Journal
JOURNAL OF THEORETICAL BIOLOGY
Volume 438, Issue -, Pages 61-77Publisher
ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
DOI: 10.1016/j.jtbi.2017.11.002
Keywords
Prisoner's dilemma game; Cooperation; Direct reciprocity; Discount factor
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Funding
- JST, ERATO, Kawarabayashi Large Graph Project, Japan [JPMJER1201]
- JST, CREST, Japan [JPMJCR1304]
- HAYAO NAKAYAMA Foundation for Science AMP
- Technology and Culture
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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 equlizers, 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. (C) 2017 The Authors. Published by Elsevier Ltd.
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