Zero-determinant strategies in iterated multi-strategy games

Self-serving, rational agents sometimes cooperate to their mutual benefit. The two-player iterated prisoner’s dilemma game is a model for including the emergence of cooperation. It is generally believed that there is no simple ultimatum strategy which a player can control the return of the other participants. The recent discovery of the powerful class of zero-determinant strategies in the iterated prisoner’s dilemma dramatically expands our understanding of the classic game by uncovering strategies that provide a unilateral advantage to sentient players pitted against unwitting opponents. However, strategies in the prisoner’s dilemma game are only two strategies. Are there these results for general multi-strategy games? To address this question, the paper develops a theory for zero-determinant strategies for multi-strategy games, with any number of strategies. The analytical results exhibit a similar yet different scenario to the case of two-strategy games. Zero-determinant strategies in iterated Prisoner’s Dilemma can be seen as degenerate case of our results. The results are also applied to the Snowdrift game, the Hawk–Dove game and the Chicken game.