Evolutionary games on the lattice: best-response dynamics

TL;DR

This paper investigates a stochastic spatial version of best-response evolutionary games on a lattice, revealing conditions for strategy stability and the influence of spatial structure on evolutionary outcomes.

Contribution

It introduces a spatial model for best-response dynamics, analyzing stability conditions and comparing spatial and non-spatial cases using coupling with bootstrap percolation.

Findings

A strategy is stable if and only if it is selfish in the non-spatial mean-field approximation.

The spatial and non-spatial models agree when at least one strategy is altruistic.

Only the most selfish strategy remains stable when both strategies are selfish in any spatial dimension.

Abstract

The best-response dynamics is an example of an evolutionary game where players update their strategy in order to maximize their payoff. The main objective of this paper is to study a stochastic spatial version of this game based on the framework of interacting particle systems in which players are located on an infinite square lattice. In the presence of two strategies, and calling a strategy selfish or altruistic depending on a certain ordering of the coefficients of the underlying payoff matrix, a simple analysis of the non-spatial mean-field approximation of the spatial model shows that a strategy is evolutionary stable if and only if it is selfish, making the system bistable when both strategies are selfish. The spatial and non-spatial models agree when at least one strategy is altruistic. In contrast, we prove that, in the presence of two selfish strategies and in any spatial dimensions, only the most selfish strategy remains evolutionary stable. The main ingredients of the proof are monotonicity results and a coupling between the best-response dynamics properly rescaled in space with bootstrap percolation to compare the infinite time limits of both systems.