Pub Date : 2025-12-13DOI: 10.1016/j.geb.2025.12.003
Hyunjun Cho , Jin Yeub Kim , Jaeok Park
We examine the attainability of marginal contributions in two models of one-sided matching and assignment. For the one-sided matching problem, where agents in a single group are matched with each other, the core may be empty, and even when nonempty, some agents may fail to attain their marginal contributions in the core. By allowing fractional matchings, however, we guarantee the non-emptiness of the core and show that every agent’s marginal contribution is attainable in the core. This implies that all the agents can receive their marginal contributions at the same time if and only if the core is a singleton. For the one-sided assignment problem, where agents are matched to objects they own, we obtain analogous results even without introducing fractional assignments. Finally, extending to linear production games, which encompass both models, we show that the attainability property may fail but is guaranteed under sufficiently many replications.
{"title":"Marginal contribution and singleton cores in one-sided matching and assignment","authors":"Hyunjun Cho , Jin Yeub Kim , Jaeok Park","doi":"10.1016/j.geb.2025.12.003","DOIUrl":"10.1016/j.geb.2025.12.003","url":null,"abstract":"<div><div>We examine the attainability of marginal contributions in two models of one-sided matching and assignment. For the one-sided matching problem, where agents in a single group are matched with each other, the core may be empty, and even when nonempty, some agents may fail to attain their marginal contributions in the core. By allowing fractional matchings, however, we guarantee the non-emptiness of the core and show that every agent’s marginal contribution is attainable in the core. This implies that all the agents can receive their marginal contributions at the same time if and only if the core is a singleton. For the one-sided assignment problem, where agents are matched to objects they own, we obtain analogous results even without introducing fractional assignments. Finally, extending to linear production games, which encompass both models, we show that the attainability property may fail but is guaranteed under sufficiently many replications.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 82-97"},"PeriodicalIF":1.0,"publicationDate":"2025-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-11DOI: 10.1016/j.geb.2025.12.001
Mustafa Oǧuz Afacan , Umut Dur , A. Arda Gitmez , Özgür Yılmaz
In school choice problems, the pursuit of student welfare (efficiency) is constrained by the requirement to respect schools’ priorities (fairness). Among fair matchings, even the welfare maximizing one, the Student-Optimal Stable Matching (SOSM), is inefficient. Moreover, any mechanism that yields welfare gains over the SOSM is manipulable by students. This paper investigates the fairness and incentive properties of efficient mechanisms. We compare matchings using the notion that one matching is less unfair than another if it generates a smaller set of students whose priorities are violated, and we define minimal unfairness accordingly. We show that the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism is minimally unfair within the class of Pareto efficient mechanisms satisfying a simple incentive requirement, top-manipulation-proofness. Moreover, the EADA satisfies a stronger property, upper-manipulation-proofness. Upper-manipulation-proofness is a broad incentive property one can maintain while improving upon the SOSM. Together, these results highlight a sharp frontier in school choice: when the objective is efficiency and improving student welfare over the SOSM, the EADA emerges as a compelling mechanism in both fairness and incentive terms.
