Zero-determinant strategies in finitely repeated games

Abstract

Direct reciprocity is a mechanism for sustaining mutual cooperation in repeated social dilemma games, where a player would keep cooperation to avoid being retaliated by a co-player in the future. So-called zero-determinant (ZD) strategies enable a player to unilaterally set a linear relationship between the player's own payoff and the co-player's payoff regardless of the strategy of the co-player. In the present study, we analytically study zero-determinant strategies in finitely repeated (two-person) prisoner's dilemma games with a general payoff matrix. Our results are as follows. First, we present the forms of solutions that extend the known results for infinitely repeated games (with a discount factor w of unity) to the case of finitely repeated games (0 < w < 1). Second, for the three most prominent ZD strategies, the equalizers, extortioners, and generous strategies, we derive the threshold value of w above which the ZD strategies exist. Third, we show that the only strategies that enforce a linear relationship between the two players' payoffs are either the ZD strategies or unconditional strategies, where the latter independently cooperates with a fixed probability in each round of the game, proving a conjecture previously made for infinitely repeated games.

arXiv comment24 pages, 2 figures

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• Five rules for friendly rivalry in direct reciprocity
• Memory-two zero-determinant strategies in repeated games
• Linear algebraic structure of zero-determinant strategies in repeated games
• Evolution of direct reciprocity in group-structured populations
• Conditions for the existence of zero-determinant strategies under observation errors in repeated games
• Single Equalizer Strategy with no Information Transfer for Conflict Escalation
• Necessary and Sufficient Condition for the Existence of Zero-Determinant Strategies in Repeated Games
• Automata representation of successful strategies for social dilemmas
• Controlling conditional expectations by zero-determinant strategies
• Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games
• On the implementation of zero-determinant strategies in repeated games
• Autocratic strategies in Cournot oligopoly game
• Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner's Dilemma game
• Zero-Determinant strategies in repeated multiplayer social dilemmas with discounted payoffs
• Exerting Control in Repeated Social Dilemmas with Thresholds