Abstract
Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need to be for cooperation to evolve. Here, we study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player’s previous move, and memory-one strategies, which take into account the own and the co-player’s previous move. In both cases, we compare deterministic strategy and stochastic strategy spaces. For reactive strategies and small costs, we find that stochasticity benefits cooperation, because it allows for generous-tit-for-tat. For memory one strategies and small costs, we find that stochasticity does not increase the propensity for cooperation, because the deterministic rule of win-stay, lose-shift works best. For memory one strategies and large costs, however, stochasticity can augment cooperation.
Highlights
Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions
We study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player’s previous move, and memory-one strategies, which take into account the own and the co-player’s previous move
Because almost all our social interactions occur repeatedly, reciprocity considerations may have played an important role for the evolution of social heuristics[18,19], which in turn helps to understand why we cooperate with strangers[20], sometimes even without considering the resulting costs to ourselves[21]
Summary
Seung Ki Baek[1], Hyeong-Chai Jeong[2], Christian Hilbe3,4 & Martin A. We study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player’s previous move, and memory-one strategies, which take into account the own and the co-player’s previous move. To model the emergence of direct reciprocity, researchers often use the example of the iterated prisoner’s dilemma In this game, two players can decide repeatedly whether to cooperate or to defect. By assuming that mutations are sufficiently rare, we can use the framework of Fudenberg & Imhof 37 to calculate how often players use each of the available strategies in the long run[38] This in turn allows us to calculate the evolving cooperation rates for each of the four strategy classes, as explained in more detail . Our results suggest that strategies with larger memory are typically beneficial for the evolution of cooperation, whereas the strategies' stochasticity can sometimes have a detrimental effect
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