Conditional imitation might promote cooperation under high temptations to defect

In this paper we introduce a conditional imitation rule into an evolutionary game, in which the imitation probabilities of individuals are determined by a function of payoff difference and two crucial parameters $\ensuremath{\mu}$ and $\ensuremath{\sigma}$. The parameter $\ensuremath{\mu}$ characterizes the most adequate goal for individuals and the parameter $\ensuremath{\sigma}$ characterizes the tolerance of individuals. By using the pair approximation method and numerical simulations, we find an anomalous cooperation enhancement in which the cooperation level shows a nonmonotonic variation with the increase of temptation. The parameter $\ensuremath{\mu}$ affects the regime of the payoff parameter which supports the anomalous cooperation enhancement, whereas the parameter $\ensuremath{\sigma}$ plays a decisive role on the appearance of the nonmonotonic variation of the cooperation level. Furthermore, to give explicit implications for the parameters $\ensuremath{\mu}$ and $\ensuremath{\sigma}$ we present an alterative form of the conditional imitation rule based on the benefit and the cost incurred to individuals during strategy updates. In this way, we also provide a phenomenological interpretation for the nonmonotonic behavior of cooperation with the increase of temptation. The results give a clue that a higher cooperation level could be obtained under adverse environments for cooperation by applying the conditional imitation rule, which is possible to be manipulated in real life. More generally, the results in this work might point out an efficient way to maintain cooperation in the risky environments to cooperators.