG}}(\rho)}{m_{\textrm{RBTS}}(\rho)}>1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.8.m1.2d">divide start_ARG italic_m start_POSTSUBSCRIPT DG end_POSTSUBSCRIPT ( italic_ρ ) end_ARG start_ARG italic_m start_POSTSUBSCRIPT RBTS end_POSTSUBSCRIPT ( italic_ρ ) end_ARG > 1</annotation></semantics></math> for all <math alttext="a,b>0" class="ltx_Math" display="inline" id="A4.SS1.6.p1.9.m2.2"><semantics id="A4.SS1.6.p1.9.m2.2a"><mrow id="A4.SS1.6.p1.9.m2.2.3" xref="A4.SS1.6.p1.9.m2.2.3.cmml"><mrow id="A4.SS1.6.p1.9.m2.2.3.2.2" xref="A4.SS1.6.p1.9.m2.2.3.2.1.cmml"><mi id="A4.SS1.6.p1.9.m2.1.1" xref="A4.SS1.6.p1.9.m2.1.1.cmml">a</mi><mo id="A4.SS1.6.p1.9.m2.2.3.2.2.1" xref="A4.SS1.6.p1.9.m2.2.3.2.1.cmml">,</mo><mi id="A4.SS1.6.p1.9.m2.2.2" xref="A4.SS1.6.p1.9.m2.2.2.cmml">b</mi></mrow><mo id="A4.SS1.6.p1.9.m2.2.3.1" xref="A4.SS1.6.p1.9.m2.2.3.1.cmml">></mo><mn id="A4.SS1.6.p1.9.m2.2.3.3" xref="A4.SS1.6.p1.9.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.9.m2.2b"><apply id="A4.SS1.6.p1.9.m2.2.3.cmml" xref="A4.SS1.6.p1.9.m2.2.3"><gt id="A4.SS1.6.p1.9.m2.2.3.1.cmml" xref="A4.SS1.6.p1.9.m2.2.3.1"></gt><list id="A4.SS1.6.p1.9.m2.2.3.2.1.cmml" xref="A4.SS1.6.p1.9.m2.2.3.2.2"><ci id="A4.SS1.6.p1.9.m2.1.1.cmml" xref="A4.SS1.6.p1.9.m2.1.1">𝑎</ci><ci id="A4.SS1.6.p1.9.m2.2.2.cmml" xref="A4.SS1.6.p1.9.m2.2.2">𝑏</ci></list><cn id="A4.SS1.6.p1.9.m2.2.3.3.cmml" type="integer" xref="A4.SS1.6.p1.9.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.9.m2.2c">a,b>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.9.m2.2d">italic_a , italic_b > 0</annotation></semantics></math>. It immediately follows that the best response <math alttext="\hat{\tau}" class="ltx_Math" display="inline" id="A4.SS1.6.p1.10.m3.1"><semantics id="A4.SS1.6.p1.10.m3.1a"><mover accent="true" id="A4.SS1.6.p1.10.m3.1.1" xref="A4.SS1.6.p1.10.m3.1.1.cmml"><mi id="A4.SS1.6.p1.10.m3.1.1.2" xref="A4.SS1.6.p1.10.m3.1.1.2.cmml">τ</mi><mo id="A4.SS1.6.p1.10.m3.1.1.1" xref="A4.SS1.6.p1.10.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.6.p1.10.m3.1b"><apply id="A4.SS1.6.p1.10.m3.1.1.cmml" xref="A4.SS1.6.p1.10.m3.1.1"><ci id="A4.SS1.6.p1.10.m3.1.1.1.cmml" xref="A4.SS1.6.p1.10.m3.1.1.1">^</ci><ci id="A4.SS1.6.p1.10.m3.1.1.2.cmml" xref="A4.SS1.6.p1.10.m3.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.6.p1.10.m3.1c">\hat{\tau}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.6.p1.10.m3.1d">over^ start_ARG italic_τ end_ARG</annotation></semantics></math> converges to 0 strictly faster under the DG mechanism than under the RBTS mechanism. ∎</p> </div> </div> </section> </section> <section class="ltx_appendix" id="A5"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix E </span>Omitted Experiments</h2> <div class="ltx_para" id="A5.p1"> <p class="ltx_p" id="A5.p1.1">We include bifurcation figures for the four-component Gaussian mixture case within the DG and OA mechanisms, as well as bifurcation figures for RBTS. Again, we vary the precision of each component in the Gaussian mixture and study how the stability and number of equilibria change across mechanisms.</p> </div> <figure class="ltx_figure" id="A5.F7.sf1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A5.F7.sf1.1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="A5.F7.sf1.1.g1" src="x9.png" width="830"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A5.F7.sf1.2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="A5.F7.sf1.2.g1" src="x10.png" width="830"/> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A5.F7.sf1.6.2.