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 $\mu$ and $\sigma$. The parameter $\mu$ characterizes the most adequate goal for individuals and the parameter $\sigma$ 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 $\mu$ affects the regime of the payoff parameter which supports the anomalous cooperation enhancement, whereas the parameter $\sigma$ plays a decisive role on the appearance of the nonmonotonic variation of the cooperation level. Furthermore, to give explicit implications for the parameters $\mu$ and $\sigma$ 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.

Figure 1

The pair approximation results for the relationship between the cooperation level ${\rho }_c$ and the payoff parameter $b$ under the conditional imitation rule. Two groups of the parameters $\sigma$ and $\mu$ are considered. In one group, the parameter $\sigma$ is held at $\sigma =0.1$ and the parameter $\mu$ varies from $\mu =0.1$ to $\mu =0.9$. In the other group, $\mu$ is held at $\mu =0.9$ and $\sigma$ increases from $\sigma =0.1$ to $\sigma =0.3$ and to $\sigma =0.5$.

Figure 2

The relationship between the cooperation level ${\rho }_c$ and the payoff parameter $b$ with the conditional imitation rule on ER networks with the average degree $\langle k\rangle =4$ and the size $N=1000$. Here we set $\sigma =0.1$ and $\mu =0.4$. As predicted by the pair approximation, the nonmonotonicity of ${\rho }_c$ against $b$ is produced with the help of the conditional imitation rule.

Figure 3

Contour plots for the cooperation level ${\rho }_c$ in the $\mu -b$ parameter space (upper plots) and in the $\sigma -b$ parameter space (lower plots) on four different types of networks under the conditional imitation rule. We set $\sigma =0.1$ in upper plots and $\mu =0.2$ in lower plots. The average degree is set as $\langle k\rangle =4$ for these four different types of networks. (a, b) Square lattices with $N=50\times 50$. (c, d) Homogenous small-world networks with $N=1000$. (e, f) ER networks with $N=1000$. (g, h) Scale-free networks with $N=1000$. To facilitate comparison, we use the same color profile encoding different values of ${\rho }_c$ from 0 [yellow (light gray)] to 1 [red (dark gray)] in these contour plots.

Figure 4

Contour plots for the cooperation level ${\rho }_c$ in (a) the ${\mu }^{\prime }-b$ parameter space and in (b) the ${\sigma }^{\prime }-b$ parameter space on square lattices under the alterative conditional imitation rule. We fix ${\sigma }^{\prime }=0.4$ in (a) and ${\mu }^{\prime }=0.2$ in (b). The population size on square lattices is $N=50\times 50$. The color profile encoding different values of ${\rho }_c$ is from 0 [yellow (light gray)] to 1 [red (dark gray)].

Figure 5

The cooperation level ${\rho }_c$ (solid squares) and the ratio of $r_{CD}$ to $r_{DC}$ (solid circles) as functions of $b$. $r_{CD}$ and $r_{DC}$ are the ratios of the average benefit over the average cost incurred during strategy updates for two competitive events, $C\to D$ and $D\to C$, respectively. The variations of $r_{CD}$ (hollow triangles) and $r_{DC}$ (hollow diamonds) against $b$ are shown in the inset.