Neighborhood size effects on the evolution of cooperation under myopic dynamics

We study the evolution of cooperation in 2×22×2 social dilemma games in which players are located on a two-dimensional square lattice. During the evolution, each player modifies her strategy by means of myopic update dynamic to maximize her payoff while composing neighborhoods of different sizes, which are characterized by the corresponding radius, 𝑟r. An investigation of the sublattice-ordered spatial structure for different values of 𝑟r reveals that some patterns formed by cooperators and defectors can help the former to survive, even under untoward conditions. In contrast to individuals who resist the invasion of defectors by forming clusters due to network reciprocity, innovators spontaneously organize a socially divisive structure that provides strong support for the evolution of cooperation and advances better social systems. As a basic research issue, how to maintain high-level cooperation has attracted great attention both theoretically and experimentally. The most commonly used theoretical framework to study the cooperation between selfish individuals is evolutionary game theory.

Here, we study the effects of myopic strategy update and neighborhood on social dilemma games. Interestingly, the evolution outcomes show a spatial order strategy distribution, which is similar to the antiferromagnetic order in the spin system. In detail, below the threshold temptation value, the distribution of cooperators is homogeneous; i.e., 𝜌𝐴=𝜌𝐵ρA=ρB. While above the threshold value, there is an ordered structure. That is, one sublattice is mainly occupied by defectors and the other sublattice is occupied by cooperators. This orderly arrangement of cooperators and defectors can provide maximum total payoff in social dilemmas. Through mean-field approximation and Monte Carlo simulation, we associate the emergence of these ordered structures with the microscopic dynamics of the evolutionary process.