Quantifying the impact of noise on macroscopic organization of cooperation in spatial games

Spatial games have been extensively studied in recent years. It is known that spatial structure could have a differing role in promoting cooperation, if comparing prisoner's dilemma to snowdrift game. Less is known about whether there also exists qualitative difference of the emerging macroscopic spatial pattern of cooperation in the two respective games, in particular when noise is present in individual strategy learning. To address this issue, we quantify and compare the impact of noise on spatial organization of cooperation in the two games. In our simulations, individuals are located in a spatial lattice with the von Neumann neighbourhood. They play games with their immediate neighbors and update their strategies based on a Fermi function-like rule, 1/[1+exp(ΔP)/K], where ΔP is the payoff difference between the focal individual and one randomly chosen neighbour, and K represents the noise level (temperature) in strategy learning. Small K values represent the tendency of imitating better performing individuals while high K values increase the likelihood that individuals adopt worse performing strategies. Our results reveal striking differences between the two types of games in regard to the resulting macroscopic pattern of cooperation. Assortment between cooperators is much weaker in the snowdrift game than in the prisoner's dilemma. Intermediate levels of noise maximize the local assortment in the prisoner's dilemma while most weakening assortment in the snowdrift game. The cluster size of cooperators has a peak in the prisoner's dilemma but shows a valley in the snowdrift game for varying level of noise. With increasing the noise level cooperators break into small clusters in the prisoner's dilemma while cooperators tend to be clustered into larger lumps in the snowdrift game. These quantitative results highlight the underlying microscopic interaction, whether being the prisoner's dilemma or the snowdrift type, nontrivially determines how noise affects the spatial pattern of cooperation.