Is Having a Unique Equilibrium Robust?

Le fichier accessible ci-dessous est une version également éditée dans les Cahiers de la Chaire "Les Particuliers face aux Risques" de l'Institut de Finance de Dauphine, cahier n° 28, décembre 2008We 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.ou