Adaptive coordination promotes collective cooperation in repeated social dilemmas

Abstract

Direct reciprocity based on the repeated prisoner's dilemma has been intensively studied. Most theoretical investigations have concentrated on memory-$1$ strategies, a class of elementary strategies just reacting to the previous-round outcomes. Though the properties of "All-or-None" strategies ($AoN_K$) have been discovered, simulations just confirmed the good performance of $AoN_K$ of very short memory lengths. It remains unclear how $AoN_K$ strategies would fare when players have access to longer rounds of history information. We construct a theoretical model to investigate the performance of the class of $AoN_K$ strategies of varying memory length $K$. We rigorously derive the payoffs and show that $AoN_K$ strategies of intermediate memory length $K$ are most prevalent, while strategies of larger memory lengths are less competent. Larger memory lengths make it hard for $AoN_K$ strategies to coordinate, and thus inhibiting their mutual reciprocity. We then propose the adaptive coordination strategy combining tolerance and $AoN_K$' coordination rule. This strategy behaves like $AoN_K$ strategy when coordination is not sufficient, and tolerates opponents' occasional deviations by still cooperating when coordination is sufficient. We found that the adaptive coordination strategy wins over other classic memory-$1$ strategies in various typical competition environments, and stabilizes the population at high levels of cooperation, suggesting the effectiveness of high level adaptability in resolving social dilemmas. Our work may offer a theoretical framework for exploring complex strategies using history information, which are different from traditional memory-$n$ strategies.