Cooperation percolation in spatial prisoner's dilemma game

TL;DR

This paper investigates the formation of large cooperator clusters in spatial prisoner's dilemma games, identifying phase transitions and universality classes that determine when cooperation prevails.

Contribution

It introduces the concept of cooperation percolation, revealing phase transition conditions and universality classes based on the temptation to defect parameter.

Findings

A giant cooperator cluster emerges when initial cooperators exceed a threshold.

Phase transition types depend on the temptation to defect, with different universality classes.

The study links percolation theory to the dynamics of cooperation in spatial games.

Abstract

The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that cooperator clusters in spatial games are effective to protect cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that assures the prevalence of cooperation in a system. Here, we study the dynamical organization of cooperator clusters in spatial prisoner's dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized by the emergence of a large spanning cooperator cluster occurs when the initial fraction of cooperators exceeds a certain threshold. Interestingly, the phase transition belongs to different universality classes of percolation determined by the temptation to defect $b$. Specifically, on square lattices, $1<b<4/3$ leads to a phase transition pertaining to the class of regular site percolation, whereas $3/2<b<2$ gives rise to a phase transition subject to invasion percolation with trapping. Our findings offer deeper understanding of the cooperative behaviors in nature and society.