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Published: 10 December 2022

Last updated: 29 January 2023

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1

PETROSJAN, L. A., and L. V. GRAUER. "STRONG NASH EQUILIBRIUM IN MULTISTAGE GAMES." International Game Theory Review 04, no. 03 (September 2002): 255–64. http://dx.doi.org/10.1142/s0219198902000689.

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Infinite multistage games G with games Γ(·) played on each stage are considered. The definition of path and trajectory in graph tree are introduced. For infinite multistage games G a regularization procedure is introduced and in the regularizied game a strong Nash Equilibrium (coalition proof) is constructed. The approach considered in this paper is similar to one used in the proof of Folk theorems for infinitely repeated games. The repeated n-person "Prisoner's Dilemma" game is considered, as a special case. For this game a strong Nash Equilibrium is found.

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2

Andersen, Garrett, and Vincent Conitzer. "Fast Equilibrium Computation for Infinitely Repeated Games." Proceedings of the AAAI Conference on Artificial Intelligence 27, no. 1 (June 30, 2013): 53–59. http://dx.doi.org/10.1609/aaai.v27i1.8573.

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It is known that an equilibrium of an infinitely repeated two-player game (with limit average payoffs) can be computed in polynomial time, as follows: according to the folk theorem, we compute minimax strategies for both players to calculate the punishment values, and subsequently find a mixture over outcomes that exceeds these punishment values. However, for very large games, even computing minimax strategies can be prohibitive. In this paper, we propose an algorithmic framework for computing equilibria of repeated games that does not require linear programming and that does not necessarily need to inspect all payoffs of the game. This algorithm necessarily sometimes fails to compute an equilibrium, but we mathematically demonstrate that most of the time it succeeds quickly on uniformly random games, and experimentally demonstrate this for other classes of games. This also holds for games with more than two players, for which no efficient general algorithms are known.

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3

Rubinstein, Ariel, and Asher Wolinsky. "Remarks on Infinitely Repeated Extensive-Form Games." Games and Economic Behavior 9, no. 1 (April 1995): 110–15. http://dx.doi.org/10.1006/game.1995.1007.

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4

Watson, Joel. "Cooperation in the Infinitely Repeated Prisoners′ Dilemma with Perturbations." Games and Economic Behavior 7, no. 2 (September 1994): 260–85. http://dx.doi.org/10.1006/game.1994.1049.

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5

Dal Bó, Pedro, and Guillaume R. Fréchette. "On the Determinants of Cooperation in Infinitely Repeated Games: A Survey." Journal of Economic Literature 56, no. 1 (March 1, 2018): 60–114. http://dx.doi.org/10.1257/jel.20160980.

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A growing experimental literature studies the determinants of cooperation in infinitely repeated games, tests different predictions of the theory, and suggests an empirical solution to the problem of multiple equilibria. To provide a robust description of the literature's findings, we gather and analyze a metadata set of experiments on infinitely repeated prisoner's dilemma games. The experimental data show that cooperation is affected by infinite repetition and is more likely to arise when it can be supported in equilibrium. However, the fact that cooperation can be supported in equilibrium does not imply that most subjects will cooperate. High cooperation rates will emerge only when the parameters of the repeated game are such that cooperation is very robust to strategic uncertainty. We also review the results regarding the effect of imperfect monitoring, changing partners, and personal characteristics on cooperation and the strategies used to support it. (JEL C71, C73, D81, D83)

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6

Lehrer, Ehud. "Finitely Many Players with Bounded Recall in Infinitely Repeated Games." Games and Economic Behavior 7, no. 3 (November 1994): 390–405. http://dx.doi.org/10.1006/game.1994.1058.

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7

Häckner, Jonas. "Endogenous product design in an infinitely repeated game." International Journal of Industrial Organization 13, no. 2 (June 1995): 277–99. http://dx.doi.org/10.1016/0167-7187(94)00451-7.

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8

Liu, Jian, Yinzhen Li, and Jun Li. "Coopetition in Intermodal Freight Transport Services." Discrete Dynamics in Nature and Society 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/680685.

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The paper studies the coopetition of the downstream different carriers by providing complementary transport services in intermodal freight transport chain. Considering different information structure, a two-stage dynamic game model with simultaneous actions on investment and price is first formulated. Equilibria show both parties have motivation to select coopetition even if the agreement for cooperation investment is reached in advance. When both firms agree on the specific allocation, the new coopetition with higher efficiency would be emerged. Moreover, we analyze the complexity and evolution of coopetition by repeated pricing game with finitely and infinitely time horizon. In the finitely repeated pricing game, both firms have incentive to reach a tacit understanding to alternate choosing price cooperation and competition after setting suitable allocation scheme; the repeated periodstare then going to be an issue. In the infinitely repeated pricing game, the perfect cooperation is realized by designing the suitable trigger strategy.

