Is Having a Unique Equilibrium Robust?

TL;DR

This paper examines the robustness of having a unique equilibrium in finite games, showing that uniqueness of correlated equilibrium is robust, unlike Nash equilibrium for more than two players, with implications for game stability.

Contribution

It establishes that the set of games with a unique correlated equilibrium is open and characterizes conditions for Nash equilibrium uniqueness and robustness.

Findings

Unique correlated equilibrium is an open set in finite games.

For n>2, Nash equilibrium uniqueness is not robust to payoff perturbations.

Generic two-player zero-sum games have a unique correlated equilibrium.

Abstract

We investigate whether having a unique equilibrium (or a given number of equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium and correlated equilibrium. We show that the set of n-player finite games with a unique correlated equilibrium is open, while this is not true of Nash equilibrium for n>2. The crucial lemma is that a unique correlated equilibrium is a quasi-strict Nash equilibrium. Related results are studied. For instance, we show that generic two-person zero-sum games have a unique correlated equilibrium and that, while the set of symmetric bimatrix games with a unique symmetric Nash equilibrium is not open, the set of symmetric bimatrix games with a unique and quasi-strict symmetric Nash equilibrium is.