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We study the effects of empty sites in the prisoner’s dilemma game based on social diversity by introducing some empty sites into a square lattice. The results reveal that the empty sites dramatically enhance the cooperation level for a wide range of temptation to defection values if two types of players coexist. By calculating the chances of different type-combinations of the players located on the square lattice, we find that there is an intermediate region where five social ranks are induced to satisfy the certain rank distributions and the cooperation level is significantly enhanced. Moreover, simulation results also show that the moderate gaps among the social ranks can favor cooperation for a larger occupation density.

Cooperation is fundamental to biological and social systems. Thus, it is a crucial issue to find and understand what kinds of factors facilitate cooperation. Over the past decades, various versions of evolutionary games have been studied extensively to explore the possibilities for enhancing the cooperative behavior among selfish individuals. Consequently, different mechanisms, for example, kin selection [

Despite that several authors [

In this paper, we preliminarily focus on spatial game. Further, we only consider how the population density and the social diversity affect the cooperation between individuals. The paper is organized as follows. At first, the details of the model are presented and the parameters of the model are defined in Section

We consider an evolutionary prisoner’s dilemma game on a square lattice with the periodic boundary conditions, where two types of players (A and B) are located on the sites of a square lattice with the different concentrations, respectively. Different from the original version of the prisoner’s dilemma based on social diversity [

Each Monte Carlo (MC) simulation procedure has two elementary steps. Firstly, all individuals play the game against all their four nearest neighbors (if present) and collect the payoffs from the combats according to the parameterized payoff matrix suggested by Li et al. [

Secondly, after each full cycle of the game, the player

In order to characterize the macroscopic behaviors of the system, we introduce a parameter

Since we are interested in the long time regime, we define a parameter

Before showing the simulation results, we would like to analyze how the parameters

Results of MC simulations displayed below are obtained on a square lattice with the periodic boundary conditions and the size of

First, we present

Relative density of cooperators

In order to examine the effects of the parameter

Relative density of cooperators

In order to examine the effects of the parameter

Relative density of cooperators

Besides, for the sake of intuitively understanding the effects of the parameter

Snapshots of typical distributions of cooperators (white), defectors (black), and empty sites (green) on the square lattice for different values of

In order to examine the effects of the parameter

Relative density of cooperators

In the above discussion, we successively examine the effects of the parameters

Relative density of cooperators

In order to investigate the relation between cooperation and social diversity, Perc and Szolnoki [

As to the reasons for some appropriate values for

Relative fraction of players belonging to rank

By virtue of these histograms, one can find that the different rank distributions in Figure

Besides, the rank distribution is also affected by

Relative fraction of players belonging to different ranks in dependence on

In summary, we have extended the model of [

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

This work was partially supported by the Langfang Teachers College Foundation, China (Grant no. LSZY201204), and the National Science Fund for Distinguished Young Scholars of China (Grant no. 11204214).