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The effect of the random drift on the evolutionary prisoner’s dilemma game is studied on regular lattices. A
new evolutionary rule is proposed, which stochastically combines the deterministic rule with the random drift rule. It is found that the random drift has an effect on the evolutionary dynamics depending on the values of the temptation-to-defect

Since the introduction of evolutionary game theory by Smith and Price [

However, cooperation is widespread in many biological, social, and economic systems [

All the game models incorporate some kind of evolutionary dynamics, which also play a crucial role in the results. As for [

The best-takes-over update rule is a nonstochastic imitation strategy. However, for real dynamical systems, external disturbances or system errors are generally inevitable. Therefore, it is worth considering the dependence of the promotion of cooperation on the evolutionary rules and its robustness against perturbations. Indeed, previous work has pursued this enquiry [

In these game models, players are viewed as rational, who update their strategy by copying, within certain constraints, the strategy of those others that are doing better, or in game theoretical terms that are obtaining higher payoffs from the game. Then, what difference will be there in the evolution of cooperation if players occasionally behave irrationally, updating strategy with no concern of their payoffs. In this paper, we will explore this problem by combining the best-takes-over rule with the random drift reproduction rule in the prisoner’s dilemma game.

We study the PDG with pure strategies: the players can either defect (D) or cooperate (C). The same as in [

Best-takes-over: in each generation, an individual node imitates the strategy of one of its neighbors (including the node itself) that received the highest payoff in the last round. The best-takes-over rule is a deterministic rule according to which the individual with the highest gain in a given neighborhood reproduces with certainty.

Random drift: whenever an individual node

To investigate how the random drift affects evolutionary games, we propose a new evolutionary rule which stochastically combines the deterministic rule with the random drift rule. A parameter of probability

Our simulations are carried out on the regular 8-neighbored square lattices. Initially, the cooperative and defective strategies are randomly distributed among the players with equal probability

Figure

The frequency of cooperators,

Despite the huge gap of the two lattices in size, the simulation results are qualitatively similar. In the following, we mainly analyze the results on the

To have a more clear knowledge of the influence of the random drift on the evolution of cooperation, the variation of

The frequency of cooperators as a function of the probability

Thus, in fact, the random drift has an influence on the evolutionary PDG depending on the values of the temptation-to-defect

How come this kind of Matthew effect arises when the random drift is applied to the deterministic imitative mechanism with small probability? The dynamics and the pattern formation of the PDG system may shed light on the explanation.

Setting

The dynamics of the PDG system with or without the influence of the random drift on a

The corresponding pattern formations of the above two evolutionary processes are displayed in Figures

The pattern formation of the PDG system without the influence of the random drift on a

The pattern formation of the PDG system with the influence of the random drift on a

The dynamics of the PDG system with temptation-to-defect

The dynamics of the PDG system with or without the influence of the random drift on a

Figures

The pattern formation of the PDG system without the influence of the random drift on a

The pattern formation of the PDG system with the influence of the random drift on a

At the beginning, the two systems have similar pattern formations as shown in Figures

Now we are ready to explain the Matthew effect of the random drift on the evolutionary PDG. Despite the population of its opponent, some particular shaped C-clusters and

Evolutionary dynamics are affected by population structure and update rules. Spatial or network structure facilitates the clustering of strategies, which represents a mechanism for the evolution of cooperation. However, whether the evolutionary dynamics is robust to disturbance deserves further studies. In this paper, the problem is explored by combining the deterministic imitative rule with the random drift reproduction rule in the prisoner’s dilemma game on regular lattices. It is found that the employed random drift has an effect on the evolutionary dynamics depending on the values of the temptation-to-defect

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Dr. Yu-Zhong Chen and Professor Ying-Cheng Lai for helpful discussions. This work was supported by the National Science Foundation of China under Grant no. 11101256 and Research and Innovation Project of Shanghai Municipal Education Commission (no. 14YZ149).