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9

Kufel, Tadeusz, Sławomir Plaskacz, and Joanna Zwierzchowska. "Strong and Safe Nash Equilibrium in Some Repeated 3-Player Games." Przegląd Statystyczny 65, no. 3 (January 30, 2019): 271–95. http://dx.doi.org/10.5604/01.3001.0014.0540.

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The paper examines an infinitely repeated 3-player extension of the Prisoner’s Dilemma game. We consider a 3-player game in the normal form with incomplete information, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage, players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players’ payoffs do not exceed the nondeviating player payoff more than by a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale’s good strategies described in Smale’s paper (1980), where the repeated Prisoner’s Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.

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10

Dargaj, Jakub, and Jakob Grue Simonsen. "Discounted Repeated Games Having Computable Strategies with No Computable Best Response under Subgame-Perfect Equilibria." ACM Transactions on Economics and Computation 10, no. 1 (March 31, 2022): 1–39. http://dx.doi.org/10.1145/3505585.

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A classic result in computational game theory states that there are infinitely repeated games where one player has a computable strategy that has a best response, but no computable best response. For games with discounted payoff, the result is known to hold for a specific class of games—essentially generalizations of Prisoner’s Dilemma—but until now, no necessary and sufficient condition is known. To be of any value, the computable strategy having no computable best response must be part of a subgame-perfect equilibrium, as otherwise a rational, self-interested player would not play the strategy. We give the first necessary and sufficient conditions for a two-player repeated game \( G \) to have such a computable strategy with no computable best response for all discount factors above some threshold. The conditions involve existence of a Nash equilibrium of the repeated game whose discounted payoffs satisfy certain conditions involving the min–max payoffs of the underlying stage game. We show that it is decidable in polynomial time in the size of the payoff matrix of \( G \) whether it satisfies these conditions.

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11

Cronshaw, Mark B., and David G. Luenberger. "Strongly Symmetric Subgame Perfect Equilibria in Infinitely Repeated Games with Perfect Monitoring and Discounting." Games and Economic Behavior 6, no. 2 (March 1994): 220–37. http://dx.doi.org/10.1006/game.1994.1012.

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12

Chandra, Prakash, and K. C. Sharma. "Non Negative Benefits of the Commons in Infinitely Repeated Game." Journal of Game Theory 1, no. 5 (January 7, 2013): 38–42. http://dx.doi.org/10.5923/j.jgt.20120105.03.

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13

Simpson, R. David. "Signaling in an infinitely repeated Cournot game with output restrictions." International Journal of Industrial Organization 9, no. 3 (September 1991): 365–88. http://dx.doi.org/10.1016/0167-7187(91)90017-f.

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14

Dal Bó, Pedro, and Guillaume R. Fréchette. "Strategy Choice in the Infinitely Repeated Prisoner’s Dilemma." American Economic Review 109, no. 11 (November 1, 2019): 3929–52. http://dx.doi.org/10.1257/aer.20181480.

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We use a novel experimental design to reliably elicit subjects’ strategies in an infinitely repeated prisoner’s dilemma experiment with perfect monitoring. We find that three simple strategies repre‑ sent the majority of the chosen strategies: Always Defect, Tit‑for‑Tat, and Grim. In addition, we identify how the strategies systematically vary with the parameters of the game. Finally, we use the elicited strategies to test the ability to recover strategies using statistical methods based on observed round‑by‑round cooperation choices and find that this can be done fairly well, but only under certain conditions. (JEL C72, C73, C92)

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15

Liu, Guiyun, Yonggui Liu, Jing Yao, Hongbin Chen, and Dong Tang. "Repeated Game for Distributed Estimation in Autonomous Clustered Wireless Sensor Networks." International Journal of Distributed Sensor Networks 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/806456.

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A commonly encountered problem in wireless sensor networks (WSNs) applications is to reconstruct the state of nature, that is, distributed estimation of a parameter of interest through WSNs’ observations. However, the distributed estimation in autonomous clustered WSNs faces a vital problem of sensors’ selfishness. Each sensor autonomously decides whether or not to transmit its observations to the fusion center (FC) and not be controlled by the fusion center (FC) any more. Thus, to encourage cooperation within selfish sensors, infinitely and finitely repeated games are firstly modeled to depict sensors’ behaviors. Then, the existences of Nash equilibriums for infinitely and finitely repeated games are discussed. Finally, simulation results show that the proposed Nash equilibrium strategies are effective.

