Uniform Topologies on Types

38 Pages Posted: 26 Oct 2009

See all articles by Yi-Chun Chen

Yi-Chun Chen

National University of Singapore (NUS) - Department of Economics

Alfredo Di Tillio

Bocconi University - Department of Economics

Eduardo Faingold

Insper Institute of Education and Research

Siyang Xiong

Rice University - Department of Economics

Date Written: October 26, 2009

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

JEL Classification: C70, C72

Suggested Citation

Chen, Yi-Chun and Di Tillio, Alfredo and Faingold, Eduardo and Xiong, Siyang, Uniform Topologies on Types (October 26, 2009). Cowles Foundation Discussion Paper No. 1734. Available at SSRN: https://ssrn.com/abstract=1494432 or http://dx.doi.org/10.2139/ssrn.1494432

Yi-Chun Chen

National University of Singapore (NUS) - Department of Economics ( email )

1 Arts Link AS2 #06-02
Singapore 117570, Singapore 119077
Singapore

Alfredo Di Tillio

Bocconi University - Department of Economics ( email )

Milan
Italy

HOME PAGE: http://mypage.unibocconi.eu/alfredoditillio

Eduardo Faingold (Contact Author)

Insper Institute of Education and Research ( email )

R Quata 300
Sao Paulo, 04542-030
Brazil

HOME PAGE: http://www.eduardofaingold.com

Siyang Xiong

Rice University - Department of Economics ( email )

6100 South Main Street
Houston, TX 77005
United States

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