COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1734 Uniform Topologies on Types Yi-Chun Chen, Alfredo Di Tillio, Eduardo Faingold and Siyang Xiong October 2009 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. |
