Percolation and cooperation with mobile agents: Geometric and strategy clusters

We study the conditions for persistent cooperation in an off-lattice model of mobile agents playing the Prisoner's Dilemma (PD) game with pure strategies: C-cooperators, D-defectors. Each agent has an exclusion radius r_P and an interaction radius r_int that defines the instantaneous contact network. The agents undergo random diffusion and strategy evolution follows the finite-population analog of replicator dynamics. We show that, differently from the r_P = 0 case, the model with finite sized agents presents a coexistence phase. Moreover, there are also two absorbing phases in which either C's or D's dominate. We provide a geometric interpretation of the transitions between phases and present a phase diagram of the PD dynamics as a function of both r_P and r_int. In analogy with lattice models, geometric percolation of the contact network enhances cooperation. More importantly, we show that the percolation of D's is an essential condition for their survival. Differently from compact clusters of C's, isolated groups of D's will eventually become extinct if not percolating, independently of their size. Our results are robust for a great range of mobilities and of the temptation parameter in the PD game.