Adaptive dynamics of memory-one strategies in the repeated donation game

Human interactions can take the form of social dilemmas: collectively, people fare best if all cooperate but each individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows them to adopt reciprocal strategies which incentivize cooperation. The most basic model for direct reciprocity is the repeated donation game, a variant of the prisoner's dilemma. Two players interact over many rounds; in each round they decide whether to cooperate or to defect. Strategies take into account the history of the play. Memory-one strategies depend only on the previous round. Even though they are among the most elementary strategies of direct reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, much previous work has relied on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-one strategies has an invariant three-dimensional subspace, generated by the memory-one counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-one strategies and a full characterization for memory-one counting strategies.

Automated summary

Human interactions can lead to social dilemmas, where individuals are tempted to free ride despite the collective benefit of cooperation. Repetition of interactions allows for the adoption of reciprocal strategies, incentivizing cooperation. The repeated donation game is a basic model for direct reciprocity, where strategies depend on the previous rounds. Analyzing the adaptive dynamics of these strategies has been challenging, but this study derived and analyzed their dynamics, characterizing the space of memory-one strategies.