Beliefs Revealed in Bayesian Equilibrium

Harsanyi in a series of papers (cf. [Harsanyi, 1967–1968]) introduced the Bayesian games framework for analyzing games with incomplete information. Part of its novelty consisted in replacing decision theoretically well motivated but unwieldy hierarchy of beliefs with a simple device, a type space, as a representation of incomplete information. The other half was the introduction of BE as the solution concept. The resulting framework—tandem of BE and type spaces—paved the way for the bulk of the contemporary literature on auctions, bargaining, insurance, principal-agent, moral hazard, rational expectations, repeated games, reputation, signaling etc.
However, type spaces and belief hierarchies are not interchangeable. Two states in two type spaces that formally correspond to the same hierarchy may support different action profiles in BE (see Bergmann and Morris [Bergmann, Morris, 2005], Battigalli and Siniscalchi [2003]). They model different environments, give different predictions of rational behavior under the very solution concept that completed Harsanyi’s framework. Belief hierarchies are not expressive enough for the BE solution concep