Mean-field interactions in evolutionary spatial games

TL;DR

This paper introduces a mean-field term into an evolutionary spatial game based on the Prisoner's dilemma, revealing new steady-state behaviors and phase transitions, and demonstrating how nonlocal interactions influence spatial chaos.

Contribution

The study extends the Nowak and May spatial game model by incorporating a self-consistent mean-field term, unveiling novel steady states and transition behaviors.

Findings

Mean-field term creates steady states with density depending on payoff.

Transitions shift from discontinuous jumps to derivative jumps.

Mean-field coupling induces spatial chaos near stationary states.

Abstract

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.