Approximate master equations for the spatial public goods game

Abstract

The spatial public goods game has been used to examine factors that promote cooperation. Owing to the complexity of the dynamics of this game, previous studies on this model neglected analytical approaches and relied entirely on numerical calculations using the Monte Carlo (MC) simulations. In this paper, we present the approximate master equations (AMEs) for this model. We report that the results obtained by the AMEs are mostly qualitatively consistent with those obtained by the MC simulations. Furthermore, we show that it is possible to obtain phase boundaries analytically in certain parameter regions. In the region where the noise in strategy decisions is very large, the phase boundary can be obtained analytically by considering perturbations from the steady state of the voter model. In the noiseless region, discontinuous phase transitions occur because of the characteristics of the function that represents strategy updating. Our approach is useful for clarifying the details of the mechanisms that promote cooperation and can be easily applied to other group interaction models.

Figures (5)

• Figure 1: Phase diagram with degree $k=4$. The solid lines and the color map are the results obtained by numerical calculations of the approximate master equations (AMEs) using the Euler method, with the time step $1/10$, computed up to $t=5\times 10^3$. We define $\rho^{\mathrm{C}} < 10^{-5}$ as the C phase (and the same for the D phase). The dotted lines indicate the phase boundary obtained from the Monte Carlo (MC) simulations. The number of nodes $N=10^5$, $5\times 10^3$MCS.
• Figure 2: Fraction of cooperators at $K=0$ (upper) and $K=1$ (lower). The orange lines and the blue lines (or marks) denote the results obtained by the AMEs and the MC simulations, respectively. The MCS were taken up to $5\times 10^3$ ($1.5\times 10^5$ in $K=0$ and $r > 5$), and the last $10^3$ MCS results were averaged. In addition, independent simulations were run $10$ times and averaged. The error bars represent the maximum and minimum values among all samples. At $K=0$, the variance is so small that the error bar overlaps the line of the average, and hence, this case is not shown.
• Figure 3: Time evolution of $\rho^{\mathrm{D}}$ in $r > 5$ and $K=0$, and the number of nodes $N=10^5$ (blue), $5 \times 10^4$ (orange), $10^4$ (green). The solid line has a slope of $-1$.
• Figure 4: Schematic showing the $\mathrm{C}_m$ node imitating the strategy of the $\mathrm{D}_{m'}$ node. The dotted enclosure represents the group containing the $\mathrm{C}_m$ node.
• Figure 5: Schematic of transitions in the AME.