Time averages, recurrence and transience in the stochastic replicator dynamics

TL;DR

This paper studies the long-term behavior of stochastic replicator dynamics in game theory, establishing conditions for stability, transience, and the relationship between time averages and Nash equilibria.

Contribution

It introduces an averaging principle linking time averages to Nash equilibria and provides criteria for transience and stability in stochastic replicator processes.

Findings

Averaging principle relates long-term averages to Nash equilibria.

Conditions for transience based on mixed equilibria and payoff matrix.

Necessary and sufficient conditions for stability of pure equilibria.

Abstract

We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for transience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria.