Uniform Topologies on Types

Yi-Chun Chen (), Alfredo Di Tillio, Eduardo Faingold () and Siyang Xiong
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Alfredo Di Tillio: IGIR and Dept. of Economics, Universita Luigi Bocconi
Eduardo Faingold: Cowles Foundation, Yale University, http://cowles.econ.yale.edu/faculty/faingold.htm
Siyang Xiong: Dept. of Economics, Rice University

No 1734, Cowles Foundation Discussion Papers from Cowles Foundation, Yale University

Abstract: 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.

Keywords: Rationalizability; Incomplete information; Higher-order beliefs; Strategic topology; Electronic mail game (search for similar items in EconPapers)
JEL-codes: C70 C72 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-gth
Date: 2009-10

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