If players are sparse social dilemmas are too: Importance of percolation for evolution of cooperation

TL;DR

This paper investigates how population density and graph percolation thresholds influence the evolution of cooperation in social dilemmas, revealing that optimal cooperation occurs near the percolation point and depends on the update rule used.

Contribution

It demonstrates the universal role of the percolation threshold in promoting cooperation across various social dilemmas and interaction graphs, highlighting the importance of population density.

Findings

Sparsity can promote cooperation near the percolation threshold.

Percolation threshold universally indicates optimal population density for cooperation.

Myopic updating diminishes the cooperation-promoting effect of sparsity.

Abstract

Spatial reciprocity is a well known tour de force of cooperation promotion. A thorough understanding of the effects of different population densities is therefore crucial. Here we study the evolution of cooperation in social dilemmas on different interaction graphs with a certain fraction of vacant nodes. We find that sparsity may favor the resolution of social dilemmas, especially if the population density is close to the percolation threshold of the underlying graph. Regardless of the type of the governing social dilemma as well as particularities of the interaction graph, we show that under pairwise imitation the percolation threshold is a universal indicator of how dense the occupancy ought to be for cooperation to be optimally promoted. We also demonstrate that myopic updating, due to the lack of efficient spread of information via imitation, renders the reported mechanism dysfunctional, which in turn further strengthens its foundations.