[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84444-en":3,"doc-seo-84444-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84444,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Learnable Mixed Nash Equilibria are Collectively Rational","The paper extends learning-in-games to learning dynamics that do not enjoy asymptotic stability, using the notion of uniform stability tied to equilibria reached by individually utility-seeking dynamics. Uniform stability is shown to correspond closely to collective rationality: a uniformly stable mixed equilibrium is weakly Pareto optimal, ruling out joint-deviation behaviors characteristic of the prisoner’s dilemma and the tragedy of the commons. The work also links uniform stability to last-iterate convergence for incremental smoothed best-response dynamics.","arXiv :2510 . 14907v2 [ cs .GT] 12 Jul 2026  \nLearnable Mixed Nash Equilibria are Collectively Rational  \nGeelon So Yi-An Ma  \nUniversity of California, San Diego  \nAbstract  \nWe extend the study of learning in games to dynamics that exhibit non-asymptotic stability. We do so through the notion of uniform stability, which is concerned with equilibria of individually utility-seeking dynamics. Perhaps surprisingly, it turns out to be closely connected to economic properties of collective rationality. Up to strategic equivalence, if a mixed equilibrium is uniformly stable, then it is weakly Pareto optimal—there is no way for all players to improve by jointly deviating from the equilibrium—a form of collective rationality that rules out the types of behaviors in the prisoner’s dilemma or the tragedy of the commons. Moreover, we show that uniform stability determines the last-iterate convergence behavior for the family of incremental smoothed best-response dynamics, used to model individual and corporate behaviors in the markets. Unlike dynamics around strict equilibria, which can stabilize to socially-inefficient solutions, individually utility-seeking behaviors near mixed Nash equilibria lead to collective rationality.  \nKeywords: algorithmic game theory, evolutionary dynamics, non-asymptotic stability, last-iterate convergence, smoothed best-response  \n1 Introduction  \nThe Nash (1951) equilibrium is a foundational solution concept in games, capturing when collective behavior or strategies may be stationary. These equilibria are meaningful to study, for once such strategies appear, they may persist for a long time. But, there is an important caveat: not all equilibria can be robustly reached by players in the game (Hart and Mas-Colell, 2003; Papadimitriou, 2007; Daskalakis et al. , 2009 , 2010; Milionis et al. , 2023) . Any equilibrium that cannot be found or sustained is unlikely to have practical relevance. This caveat leads to the question on learnability: which Nash equilibria can players eventually learn to play from repeated interactions?  \nTo grasp individual and corporate behaviors, literature in classical economics considers a model of learning where players: (i) take an evolutionary approach and incrementally update their strategies; (ii) are utilityseeking, which means that they aim to improve their own payoffs; and (iii) are uncoupled, where players are unaware of the other players’ utilities or methods to improve them (Alchian, 1950; Winter, 1971) . The goal of the players in this model is not to compute any pre-determined Nash equilibrium, at least not explicitly. Rather, the equilibrium is to emerge out of their joint, but individually utility-seeking, behavior.  \nWithin this model of learning, the impossibility result of Hart and Mas-Colell (2003) shows that nouncoupled and asymptotically-stable learning dynamics can converge to all Nash equilibria. Uncoupled means that players learn in uncoordinated and decentralized ways. Asymptotic stability means that the convergence is robust to small perturbations in the dynamics. Simply put, not all equilibria admit such a strong notion of learnability, viz. dynamical stability. The ones that do, however, have been characterized for certain classes of learning dynamics in standard normal-form games: an equilibrium is ‘asymptotically learnable’ in this way if and only if it is strict, where every player has a single, deterministic strategy that is clearly locally optimal (Samuelson and Zhang, 1992; Vlatakis-Gkaragkounis et al. , 2020; Giannou et al. , 2021) .  \nA significant gap remains for the learnability of mixed Nash equilibria, where players may use randomized strategies. On the one hand, mixed equilibria are not strict, and as a result, they are not asymptotically stable under these learning dynamics. Observations corroborating this finding demonstrate that many dynamics are not able to generically learn mixed Nash equilibria. This has been a significant source of crit","cbCaioaLHBicvFEZ","https://ap.wps.com/l/cbCaioaLHBicvFEZ","pdf",1133604,1,31,"English","en",105,"# Introduction\n## Learnability and dynamical stability\n## Learning model assumptions (evolutionary, utility-seeking, uncoupled)\n## Strict vs mixed Nash equilibria\n## Extending stability to non-asymptotic settings","[{\"question\":\"What notion of stability does the paper introduce to study learning dynamics?\",\"answer\":\"The paper uses uniform stability, focusing on equilibria arising from individually utility-seeking dynamics even when asymptotic stability is not present.\"},{\"question\":\"How does uniform stability relate to collective rationality for mixed Nash equilibria?\",\"answer\":\"Up to strategic equivalence, a uniformly stable mixed equilibrium is weakly Pareto optimal, meaning no joint deviation can make all players better off.\"},{\"question\":\"Why does the paper go beyond asymptotic stability?\",\"answer\":\"Mixed Nash equilibria are not strict and therefore are not asymptotically stable under the standard learning dynamics, so asymptotic stability is too stringent to capture learnability.\"}]",1784195666,78,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"learnable-mixed-nash-equilibria-are-collectively-rational","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/learnable-mixed-nash-equilibria-are-collectively-rational/84444/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What notion of stability does the paper introduce to study learning dynamics?","Question",{"text":74,"@type":75},"The paper uses uniform stability, focusing on equilibria arising from individually utility-seeking dynamics even when asymptotic stability is not present.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does uniform stability relate to collective rationality for mixed Nash equilibria?",{"text":79,"@type":75},"Up to strategic equivalence, a uniformly stable mixed equilibrium is weakly Pareto optimal, meaning no joint deviation can make all players better off.",{"name":81,"@type":72,"acceptedAnswer":82},"Why does the paper go beyond asymptotic stability?",{"text":83,"@type":75},"Mixed Nash equilibria are not strict and therefore are not asymptotically stable under the standard learning dynamics, so asymptotic stability is too stringent to capture learnability.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]