Evolutionary dynamics on stochastic evolving networks for multiple-strategy games

Evolutionary game theory on dynamical networks has received much attention. Most of the work has been focused on $2\times 2$ games such as prisoner's dilemma and snowdrift, with general $n\times n$ games seldom addressed. In particular, analytical methods are still lacking. Here we generalize the stochastic linking dynamics proposed by Wu, Zhou, Fu, Luo, Wang, and Traulsen [PLoS ONE 5, e11187 (2010)] to $n\times n$ games. We analytically obtain that the fast linking dynamics results in the replicator dynamics with a rescaled payoff matrix. In the rescaled matrix, intuitively, each entry is the product of the original entry and the average duration time of the corresponding link. This result is shown to be robust to a wide class of imitation processes. As applications, we show both analytically and numerically that the biodiversity, modeled as the stability of a zero-sum rock-paper-scissors game, cannot be altered by the fast linking dynamics. In addition, we show that the fast linking dynamics can stabilize tit-for-tat as an evolutionary stable strategy in the repeated prisoner's dilemma game provided the interaction between the identical strategies happens sufficiently often. Our method paves the way for an analytical study of the multiple-strategy coevolutionary dynamics.

Figure 1

The coevolutionary dynamics of strategies and links. With probability $w$, the strategy update happens. In this case, select a random individual (e.g., a strategy 1 player) in the population and it imitates the strategy of another randomly selected neighbor (e.g., a strategy 2 player) based on the payoff difference. The richer the strategy 2 player is, the more likely it is to be imitated by the strategy 1 player. Otherwise, the linking dynamics happens with probability $1-w$. A random link is selected, and it breaks with a prescribed probability dependent on the link type. If it indeed breaks, randomly select an extreme of the broken link and then switch to a random individual in the population who is not its current neighbor. As an illustration, the dashed link is selected and broken with probability $k_{12}$, where 12 indicates the link type, and the strategy 1 player of the broken link is selected, eventually it connects with another strategy 2 player. In this way, a 12 links becomes another 12 link.

Figure 2

The internal fixed points and flow fields of the replicator equations for the rock-paper-scissors game with active linking dynamics in large populations. Each of the three plots has an internal equilibrium. The fixed points are all stable, which is in agreement with our theoretical prediction. But the internal fixed point moves away from P as $k_{\text{RS}}$ increases. In other words, the more fragile the RS link is, the less likely P is to be prosperous. Intuitively, increasing $k_{\text{RS}}$ decreases the interaction rate between R and S. Thus the dynamic flow around the edge RS is slower. This paves the way for the equilibrium to move toward the RS link or far away from P. For all the plots, $k_{\text{PR}}=k_{\text{SP}}=0.5$, $a=0.7$. (Images are generated by a modified version of the dynamo package [51].)

Figure 3

Frequencies for rock and scissors at the stationary regime. The frequency of paper can be given by $1-x_{\text{R}}-x_{\text{S}}$. For different initial conditions and the same $k_{\text{RS}}$, the stationary equilibria obtained by the simulations are almost the same. This indicates the stability of the internal equilibrium. Concerning the stationary frequencies of rock, $x_{\text{R}}$, and scissors, $x_{\text{S}}$, simulation results show that they both increases with the interaction rate between rock and scissors, $k_{\text{RS}}$, as predicted by Eq. (14). The simulation results agree well with the analytical predictions. Here the population size $N=400$, selection intensity $\beta =10$, and strategy updating frequency $w=0.01$. Besides, $k_{\text{PR}}=k_{\text{SP}}=0.5$ and $a=0.7$, which are the same as that in Fig. 2. For each data point, it is averaged over 50 independent runs. In each run, for a specified strategy, we set the mean value over a time window of ${10}^3$ generations to be the final frequency after a transient time of ${10}^6$ generations.

Figure 4

The frequency for TFT at the stationary regime. The simulation result shows when the population is large in the frequency of TFT (or small in the frequencies of ALLC and ALLD); TFT finally takes over the whole population for $k<0.5$. In other words, it resists the invasion of cooperators and defectors. This implies that TFT becomes an evolutionary stable strategy for $k<0.5$. This is consistent with the analytical prediction by the inequality (17) with our simulation parameters $b=5$, $c=1$, $m=5$, and $\gamma =0.5$. Other parameters are the same as those in Fig. 3. For $k>0.5$, the simulation results do not perfectly agree with the analytical one. This is due to the finite population effect as well as the inaccuracy of the transition matrix for the linking dynamics [38].