Uniform Topology on Types and Strategic Convergence∗

We study the continuity of the correspondence of interim ε-rationalizable actions in incomplete information games. We introduce a topology on types, called uniformweak topology, under which two types of a player are close if they have similar firstorder beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This notion of proximity of types is an extension of the concept of common p-belief due to Monderer and Samet (1989). We show that, given any finite game, every action that is interim rationalizable for a finite type t remains interim εrationalizable for all types sufficiently close to t in the uniform-weak topology. Conversely, given any finite type t there exist ε > 0 and a finite game such that some interim rationalizable action for t fails to be interim ε-rationalizable for every type that is not close to t in the uniform-weak topology. Our results thus establish the equivalence between the uniform-weak topology and the strategic topology of Dekel, Fudenberg, and Morris (2006) around finite types.