Uniform topologies on types

• 成果类型: Article
• 署名作者: Chen, Yi-Chun; Di Tillio, Alfredo; Faingold, Eduardo; Xiong, Siyang
• 署名单位: National University of Singapore; Bocconi University; Bocconi University; Yale University; Rice University
• 刊物名称: THEORETICAL ECONOMICS
• ISSN/ISSBN: 1933-6837
• DOI: 10.3982/TE462
• 发表日期: 2010-09-01
• 页码: 445-478
• 关键词: Rationalizability; incomplete information; higher-order beliefs; strategic topology; Electronic Mail game

摘要

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 et al. 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.