Memory-n strategies of direct reciprocity

Significance Direct reciprocity is one of the fundamental mechanisms for cooperation. It is based on the idea that individuals are more likely to cooperate if they can expect their beneficiaries to remember and to return their cooperative acts in future. Previous computational models, however, often had to restrict the number of past rounds subjects can memorize. Herein we suggest an alternative approach. We propose general properties that robust cooperative strategies ought to have. Then we characterize all memory-n strategies that meet these properties, and we show that such strategies naturally emerge across different evolutionary scenarios. Our results are applicable to general social dilemmas of arbitrary size. For some dilemmas, longer memory is all it takes for cooperation to evolve. Humans routinely use conditionally cooperative strategies when interacting in repeated social dilemmas. They are more likely to cooperate if others cooperated before, and are ready to retaliate if others defected. To capture the emergence of reciprocity, most previous models consider subjects who can only choose from a restricted set of representative strategies, or who react to the outcome of the very last round only. As players memorize more rounds, the dimension of the strategy space increases exponentially. This increasing computational complexity renders simulations for individuals with higher cognitive abilities infeasible, especially if multiplayer interactions are taken into account. Here, we take an axiomatic approach instead. We propose several properties that a robust cooperative strategy for a repeated multiplayer dilemma should have. These properties naturally lead to a unique class of cooperative strategies, which contains the classical Win–Stay Lose–Shift rule as a special case. A comprehensive numerical analysis for the prisoner’s dilemma and for the public goods game suggests that strategies of this class readily evolve across various memory-n spaces. Our results reveal that successful strategies depend not only on how cooperative others were in the past but also on the respective context of cooperation.