When "Better" is better than "Best"

TL;DR

This paper demonstrates that in symmetric two-player games with random payoffs, better-response dynamics reliably find pure Nash equilibria, unlike best-response dynamics which often get stuck, highlighting a significant difference in convergence behavior.

Contribution

It reveals that better-response dynamics almost surely converge to pure Nash equilibria in symmetric games with random payoffs, unlike best-response dynamics which can fail to converge.

Findings

Better-response dynamics converge to pure Nash equilibria with high probability.

Best-response dynamics often fail to converge due to being trapped.

Symmetric games with i.i.d. payoffs exhibit these convergence properties.

Abstract

We consider two-player normal form games where each player has the same finite strategy set. The payoffs of each player are assumed to be i.i.d. random variables with a continuous distribution. We show that, with high probability, the better-response dynamics converge to pure Nash equilibrium whenever there is one, whereas best-response dynamics fails to converge, as it is trapped.