{"title":"Improving the deferred acceptance with minimal compromise","authors":"Mustafa Oǧuz Afacan , Umut Dur , A. Arda Gitmez , Özgür Yılmaz","doi":"10.1016/j.geb.2025.12.001","DOIUrl":"10.1016/j.geb.2025.12.001","url":null,"abstract":"<div><div>In school choice problems, the pursuit of student welfare (efficiency) is constrained by the requirement to respect schools’ priorities (fairness). Among fair matchings, even the welfare maximizing one, the Student-Optimal Stable Matching (<em>SOSM</em>), is inefficient. Moreover, any mechanism that yields welfare gains over the <em>SOSM</em> is manipulable by students. This paper investigates the fairness and incentive properties of efficient mechanisms. We compare matchings using the notion that one matching is <em>less unfair</em> than another if it generates a smaller set of students whose priorities are violated, and we define <em>minimal unfairness</em> accordingly. We show that the Efficiency-Adjusted Deferred Acceptance (<em>EADA</em>) mechanism is minimally unfair within the class of Pareto efficient mechanisms satisfying a simple incentive requirement, <em>top-manipulation-proofness</em>. Moreover, the <em>EADA</em> satisfies a stronger property, <em>upper-manipulation-proofness</em>. Upper-manipulation-proofness is a broad incentive property one can maintain while improving upon the <em>SOSM</em>. Together, these results highlight a sharp frontier in school choice: when the objective is efficiency and improving student welfare over the <em>SOSM</em>, the <em>EADA</em> emerges as a compelling mechanism in both fairness and incentive terms.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 64-81"},"PeriodicalIF":1.0,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-04DOI: 10.1016/j.geb.2025.11.012
Álvaro Cartea , Patrick Chang , José Penalva , Harrison Waldon
We prove a Folk theorem when players with bounded rationality learn as they play a repeated potential game. We use a dynamic generalization of smooth fictitious play with bounded m-recall strategies to model learning with bounded rationality that is consistent with learning by algorithms. In a repeated potential game with perfect monitoring, we use this learning model to show that for any feasible and individually rational payoff profile, if players have sufficient recall, are sufficiently patient, and best respond with sufficiently few mistakes, then the players have a non-zero probability of learning an m-recall strategy profile that achieves an average payoff close to the specified payoff profile for an appropriate continuation game. Moreover, the strategy profile learned is an m-recall ϵ-subgame perfect equilibrium of the repeated game. This finding demonstrates that competition authorities are correct in their concern about algorithmic collusion.
{"title":"Algorithmic collusion and a folk theorem from learning with bounded rationality","authors":"Álvaro Cartea , Patrick Chang , José Penalva , Harrison Waldon","doi":"10.1016/j.geb.2025.11.012","DOIUrl":"10.1016/j.geb.2025.11.012","url":null,"abstract":"<div><div>We prove a Folk theorem when players with bounded rationality learn as they play a repeated potential game. We use a dynamic generalization of smooth fictitious play with bounded <em>m</em>-recall strategies to model learning with bounded rationality that is consistent with learning by algorithms. In a repeated potential game with perfect monitoring, we use this learning model to show that for any feasible and individually rational payoff profile, if players have sufficient recall, are sufficiently patient, and best respond with sufficiently few mistakes, then the players have a non-zero probability of learning an <em>m</em>-recall strategy profile that achieves an average payoff close to the specified payoff profile for an appropriate continuation game. Moreover, the strategy profile learned is an <em>m</em>-recall ϵ-subgame perfect equilibrium of the repeated game. This finding demonstrates that competition authorities are correct in their concern about algorithmic collusion.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"157 ","pages":"Pages 1-21"},"PeriodicalIF":1.0,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-02DOI: 10.1016/j.geb.2025.11.011
Ata Atay , Marina Núñez , Tamás Solymosi
This paper analyzes the extent to which well-known results on the relationship between the bargaining set, the core, and the kernel in one-to-one assignment games generalize to many-to-one assignment markets, and by extension, many-to-many markets. Using a minimal counterexample, we show that the bargaining set does not necessarily coincide with the core and that the kernel may not be contained within the core. We would like to highlight that the failure of the coincidence between the core and the bargaining set, as observed in the many-to-one assignment game, is quite notable. This is especially true when compared to various other highly structured games, many of which emerge from combinatorial optimization problems, such as the one underlying many-to-one assignment games.
