On the existence of fair zero-determinant strategies in the periodic prisoner’s dilemma game

Stochastic games are a framework for investigating long-term interdependence of multi-agent systems with environmental feedback. When the number of environmental states is one, they are reduced to repeated games. In repeated games, zero-determinant (ZD) strategies attract much attention in evolutionary game theory, since they can unilaterally control payoffs. Especially, fair ZD strategies unilaterally equalize the payoff of the focal player and the average payoff of the opponents, and they were found in several games including the social dilemma games. Although the existence condition of ZD strategies in repeated games was specified, its extension to stochastic games remains largely unclear. Here, we investigate the existence condition of fair ZD strategies in the periodic prisoner's dilemma game, which is one of the simplest stochastic games. The periodic prisoner's dilemma game consists of two environmental states and the two states alternate deterministically. Whereas each stage game is not necessarily the prisoner's dilemma game, the whole game can be regarded as the prisoner's dilemma game on average. We show that fair ZD strategies do not necessarily exist in the periodic prisoner's dilemma game, in contrast to the repeated prisoner's dilemma game. Furthermore, we also prove that the Tit-for-Tat strategy, which imitates the opponent's action, is not necessarily a fair ZD strategy in the periodic prisoner's dilemma game, whereas the Tit-for-Tat strategy is always a fair ZD strategy in the repeated prisoner's dilemma game. Our results highlight difference between ZD strategies in the periodic prisoner's dilemma game and those in the standard repeated prisoner's dilemma game.