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Abstract

This paper constructs a universal type space on an arbitrary measurable space of states of nature, without any topological assumption. The construction is explicit: the universal type space is the set of belief hierarchies that satisfy a coherency condition. This condition is stronger than the standard one. It requires a belief hierarchy (consisting of finite-order beliefs) to extend to all countable orders of beliefs, with no two orders in conflict. A topological assumption delivers such an extension automatically, and without one, the extension may fail. Under the stronger coherency condition, the universal type space also coincides with the implicit construction: the set of belief hierarchies induced by some type in some type space. Hence, the stronger coherency condition unifies the two known constructions. Finally, the need to track countable orders of beliefs has a game-theoretic counterpart: iterated elimination of strictly dominated actions may take transfinitely many rounds.