[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83513-en":3,"doc-seo-83513-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83513,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Positive and Negative Determinant Strategies in Repeated Games with Behavior-Value Inconsistency","Direct reciprocity is a core mechanism for sustaining cooperation in repeated interactions, and zero-determinant (ZD) strategies enable unilateral payoff control. Existing work overlooks internal costs arising when behavior differs from an agent’s internal thoughts. A game-theoretic framework assumes an agent pays such an internal cost under behavior–value inconsistency. It is proven that ZD strategies cannot exist under this cost, and a new positive/negative determinant strategy class is derived to enforce unilateral affine control over opponents’ average payoffs, with improved control over ZD.","arXiv :2607 .00625v 1 [ cs .GT] 1 Jul 2026  \nPositive and Negative Determinant Strategies in Repeated Games with Behavior-Value Inconsistency  \nYuan Liu 1 ,2 ,3 , Yakun Wang 1 ,2 ,4 , Bin Wu 1 ,2 ,∗  \n1 School of Mathematical Sciences, Beijing University of Posts and Telecommunications, Beijing, China  \n2 Key Laboratory of Mathematics and Information Networks (Beijing University of Posts and  \nTelecommunications), Ministry of Education, Beijing, China  \n3 Department of Theoretical Biology, Max Planck Institute for Evolutionary Biology, Pl¨on, Germany  \n4 Department of Engineering, Universitat Pompeu Fabra, Barcelona, Spain  \n∗ Bin Wu: [bin.wu@bupt.edu.cn](bin.wu@bupt.edu.cn)  \nDirect reciprocity, based on the repeated interactions, is a fundamental mechanism to promote cooperation. Zero-determinant (ZD) strategies have opened an avenue for unilateral payoff control. However, previous studies neglect internal costs provided what agents do differ from what agents think, which is crucial for decision making of intelligent agents. Motivated by this, we establish a game theoretical framework by assuming that an individual pays the internal cost if the behavior is inconsistent with the internal thought. We prove that ZD strategy does not exist if the cost via behavior-value inconsistency is present. Instead, we find a new class of repeated strategies that enforce a unilateral payoff control, which is termed as positive/negative determinant strategy. The found strategy allows an individual to enforce an affine combination of two individuals’ average payoffs above/below zero. Consequently, a focal individual is able to unilaterally control the opponent’spayoff below a given value via negative determinant strategy, and a focal individual is able to get more payoff than the opponent via positive determinant strategy. We also find that the control ability of positive/negative determinant strategies is better off than that of ZD strategies. Our work highlights the importance of inconsistency between the behavior and value on payoff control, which is typically absent in classic ZD strategies.  \n1 Introduction  \nCooperation is a non trivial phenomenon in the sight of evolutionary theory. Recent decades have seen a progress in the mechanisms to promote cooperation. Direct reciprocity, based on the repeated encounters, is a fundamental mechanism to promote  \ncooperation [36, 39, 35, 32, 6] . The repeated Prisoner’s Dilemma game is typically used to model the direct reciprocity. Axelrod discovered the “winning strategy” that an individual cooperates in the first round and repeats the opponent’s behavior in the previous round, i.e., tit-for-tat [7] . Tit-for-tat strategy has the property that if a player adopts such strategy, both players have the same payoff in the long run, regardless of the strategy chosen by the opponent. Boerlijst et al. discovered a general class of such strategies named by equalizer strategies [8] . In 2012, Press and Dyson discovered a more general class of strategies, which allow a player to enforce an affine relationship between her own payoff and the opponent’s payoff, regardless of the opponent’s strategy [41] . This is referred as zero-determinant strategies (ZD strategies), since they are obtained via the manipulation of matrices without changing its determinant.  \nZD strategy opens an avenue for unilateral payoff control [2, 24, 40, 33, 9, 10, 48, 1, 11] . Using ZD strategies, the focal individual is able to enforce an extortionate share of payoffs. Furthermore, the control can be unilateral. Last but not the least, the unilateral payoff control via ZD strategy makes use of almost the least information, that is, only the focal and opponent’s behavior in the current round without payoff evaluation and without recursively updating one’s response. It is a once-for-all strategy designed atthe beginning of the game. Beyond expectations, such counterintuitive unilateral payoff control is ubiquitous ranging from pairwis","cbCais1KM7MgLt7J","https://ap.wps.com/l/cbCais1KM7MgLt7J","pdf",945056,1,20,"English","en",105,"# Introduction\n## Direct reciprocity and repeated Prisoner’s Dilemma\n## Zero-determinant (ZD) strategies and payoff control\n## Motivation from behavior–value inconsistency","[{\"question\":\"What internal inconsistency is modeled in the framework?\",\"answer\":\"An agent incurs an internal cost when its observable behavior is inconsistent with its internal thought or value.\"},{\"question\":\"Why do zero-determinant (ZD) strategies fail under behavior–value inconsistency?\",\"answer\":\"The paper proves that ZD strategies do not exist when the cost associated with behavior–value inconsistency is present.\"},{\"question\":\"What do positive and negative determinant strategies achieve?\",\"answer\":\"They enforce unilateral payoff control: a focal individual can keep the opponent’s payoff above or below a chosen level via an affine combination of average payoffs, depending on the sign of the determinant.\"}]",1784188552,50,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"positive-and-negative-determinant-strategies-in-repeated-games-with-behavior-value-inconsistency","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/positive-and-negative-determinant-strategies-in-repeated-games-with-behavior-value-inconsistency/83513/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What internal inconsistency is modeled in the framework?","Question",{"text":74,"@type":75},"An agent incurs an internal cost when its observable behavior is inconsistent with its internal thought or value.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why do zero-determinant (ZD) strategies fail under behavior–value inconsistency?",{"text":79,"@type":75},"The paper proves that ZD strategies do not exist when the cost associated with behavior–value inconsistency is present.",{"name":81,"@type":72,"acceptedAnswer":82},"What do positive and negative determinant strategies achieve?",{"text":83,"@type":75},"They enforce unilateral payoff control: a focal individual can keep the opponent’s payoff above or below a chosen level via an affine combination of average payoffs, depending on the sign of the 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