[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84190-en":3,"doc-seo-84190-105":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},84190,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Simple Nash Equilibria for Qualitative Multiplayer Games","Investigates memory requirements for Nash and subgame-perfect equilibria in turn-based deterministic games with ω-regular objectives. Proves that memoryless randomized (stationary) subgame-perfect equilibria always exist for games with reachability, safety, and 0–2 Muller objectives, and any combination of these objectives. Presents an algorithm constructing such an equilibrium and shows randomisation can be required for memoryless equilibria under reachability/Büchi and safety/CoBüchi, while memoryless equilibria fail to exist for other Muller classes in the Mostowski hierarchy.","arXiv :2607 .07 15 1v 1 [ cs .GT] 8 Jul 2026  \nSimple Nash Equilibria for Qualitative Multiplayer Games  \nMona Alluwaym   \nUniversity of Liverpool, United Kingdom  \nJames C. A. Main   \nUniversity of Oxford, United Kingdom  \nSven Schewe   \nUniversity of Liverpool, United Kingdom  \n~~ Abstract ~~  \nWe investigate memory requirements for Nash and subgame-perfect equilibria in turn-based deterministic games with ω-regular objectives. We prove that memoryless randomised (i.e., stationary) subgame-perfect equilibria always exist in games with reachability, safety, and 0-2 Muller objectives (i.e., Muller objectives for which accepting sets are either up-or downward closed), and any combination of these objectives. We provide an algorithm to construct such an equilibrium. We also show that randomisation may be required to construct memoryless equilibria in games with reachability or Büchi as well as safety or CoBüchi objectives, and that memoryless equilibria need not exist for any other class of Muller objectives (with respect to the Mostowski hierarchy) .  \n2012 ACM Subject Classification Theory of computation → Solution concepts in game theory  \nKeywords and phrases games on graphs, multiplayer games, Nash equilibria, subgame-perfect equilibria, memoryless strategies  \nFunding This work has been supported by the EPSRC projects EP/Z536179/1, EP/X03688X/1, and EP/X042596/1 .  \n 1  Introduction  \nGames on graphs. We study turn-based games played on finite graphs (e.g., [24, 20]) . The structure on which the game is played is called a turn-based deterministic arena (or simply an arena); it is described by a directed graph and a partition of the vertices among the players. At the beginning of a play, a token is placed on an initial vertex and, in each round, the player in control of the current vertex moves the token along an outgoing edge. The players continue this interaction for infinitely many rounds and construct a play, i.e., an infinite path of the graph. The goal of a player is formalised by an objective, i.e., a subset of plays. Players follow strategies describing how to select moves. Formally, strategies are functions assigning to any play history the decisions to be made. In full generality, strategies may use memory and randomisation. A strategy is called pure if it deterministically assigns moves to histories, memoryless (or stationary) if its decisions depend only on the current vertex and not the full history, and positional if it is both memoryless and pure.  \nGames on graphs are notably useful to solve the controller synthesis problem (e.g., [2]) . Given a reactive system, the goal of controller synthesis is to automatically construct a controller for the system that enforces some specification no matter the behaviour of the environment of the system. It can be solved by modelling the interaction of the system and its environment as a zero-sum game on a graph: the goal of the system player is to enforce their objective (which formalises the specification) and the environment player aims to falsify this objective. Strategies of the system player correspond to controllers; thus the goal is to find a winning strategy, i.e., a strategy enforcing the objective of the system against all  \n2 Simple Nash Equilibria for Qualitative Multiplayer Games  \nstrategies of its adversary. In general, we are interested in simple winning strategies (e.g. , using limited memory), as they yield simpler controllers (see [37] and references therein) .  \nMultiplayer games. For systems consisting of several components interacting together, treating each component as competing with the others may be too restrictive (e.g., it precludes cooperation between components) . Instead, we consider games in which each component is a player. Each player has their own objective and these are not necessarily incompatible. Such games are called multiplayer non-zero-sum games.  \nNash equilibria [34](NE) are the most classical formalisation of rational behaviou","cbCaiezbYy5HHXMz","https://ap.wps.com/l/cbCaiezbYy5HHXMz","pdf",513681,4,1,18,"English","en",105,"# Introduction\n## Games on graphs\n## Multiplayer games\n## When is memory needed?","[{\"question\":\"What problem does the paper address about equilibria in qualitative multiplayer games?\",\"answer\":\"It studies how much memory is needed to realize Nash and subgame-perfect equilibria in turn-based deterministic games with ω-regular objectives.\"},{\"question\":\"When do memoryless randomized (stationary) subgame-perfect equilibria always exist?\",\"answer\":\"They always exist for games whose ω-regular objectives are reachability, safety, and 0–2 Muller objectives, including any combination of these objective types.\"},{\"question\":\"Is randomisation always unnecessary for memoryless equilibria?\",\"answer\":\"No. Randomisation may be required to construct memoryless equilibria for games with reachability or Büchi together with safety or CoBüchi objectives.\"}]",1784193821,45,{"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},"simple-nash-equilibria-for-qualitative-multiplayer-games","",{"@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/simple-nash-equilibria-for-qualitative-multiplayer-games/84190/",{"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","2026-07-27","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 equilibria in qualitative multiplayer games?","Question",{"text":75,"@type":76},"It studies how much memory is needed to realize Nash and subgame-perfect equilibria in turn-based deterministic games with ω-regular objectives.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"When do memoryless randomized (stationary) subgame-perfect equilibria always exist?",{"text":80,"@type":76},"They always exist for games whose ω-regular objectives are reachability, safety, and 0–2 Muller objectives, including any combination of these objective types.",{"name":82,"@type":73,"acceptedAnswer":83},"Is randomisation always unnecessary for memoryless equilibria?",{"text":84,"@type":76},"No. Randomisation may be required to construct memoryless equilibria for games with reachability or Büchi together with safety or CoBüchi 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