Controlled Iterated Function Systems for Minority Dynamics and Diversity-Preserving Ergodic Control

Controlled Iterated Function Systems for Minority Dynamics and Diversity-Preserving Ergodic Control | 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 Controlled Iterated Function Systems for Minority Dynamics and Diversity-Preserving Ergodic Control Ramen Ghosh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7411800/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We study \emph{ergodic regulation} of large-agent strategic systems in which agents prefer to be in the minority. The population dynamics are modeled as a controlled iterated function system (IFS) on a Polish space; the control reshapes the transition kernel and hence the stationary law of the ensemble. We formulate a distribution-shaping problem that penalizes collapse into majority states and rewards \emph{diversity} by aligning the controlled invariant measure with a dispersed target in Wasserstein geometry. Our contributions are fourfold. (i) We prove Fréchet differentiability of the stationary law with respect to control and obtain a \emph{linear-response} (resolvent) representation. (ii) We derive an adjoint \emph{Poisson equation} yielding an explicit gradient formula for the Wasserstein objective, providing a measure-valued Pontryagin-type optimality condition suitable for feedback design. (iii) We establish non-asymptotic convergence of projected stochastic gradient with \emph{dependent} samples, with explicit bias--variance tradeoffs induced by finite trajectory length and truncated Poisson solves. (iv) We quantify \emph{robustness} of both the invariant law and the optimal value under kernel misspecification. Numerical studies on minority-game stylizations in low- and moderate-dimensional IFS illustrate suppression of informational cascades, preservation of multi-modality, and favorable sample–compute profiles consistent with theory. Ergodic control Iterated function systems Distribution shaping Wasserstein geometry Linear response Stochastic approximation Minority games Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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-7411800","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":513897119,"identity":"4915c00c-29b7-448e-9ca5-7faaf039cedd","order_by":0,"name":"Ramen Ghosh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYBCDBAYGHsYHIBYbKVqYDUAUSVrYJEAUQWDefjrxcQFDbR4//9ljFT9/HGbgY2/Ar0XmTO5m4xkMx4slZ+Sl3exJOMzAxnMAvxYJhtxt0jwMxxI33OAxu8ED0iJBwHUS/G+3/wZrOX/GrPAPSIv8AwJaJHK3MfMw1CRuOJBjxgyxBb8OoJa3m6V5DA4kzpyRYywtk5bOw8ZD0GG5Gz/zVNQl9vOfMfz4xsZaTr79AAFrwMDgMJzJQ4x6EKgjVuEoGAWjYBSMRAAAYpk9dkX9KmUAAAAASUVORK5CYII=","orcid":"","institution":"Atlantic Technological University","correspondingAuthor":true,"prefix":"","firstName":"Ramen","middleName":"","lastName":"Ghosh","suffix":""}],"badges":[],"createdAt":"2025-08-19 21:38:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7411800/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7411800/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101432132,"identity":"dc57a1fe-58d9-43e0-9588-6125924b6291","added_by":"auto","created_at":"2026-01-29 15:45:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":337645,"visible":true,"origin":"","legend":"","description":"","filename":"Controlled.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7411800/v1_covered_2e734b83-1871-4dbe-9d79-b0f01100c033.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Controlled Iterated Function Systems for Minority Dynamics and Diversity-Preserving Ergodic Control","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Ergodic control, Iterated function systems, Distribution shaping, Wasserstein geometry, Linear response, Stochastic approximation, Minority games","lastPublishedDoi":"10.21203/rs.3.rs-7411800/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7411800/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We study \\emph{ergodic regulation} of large-agent strategic systems in which agents prefer to be in the minority. The population dynamics are modeled as a controlled iterated function system (IFS) on a Polish space; the control reshapes the transition kernel and hence the stationary law of the ensemble. We formulate a distribution-shaping problem that penalizes collapse into majority states and rewards \\emph{diversity} by aligning the controlled invariant measure with a dispersed target in Wasserstein geometry. Our contributions are fourfold. (i) We prove Fréchet differentiability of the stationary law with respect to control and obtain a \\emph{linear-response} (resolvent) representation. (ii) We derive an adjoint \\emph{Poisson equation} yielding an explicit gradient formula for the Wasserstein objective, providing a measure-valued Pontryagin-type optimality condition suitable for feedback design. (iii) We establish non-asymptotic convergence of projected stochastic gradient with \\emph{dependent} samples, with explicit bias--variance tradeoffs induced by finite trajectory length and truncated Poisson solves. (iv) We quantify \\emph{robustness} of both the invariant law and the optimal value under kernel misspecification. 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