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

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  • Philip LaPorte
  • Christian Hilbe
  • Martin A Nowak

Abstract

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.Author summary: Direct 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-one 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-one 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.

Suggested Citation

  • Philip LaPorte & Christian Hilbe & Martin A Nowak, 2023. "Adaptive dynamics of memory-one strategies in the repeated donation game," PLOS Computational Biology, Public Library of Science, vol. 19(6), pages 1-31, June.
    • Handle: RePEc:plo:pcbi00:1010987
    DOI: 10.1371/journal.pcbi.1010987
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