Spatially Inhomogeneous Evolutionary Games

TL;DR

This paper develops a mean-field model for spatially distributed evolutionary games, describing how strategies evolve and influence player movement, with rigorous proofs of existence, uniqueness, and convergence from finite to infinite agent systems.

Contribution

It introduces a novel mean-field framework for spatial evolutionary games, including a measure-based description and equivalence of Lagrangian and Eulerian formulations, with stability and convergence results.

Findings

Established existence and uniqueness of solutions.

Proved convergence of finite agent models to the mean-field limit.

Developed functional analytic tools for interaction dynamics in Banach spaces.

Abstract

We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represented by an evolving probability measure $\Sigma$ on an infinite dimensional state space (pairs $(x,\sigma)$ of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number $N$ of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces, that could be of independent interest.