Coalition-Proof Correlated Equilibrium : A Definition

We refine the notion of correlated equilibrium capturing the essence of coalition-proof Nash equilibrium concept. We define coalition-proof correlated equilibrium of a game as a pair consisting of a correlation device and a coalition-proof Nash equilibrium of the game extended by the correlation device. A direct coalition-proof correlated equilibrium is a canonical device such that the obedient strategy is a coalition-proof Nash equilibrium of the canonical extended game. The "revelation principle" does not hold. Even in case of two-person games, there exists coalition-proof correlated equilibrium for which the corresponding induced distribution is not a direct coalition-proof correlated equilibrium. The set of direct coalition-proof correlated equilibria is not convex unlike the set of correlated equilibria. For any game, a (pure) coalition-proof Nash equilibrium is always a direct coalition-proof correlated equilibrium. We compare our notion with other existing concepts through several examples.