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.

Social interactions often take the form of a social dilemma: collectively, individuals fare best if everybody cooperates, yet each single individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows individuals to adopt reciprocal strategies which incentivize cooperation. The most basic model to study reciprocity is the repeated donation game, a variant of the repeated prisoner’s dilemma. Two players interact over many rounds, in which they repeatedly decide whether to cooperate or to defect. To make their decisions, they need a strategy that tells them what to do depending on the history of previous play. Memory-1 strategies depend on the previous round only. Even though memory-1 strategies are among the most elementary strategies of reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, most previous work relies on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-1 strategies has an invariant three-dimensional subspace, generated by the memory-1 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-1 strategies and a full characterization for memory-1 counting strategies.Author summaryDirect reciprocity is a mechanism for evolution of cooperation based on the repeated interaction of the same players. In the most basic setting, we consider a game between two players and in each round they choose between cooperation and defection. Hence, there are four possible outcomes: (i) both cooperate; (ii) I cooperate, you defect; (ii) I defect, you cooperate; (iv) both defect. A memory-1 strategy for playing this game is characterized by four quantities which specify the probabilities to cooperate in the next round depending on the outcome of the current round. We study evolutionary dynamics in the space of all memory-1 strategies. We assume that mutant strategies are generated in close proximity to the existing strategies, and therefore we can use the framework of adaptive dynamics, which is deterministic.