On redundant types and Bayesian formulation of incomplete information

A type structure is non-redundant if no two types of a player represent the same hierarchy of beliefs over the given set of basic uncertainties, and it is redundant otherwise. Under a mild necessary and sufficient condition termed separativity, we show that any redundant structure can be identified with a non-redundant structure with an extended space of basic uncertainties. The belief hierarchies induced by the latter structure, when “marginalized,” coincide with those induced by the former. We argue that redundant structures can provide different Bayesian equilibrium predictions only because they reflect a richer set of uncertainties entertained by players but unspecified by the analyst. The analyst shall make use of a non-redundant structure, unless he believes that he misspecified the players' space of basic uncertainties. We also consider bounding the extra uncertainties by the action space for Bayesian equilibrium predictions.