{"title":"A note on the non-coincidence of the core and the bargaining set in many-to-one assignment markets","authors":"Ata Atay , Marina Núñez , Tamás Solymosi","doi":"10.1016/j.geb.2025.11.011","DOIUrl":"10.1016/j.geb.2025.11.011","url":null,"abstract":"<div><div>This paper analyzes the extent to which well-known results on the relationship between the bargaining set, the core, and the kernel in one-to-one assignment games generalize to many-to-one assignment markets, and by extension, many-to-many markets. Using a minimal counterexample, we show that the bargaining set does not necessarily coincide with the core and that the kernel may not be contained within the core. We would like to highlight that the failure of the coincidence between the core and the bargaining set, as observed in the many-to-one assignment game, is quite notable. This is especially true when compared to various other highly structured games, many of which emerge from combinatorial optimization problems, such as the one underlying many-to-one assignment games.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 58-63"},"PeriodicalIF":1.0,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145750259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze risk-taking behaviors of financial institutions linked through cross-shareholding relationships. We endogenize risk exposure accounting for default risk by modeling Value-at-Risk-based risk-management – that is targeted default probabilities – in the presence of extreme risk on asset return. We relate risk-taking behaviors to a centrality measure that captures the propagation of losses within the network, and show that network integration increases risk-taking levels and expected return. However, we show that network integration also results in larger expected shortfall, indicating greater exposure to losses for creditors. We explore the impact of the cross-shareholding network on the implementation of regulation, particularly through capital requirements, and identify key institutions, those with the highest influence on aggregate investments in risky assets.
{"title":"Risk-taking in financial networks","authors":"Mohamed Belhaj , Renaud Bourlès , Frédéric Deroïan","doi":"10.1016/j.geb.2025.11.006","DOIUrl":"10.1016/j.geb.2025.11.006","url":null,"abstract":"<div><div>We analyze risk-taking behaviors of financial institutions linked through cross-shareholding relationships. We endogenize risk exposure accounting for default risk by modeling Value-at-Risk-based risk-management – that is targeted default probabilities – in the presence of extreme risk on asset return. We relate risk-taking behaviors to a centrality measure that captures the propagation of losses within the network, and show that network integration increases risk-taking levels and expected return. However, we show that network integration also results in larger expected shortfall, indicating greater exposure to losses for creditors. We explore the impact of the cross-shareholding network on the implementation of regulation, particularly through capital requirements, and identify key institutions, those with the highest influence on aggregate investments in risky assets.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 35-57"},"PeriodicalIF":1.0,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.geb.2025.11.010
Mauricio Ribeiro
Economists often rely on people’ s choices to infer their preferences. However, inferring preferences from choices becomes problematic when people face unobserved constraints. In this paper, I study how to (cautiously) infer preferences from choices when the choices of the members of a social group are subject to a constraint that we only imperfectly know. When this happens, heterogeneity in the group’ s choices, along with more information about the constraint, helps recover the preferences of group members, whereas homogeneity in choices hinders it. I further argue that standard approaches to choice-based welfare analysis can lead to misleading inferences about welfare when there are unobserved social constraints.
{"title":"Choice and welfare under social constraints","authors":"Mauricio Ribeiro","doi":"10.1016/j.geb.2025.11.010","DOIUrl":"10.1016/j.geb.2025.11.010","url":null,"abstract":"<div><div>Economists often rely on people’ s choices to infer their preferences. However, inferring preferences from choices becomes problematic when people face unobserved constraints. In this paper, I study how to (cautiously) infer preferences from choices when the choices of the members of a social group are subject to a constraint that we only imperfectly know. When this happens, heterogeneity in the group’ s choices, along with more information about the constraint, helps recover the preferences of group members, whereas homogeneity in choices hinders it. I further argue that standard approaches to choice-based welfare analysis can lead to misleading inferences about welfare when there are unobserved social constraints.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 1-13"},"PeriodicalIF":1.0,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145659081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.geb.2025.11.009
Korhan Kocak
Parties in primary elections often choose between candidates who appeal to their base and those who appeal to the broader electorate. I present a model of primaries where incumbents can exploit this trade-off the opposition faces between ideological congruence and electability. In the model, incumbents reduce their appeal to the median voter by moving away from the center. In doing so, they provoke the opposition into nominating extremists, improving their reelection prospects. This mechanism generates elite polarization as politicians leapfrog voters — not despite electoral concerns, but because of them. The analysis fits the observation that incumbents sometimes move away from the center near the end of their term. Provoking the opposition relies on two conditions: divergence of primary and general electorates and a limited set of potential nominees. I argue that partisan sorting and changes in nomination procedures over the last decades made this strategy viable.
