[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86519-en":3,"doc-seo-86519-105":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},86519,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","The Complexity of Computing Coarse Correlated Equilibria in Markov Games with a Single Controller","Study the computational complexity of stationary Markov coarse correlated equilibria (CCE) in discounted single-controller stochastic games, where all players affect rewards but only one player controls state transitions. Prior work showed PPAD-hardness for stationary Markov CCE in two-player general-sum games, using turn-based alternation that forces CCE to collapse to Nash equilibria. This paper resolves whether hardness persists when a single player controls all transitions, proving PPAD-completeness for approximate stationary Markov CCE.","arXiv :2607 . 10897v 1 [ cs .GT] 12 Jul 2026  \nThe Complexity of Computing Coarse Correlated Equilibriain Markov Games with a Single Controller  \nGabriele Farina 1 , Andreas Kontogiannis†2,3, Ioannis Panageas4 , and Vasilis Pollatos†3,5  \n1 Massachusetts Institute of Technology  \n2 National Technical University of Athens  \n3 Archimedes, Athena Research Center, Greece  \n4 University of California, Irvine  \n5 National and Kapodistrian University of Athens  \nJuly 14, 2026  \nAbstract  \nWe study the complexity of computing stationary Markov coarse correlated equilibria (CCE) in discounted single-controller stochastic (Markov) games [PR81, FV97], a fundamental subclass of stochastic games in which all players may affect rewards, but only one player controls the state transitions. Prior work [DGZ23, JMS23 , HN25] established PPAD-hardness for computing stationary Markov CCE in two-player general-sum stochastic games via turn-based constructionsin which each state is controlled by a single player, with control alternating across states. This structure forces every Markov CCE to collapse to a Nash equilibrium (NE), so hardness for NE transfers immediately to CCE. It remained open whether hardness persists when a single player controls all transitions—a setting where no such collapse occurs.  \nWe resolve this question: computing an approximate stationary Markov CCE in two-player single-controller stochastic games is PPAD-complete, even with a fixed discount factor and binary actions. For the perfect notion (equilibrium constraints at every state) this holds unconditionally at constant accuracy; for the non-perfect notion, we prove constant-accuracy hardness under the PCP-for-PPAD hypothesis [BPR16, DFHM26] and inverse-polynomial-accuracy hardness unconditionally. To the best of our knowledge, our result is the first to show hardness for computing CCE without relying on equilibrium collapse phenomena or other routes through Nash-like structure [FGK23 , AKSZ24 , PR24] . Instead, we construct single-controller gadgets whose local incentive constraints force a solution of a Pure-Circuit instance even under strongly correlated stationary policies.  \n†Part of this work was completed during a research visit to University of California, Irvine.  \nCorresponding authors: [gfarina@mit. edu](gfarina@mit. edu), [andreask ontogianni s@mail.ntua.gr](andreask ontogianni s@mail.ntua.gr), [ipanagea@ics.uci. edu](ipanagea@ics.uci. edu), [vaspoll@math.uoa.gr](vaspoll@math.uoa.gr).  \nContents  \n1 Introduction 1  \n1.1 Contribution and significance ............................... 2  \n1.2 Where attempts for a positive result break down . . . . . . . . . . . . . . . . . . . . 3  \n1.3 Technical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2 Preliminaries 6  \n2.1 Markov games, stationary policies, and value functions ................. 6  \n2.2 Markov Coarse Correlated Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.3 Performance-difference lemma for one-player deviations ................ 8  \n2.4 Pure-Circuit and the PCP-for-PPAD hypothesis ................... 8  \n3 The single-controller game construction 9  \n3. 1 Game definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n3.2 Single-controller gadgets for implementing the gates .................. 11  \n4 Perfect Markov CCE hardness 15  \n4.1 Proving the single-controller gadget conditions . . . . . . . . . . . . . . . . . . . . . 15  \n4.2 Perfect Markov CCE is PPAD-hard . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  \n5 Non-perfect Markov CCE hardness 17  \n5.1 Average copy ........................................ 18  \n5.2 Average controller-forcing ................................. 20  \n5.3 Non-Perfect Markov CCE is PPAD-hard . . . . . . . . . . . . . . . . . . . . . . . . . 23  \n6 Example: Pure-Circuit solution with no equilibrium collapse 25  \n7 Conclusion 29  \nA Extended Related Work 34  \nB Proof of ","cbCaio0Y7aVKbNDB","https://ap.wps.com/l/cbCaio0Y7aVKbNDB","pdf",728485,4,1,38,"English","en",105,"# Introduction\n## Contribution and significance\n## Where attempts for a positive result break down\n## Technical overview\n# Preliminaries\n## Markov games, stationary policies, and value functions\n## Markov Coarse Correlated Equilibria\n## Performance-difference lemma for one-player deviations\n## Pure-Circuit and the PCP-for-PPAD hypothesis\n# The single-controller game construction\n## Game definition\n## Single-controller gadgets for implementing the gates\n# Perfect Markov CCE hardness\n## Proving the single-controller gadget conditions\n## Perfect Markov CCE is PPAD-hard\n# Non-perfect Markov CCE hardness\n## Average copy\n## Average controller-forcing\n## Non-Perfect Markov CCE is PPAD-hard\n# Example: Pure-Circuit solution with no equilibrium collapse\n# Conclusion\n# Extended Related Work","[{\"question\":\"What problem does the paper address about coarse correlated equilibria in Markov games?\",\"answer\":\"It analyzes the complexity of computing stationary Markov coarse correlated equilibria (CCE) in discounted single-controller stochastic (Markov) games, focusing on how hard approximate computation is.\"},{\"question\":\"How is the single-controller setting different from previous two-player constructions?\",\"answer\":\"Previous hardness used turn-based constructions where each state is controlled by a single player and control alternates, forcing every Markov CCE to collapse to a Nash equilibrium. The paper studies the case where one player controls all transitions, so that collapse does not occur.\"},{\"question\":\"What complexity result is proved for computing an approximate stationary Markov CCE?\",\"answer\":\"Computing an approximate stationary Markov CCE in two-player single-controller stochastic games is PPAD-complete, even with a fixed discount factor and binary actions, with separate guarantees for perfect and non-perfect notions of equilibrium constraints.\"}]",1784212342,96,{"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},"the-complexity-of-computing-coarse-correlated-equilibria-in-markov-games-with-a-single-controller","",{"@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/the-complexity-of-computing-coarse-correlated-equilibria-in-markov-games-with-a-single-controller/86519/",{"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","2026-07-25","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},"What problem does the paper address about coarse correlated equilibria in Markov games?","Question",{"text":75,"@type":76},"It analyzes the complexity of computing stationary Markov coarse correlated equilibria (CCE) in discounted single-controller stochastic (Markov) games, focusing on how hard approximate computation is.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the single-controller setting different from previous two-player constructions?",{"text":80,"@type":76},"Previous hardness used turn-based constructions where each state is controlled by a single player and control alternates, forcing every Markov CCE to collapse to a Nash equilibrium. The paper studies the case where one player controls all transitions, so that collapse does not occur.",{"name":82,"@type":73,"acceptedAnswer":83},"What complexity result is proved for computing an approximate stationary Markov CCE?",{"text":84,"@type":76},"Computing an approximate stationary Markov CCE in two-player single-controller stochastic games is PPAD-complete, even with a fixed discount factor and binary actions, with separate guarantees for perfect and non-perfect notions of equilibrium constraints.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]