Migration based on historical payoffs promotes cooperation in continuous two-dimensional space

Abstract

Migration is a common behavior in both biology and human societies for seeking more suitable environments or driven by greater benefits. Under the theoretical framework of evolutionary games, migration has been proved to be an effective mechanism for promoting cooperation. In this work, we introduce the migration based on historical payoffs into an evolutionary public goods game in two-dimensional space. Specifically, we introduce a parameter α to account for the ambition of individuals. During the evolution, individuals compare their current payoffs with the threshold values determined by the historical payoffs obtained in last round and the parameter α, and then decide whether to migrate. In particular, if the current payoff is greater than the threshold value, the individual will remain in his position, otherwise he will move in a randomly chosen direction with the migration speed v. The results show that cooperation could be promoted by the historical-payoff–dependent migration when compared with the situation of random migration, and there exists an optimal α which is most favorable to cooperation. Moreover, we provide some phenomenological explanations based on the characteristic snapshots of strategy pattern during the evolution. We also study the effects of the synergy factor r and the migration speed v on cooperation, and find that the moderate values of v could facilitate cooperation best. Besides, the robustness of the results has been tested by investigating the model in a sparse population and in a larger range of α, respectively.