Adaptive dynamics of memory-1 strategies in the repeated donation game

Abstract

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.

A geometric process of evolutionary game dynamics

“…(For the asynchronous donation game or Prisoner’s Dilemma, see [43–47]. ) Cooperation incurs a cost c while conferring a benefit b on the other player [17,48–52]. The pay-off matrix is given by4pt1em 4pt1emCD4pt1emCD(4pt1emb−c−cb0).The game is a Prisoner’s Dilemma if b > c > 0 [26].…”

“…The resulting process is a deterministic differential equation on the space of strategies [1–3]. Adaptive dynamics has been applied to many different games and questions that arise in evolutionary biology including infectious diseases [4], evolutionary dynamics with interaction structure [5], altruism in spatial models [6], evolution of genetic polymorphism [7], speciation in patch models [8], evolutionary branching [9–11], physiologically structured populations [12], the Snowdrift game [13,14], the ultimatum game [15] and memory-1 strategies of repeated games [16,17]. The theory of adaptive dynamics is developed and extended in [1,3,18–21].…”