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16

Taha, Mohammad A., and Ayman Ghoneim. "Zero-determinant strategies in infinitely repeated three-player prisoner's dilemma game." Chaos, Solitons & Fractals 152 (November 2021): 111408. http://dx.doi.org/10.1016/j.chaos.2021.111408.

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17

JOOSTEN, REINOUD. "A NOTE ON REPEATED GAMES WITH VANISHING ACTIONS." International Game Theory Review 07, no. 01 (March 2005): 107–15. http://dx.doi.org/10.1142/s0219198905000430.

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A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ∈ {1,2} at least once in every rk ∈ ℕ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon. A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play. We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ≥ 2, r2 > n ≥ 2.

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18

McKelvey, Richard D., and Raymond Riezman. "Seniority in Legislatures." American Political Science Review 86, no. 4 (December 1992): 951–65. http://dx.doi.org/10.2307/1964347.

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We construct a stochastic model of a legislature with an endogenously determined seniority system. We model the behavior of the legislators as well as their constituents as an infinitely repeated divide-the-dollar game. The game has a stationary equilibrium with the property that the legislature imposes on itself a non-trivial seniority system, and that incumbent legislators are always reelected.

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19

Dong, Gang, and Dandan Zhong. "Tacit Collusion of Pricing Strategy Game between Regional Ports: The Case of Yangtze River Economic Belt." Sustainability 11, no. 2 (January 12, 2019): 365. http://dx.doi.org/10.3390/su11020365.

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We develop a game model to analyze the tacit collusion between regional ports under three different scenarios. In the first scenario, there is simultaneous pricing game between regional ports; this intends to depict pricing strategy adopted independently. In the second, we consider two competing ports that make sequential pricing decisions. Thirdly, an infinitely repeated game model is then formulated for regional ports to test the stability of Nash equilibrium. Our main finding is that there is a certain degree of tacit collusion of pricing strategy between regional ports in the competitive environment; in particular, the tacit collusion of pricing strategy will gradually stabilize with the increasing number regional ports games. A case study of Yangtze River Economic Belt is provided to illustrate the results.

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20

Boot, Arnoud W. A., and Anjan V. Thakor. "Moral Hazard and Secured Lending in an Infinitely Repeated Credit Market Game." International Economic Review 35, no. 4 (November 1994): 899. http://dx.doi.org/10.2307/2527003.

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21

Graham, Edward M. "Exchange of threat between multinational firms as an infinitely repeated noncooperative game." International Trade Journal 4, no. 3 (March 1990): 259–77. http://dx.doi.org/10.1080/08853909008523695.

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22

Schneider, Mark, and Jonathan W. Leland. "Reference dependence, cooperation, and coordination in games." Judgment and Decision Making 10, no. 2 (March 2015): 123–29. http://dx.doi.org/10.1017/s1930297500003909.

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AbstractThe problems of how self-interested players can cooperate despite incentives to defect, and how players can coordinate despite the presence of multiple equilibria, are among the oldest and most fundamental in game theory. In this report, we demonstrate that a plausible and even natural specification of the reference outcome in a game simultaneously predicts systematic cooperation and defection in the Prisoner’s Dilemma, as well as equilibrium selection and out-of-equilibrium play in coordination games. The predictions hold even if players are purely self-interested, there are no salient labels, the game is played only once, and there is no communication of any kind. Furthermore, the predictions are unique, as opposed to the multiplicity of equilibria in the infinitely repeated Prisoner’s Dilemma and in coordination games. We apply experimental results to test the predictions of the model.

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23

Baklanov, Artem. "Reactive Strategies: An Inch of Memory, a Mile of Equilibria." Games 12, no. 2 (May 8, 2021): 42. http://dx.doi.org/10.3390/g12020042.

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We explore how an incremental change in complexity of strategies (“an inch of memory”) in repeated interactions influences the sets of Nash equilibrium (NE) strategy and payoff profiles. For this, we introduce the two most basic setups of repeated games, where players are allowed to use only reactive strategies for which a probability of players’ actions depends only on the opponent’s preceding move. The first game is trivial and inherits equilibria of the stage game since players have only unconditional (memory-less) Reactive Strategies (RSs); in the second one, players also have conditional stochastic RSs. This extension of the strategy sets can be understood as a result of evolution or learning that increases the complexity of strategies. For the game with conditional RSs, we characterize all possible NE profiles in stochastic RSs and find all possible symmetric games admitting these equilibria. By setting the unconditional benchmark as the least symmetric equilibrium payoff profile in memory-less RSs, we demonstrate that for most classes of symmetric stage games, infinitely many equilibria in conditional stochastic RSs (“a mile of equilibria”) Pareto dominate the benchmark. Since there is no folk theorem for RSs, Pareto improvement over the benchmark is the best one can gain with an inch of memory.

