On the Approximation Performance of Fictitious Play in Finite Games

TL;DR

This paper analyzes the limitations of Fictitious Play in approximating Nash equilibria in finite 2-player games, demonstrating it cannot guarantee better than a 1/2 approximation in worst-case scenarios.

Contribution

The paper constructs specific game examples showing Fictitious Play's approximation guarantee cannot surpass 1/2, establishing fundamental limitations of this heuristic.

Findings

Fictitious Play fails to achieve better than 1/2 approximation in certain 2-player games.

Worst-case strategies can fall short of best responses by approximately 1/2.

Theoretical bounds match the constructed examples, confirming the limitations.

Abstract

We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a 2-player game. We exhibit a class of 2-player games having payoffs in the range [0,1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1/2. Our construction shows that for n times n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1/2 - O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially matching upper bound of 1/2 - O(1/n).