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J Theor Biol. 2018 Feb 7;438:61-77. doi: 10.1016/j.jtbi.2017.11.002. Epub 2017 Nov 14.

Zero-determinant strategies in finitely repeated games.

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Department of Mathematical and Systems Engineering, Shizuoka University, 3-5- 1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan.
Department of Engineering Mathematics, University of Bristol, Merchant Venturers Building, Woodland Road, Clifton, Bristol BS8 1UB, United Kingdom. Electronic address:


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.


Cooperation; Direct reciprocity; Discount factor; Prisoner’s dilemma game

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