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="A5.F7.sf1.4.1" style="font-size:90%;">Bifurcation graphs for OA (left) and DG (right) in the setting of a four-component Gaussian mixture with means (<math alttext="-4,-2,2,4" class="ltx_Math" display="inline" id="A5.F7.sf1.4.1.m1.4"><semantics id="A5.F7.sf1.4.1.m1.4b"><mrow id="A5.F7.sf1.4.1.m1.4.4.2" xref="A5.F7.sf1.4.1.m1.4.4.3.cmml"><mrow id="A5.F7.sf1.4.1.m1.3.3.1.1" xref="A5.F7.sf1.4.1.m1.3.3.1.1.cmml"><mo id="A5.F7.sf1.4.1.m1.3.3.1.1b" xref="A5.F7.sf1.4.1.m1.3.3.1.1.cmml">−</mo><mn id="A5.F7.sf1.4.1.m1.3.3.1.1.2" xref="A5.F7.sf1.4.1.m1.3.3.1.1.2.cmml">4</mn></mrow><mo id="A5.F7.sf1.4.1.m1.4.4.2.3" xref="A5.F7.sf1.4.1.m1.4.4.3.cmml">,</mo><mrow id="A5.F7.sf1.4.1.m1.4.4.2.2" xref="A5.F7.sf1.4.1.m1.4.4.2.2.cmml"><mo id="A5.F7.sf1.4.1.m1.4.4.2.2b" xref="A5.F7.sf1.4.1.m1.4.4.2.2.cmml">−</mo><mn id="A5.F7.sf1.4.1.m1.4.4.2.2.2" xref="A5.F7.sf1.4.1.m1.4.4.2.2.2.cmml">2</mn></mrow><mo id="A5.F7.sf1.4.1.m1.4.4.2.4" xref="A5.F7.sf1.4.1.m1.4.4.3.cmml">,</mo><mn id="A5.F7.sf1.4.1.m1.1.1" xref="A5.F7.sf1.4.1.m1.1.1.cmml">2</mn><mo id="A5.F7.sf1.4.1.m1.4.4.2.5" xref="A5.F7.sf1.4.1.m1.4.4.3.cmml">,</mo><mn id="A5.F7.sf1.4.1.m1.2.2" xref="A5.F7.sf1.4.1.m1.2.2.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.F7.sf1.4.1.m1.4c"><list id="A5.F7.sf1.4.1.m1.4.4.3.cmml" xref="A5.F7.sf1.4.1.m1.4.4.2"><apply id="A5.F7.sf1.4.1.m1.3.3.1.1.cmml" xref="A5.F7.sf1.4.1.m1.3.3.1.1"><minus id="A5.F7.sf1.4.1.m1.3.3.1.1.1.cmml" xref="A5.F7.sf1.4.1.m1.3.3.1.1"></minus><cn id="A5.F7.sf1.4.1.m1.3.3.1.1.2.cmml" type="integer" xref="A5.F7.sf1.4.1.m1.3.3.1.1.2">4</cn></apply><apply id="A5.F7.sf1.4.1.m1.4.4.2.2.cmml" xref="A5.F7.sf1.4.1.m1.4.4.2.2"><minus id="A5.F7.sf1.4.1.m1.4.4.2.2.1.cmml" xref="A5.F7.sf1.4.1.m1.4.4.2.2"></minus><cn id="A5.F7.sf1.4.1.m1.4.4.2.2.2.cmml" type="integer" xref="A5.F7.sf1.4.1.m1.4.4.2.2.2">2</cn></apply><cn id="A5.F7.sf1.4.1.m1.1.1.cmml" type="integer" xref="A5.F7.sf1.4.1.m1.1.1">2</cn><cn id="A5.F7.sf1.4.1.m1.2.2.cmml" type="integer" xref="A5.F7.sf1.4.1.m1.2.2">4</cn></list></annotation-xml><annotation encoding="application/x-tex" id="A5.F7.sf1.4.1.m1.4d">-4,-2,2,4</annotation><annotation encoding="application/x-llamapun" id="A5.F7.sf1.4.1.m1.4e">- 4 , - 2 , 2 , 4</annotation></semantics></math>). Following a pattern, we have seven equilibria in OA and five in DG for large enough precision. </span></figcaption> </figure> <div class="ltx_pagination ltx_role_newpage"></div> <figure class="ltx_figure" id="A5.F9.sf3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A5.F9.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="A5.F9.sf1.g1" src="x11.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A5.F9.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="A5.F9.sf1.3.2" style="font-size:90%;">3 component mixture with means (-2, 0, 2)</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="A5.F9.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="498" id="A5.F9.sf2.g1" src="x12.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A5.F9.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="A5.F9.sf2.3.2" style="font-size:90%;">4 component mixture with means (-3, -1, 1, 3)</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A5.F9.sf3.2.1.1" style="font-size:90%;">(c)</span> </span><span class="ltx_text" id="A5.F9.sf3.3.2" style="font-size:90%;">Bifurcation graphs for RBTS in the setting of a three and four-component Gaussian mixture. 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