Zero-determinant strategies in repeated prisoner's dilemma games

Direct reciprocity is one of the mechanisms for sustaining mutual cooperation in repeated social dilemma games. Zero-determinant (ZD) strategies allow 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. The original ZD strategies were derived for infinitely repeated games. Here, we analytically search for ZD strategies in finitely repeated prisoner's dilemma games. Our results can be summarized as follows. First, we present the forms of ZD in finitely repeated games, which are directly extended from the known results for infinitely repeated games. Second, for the three most notable ZD strategies, the equalizers, extortioners, and generous strategies, we derive the threshold discount factor value above which the ZD strategies exist. Finally, we show that the only strategy sets that enforce a linear payoff relationship are either the ZD strategies or unconditional strategies.