On the Exponential Rate of Convergence of Fictitious Play in Potential Games

TL;DR

This paper demonstrates that in regular potential games, fictitious play converges exponentially fast to Nash equilibria from almost all initial conditions, highlighting a rapid convergence property in these strategic settings.

Contribution

It proves that in regular potential games, fictitious play exhibits exponential convergence to Nash equilibria from almost every initial condition.

Findings

Fictitious play converges exponentially in regular potential games.

Almost all potential games are regular, making the results widely applicable.

Convergence occurs from almost every initial condition.

Abstract

The paper studies fictitious play (FP) learning dynamics in continuous time. It is shown that in almost every potential game, and for almost every initial condition, the rate of convergence of FP is exponential. In particular, the paper focuses on studying the behavior of FP in potential games in which all equilibria of the game are regular, as introduced by Harsanyi. Such games are referred to as regular potential games. Recently it has been shown that almost all potential games (in the sense of the Lebesgue measure) are regular. In this paper it is shown that in any regular potential game (and hence, in almost every potential game), FP converges to the set of Nash equilibria at an exponential rate from almost every initial condition.