Deterministic Equations for Stochastic Spatial Evolutionary Games

TL;DR

This paper derives integro-differential equations as deterministic models for spatial evolutionary games, enabling analysis of complex spatial phenomena like waves and pattern formation in strategy evolution.

Contribution

It introduces a generalization of mean-field equations to include spatial effects, providing a new analytical tool for population dynamics in evolutionary games.

Findings

Identification of standing and traveling waves

Analysis of pattern formation in strategy profiles

Extension of mean-field models to spatial contexts

Abstract

Spatial evolutionary games model individuals who are distributed in a spatial domain and update their strategies upon playing a normal form game with their neighbors. We derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. This generalizes the known mean-field ordinary differential equations and provide a powerful tool to investigate the spatial effects in populations evolution. The deterministic equations allow to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions.