Strong robustness to incomplete information and the uniqueness of a correlated equilibrium

We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game $$\mathbf{u}$$ u is strongly robust if, given that each player knows that his payoffs are those in $$\mathbf{u}$$ u with high probability, all Bayesian–Nash equilibria in the corresponding incomplete-information game are close—in terms of action distribution—to that equilibrium of $$\mathbf{u}$$ u . We prove, under some continuity requirements on payoffs, that a Nash equilibrium is strongly robust if and only if it is the unique correlated equilibrium. We then review and extend the conditions that guarantee the existence of a unique correlated equilibrium in games with a continuum of actions. The existence of a strongly robust Nash equilibrium is thereby established for several domains of games, including those that arise in economic environments as diverse as Tullock contests, all-pay auctions, Cournot and Bertrand competitions, network games, patent races, voting problems and location games.