Cooperation and control in asymmetric repeated games

Recently, a new class of memory-one strategies, zero-determinant (ZD), has been proposed in the context of repeated Prisoner's Dilemma game. A player using a ZD strategy can unilaterally enforce a linear relation between the two players’ payoffs. In this paper, we study ZD strategies in arbitrary 2 × 2 asymmetric games. We show that the existence of three typical subclasses of ZD strategies, equalizer, extortionate, and generous strategies, depends crucially on the payoff structure of the games. More interestingly, we find that extortionate and generous strategies can be expressed in a unified form of ZD strategies, so-called “multiplicative” ZD strategies, where this type of strategy can establish a multiplicative relation between the net payoffs of two players. Then, we identify the range of multiplicative factors when two players have different payoff functions or identities and discuss how this type of strategy implies extortion or generosity in different asymmetric games. Finally, we apply the above results to some specific asymmetric games, including the asymmetric PD game, the ultimatum game, the trust game as well as the generalized asymmetric social dilemma games, and analyze the effects of ZD strategies on cooperation and payoff control in these games.