The emergence of rational behavior in the presence of stochastic perturbations

TL;DR

This paper investigates how players adapt their strategies in repeated games with stochastic payoffs, showing that dominated strategies vanish and strict Nash equilibria remain stable under stochastic replicator dynamics.

Contribution

It introduces a stochastic version of replicator dynamics driven by exponential learning, highlighting stability properties in noisy environments, unlike traditional aggregate shocks models.

Findings

Dominated strategies become extinct regardless of perturbation size.

Strict Nash equilibria are stochastically asymptotically stable.

Results are illustrated through congestion game examples.

Abstract

We study repeated games where players use an exponential learning scheme in order to adapt to an ever-changing environment. If the game's payoffs are subject to random perturbations, this scheme leads to a new stochastic version of the replicator dynamics that is quite different from the "aggregate shocks" approach of evolutionary game theory. Irrespective of the perturbations' magnitude, we find that strategies which are dominated (even iteratively) eventually become extinct and that the game's strict Nash equilibria are stochastically asymptotically stable. We complement our analysis by illustrating these results in the case of congestion games.