The Frequency of Convergent Games under Best-Response Dynamics

TL;DR

This paper investigates how often random normal-form games with multiple players and strategies converge under best-response dynamics, revealing that convergence becomes rare as game size grows, with surprising prevalence of multiple equilibria in large 2-player games.

Contribution

It introduces a novel n-partite graph approach to analyze the frequency of convergence in normal-form games and extends understanding of equilibrium convergence in large games.

Findings

Frequency of convergent games approaches zero as players or strategies increase.

In large 2-player games with many strategies, multiple equilibria are more common than unique ones.

Convergence is predictable in simple cases but rare in complex, large games.

Abstract

Generating payoff matrices of normal-form games at random, we calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of $n$-player, $m$-strategy games. These are perfectly predictable as they must converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games goes to zero as the number of players or the number of strategies goes to infinity. In the $2$-player case, we show that for large games with at least $10$ strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an $n$-partite graph to describe games.