Abstract
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.
Highlights
Repetition is one of the key mechanisms to maintain cooperation
Evolution often occurs in dynamical cycles (Boyd and Lorberbaum, 1987; Nowak and Sigmund, 1993a; van Veelen et al, 2012): unconditional defectors (ALLD) can be invaded by reciprocal strategies like Tit-for-Tat (TFT), which in turn often catalyze the evolution of more cooperative strategies like generous Tit-for-Tat and unconditional cooperators (ALLC)
The multiplayer-version of TFT in a repeated public goods game is proportional Tit-for-Tat: if j of the other group members cooperated in the previous round, a pTFT-player cooperates with probability j=ðn À 1Þ in the round, with n being the size of the group
Summary
Reciprocal giving has been documented in numerous behavioral experiments (e.g., Wedekind and Milinski, 1996; Keser and van Winden, 2000; Fischbacher et al, 2001; Dreber et al, 2008; Grujic et al, 2014). Zero-determinant strategies are not confined to pairwise games; they exist in the iterated public goods game (Pan et al, 2014), and in any repeated social dilemma, with an arbitrary number of involved players (Hilbe et al, 2014b). In this way, it has become possible to identify the multiplayer-game analogues of the above mentioned strategies. The possibilities of a single player to generate a positive group dynamics diminishes with group size, irrespective of the strategy applied by the focal player Taken together, these results suggest that larger groups make it more difficult to sustain cooperation. We will argue that there are three potential mechanisms that can help individuals solving their multiplayer social dilemmas: they can either provide additional incentives on a pairwise basis (Rand et al, 2009; Rockenbach and Milinski, 2006); they can coordinate their actions and form alliances (Hilbe et al, 2014b); or they can implement central institutions which enforce mutual cooperation (Ostrom, 1990; Sigmund et al, 2010; Sasaki et al, 2012; Cressman et al, 2012; Traulsen et al, 2012; Zhang and Li, 2013; Schoenmakers et al, 2014)
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