Evolution of cooperation driven by self-recommendation

Cooperators increase the fitness of others at a cost to themselves. Thus cooperation should not be favored by natural selection in a well-mixed population. It challenges the evolutionists since cooperation is widespread. Information spreading has been revealed to play a key role in the emergence of cooperation. Individuals, however, are typically assumed to be passive in the information spreading. Here we assume that individuals self-recommend themselves to those that are about to have new neighbors. Individuals with higher tendencies of self-recommendation are likely to have more neighbors. In this way, individuals are active to spread the information. We analytically obtain a critical cost-to-benefit ratio, below which cooperation emerges. It reveals quantitatively how eloquent cooperators have to be compared with defectors to ensure that cooperation takes over the population. It also indicates that individuals need to be open enough to the self-recommendation to enhance cooperation level. In addition, the critical cost-to-benefit ratio represents the viscosity of the population, measuring how close cooperators are to each other. Our results highlight the role self-recommendation plays in cooperation.

Figure 1

The final fraction of cooperators as a function of initial fraction of cooperators. The symbols indicate the simulation results and the dashed lines represent the internal unstable fixed point $x_C^{*}$ given by Eq. (7). The simulation results are in agreement with our theoretical predictions. If the initial value is smaller than the critical fraction of cooperators $x_C^{*}$, defection dominates the population. Otherwise, cooperation is taking over the population. This mirrors a coordination game with cooperation a stable Nash equilibrium. Each data point in this figure is averaged over 500 independent runs. In each simulation, we run time up to ${10}^6$ generations. We have checked that all realizations end up with homogeneous populations. The final fraction of cooperators (in the simulation) is the estimation of fixation probability of cooperation starting from a given initial fraction of cooperators. Parameters: Prisoners' Dilemma with $b=3$ and $c=1$; the tendencies of self-recommendation of cooperators and defectors, $R_C=3$ and $R_D=1$, respectively; population size $N=1000$; average degree $\langle d\rangle =20$; probability of a strategy update $\omega ={10}^{-3}$; link-breaking probability $k=1$; and selection intensity $\beta =10$.

Figure 2

The final fraction of cooperators as a function of $R_C/R_D$ and stubbornness $p$. It is shown that cooperation prevails if the cooperators are more eloquent, i.e., $R_C/R_D$ is sufficiently large, and if individuals are open to accept the self-recommendation, i.e., $p$ is small. The simulation, i.e., the heat map, is in agreement with our theoretical prediction, i.e., the black line determined by Eq (7). Parameters: Prisoner's Dilemma with $b=3$ and $c=1$, $N=1000$, $\langle d\rangle =20$, $\omega ={10}^{-3}$, $k=1$, and $\beta =10$; the initial fraction of cooperators is 0.5.

Figure 3

The final fraction of cooperators as a function of $R_C/R_D$ and cost-to-benefit ratio $c/b$. The heat map represents the simulation result and the black line indicates the theoretical boundary predicted by $x^{*}=c{R}_D/\left[(b-c)(R_C-R_D)\right]=0.5$. We can see that the simulation is in agreement with our theoretical prediction. It is shown that a small cost-to-benefit ratio facilitates the cooperation. Parameters: $N=1000$, $\langle d\rangle =20$, $\omega ={10}^{-3}$, $k=1$, and $\beta =10$; the initial fraction of cooperators is 0.5 and the stubbornness $p=0$.

Figure 4

Illustration of the robustness of our theoretical result. (a) Simulation results for different population sizes; the critical transition from all defection to all cooperation is shifted to the right with the decrease of $N$. (b) Simulation results for different average degrees; the case with smaller average degree shows more fluctuations than the one with larger average degree. (c) Simulation results are quite sensitive to the frequency of the strategy updates $\omega$; the critical transition is shifted to the right with the increase of $\omega$; the default parameters are the same as in Fig. 1.

Figure 5

Degree distributions of the network. From the initial to final states during the coevolutionary dynamics of network and strategy, the degree distributions change and strongly depend on the cooperator frequency. We measure the degree distributions whenever the level of cooperation ($x_C$) reaches certain levels ($x_C=0.3$, 0.5, and 0.8). In each panel, the degree distributions of both cooperators (blue) and defectors (red) are shown. The degree distribution of the whole population (black line) shows a mixture distribution of red and blue. The distributions for 500 realizations at each given level of cooperation are averaged. The default parameters are the same as in Fig. 1 with the stubbornness $p=0.3$ and initial state of $x_C=0.4$. The average degree is 20 which keeps constant during the simulation.