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, unconditional strategies (C: cooperate, D: defect)$\,$[1]. Each agent has an exclusion radius $r_P$ that accounts for the population viscosity, and an interaction radius $r_{int}$ that defines the instantaneous contact network for the game dynamics. The agents undergo random diffusion and the strategy evolution follows the finite-population analog of the replicator dynamics. We show that, differently from the $r_P = 0$ case (pointlike agents), the model with finite sized agents presents a coexistence phase with both cooperators and defectors. Moreover, there are also two absorbing phases in which either cooperators or defectors dominate. We provide, in addition, a geometric interpretation of the transitions between phases and present a phase diagram of the PD dynamics as a function of both parameters, $r_P$ and $r_{int}$. To determine the phases, we performed a finite-size analysis and studied the probability of percolation of D clusters as a function of time. In analogy with lattice models, the geometric percolation of the contact network (i.e., irrespective of the strategy) enhances cooperation. More importantly, we show that the percolation of defectors is an essential condition for their survival. Differently from compact clusters of cooperators, isolated groups of defectors 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.

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[1] M. H. Vainstein, C. Brito and J. J. Arenzon, \emph{Phys. Rev. E.}, {\textbf 90}, 022132 (2014).