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24

Andersson, Lina. "Cooperation between Emotional Players." Games 11, no. 4 (October 15, 2020): 45. http://dx.doi.org/10.3390/g11040045.

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This paper uses the framework of stochastic games to propose a model of emotions in repeated interactions. An emotional player can be in either a friendly, a neutral, or a hostile state of mind. The player transitions between the states of mind as a response to observed actions taken by the other player. The state of mind determines the player’s psychological payoff which together with a material payoff constitutes the player’s utility. In the friendly (hostile) state of mind the player has a positive (negative) concern for other players’ material payoffs. This paper shows how emotions can both facilitate and obstruct cooperation in a repeated prisoners’ dilemma game. In finitely repeated games a player who cares only for their own material payoffs can have an incentive to manipulate an emotional player into the friendly state of mind. In infinitely repeated games with two emotional players less patience is required to sustain cooperation. However, emotions can also obstruct cooperation if they make the players unwilling to punish each other, or if the players become hostile when punished.

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25

Jensen, Henrik. "Efficient Bargaining and Accommodation Policies." Recherches économiques de Louvain 59, no. 4 (1993): 463–84. http://dx.doi.org/10.1017/s0770451800006618.

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Summary We show that accommodation policies may render efficient bargaining in the labour market unsustainable as a perfect Nash equilibrium of the infinitely repeated game. In fact, in the eyes of the labour market participants, efficient bargaining may be Pareto-dominated by the monopoly-union solution when the government places sufficiently high weight on employment targets.

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26

Aoyagi, Masaki, V. Bhaskar, and Guillaume R. Fréchette. "The Impact of Monitoring in Infinitely Repeated Games: Perfect, Public, and Private." American Economic Journal: Microeconomics 11, no. 1 (February 1, 2019): 1–43. http://dx.doi.org/10.1257/mic.20160304.

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This paper uses a laboratory experiment to study the effect of the monitoring structure on the play of the infinitely repeated prisoner’s dilemma. Keeping the strategic form of the stage game fixed, we examine the behavior of subjects when information about past actions is perfect (perfect monitoring), noisy but public (public monitoring), and noisy and private (private monitoring). We find that the subjects sustain cooperation in every treatment, but that their strategies differ across the three treatments. Specifically, the strategies under imperfect monitoring are both more complex and more lenient than those under perfect monitoring. The results show how the changes in strategies across monitoring structures mitigate the effect of noise in monitoring on efficiency. (JEL C72, C73, C92, D82, D83)

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27

Raigorodskaya, A. V. "Choosing the equilibrium behavior in an infinitely repeated 2 × 2 game for conservative and innovative gamers." Moscow University Computational Mathematics and Cybernetics 36, no. 1 (February 26, 2012): 14–22. http://dx.doi.org/10.3103/s0278641912010062.

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28

Wang, Jin, Jiahao Yang, Yingfeng Zhang, Shan Ren, and Yang Liu. "Infinitely repeated game based real-time scheduling for low-carbon flexible job shop considering multi-time periods." Journal of Cleaner Production 247 (February 2020): 119093. http://dx.doi.org/10.1016/j.jclepro.2019.119093.

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29

Sartori, Anne E. "The Might of the Pen: A Reputational Theory of Communication in International Disputes." International Organization 56, no. 1 (2002): 121–49. http://dx.doi.org/10.1162/002081802753485151.

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I present a theory of effective diplomacy based on honesty and reputations. I model diplomacy as a form of “cheap talk” and international interactions as an infinitely repeated game in which similar states find themselves in disparate situations over time. The theory explains the success and failure of diplomacy. Reputations for honesty make honest communication possible. A state caught bluffing is less able to communicate and less likely to attain its goals in the near future. These findings imply that domestic audience costs are unnecessary for international signaling and that military strength is not the only way to build credibility.

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30

Li, Jin, Niko Matouschek, and Michael Powell. "Power Dynamics in Organizations." American Economic Journal: Microeconomics 9, no. 1 (February 1, 2017): 217–41. http://dx.doi.org/10.1257/mic.20150138.

