Basins of Attraction in Two-Player Random Ordinal Potential Games

TL;DR

This paper analyzes the distribution of basin sizes of pure Nash equilibria in two-player ordinal potential games with many actions, providing asymptotic expected basin sizes as the number of actions grows large.

Contribution

It offers the first asymptotic analysis of the expected basin of attraction sizes for pure Nash equilibria in large two-player ordinal potential games.

Findings

Expected basin size converges as number of actions increases

Distribution of basin sizes is characterized asymptotically

Pure Nash equilibria have quantifiable attraction basins in large games

Abstract

We consider the class of two-person ordinal potential games where each player has the same number of actions $K$. Each game in this class admits at least one pure Nash equilibrium and the best-response dynamics converges to one of these pure Nash equilibria; which one depends on the starting point. So, each pure Nash equilibrium has a basin of attraction. We pick uniformly at random one game from this class and we study the joint distribution of the sizes of the basins of attraction. We provide an asymptotic exact value for the expected basin of attraction of each pure Nash equilibrium, when the number of actions $K$ goes to infinity.