Spatial prisoner’s dilemma games with increasing size of the interaction neighborhood on regular lattices

Abstract

We studied the evolution of cooperation in the prisoner's dilemma game on a square lattice where the size of the interaction neighborhood is considered. Firstly, the effects of noise and the cost-to-benefit ratio on the maintenance of cooperation were investigated. The results indicate that the cooperation frequency depends on the noise and cost-to-benefit ratio: cooperation reaches a climax as noise increases, but it monotonously decreases and even vanishes with the ratio increasing. Furthermore, we investigated how the size of the interaction neighborhood affects the emergence of cooperation in detail. Our study demonstrates that cooperation is remarkably enhanced by an increase in the size of the interaction neighborhood. However, cooperation died out when the size of the interaction neighborhood became too large since the system was similar to the mean-field system. On this basis, a cluster-forming mechanism acting among cooperators was also explored, and it showed that the moderate range of the neighborhood size is beneficial for forming larger cooperative clusters. Finally, large-scale Monte Carlo simulations were carried out to visualize and interpret these phenomena explicitly. [11][12][13][14][15][16]. As a paradigm of pair-wise interaction games, the prisoner's dilemma game (PDG) has attracted much attention [17][18][19][20][21]. In the original form of the PDG, there are two behavior options: each of two players must simultaneously choose to cooperate (C) or to defect (D). If both players choose C, they will receive the reward R separately, but only the punishment P for mutual defection. If two players take different strategies, then the defector will get the highest payoff of temptation T, while the cooperator will be left with the lowest sucker's payoff S. These payoffs usually satisfy the elementary payoff ranking T > R > P > S and the additional required condition (T + S) < 2R in repeated interactions. Henceforth, defection is the optimal choice for each player irrespective of the decision of his opponent, and defection will become widespread. Every player will end up with the payoff P instead of the payoff R, which yields the social dilemmas as depicted in [22]. According to the motivation of the non-equilibrium kinetic Ising model in statistical physics, the PDG has been well studied for a spatially structured population in past decades [11][12][13][14][15][16][17][18][19][20][21][22][23]. In these models, players are distributed at the sites of different lattices or graphs, and each player can update his/her strategy and has a given probability of adopting his/her neighbor's strategies at each time step. Recently, many real systems have been found to exhibit

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