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We examine an infinitely repeated game between a principal, who has the formal authority to decide on a project, and a biased agent, who is privately informed about what projects are available. The optimal relational contract speaks to how power is earned, lost, and retained. It shows that entrenched power structures are consistent with optimal administration of power. And it provides new perspectives on why similar firms organize differently, even when those organizational differences lead to persistent differences in performance, and why established firms fail to exploit new opportunities, even when they are publicly observable. (JEL C73, D23, D82, D86)

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31

Schmid, Laura, Christian Hilbe, Krishnendu Chatterjee, and Martin A. Nowak. "Direct reciprocity between individuals that use different strategy spaces." PLOS Computational Biology 18, no. 6 (June 14, 2022): e1010149. http://dx.doi.org/10.1371/journal.pcbi.1010149.

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In repeated interactions, players can use strategies that respond to the outcome of previous rounds. Much of the existing literature on direct reciprocity assumes that all competing individuals use the same strategy space. Here, we study both learning and evolutionary dynamics of players that differ in the strategy space they explore. We focus on the infinitely repeated donation game and compare three natural strategy spaces: memory-1 strategies, which consider the last moves of both players, reactive strategies, which respond to the last move of the co-player, and unconditional strategies. These three strategy spaces differ in the memory capacity that is needed. We compute the long term average payoff that is achieved in a pairwise learning process. We find that smaller strategy spaces can dominate larger ones. For weak selection, unconditional players dominate both reactive and memory-1 players. For intermediate selection, reactive players dominate memory-1 players. Only for strong selection and low cost-to-benefit ratio, memory-1 players dominate the others. We observe that the supergame between strategy spaces can be a social dilemma: maximum payoff is achieved if both players explore a larger strategy space, but smaller strategy spaces dominate.

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32

Baron, David P. "Comparative Dynamics of Parliamentary Governments." American Political Science Review 92, no. 3 (September 1998): 593–609. http://dx.doi.org/10.2307/2585483.

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This article presents a dynamic theory of parliamentary governments that incorporates attributes of the institutional system in a country, exogenous events that shape parliamentary and electoral opportunities, and the strategies of the government and the opposition as structured by institutions and preferences. The dynamics are investigated in an infinitely repeated game in which events in the form of shocks to income or government resources occur and the government responds with a legislative proposal that is subject to a confidence or censure procedure and may lead to government continuation, reorganization, or dissolution. With a majority confidence procedure, governments are stable, and if parties are politically patient, voting cohesion within the government is high. A censure motion initiated by the opposition can result in voluntary dissolution of government, and the approach of required elections increases the likelihood of dissolution. If the events represent fluctuations in aggregate income, governnment dissolution occurs in good times for the government leader and bad times for the other parties.

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33

Zheng, Xiaona, Luping Sun, and Andy A. Tsay. "Distribution Channel Strategies and Retailer Collusion in a Supply Chain with Multiple Retailers." Asia-Pacific Journal of Operational Research 35, no. 03 (May 31, 2018): 1850014. http://dx.doi.org/10.1142/s0217595918500148.

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Previous literature suggests that without regulations firms have incentives to collude by fixing price or reducing quantity. This paper sets up an infinitely repeated game to examine the interplay between the manufacturer’s channel strategy and the downstream retailers’ collusive behavior. The results show that the manufacturer can deter retailer collusion by strategically changing its channel strategy. This effect occurs when the discount rate (used to calculate the present value of future profits) is relatively large and the manufacturer’s direct selling efficiency is relatively high (i.e., the variable cost of direct selling is relatively low). With the deterrence of direct selling, retailers abandon collusion and “no collusion” is a win-win strategy for both levels in the supply chain. However, when the manufacturer is not efficient in direct selling or the discount rate is small, direct selling is not effective in deterring retailer collusion and the manufacturer is worse off. These findings provide insights into channel strategies and supply chain management.

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34

McGillivray, Fiona, and Alastair Smith. "Trust and Cooperation Through Agent-specific Punishments." International Organization 54, no. 4 (2000): 809–24. http://dx.doi.org/10.1162/002081800551370.

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Using the infinitely repeated prisoners' dilemma game as a modeling platform, we examine how domestic political institutions affect the ability of nations to trust and cooperate with each other. We propose a strategy, the agent-specific grim trigger, in which national leaders direct punishments for past defections at the leader of the nation responsible rather than at the nation itself. Leaders refuse to cooperate with those leaders who have cheated them in the past. However, by being prepared to cooperate with new leaders, cooperation can be restored. The focus of punishment on specific agents of the people (leaders), rather than the nation itself, means that citizens want to remove leaders who defect. Hence, domestically accountable leaders pay audience costs for failing to cooperate. These costs make accountable leaders more trustworthy and foster greater cooperation. In contrast, when replacing leaders is difficult, cooperation is less robust; and once cooperation falters, agent-specific punishment policies often lead to prolonged hostilities and periods of acrimonious relations between states.

