Metastability in Stochastic Replicator Dynamics

TL;DR

This paper introduces a new stochastic replicator dynamics model for potential games, analyzing metastability, mean exit times, and quasi-stationary measures, with applications to graph-based replicator dynamics supported by numerical experiments.

Contribution

It presents a novel stochastic replicator model that is ill-posed but admits a natural selection principle, enabling analysis of metastable states and validation through numerical experiments.

Findings

Analysis of small noise asymptotics for mean exit times

Characterization of quasi-stationary measures in the model

Numerical experiments support theoretical predictions

Abstract

We consider a novel model of stochastic replicator dynamics for potential games that converts to a Langevin equation on a sphere after a change of variables. This is distinct from the models studied earlier. In particular, it is ill-posed due to non-uniqueness of solutions, but is amenable to a natural selection principle that picks a unique solution. The model allows us to make specific statements regarding metastable states such as small noise asymptotics for mean exit times from their domain of attraction, and quasi-stationary measures. We illustrate the general results by specializing them to replicator dynamics on graphs and demonstrate that the numerical experiments support theoretical predictions.