Mean-field interactions in evolutionary spatial games

We introduce a mean-field term to an evolutionary spatial game model. Namely, we consider the game of Nowak and May, based on the Prisoner's dilemma, and augment the game rules by a self-consistent mean-field term. This way, an agent operates based on local information from its neighbors and nonlocal information via the mean-field coupling. We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term: while for the game of Nowak and May, steady states are characterized by a constant mean density of cooperators, the mean-field game contains steady states with a continuous dependence of the density on the payoff parameter. Moreover, the mean-field term changes the nature of transitions from discontinuous jumps in the steady-state density to jumps in the first derivative. The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos. Our approach can be readily generalized to a broad class of spatial evolutionary games with deterministic and stochastic decision rules.

Figure 1

(a) Steady-state mean density of cooperators $\langle f_c\rangle$ for the NM game (open blue circles) and MF game (full red squares) as a function of the payoff parameter $b$. Errorbars are shown for MF game at all points and are typically less than symbol size. We also show hyperbolas, Eq. (8) for $m,n=(8,5)$ (green dashed line) and $m,n=(5,3)$ (cyan dash-dotted line). In these simulations, $f_0=0.9$, $L=400$, $\tau ={10}^4$, and $T=1.6\times {10}^4$. Inset shows the zoom-in of the region $1.51<b<1.66$. (b) Steady-state persistence, $P(\tau ,\infty )$, for the NM game (open blue circles) and the MF game (red squares). We also show $P(0,\infty )$ for the MF game (yellow crosses). See text for discussion.

Figure 2

Typical snapshots of the steady-state configurations of players for the MF game. Here we show $100\times 100$ part of the field with $L=400$, and values of $b=1.2$ (a), 1.6 (b), 1.62 (c), and 1.63 (d). Blue (light grey) color corresponds to the cooperator state $\mathcal{C}$, red (dark grey) color is the defector state $\mathcal{D}$, green color is a $\mathcal{C}$ which was a $\mathcal{D}$ at the previous time step, and yellow is a $\mathcal{D}$ which was a $\mathcal{C}$.

Figure 3

Distribution of the cluster sizes (i.e., the number of agents in a cluster) of $\mathcal{D}$ (red) and $\mathcal{C}$ (blue) in the steady states for $L=200$, and values of $b=1.2$ (a), 1.55 (b), 1.59 (c), and 1.63 (d). We only show clusters of up to 250 agents.