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35

Forges, Françoise. "INTERVIEW WITH JEAN-FRANÇOIS MERTENS (1946–2012)." Macroeconomic Dynamics 18, no. 8 (June 12, 2013): 1832–53. http://dx.doi.org/10.1017/s1365100513000114.

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Every game theorist knows of Mertens and Zamir (1985)'s universal beliefs space, which gives deep foundations to Harsanyi's model of Bayesian games, and Kohlberg and Mertens (1986)'s strategic stability, which is the first stone of a complete, axiomatic theory of selection among Nash equilibria. Some French mathematicians refer to the “Mertens–Zamir operator” when using techniques that Mertens and Zamir (1971) introduced to solve a class of repeated games with incomplete information. Readers of Macroeconomic Dynamics may instead have seen Mertens and Rubinchik's 2012 article “Intergenerational Equity and the Discount Rate for Policy Analysis.”The previous examples give just a slight idea of the scope of Jean-François Mertens's contributions, which also deal with general equilibrium, stochastic games, nonatomic cooperative games, and the strategic foundations of microeconomic theory. In his 2005 MD interview, Robert Aumann says, “A [. . .] person at CORE who has had a tremendous influence on game theory [. . .] is Jean-François Mertens. Mertens has done some of the deepest work in the discipline, some of it in collaboration with Israelis like my students Kohlberg, Neyman, and Zamir; he established a Belgian school of mathematical game theory that is marked by its beauty, depth, and sophistication.” The short interview that follows will definitely not account for the variety and the relevance of Jean-François's research achievements, but is typical of the way in which he talked about his work.Jean-François asked me to interview him for MD during the spring of 2010. We discussed by e-mail the topics that would be covered and on July 6, 2010, I came to Louvain-la-Neuve with a tape recorder. After lunch, Jean-François suggested that we have coffee on a terrace near the golf course and there, he patiently answered my questions, sometimes in French, sometimes in English, for about two hours. We planned to go on for at least another round but kept postponing the project. . . . When I saw Jean-François for the last time, in February 2012, I gave him the transcript of the July 2010 interview, but he hardly commented on it. He rather told me about an ongoing research article, “A Random Partitions Approach to the Value,” to be presented (by Abraham Neyman) as a “von Neumann lecture” at the World Congress of Game Theory in Istanbul in July 2012. At the same time, he was also completing, with Anna Rubinchik, the revision of a companion paper to the MD article referred to previously (“Equilibria in an Overlapping Generations Model with Transfer Policies and Exogenous Growth,” forthcoming in Economic Theory).Even if Jean-François did not proofread the transcript that follows, I cannot keep this material for myself. I am confident that those who have known Jean-François will take the interview, even incomplete, as an opportunity to remember his enthusiasm and his patience when he was talking about research. He would often start by identifying holes in obvious or well-known solutions to basic problems, and after a few audacious but illuminating shortcuts, would describe the most surprising achievements in everyday words. I hope that the interview will give an idea of Jean-François's approach to those who did not know him.Quite naturally, because MD was the planned outlet of the interview, we started by talking about the paper on the discount rate for policy analysis, which was already mentioned in the preceding. Jean-François made a number of informal comments, which usefully complement the MD article. He also explained how this paper led him and his coauthor to undertake a thorough analysis of overlapping generations economies in continuous time. This made a perfect transition to Jean-François's views on general equilibrium theory, his own work in this area, and his early career.The next step would be Jean-François's meeting with Bob Aumann, who introduced him to game theory. Jean-François pursued Aumann and Maschler's seminal work on infinitely repeated games with incomplete information, mostly with Shmuel Zamir. He went on with the existence of a value in stochastic games, another model of infinitely long games, which was introduced by Shapley in 1953. This research was undertaken with Abraham Neyman at the Institute of Advanced Studies in Jerusalem in 1980. Soon after, Mertens and Zamir started to review and complete all available results on repeated games in order to prepare a reference book on this topic. The material kept growing. Sylvain Sorin joined the team in the nineties and a draft appeared as a 1994 CORE discussion paper. However, in 2010, the book was still unpublished. . . the interview ends up with Jean-François's feelings about the project.As shown by the list of publications at the end, many important contributions of Jean-François Mertens to game theory and microeconomics are not even mentioned in the interview. During his stay at the Institute of Advanced Studies in Jerusalem in 1980, Jean-François not only worked with Abraham Neyman on stochastic games, but also had his first discussions with Elon Kohlberg on refinements of Nash equilibria. These would be followed by many others, at CORE and Harvard, until the famous Econometrica paper appeared in 1986. For the next 15 years or so, Jean-François further developed the theory of strategic stability, by himself and with his students.During the same period, Jean-François was also making progress on a completely different problem, the extension of the Shapley value to nonatomic cooperative games. Aumann and Shapley (1974) had made the first steps by proposing a value for smooth games. Jean-François proposed a complete answer to the problem in the eighties and, as already pointed out above, kept working on related topics until the very end.Even without entering into details, a description of Jean-François's more recent contributions would be beyond the scope of this short introduction. As the interview makes clear, Jean-François became more and more interested in the foundations of microeconomic theory. A typical example is his “limit price mechanism,” which can be loosely described as a double auction with several goods or as an extension of Shapley and Shubik's strategic market games. Another example is “relative utilitarianism,” which, as Jean-François explains in the interview, plays a crucial role in the determination of an appropriate social discount rate for the evaluation of long-term economic policies. Let us listen to him.

