Regular Potential Games

TL;DR

This paper demonstrates that almost all potential games are regular, meaning their equilibria are isolated and robust, and shows that such games typically have a finite, odd number of Nash equilibria, with implications for learning.

Contribution

It proves that nearly all potential games are regular, establishing their equilibria's robustness and simplicity, and derives an oddness property for the number of Nash equilibria.

Findings

Almost all potential games are regular.

In almost all potential games, the number of Nash equilibria is finite and odd.

Specialized results are provided for weighted, exact, and identical payoff potential games.

Abstract

A fundamental problem with the Nash equilibrium concept is the existence of certain "structurally deficient" equilibria that (i) lack fundamental robustness properties, and (ii) are difficult to analyze. The notion of a "regular" Nash equilibrium was introduced by Harsanyi. Such equilibria are isolated, highly robust, and relatively simple to analyze. A game is said to be regular if all equilibria in the game are regular. In this paper it is shown that almost all potential games are regular. That is, except for a closed subset with Lebesgue measure zero, all potential games are regular. As an immediate consequence of this, the paper also proves an oddness result for potential games: in almost all potential games, the number of Nash equilibrium strategies is finite and odd. Specialized results are given for weighted potential games, exact potential games, and games with identical payoffs. Applications of the results to game-theoretic learning are discussed.