Non-Chaotic Limit Sets in Multi-Agent Learning

Non-Chaotic Limit Sets in Multi-Agent Learning | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Non-Chaotic Limit Sets in Multi-Agent Learning Aleksander Czechowski, Georgios Piliouras This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2188216/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 12 Jul, 2023 Read the published version in Autonomous Agents and Multi-Agent Systems → Version 1 posted 4 You are reading this latest preprint version Abstract Non-convergence is an inherent aspect of adaptive multi-agent systems, and even basic learning models, such as the replicator dynamics, are not guaranteed to equilibriate. Limit cycles, and even more complicated chaotic sets are in fact possible even in rather simple games, including variants of the Rock-Paper-Scissors game. A key challenge of multi-agent learning theory lies in characterization of these limit sets, based on qualitative features of the underlying game. Although chaotic behavior in learning dynamics can be precluded by the celebrated Poincar'e-Bendixson theorem, it is only applicable directly to low-dimensional settings. In this work, we attempt to find other characteristics of a game that can force regularity in the limit sets of learning. We show that behavior consistent with the Poincaré-Bendixson theorem (limit cycles, but no chaotic attractor) follows purely from the topological structure of the interaction graph, even for high-dimensional settings with an arbitrary number of players, and arbitrary payoff matrices. We prove our result for a wide class of follow-the-regularized leader (FoReL) dynamics, which generalize replicator dynamics, for binary games characterized interaction graphs where the payoffs of each player are only affected by one other player (i.e., interaction graphs of indegree one). Moreover, we provide simple conditions under which such behavior translates into efficiency guarantees, implying that FoReL learning achievestime-averaged sum of payoffs at least as good as that of a Nash equilibrium, thereby connecting the topology of the dynamics to social-welfare analysis. Replicator Dynamics Follow-the-Regularized Leader Polymatrix Games Poincar´e-Bendixson Theorem Regret Minimization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 12 Jul, 2023 Read the published version in Autonomous Agents and Multi-Agent Systems → Version 1 posted Editorial decision: Major revision 02 Nov, 2022 Editor assigned by journal 02 Nov, 2022 Submission checks completed at journal 21 Oct, 2022 First submitted to journal 20 Oct, 2022 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-2188216","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":146136097,"identity":"f2944d43-5efa-459a-96f3-1e3359b180a5","order_by":0,"name":"Aleksander Czechowski","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVRIie3RMWsCMRTA8XcctEvx1tzifYV3CO1y+FkigZusCi5OYhenQFf7LQ4KBbcnN9wS6iq4nIuzt8gNB5q4CU17Y4f8IQRCfiQhAC7Xfw718BaY6Il7NNNTx7rXvyMp0wRIcYCHdgTyv0mwCAjqJoleHmWvWk22cywE0aaGkY0w8sGTT2m8luo5zHDPUB05EYep9WKa+MByjrvXL680ZDfE/MRhsLSI6EbwYshnVeK3JuOTOcVK8EY4GZLpi5E5BX4lce7jRpKIM7U9hysU4Yc6IlHKrKRbvB3KuulHWCxFJZt+0CnEoaQkGbzbnq+/hX5aZzbgcrlcrhZdAQagYjBLucCrAAAAAElFTkSuQmCC","orcid":"","institution":"Delft University of Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Aleksander","middleName":"","lastName":"Czechowski","suffix":""},{"id":146136098,"identity":"2716ac6f-1849-4fb9-8ec9-6dd8af36a5c9","order_by":1,"name":"Georgios Piliouras","email":"","orcid":"","institution":"DeepMind (United Kingdom)","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Georgios","middleName":"","lastName":"Piliouras","suffix":""}],"badges":[],"createdAt":"2022-10-20 20:44:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2188216/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2188216/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10458-023-09612-x","type":"published","date":"2023-07-13T01:08:10+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":28195019,"identity":"2992ed5f-3c02-4041-94ea-0c842997bf87","added_by":"auto","created_at":"2022-10-24 18:25:12","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":706174,"visible":true,"origin":"","legend":"","description":"","filename":"JAAMAS223.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2188216/v1_covered.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Non-Chaotic Limit Sets in Multi-Agent Learning","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-2188216/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"autonomous-agents-and-multi-agent-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"agnt","sideBox":"Learn more about [Autonomous Agents and Multi-Agent Systems](http://link.springer.com/journal/10458)","snPcode":"10458","submissionUrl":"https://submission.nature.com/new-submission/10458/3","title":"Autonomous Agents and Multi-Agent Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Replicator Dynamics, Follow-the-Regularized Leader, Polymatrix Games, Poincar´e-Bendixson Theorem, Regret Minimization","lastPublishedDoi":"10.21203/rs.3.rs-2188216/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2188216/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Non-convergence is an inherent aspect of adaptive multi-agent systems, and even basic learning models, such as the replicator dynamics, are not guaranteed to equilibriate. Limit cycles, and even more complicated chaotic sets are in fact possible even in rather simple games, including variants of the Rock-Paper-Scissors game. A key challenge of multi-agent learning theory lies in characterization of these limit sets, based on qualitative features of the underlying game. Although chaotic behavior in learning dynamics can be precluded by the celebrated Poincar\\'e-Bendixson theorem, it is only applicable directly to low-dimensional settings. In this work, we attempt to find other characteristics of a game that can force regularity in the limit sets of learning. We show that behavior consistent with the Poincaré-Bendixson theorem (limit cycles, but no chaotic attractor) follows purely from the topological structure of the interaction graph, even for high-dimensional settings with an arbitrary number of players, and arbitrary payoff matrices. We prove our result for a wide class of follow-the-regularized leader (FoReL) dynamics, which generalize replicator dynamics, for binary games characterized interaction graphs where the payoffs of each player are only affected by one other player (i.e., interaction graphs of indegree one). 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