Hierarchies of beliefs and the belief-invariant Bayesian solution

The belief-invariant Bayesian solution is a notion of correlated equilibrium in games with incomplete information proposed by Forges (1993), and hierarchy of beliefs over conditional beliefs is introduced by Ely and Peski (2006) in their study of interim independent rationalizability. We study the connection between the two concepts. We partially characterize the correlations embedded among type spaces with the same set of hierarchies of beliefs over conditional beliefs with partially correlating devices, which send correlated signals to players in a way that preserves each player's belief about others' types. Since the belief-invariant Bayesian solution is also implemented by such correlating devices, we then establish that it is invariant on equivalent type space.