Evolutionary performance of zero-determinant strategies in multiplayer games

Repetition is one of the key mechanisms to maintain cooperation. In long-term relationships, in which individuals can react to their peers׳ past actions, evolution can promote cooperative strategies that would not be stable in one-shot encounters. The iterated prisoner׳s dilemma illustrates the power of repetition. Many of the key strategies for this game, such as ALLD, ALLC, Tit-for-Tat, or generous Tit-for-Tat, share a common property: players using these strategies enforce a linear relationship between their own payoff and their co-player׳s payoff. Such strategies have been termed zero-determinant (ZD). Recently, it was shown that ZD strategies also exist for multiplayer social dilemmas, and here we explore their evolutionary performance. For small group sizes, ZD strategies play a similar role as for the repeated prisoner׳s dilemma: extortionate ZD strategies are critical for the emergence of cooperation, whereas generous ZD strategies are important to maintain cooperation. In large groups, however, generous strategies tend to become unstable and selfish behaviors gain the upper hand. Our results suggest that repeated interactions alone are not sufficient to maintain large-scale cooperation. Instead, large groups require further mechanisms to sustain cooperation, such as the formation of alliances or institutions, or additional pairwise interactions between group members.