Zero-Determinant Strategies in Noisy Repeated Games

Press and Dyson has recently revealed a new set of “zero-determinant” (ZD) strategies for repeated games, which grant a player with power of manipulation on the game’s payoffs. A ZD strategy player can unilaterally enforce a fixed linear relationship between his payoff and that of his opponent. In particular it can (i) deterministically set his opponent’s payoff to a fixed value, regardless of how the opponent responds; or (ii) extort his opponent such that when the opponent tries to increase his payoff, he will increase the ZD strategy player’s even more, and the opponent can only maximize his payoff by unconditional cooperating; or even (iii) always obtain an advantageous proportion of two players’ total payoff. ZD strategies exists in all kinds of two-player two-action repeated games without taking into consideration the influence of noise. However, outside the laboratory, interactions are often involved with errors. Perception error occurs when a player wrongly observes the action of the opponent and implementation error occurs when a player behaves differently as what he intended to do. In this work, we derive the generalized form of ZD strategies for noisy repeated games, and also study the conditions, feasible regions and corresponding payoffs for two specifications of ZD strategies, which are called the pinning strategies and extortion strategies. It is found that the ZD strategies still widely exist in noisy repeated games with reasonable level of noise, although the noise has strong impact on the existence and performance of ZD strategies. The noises will expose the ZD strategy player to uncertainty and risk, however, it is still possible for him to set the opponent’s payoff to a fixed value, or to extort the opponent.