Non-exchangeable evolutionary and mean field games and their applications

TL;DR

This paper develops a replicator dynamic for non-exchangeable agents in continuous action spaces, linking it to mean field games and applying it to models like the q-voter, with computational examples illustrating the theory.

Contribution

It introduces a well-posed replicator dynamic for non-exchangeable agents and connects it to stationary mean field games, expanding the analysis of heterogeneous evolutionary games.

Findings

Replicator dynamic formulated and proven well-posed in probability measure space.

Connection established between replicator dynamics and stationary mean field games.

Computational examples demonstrate applications to tragedy of the commons and q-voter models.

Abstract

A replicator dynamic for non-exchangeable agents in a continuous action space is formulated and its well-posedness is proven in a space of probability measures. The non-exchangeability allows for the analysis of evolutionary games involving agents with distinct (and possibly infinitely many) types. We also explicitly connect this replicator dynamic to a stationary mean field game, which determines the pairwise actions of the heterogeneous agents. Moreover, as a byproduct of our theoretical results, we show that a class of nonlinear voter models, recently the subject of increasing interest, called q-voter models, can be viewed as a replicator dynamic driven by a utility that is a power of the probability density. This implies that non-exchangeable and/or mean-field game formulations of these models can also be constructed. We also present computational examples of evolutionary and mean field game models using a finite difference method, focusing on tragedy of the commons and the q-voter model with non-exchangeable agents, of which are interesting cases from theoretical and computational perspectives.