[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83659-en":3,"doc-seo-83659-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83659,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Complex dynamics in the Sherrington-Kirkpatrick game","We study adaptive learning outcomes for many players participating in two-strategy, two-player games where the payoff matrices are randomly generated at the outset and then held fixed. The work provides a game-theoretic foundation for the Sherrington–Kirkpatrick (SK) model, focusing on learning stability under general random strategy bias and the role of random fields in shaping stable states. A grand-canonical variant, allowing abstention, is analyzed, highlighting when learning converges to unique fixed points versus persistent volatility in multi-agent settings.","arXiv :2607 .02422v1 [ cond-mat .dis-nn] 2 Jul 2026  \nComplex dynamics in the Sherrington–Kirkpatrick game  \nDesmond Chan  \nDepartment of Informatics, King’s College, London WC2B 4BG, UK∗  \nTobias Galla  \nInstituto de F´ısica Interdisciplinar y Sistemas Complejos,  \nIFISC, Edifici Instituts Universitaris de Recerca,  \nCampus Universitat de les Illes Balears E-07122 Palma, Mallorca, Spain†  \n(Dated: July 3, 2026)  \nWe study the outcome of adaptive learning of a large number of players engaging in sets of twostrategy two-player games. We are interested in typical games, and generate the payoff matrices at random at the beginning. The payoff matrices then remain fixed during the learning process. This provides a game theoretic foundation for the Sherrington-Kirkpatrick (SK) game, recently introduced by Garnier-Brun, Benzaquen and Bouchaud. The original model by these authors is a special case, with no bias towards any strategy. We here determine stability of learning for SK games with general random bias, and find that the nature of the stable state is affected by random fields. We also introduce a grand-canonical version of the SK game, in which players can choose to abstain. We determine the stability of learning for this game. Our analysis confirms that complex situations involving many players are frequently unlearnable, even if each player only chooses between two different actions. The rate with which players lose memory of past payoffs and the competitiveness of the game emerge as key parameters determining whether learning converges to a unique fixed point, whether there are many fixed points, or if the dynamics remains persistently volatile.  \nI. INTRODUCTION  \nWhile there is no unique definition of a complex system, complexity is generally understood to involve the emergence of unexpected and perhaps unpredictable behaviour from the interaction of relatively simple agents. What these agents represent depends on the context. They could be members of a population in a model of the spread of an epidemic, genes and proteins in developmental biology, or traders ina financial market. Many of these systems involve competition for limited resources, strategic decision making and adaptation. Since its inception in the 1940s [1], game theory has been an important ingredient of modelling these processes. There has been significant interest in the physics community in evolutionary games in particular, and in the emergence of cooperation and the general phenomenology of different games, see for example [2, 3] .  \nThe bulk of existing work in game theory, both in physics and in other disciplines, focuses on simple games. By this we mean games with a small number of players who each only have a small number of ‘actions’ (pure strategies) at their disposal. Classical examples are the prisoners’ dilemma, or cyclic games such as rock-paper-scissors. In these games only two players interact, and they choose between two actions in the prisoner’s dilemma, or three actions in the rock-paper-scissors game.  \nMuch of the modelling work on strategic decision  \n∗ Electronic address: [desmond.chan@kcl.ac.uk](desmond.chan@kcl.ac.uk)  \n†Electronic address: [tobias.galla@ifisc.uib-csic.es](tobias.galla@ifisc.uib-csic.es)  \nmaking in economics is based on the assumption of perfectly rational players who have full information about the game, and who assume that all other players are also fully rational and fully informed. Under these assumptions, the natural outcomes of a game are the so-called Nash equilibria [4] . These are points in strategy space such that no player can improve their payoff by unilaterally changing their strategy. Nash proved that such points always exist in sufficiently simple games – broadly when the number of players is finite and when each player only has a finite number of actions to choose from. Importantly, Nash equilibria can be probabilistic combinations of pure strategies. For example the Nash point for the rock-paper-sc","cbCaiaXAJKTqg0UG","https://ap.wps.com/l/cbCaiaXAJKTqg0UG","pdf",13068595,4,1,24,"English","en",105,"# Introduction\n## Complex systems and game theory background\n## Nash equilibria and limitations in complex games\n## Sources of complexity in normal-form games\n## Two routes to complexity: more actions vs more players","[{\"question\":\"How are the games and payoff matrices defined in this study?\",\"answer\":\"Players repeatedly engage in two-strategy two-player games. The payoff matrices are generated randomly at the beginning and then remain fixed during the learning process.\"},{\"question\":\"What determines the stability of learning in the SK game here?\",\"answer\":\"Stability is analyzed for SK games with general random bias, and the nature of the stable state is shown to depend on random fields.\"},{\"question\":\"How does the grand-canonical version of the SK game change the learning dynamics?\",\"answer\":\"Players can choose to abstain. The study determines learning stability for this setup and finds that many-player situations can remain unlearnable, with convergence versus volatility controlled by memory-loss rate and competitiveness.\"}]",1784189578,60,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"complex-dynamics-in-the-sherrington-kirkpatrick-game","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/complex-dynamics-in-the-sherrington-kirkpatrick-game/83659/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How are the games and payoff matrices defined in this study?","Question",{"text":75,"@type":76},"Players repeatedly engage in two-strategy two-player games. The payoff matrices are generated randomly at the beginning and then remain fixed during the learning process.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What determines the stability of learning in the SK game here?",{"text":80,"@type":76},"Stability is analyzed for SK games with general random bias, and the nature of the stable state is shown to depend on random fields.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the grand-canonical version of the SK game change the learning dynamics?",{"text":84,"@type":76},"Players can choose to abstain. The study determines learning stability for this setup and finds that many-player situations can remain unlearnable, with convergence versus volatility controlled by memory-loss rate and competitiveness.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]