Social diversity and promotion of cooperation in the spatial prisoner’s dilemma game

The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determine the social diversity of players engaging in the prisoner’s dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game payoffs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrinsic factors that determine the social diversity of players, and find that the power-law distribution enables the best promotion of cooperation. The facilitation of the cooperative strategy relies mostly on the inhomogeneous social state of players, resulting in the formation of cooperative clusters which are ruled by socially high-ranking players that are able to prevail against the defectors even when there is a large temptation to defect. To confirm this, we also study the impact of spatially correlated social diversity and find that cooperation deteriorates as the spatial correlation length increases. Our results suggest that the distribution of wealth and social status might have played a crucial role by the evolution of cooperation amongst egoistic individuals.

Figure 1

Distributions of scaling factors $\left(\xi \right)$ originating from the uniformly distributed random variable $\left(\chi \right)$. The three functions correspond to uniform (u), exponential (e), and power-law (p) distribution when $a=1$. For easier comparison only the $[-1,1]$ interval of $\xi$ is plotted.

Figure 2

Spatial distributions of social diversity obtained for $a=1$ when uniform (u), exponential (e), and power-law (p) distributed values of scaling factors were used. Lines show an exemplary cross section of the two-dimensional grid.

Figure 3

Spatial distributions of social diversity constituted by randomly drawn scaling factors from a uniform distribution by $a=1$ and spatial correlation length $\lambda =1$ (bottom), $\lambda =3$ (middle), and $\lambda =7$ (top). Lines show an exemplary cross section of the two-dimensional grid.

Figure 4

Color-coded $F_C$ for the uniform (left), exponential (middle), and power-law (right) distributed social diversity on the $b\text{−}a$ parameter space. The color scale is linear, white depicting $0.0$ and black $1.0$ values of $F_C$.

Figure 5

$F_C$ as a function of $b$ for different values of $\lambda$. The $a=0$ curve is depicted for reference.