{"title":"Provoking the opposition","authors":"Korhan Kocak","doi":"10.1016/j.geb.2025.11.009","DOIUrl":"10.1016/j.geb.2025.11.009","url":null,"abstract":"<div><div>Parties in primary elections often choose between candidates who appeal to their base and those who appeal to the broader electorate. I present a model of primaries where incumbents can exploit this trade-off the opposition faces between ideological congruence and electability. In the model, incumbents reduce their appeal to the median voter by moving away from the center. In doing so, they provoke the opposition into nominating extremists, improving their reelection prospects. This mechanism generates elite polarization as politicians leapfrog voters — not despite electoral concerns, but because of them. The analysis fits the observation that incumbents sometimes move away from the center near the end of their term. Provoking the opposition relies on two conditions: divergence of primary and general electorates and a limited set of potential nominees. I argue that partisan sorting and changes in nomination procedures over the last decades made this strategy viable.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"156 ","pages":"Pages 14-34"},"PeriodicalIF":1.0,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.geb.2025.11.008
Mitsunori Noguchi
This note challenges the belief that mining processes are too complex for traditional game-theoretic analysis. We provide Nash-like equilibrium solutions for two-player gap games, focusing on the strategic delay of mining rig activation in Bitcoin’s Proof-of-Work (PoW) system. By delaying activation, miners can optimize profits, particularly as transaction fees become a more significant portion of their revenues (Carlsten et al., 2016). Building on the game-theoretic model of Tsabary and Eyal (2018), we offer a rigorous, proof-based analysis of this behavior, highlighting the importance of verifying simulation results through precise mathematical methods. The primary technical contribution of this note is a novel method for deriving closed-form equilibrium solutions for potentially non-differentiable ex-ante expected payoff functions Ui, defined as integrals in which the integrands—representing profit at time t—depend discontinuously on miners’ rig start times s.
这篇文章挑战了传统博弈论分析认为采矿过程过于复杂的观点。我们为两人博弈提供了类似纳什的均衡解决方案，重点关注比特币工作量证明（PoW）系统中矿机激活的战略延迟。通过延迟激活，矿工可以优化利润，特别是当交易费用成为其收入中更重要的一部分时（Carlsten et al., 2016）。在Tsabary和Eyal（2018）的博弈论模型的基础上，我们对这种行为进行了严格的、基于证据的分析，强调了通过精确的数学方法验证模拟结果的重要性。本文的主要技术贡献是一种新方法，用于推导潜在不可微的预先预期收益函数Ui的封闭形式均衡解，该函数定义为积分，其中表示时间t的利润的积分不连续地依赖于矿工的钻机启动时间s。
{"title":"Exact Nash-like solutions of blockchain gap games","authors":"Mitsunori Noguchi","doi":"10.1016/j.geb.2025.11.008","DOIUrl":"10.1016/j.geb.2025.11.008","url":null,"abstract":"<div><div>This note challenges the belief that mining processes are too complex for traditional game-theoretic analysis. We provide Nash-like equilibrium solutions for two-player gap games, focusing on the strategic delay of mining rig activation in Bitcoin’s Proof-of-Work (PoW) system. By delaying activation, miners can optimize profits, particularly as transaction fees become a more significant portion of their revenues (Carlsten et al., 2016). Building on the game-theoretic model of Tsabary and Eyal (2018), we offer a rigorous, proof-based analysis of this behavior, highlighting the importance of verifying simulation results through precise mathematical methods. The primary technical contribution of this note is a novel method for deriving closed-form equilibrium solutions for potentially non-differentiable ex-ante expected payoff functions <em>U<sub>i</sub></em>, defined as integrals in which the integrands—representing profit at time <em>t</em>—depend discontinuously on miners’ rig start times <em>s</em>.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"155 ","pages":"Pages 310-320"},"PeriodicalIF":1.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.geb.2025.11.004
