Reciprocity and repeated games have been at the center of attention when studying the evolution of human cooperation. Direct reciprocity is considered to be a powerful mechanism for the evolution of cooperation, and it is generally assumed that it can lead to high levels of cooperation. Here we explore an openended, infinite strategy space, where every strategy that can be encoded by a finite state automaton is a possible mutant. Surprisingly, we find that direct reciprocity alone does not lead to high levels of cooperation. Instead we observe perpetual oscillations between cooperation and defection, with defection being substantially more frequent than cooperation. The reason for this is that "indirect invasions" remove equilibrium strategies: every strategy has neutral mutants, which in turn can be invaded by other strategies. However, reciprocity is not the only way to promote cooperation. Another mechanism for the evolution of cooperation, which has received as much attention, is assortment because of population structure. Here we develop a theory that allows us to study the synergistic interaction between direct reciprocity and assortment. This framework is particularly well suited for understanding human interactions, which are typically repeated and occur in relatively fluid but not unstructured populations. We show that if repeated games are combined with only a small amount of assortment, then natural selection favors the behavior typically observed among humans: high levels of cooperation implemented using conditional strategies. repeated prisoner's dilemma | game theory T he problem of cooperation in its simplest and most challenging form is captured by the Prisoners' Dilemma. Two people can choose between cooperation and defection. If both cooperate, they get more than if both defect, but if one defects and the other cooperates, the defector gets the highest payoff and the cooperator gets the lowest. In the one-shot Prisoners' Dilemma, it is in each person's interest to defect, even though both would be better off had they cooperated. This game illustrates the tension between private and common interest.However, people often cooperate in social dilemmas. Explaining this apparent paradox has been a major focus of research across fields for decades. Two important explanations for the evolution of cooperation that have emerged are reciprocity (1-19) and population structure (20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32). If individuals find themselves in a repeated Prisoner's Dilemma-rather than a one-shot version-then there are Nash equilibria where both players cooperate under the threat of retaliation in future rounds (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). The existence of such equilibria is a cornerstone result in economics (1-3), and the evolution of cooperation in repeated games is of shared interest for biology (4-10), economics (11)(12)(13)(14), psychology (15), and sociology (16), with applications that range from antitrust laws (17) to sticklebacks (18), alt...

In evolutionary games, reproductive success is determined by payoffs. Weak selection means that even large differences in game outcomes translate into small fitness differences. Many results have been derived using weak selection approximations, in which perturbation analysis facilitates the derivation of analytical results. Here, we ask whether results derived under weak selection are also qualitatively valid for intermediate and strong selection. By “qualitatively valid” we mean that the ranking of strategies induced by an evolutionary process does not change when the intensity of selection increases. For two-strategy games, we show that the ranking obtained under weak selection cannot be carried over to higher selection intensity if the number of players exceeds two. For games with three (or more) strategies, previous examples for multiplayer games have shown that the ranking of strategies can change with the intensity of selection. In particular, rank changes imply that the most abundant strategy at one intensity of selection can become the least abundant for another. We show that this applies already to pairwise interactions for a broad class of evolutionary processes. Even when both weak and strong selection limits lead to consistent predictions, rank changes can occur for intermediate intensities of selection. To analyze how common such games are, we show numerically that for randomly drawn two-player games with three or more strategies, rank changes frequently occur and their likelihood increases rapidly with the number of strategies . In particular, rank changes are almost certain for , which jeopardizes the predictive power of results derived for weak selection.

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