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Heterogeneous aspirations promote cooperation in the prisoner's dilemma game

To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter $u$ that interpolates between these two extreme options. Setting $u$ equal to zero returns the random selection of the opponent, while positive $u$ favor the fitter players. In addition, we divide the population into two groups. Players from group $A$ select their opponents as dictated by the parameter $u$, while players from group $B$ do so randomly irrespective of $u$. We denote the fraction of players contained in groups $A$ and $B$ by $v$ and $1-v$, respectively. The two parameters $u$ and $v$ allow us to analyze in detail how aspirations in the context of the prisoner's dilemma game influence the evolution of cooperation. We find that for sufficiently positive values of $u$ there exist a robust intermediate $v \approx 0.5$ for which cooperation thrives best. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process $K$ and for different interaction networks. We also provide complete phase diagrams depicting the dependence of the impact of $u$ and $v$ for different values of $K$, and contrast the validity of our conclusions by means of an alternative model where individual aspiration levels are subject to evolution as well. Our study indicates that heterogeneity in aspirations may be key for the sustainability of cooperation in structured populations.