Lars-Gunnar Svensson , Tommy Andersson , Lars Ehlers
We consider a market with indivisible objects, called houses, and monetary transfers. Each house is initially occupied by one agent and each agent demands exactly one house. The problem is to identify the complete set of direct allocation mechanisms that can be used to reallocate the houses among the agents. On the one hand, for price eq1uilibrium mechanisms, we show that the only non-manipulable mechanism is one with a minimum equilibrium price vector. The result is not true on the classic or the quasi-linear domains, but on reduced domains of preference profiles containing “almost all” profiles in the classic or the quasi-linear domain, respectively. On the other hand, while minimum price equilibrium mechanisms are not necessarily efficient (as prices are not zero), we show that no non-manipulable mechanism Pareto dominates a minimum price equilibrium mechanism, making them constrained efficient in the class of non-manipulable mechanisms.
{"title":"Non-manipulable house exchange under (minimum) equilibrium prices","authors":"Lars-Gunnar Svensson , Tommy Andersson , Lars Ehlers","doi":"10.1016/j.geb.2025.11.004","DOIUrl":"10.1016/j.geb.2025.11.004","url":null,"abstract":"<div><div>We consider a market with indivisible objects, called houses, and monetary transfers. Each house is initially occupied by one agent and each agent demands exactly one house. The problem is to identify the complete set of direct allocation mechanisms that can be used to reallocate the houses among the agents. On the one hand, for price eq1uilibrium mechanisms, we show that the only non-manipulable mechanism is one with a minimum equilibrium price vector. The result is not true on the classic or the quasi-linear domains, but on reduced domains of preference profiles containing “almost all” profiles in the classic or the quasi-linear domain, respectively. On the other hand, while minimum price equilibrium mechanisms are not necessarily efficient (as prices are not zero), we show that no non-manipulable mechanism Pareto dominates a minimum price equilibrium mechanism, making them constrained efficient in the class of non-manipulable mechanisms.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"155 ","pages":"Pages 270-286"},"PeriodicalIF":1.0,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-12DOI: 10.1016/j.geb.2025.11.002
Inés Macho-Stadler, David Pérez-Castrillo, David Wettstein
Theorem 1 in Macho-Stadler et al. (2018) is incorrect. In this note, we correct the theorem by introducing a new axiom, the non-negative payoffs axiom, and adjusting the proof accordingly. We also discuss the implications of this correction for other results presented in the paper.
Macho-Stadler et al.（2018）中的定理1是不正确的。在这篇笔记中，我们通过引入一个新的公理——非负收益公理来修正定理，并相应地调整证明。我们还讨论了这一修正对论文中提出的其他结果的影响。
{"title":"Corrigendum to “Values for environments with externalities – the average approach” [Games Econ. Behav. 108 (2018) 49–64.]","authors":"Inés Macho-Stadler, David Pérez-Castrillo, David Wettstein","doi":"10.1016/j.geb.2025.11.002","DOIUrl":"10.1016/j.geb.2025.11.002","url":null,"abstract":"<div><div>Theorem 1 in Macho-Stadler et al. (2018) is incorrect. In this note, we correct the theorem by introducing a new axiom, the non-negative payoffs axiom, and adjusting the proof accordingly. We also discuss the implications of this correction for other results presented in the paper.</div></div>","PeriodicalId":48291,"journal":{"name":"Games and Economic Behavior","volume":"155 ","pages":"Pages 267-269"},"PeriodicalIF":1.0,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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