Approximate master equations for the spatial public goods game

TL;DR

This paper introduces approximate master equations for the spatial public goods game, providing analytical insights into phase boundaries and cooperation mechanisms, complementing previous numerical studies.

Contribution

It develops an analytical framework using AMEs for the spatial public goods game, enabling phase boundary analysis and understanding of cooperation dynamics.

Findings

AME results align qualitatively with Monte Carlo simulations.

Analytical phase boundaries are derived in certain parameter regions.

Discontinuous phase transitions occur in noiseless regions.

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.