We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar ﬁrst-order 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 topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet (1989)). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris (2006)). Moreover, when the limit is a ﬁnite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of ﬁnite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.
Chen, Yi-Chun; Di Tillio, Alfredo; Faingold, Eduardo; and Xiong, Siyang, "Uniform Topologies on Types" (2009). Cowles Foundation Discussion Papers. 2058.