[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83256-en":3,"doc-seo-83256-105":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},83256,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Stability and Convergence of Optimistic Exponential Weights with Asymmetric Step Sizes in Bimatrix Games","Study of two-player bimatrix games under optimistic exponential weights (optEW) focuses on last-iterate convergence and equilibrium stability when players use different step sizes ηx and ηy. A first theorem gives a sufficient global last-iterate convergence condition for zero-sum games, assuming finitely many fixed points, depending only on the product ηxηy. A second theorem provides an almost-tight threshold for asymptotic stability or instability for general bimatrix games, also expressed via ηxηy. Known results and step-size bounds are recovered and numerical experiments illustrate the theory.","arXiv :2607 .075 17v 1 [ cs .MA] 8 Jul 2026  \nStability and Convergence of Optimistic Exponential Weights with Asymmetric Step Sizes in Bimatrix Games  \nHédi Hadiji 1 and Sarah Sachs2  \n1 Laboratoire des Signaux et Systèmes, CentraleSupélec, Paris, France  \n2 School of Mathematics, University of Bristol, Bristol, United Kingdom  \nAbstract  \nWe study bimatrix two-player games and investigate the last-iterate convergence and stability of equilibria for the iterates generated by the optimistic exponential weights method. In contrast to prior work, we allow the step sizes ηx and ηy to differ. Our first main result establishes, under the assumption that the set of fixed points is finite, a sufficient condition for global last-iterate convergence in the special case of zero-sum games, which constrains only the product ηx ηy of the step sizes. This condition is practically relevant and partially explains empirically observed behavior. Our second main result provides an almost-tight threshold for asymptotic stability and instability, again in terms of products of the step sizes, for general bimatrix games. This result is primarily of theoretical interest. We derive several known results and practically relevant step size bounds for special cases and illustrate our results by experiments.  \nKeywords: Optimistic Exponential Weights, Bimatrix Games, Stability, Convergence.  \n1 Introduction  \nWe consider bimatrix games ΓpA, Bq with payoff matrices A, BJ P Rdx ˆdy . Denote by ∆dx and ∆dy the probability simplices in Rdx (respectively Rdy ) . The x-player chooses x P ∆dx to maximize xJAy, while the y-player chooses y P ∆dy to maximize yJ Bx. A Nash equilibrium rx‹, y‹s satisfies  \nx‹ P arg max xJAy‹, y‹ P arg max y JBx‹.  \nxP∆dx yP∆dy  \nA central question in game theory and learning in games is whether simple, efficient iterative algorithms converge to such equilibrium points. Classical dynamics such as Gradient Descent-Ascent and Multiplicative Weights Update are known to exhibit cycling and fail to converge even in simple bilinear games (see, e.g. , Bailey and Piliouras [2018], Cheung and Piliouras [2019], Mertikopoulos et al. [2018b]) . This has motivated significant interest in simple modifications of these algorithms that mitigate this issue. A prominent approach is the optimism framework [Chiang et al. , 2012 , Rakhlin and Sridharan, 2013 , Syrgkanis et al. , 2015], which underlies algorithms such as Optimistic Gradient Descent–Ascent and optimistic exponential weights (optEW) .  \nIn this paper, we study the stability of Nash equilibria under optimistic exponential weights dynamics (and variants thereof) in two-player bimatrix games. That is, all coordinates i P t1, . . . , dxu and j P t1, . . . , dyu are updated as  \nxt`1,i9 xt,i exppηx rAp2yt ´yt´ 1 qsiq and yt`1,j9 yt,j exppηy rBp2xt ´xt´ 1 qsj q , (1) where ηx , ηy ą 0 are the step sizes. There are two fundamentally different settings for the step size choices: (1) constant step sizes and (2) time-dependent step sizes. We study the convergence behavior of pzt qtPN “ ppxt , ytqqtPN under constant, potentially unequal step sizes. We refer to this setting as using asymmetric step sizes, noting that related literature sometimes describes it as ‘two-time-scale’ step sizes [Lin et al., 2025] . We adopt the former terminology to clearly distinguish our setting from time-varying step size schemes for saddle-point problems with stochastic feedback. In the classical two-time-scale framework, the step size sequences pηx,t q and pηy,tq are assumed to be not summable but square summable and satisfy ηx,t{ηy,t Ñ 0 as t Ñ 8 [Borkar, 1997 , 2025] .  \nExisting analyses often rely on simultaneously controlling both step sizes, which results in a step size requirement ηx “ ηy , or controlling the individual step sizes, which results in step size requirements on maxtηx , ηyu. Such conditions do not capture the product dependence as suggested by empirical observations (see Figure 1) . Our results align ","cbCaiol6e0vLAKC7","https://ap.wps.com/l/cbCaiol6e0vLAKC7","pdf",7240645,2,1,54,"English","en",105,"# Introduction\n## Contributions\n## Motivation and Broader Context","[{\"question\":\"What does the paper study about optimistic exponential weights in bimatrix games?\",\"answer\":\"It analyzes last-iterate convergence and the stability of equilibria produced by optimistic exponential weights dynamics in two-player bimatrix games, especially under asymmetric step sizes.\"},{\"question\":\"How do the main convergence and stability conditions depend on the step sizes?\",\"answer\":\"Both the global convergence condition for zero-sum games and the stability/instability threshold for general games are characterized in terms of the product ηxηy rather than requiring ηx = ηy.\"},{\"question\":\"What are the key limitations or open cases mentioned regarding the product ηxηy?\",\"answer\":\"For general games, the threshold is almost tight: stability holds for small products and instability for large products, while the boundary case ηxηy = c is left indeterminate.\"}]",1784186305,136,{"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},"stability-and-convergence-of-optimistic-exponential-weights-with-asymmetric-step-sizes-in-bimatrix-games","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/stability-and-convergence-of-optimistic-exponential-weights-with-asymmetric-step-sizes-in-bimatrix-games/83256/",4,{"url":51,"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","2026-07-24","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 does the paper study about optimistic exponential weights in bimatrix games?","Question",{"text":75,"@type":76},"It analyzes last-iterate convergence and the stability of equilibria produced by optimistic exponential weights dynamics in two-player bimatrix games, especially under asymmetric step sizes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the main convergence and stability conditions depend on the step sizes?",{"text":80,"@type":76},"Both the global convergence condition for zero-sum games and the stability/instability threshold for general games are characterized in terms of the product ηxηy rather than requiring ηx = ηy.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the key limitations or open cases mentioned regarding the product ηxηy?",{"text":84,"@type":76},"For general games, the threshold is almost tight: stability holds for small products and instability for large products, while the boundary case ηxηy = c is left 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