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36

Gilboa, Itzhak, and David Schmeidler. "Infinite Histories and Steady Orbits in Repeated Games." Games and Economic Behavior 6, no. 3 (May 1994): 370–99. http://dx.doi.org/10.1006/game.1994.1022.

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37

Sandomirskaia, Marina. "Repeated Bidding Games with Incomplete Information and Bounded Values: On the Exponential Speed of Convergence." International Game Theory Review 19, no. 01 (March 2017): 1650017. http://dx.doi.org/10.1142/s0219198916500171.

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We consider the repeated zero-sum bidding game with incomplete information on one side with non-normalized total payoff. De Meyer and Marino [(2005) Continuous versus discrete market game, Cowles Foundation Discussion Paper 1535] and Domansky and Kreps [(2005) Repeated games with asymmetric information and random price fluctuations at finance markets, Proc. Appl. Ind. Math. 12(4), 950–952 (in Russian)] investigated a game [Formula: see text] modeling multistage bidding with asymmetrically informed agents and proved that for this game [Formula: see text] converges to a finite limit [Formula: see text], i.e., the error term is [Formula: see text]. In this paper, we show that for this example [Formula: see text] converges to the limit exponentially fast. For this purpose we apply the optimal strategy [Formula: see text] of insider in the infinite-stage game obtained by Domansky [(2007) Repeated games with asymmetric information and random price fluctuations at finance markets, Int. J. Game Theor. 36(2), 241–257] to the [Formula: see text]-stage game and deduce that it is [Formula: see text]-optimal with [Formula: see text] exponentially small.

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38

Neilson, William S., and Harold Winter. "Infinitely-repeated games with endogenous discounting." Economics Letters 52, no. 2 (August 1996): 163–69. http://dx.doi.org/10.1016/s0165-1765(96)00861-0.

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39

Xue, Licun. "Stable agreements in infinitely repeated games." Mathematical Social Sciences 43, no. 2 (March 2002): 165–76. http://dx.doi.org/10.1016/s0165-4896(01)00089-0.

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40

Breitmoser, Yves. "Infinitely repeated games of reciprocal players." Economics Letters 89, no. 3 (December 2005): 323–27. http://dx.doi.org/10.1016/j.econlet.2005.06.005.

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41

Lipman, Barton L., and Ruqu Wang. "Switching costs in infinitely repeated games." Games and Economic Behavior 66, no. 1 (May 2009): 292–314. http://dx.doi.org/10.1016/j.geb.2008.04.018.

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42

Dal Bó, Pedro, and Guillaume R. Fréchette. "The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence." American Economic Review 101, no. 1 (February 1, 2011): 411–29. http://dx.doi.org/10.1257/aer.101.1.411.

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A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner's dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance. (JEL C71, C73)

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43

Wang, Lei, Cui Liu, Juan Xue, and Hongwei Gao. "A Note on Strategic Stability of Cooperative Solutions for Multistage Games." Discrete Dynamics in Nature and Society 2018 (November 1, 2018): 1–6. http://dx.doi.org/10.1155/2018/3293745.

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The problem of strategic stability of cooperative solutions for multistage games is studied. The sufficient conditions related to discount factors are presented, which guarantee the existence of Nash or strong Nash equilibria in such games and therefore guarantee the strategic stability of cooperative solutions. The deviating payoffs of players are given directly, which are related closely to these conditions and avoid the loss of super-additivity of a class of general characteristic functions. As an illustration, Nash and strong Nash equilibria are found for the repeated infinite stage Prisoner’s dilemma game.

