Performance of Zero-determinant Strategies in Repeated Games without Discounting

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.

Imitation sometimes achieves success in multi-agent situations even though it is very simple. In repeated games, success of imitation has been characterized by unbeatability against other agents. Previous studies specified conditions under which imitation is unbeatable against co-players in repeated symmetric games, and clarified that the existence of unbeatable imitation is strongly related to the existence of payoff-controlling strategies, called zero-determinant strategies. However, these results were on the situations in which one repeated game is played by one group of players. It was pointed out that imitation of other players in the same group and imitation of other players in the same role in other groups generally result in different outcomes. Here, we investigate the existence condition of unbeatable imitation in the latter situations, described by two concurrent repeated games played by two different groups of players. We find that the existence condition of unbeatable imitation is stronger than the existence condition of unbeatable zero-determinant strategies, whereas both are very limited. Furthermore, we also find that adding exploration can make an imitation strategy unbeatable even in games where imitation without exploration can be beaten. Our findings suggest a strong relation between unbeatable imitation and payoff control even in the two concurrent repeated game situations.

A stochastic game is an optimal control problem for multiple agents in a stochastic environment. Stochastic games contain important practical problems, related to environmental issues or traffic flows. If the number of environmental states is one, such stochastic games are called repeated games. In repeated games, several strategies which unilaterally control payoffs have been proposed, and these strategies work in a mechanism similar to feedback control. In this paper, we generalize these strategies as feedback strategies in stochastic games, and investigate their properties. In particular, we provide a necessary and sufficient condition for such feedback strategies to work as controllers of payoffs under some assumptions. We demonstrate the performance of such feedback strategies in several examples related to social dilemmas. Our findings connect the theory of payoff control in repeated games to the theory of feedback control in stochastic games.