Zero-determinant strategies of multi-player multi-action repeated games with multiple memories

Abstract

This paper investigates zero-determinant (ZD) strategies for iterated multi-player multi-action games with multiple memories. First, using the semi-tensor product (STP) of matrices, an equivalent algebraic model for the iterated game is proposed. Based on the algebraic expression, ZD strategies are derived for repeated finite games with multiple memories, allowing players to unilaterally apply a linear relationship among their augmented expected payoffs regardless of their opponents’ strategies. Furthermore, ZD strategies for repeated symmetric and skew-symmetric games are considered, based on which the coupling correlations among players’ payoffs, as well as the structure of the payoff vectors, are clarified. It is found that in repeated symmetric games, any player can get the same augmented expected payoff by employing a ZD strategy. In repeated skew-symmetric games, it is possible for all participants to achieve a zero-sum outcome. To execute the aforementioned ZD strategies, some identical elements are required in each augmented payoff vector for symmetric games, whereas the payoff vector for skew-symmetric games must contain zero elements.