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 game with pure, unconditional strategies. Each agent has an exclusion radius $r_P$, which accounts for the population viscosity, and an interaction radius $r_{\mathrm{int}}$, which defines the instantaneous contact network for the game dynamics. We show that, differently from the $r_P=0$ case, the model with finite-sized agents presents a coexistence phase with both cooperators and defectors, besides the two absorbing phases, in which either cooperators or defectors dominate. We provide, in addition, a geometric interpretation of the transitions between phases. 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.

Figure 1

Geometric parameters of the model, the hard-core radius $r_P$, and the interaction radius $r_{\mathrm{int}}$. While the former sets an exclusion area around each agent, the latter defines its contact network. The characteristic length $d$, as indicated at the right, is defined by dividing the available area, $L^2$, by the area of the square box around each of the $N$ particles of diameter $d$, that is, $L^2/d^2=N$.

Figure 2

Average fraction of cooperators as a function of time (in Monte Carlo steps). $N={32}^2$, $T=1.1$, $r_{\mathrm{int}}=\sqrt{3.5}d$, and $\mu =0.01$ for various area fractions $\varphi$. Below $\varphi \simeq 0.37$, cooperators eventually invade the whole system. As $\varphi$ approaches this threshold, the time spent close to the critical plateau at ${\rho }_{\mathrm{C}}\simeq 0.85$ also increases.

Figure 3

Phase diagram in the plane $r_{\mathrm{int}}/d$ and $\varphi$ for $N={32}^2$, $\mu =0.01$, and $T=1.1$. The color code is the average fraction of cooperators, ${\rho }_{\mathrm{C}}$. Besides the phases corresponding to the absorbing states, C (${\rho }_{\mathrm{C}}=f_{\mathrm{C}}=1$) and D (${\rho }_{\mathrm{C}}=f_{\mathrm{C}}=0$), there are two coexistence phases, C and D.

Figure 4

Probability of percolation of D clusters as a function of time, $P_{\mathrm{D}}\left(t\right)$, for several area fractions $\varphi$ and the same parameters as in Fig. 2. Inset: Asymptotic limit of $P_{\mathrm{D}}\left(t\right)$ as a function of $\varphi$. The linear fit indicates that the percolation probability goes to 0 at $\varphi \simeq 0.37$. The small deviation seen close to this point is due to the long time of convergence.

Figure 5

Fraction of initial states that go to the ${\rho }_{\mathrm{C}}=1$ absorbing state as a function of $r_{\mathrm{int}}/d$ for several system sizes and two area fractions, $\varphi =0$ and 0.28. For large values of $r_{\mathrm{int}}/d$, as $f_{\mathrm{C}}$ increases for larger system sizes, the existence of the ${\rho }_{\mathrm{C}}=0$ state for point particles ($\varphi =0$) is due to finite-size fluctuations. For $\varphi \simeq 0.28$ the behavior is the opposite and the transition becomes sharper when the system size increases.

Figure 6

Average asymptotic fraction of cooperators as a function of $r_{\mathrm{int}}/d$ for $\varphi \simeq 0.28$ and several values of $\mu$ (top), including $\mu =0$ (open symbols), and $T$ (bottom). In both cases, $N={32}^2$. Top, $T=1.1$; bottom, $\mu =0.01$.