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44

Bó, Pedro Dal. "Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games." American Economic Review 95, no. 5 (November 1, 2005): 1591–604. http://dx.doi.org/10.1257/000282805775014434.

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While there is an extensive literature on the theory of infinitely repeated games, empirical evidence on how “the shadow of the future” affects behavior is scarce and inconclusive. I simulate infinitely repeated prisoner's dilemma games in the lab with a random continuation rule. The experimental design represents an improvement over the existing literature by including sessions with finite repeated games as controls and a large number of players per session (which allows for learning without contagion effects). I find that the shadow of the future matters not only by significantly reducing opportunistic behavior, but also because its impact closely follows theoretical predictions.

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45

Cigler, Ludek, and Boi Faltings. "Symmetric Subgame Perfect Equilibria in Resource Allocation." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (September 20, 2021): 1326–32. http://dx.doi.org/10.1609/aaai.v26i1.8233.

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We analyze symmetric protocols to rationally coordinate on an asymmetric, efficient allocation in an infinitely repeated N-agent, C-resource allocation problems. (Bhaskar 2000) proposed one way to achieve this in 2-agent, 1-resource allocation games: Agents start by symmetrically randomizing their actions, and as soon as they each choose different actions, they start to follow a potentially asymmetric "convention" that prescribes their actions from then on. We extend the concept of convention to the general case of infinitely repeated resource allocation games with N agents and C resources. We show that for any convention, there exists a symmetric subgame perfect equilibrium which implements it. We present two conventions: bourgeois, where agents stick to the first allocation; and market, where agents pay for the use of resources, and observe a global coordination signal which allows them to alternate between different allocations. We define price of anonymity of a convention as the ratio between the maximum social payoff of any (asymmetric) strategy profile and the expected social payoff of the convention. We show that while the price of anonymity of the bourgeois convention is infinite, the market convention decreases this price by reducing the conflict between the agents.

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46

Abreu, Dilip. "On the Theory of Infinitely Repeated Games with Discounting." Econometrica 56, no. 2 (March 1988): 383. http://dx.doi.org/10.2307/1911077.

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Goldlücke, Susanne, and Sebastian Kranz. "Infinitely repeated games with public monitoring and monetary transfers." Journal of Economic Theory 147, no. 3 (May 2012): 1191–221. http://dx.doi.org/10.1016/j.jet.2012.01.008.

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48

Aoyagi, Masaki. "Reputation and Dynamic Stackelberg Leadership in Infinitely Repeated Games." Journal of Economic Theory 71, no. 2 (November 1996): 378–93. http://dx.doi.org/10.1006/jeth.1996.0126.

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49

Cigler, L., and B. Faltings. "Symmetric Subgame-Perfect Equilibria in Resource Allocation." Journal of Artificial Intelligence Research 49 (February 26, 2014): 323–61. http://dx.doi.org/10.1613/jair.4166.

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We analyze symmetric protocols to rationally coordinate on an asymmetric, efficient allocation in an infinitely repeated N-agent, C-resource allocation problems, where the resources are all homogeneous. Bhaskar proposed one way to achieve this in 2-agent, 1-resource games: Agents start by symmetrically randomizing their actions, and as soon as they each choose different actions, they start to follow a potentially asymmetric "convention" that prescribes their actions from then on. We extend the concept of convention to the general case of infinitely repeated resource allocation games with N agents and C resources. We show that for any convention, there exists a symmetric subgame-perfect equilibrium which implements it. We present two conventions: bourgeois, where agents stick to the first allocation; and market, where agents pay for the use of resources, and observe a global coordination signal which allows them to alternate between different allocations. We define price of anonymity of a convention as a ratio between the maximum social payoff of any (asymmetric) strategy profile and the expected social payoff of the subgame-perfect equilibrium which implements the convention. We show that while the price of anonymity of the bourgeois convention is infinite, the market convention decreases this price by reducing the conflict between the agents.

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50

Niu, Gen Yi. "A Game Theory Based Analysis of the Tacit Knowledge Sharing and Incentive Mechanism." Advanced Materials Research 601 (December 2012): 564–69. http://dx.doi.org/10.4028/www.scientific.net/amr.601.564.

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Tacit knowledge sharing is the basis and prerequisite of knowledge innovation. In this paper the incentive mechanism of tacit knowledge sharing is proposed to improve tacit knowledge sharing. Firstly, game theory is introduced, and then we analyze the knowledge sharing and mechanism. Finally, Static and repeated game models of knowledge sharing are constructed. The results indicate that infinite repeated game can solve “Prisoner's dilemma” of knowledge sharing in the one-time game.

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