Best-response dynamics, playing sequences, and convergence to equilibrium in random games

TL;DR

This paper investigates how the order of player updates affects the convergence of best-response dynamics to Nash equilibria in large random games, showing that random sequences almost always lead to convergence, unlike fixed cyclic orders.

Contribution

It provides a comprehensive analysis of best-response dynamics in large random games, highlighting the critical role of playing sequence in convergence behavior.

Findings

Convergence is rare with fixed cyclic order in large games.

Random playing sequences almost always lead to convergence.

Sequence choice drastically affects dynamic outcomes in large games.

Abstract

We analyze the performance of the best-response dynamic across all normal-form games using a random games approach. The playing sequence -- the order in which players update their actions -- is essentially irrelevant in determining whether the dynamic converges to a Nash equilibrium in certain classes of games (e.g. in potential games) but, when evaluated across all possible games, convergence to equilibrium depends on the playing sequence in an extreme way. Our main asymptotic result shows that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all (large) games when players take turns according to a fixed cyclic order. By contrast, when the playing sequence is random, the dynamic converges to a pure Nash equilibrium if one exists in almost all (large) games.