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.

24 pages, 2 figures

References in corpus (3)

• Zero-determinant alliances in multiplayer social dilemmas
• Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations
• Boosting cooperation by involving extortion in spatial Prisoner's dilemma

Cited by in corpus (4)

• Evolution of direct reciprocity in group-structured populations
• Conditions for the existence of zero-determinant strategies under observation errors in repeated games
• Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner's Dilemma game
• Exerting Control in Repeated Social Dilemmas with Thresholds