The robustness of zero-determinant strategies in Iterated Prisoner's Dilemma games

Press and Dyson (2012) discovered a special set of strategies in two-player Iterated Prisoner's Dilemma games, the zero-determinant (ZD) strategies. Surprisingly, a player using such strategies can unilaterally enforce a linear relation between the payoffs of the two players. In particular, with a subclass of such strategies, the extortionate strategies, the former player obtains an advantageous share of the total payoff of the players, and the other player׳s best response is to always cooperate, by doing which he maximizes the payoff of the extortioner as well. When an extortionate player faces a player who is not aware of the theory of ZD strategies and improves his own payoff by adaptively changing his strategy following some unknown dynamics, Press and Dyson conjecture that there always exist adapting paths for the latter leading to the maximum possible scores for both players. In this work we confirm their conjecture in a very strong sense, not just for extortionate strategies, but for all ZD strategies that impose positive correlations between the players' payoffs. We show that not only the conjectured adapting paths always exist, but that actually every adapting path leads to the maximum possible scores, although some paths may not lead to the unconditional cooperation by the adapting player. This is true even in the rare cases where the setup of Press and Dyson is not directly applicable. Our result shows that ZD strategies are even more powerful than as pointed out by their discoverers. Given our result, the player using ZD strategies is assured that she will receive the maximum payoff attainable under the desired payoff relation she imposes, without knowing how the other player will evolve. This makes the use of ZD strategies even more desirable for sentient players.