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Uniform Topologies on Types

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.

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File URL: http://cowles.econ.yale.edu/P/cd/d17a/d1734.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1734.

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Length: 37 pages
Date of creation: Oct 2009
Date of revision:
Handle: RePEc:cwl:cwldpp:1734
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Web page: http://cowles.econ.yale.edu/
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Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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  1. Eddie Dekel & Drew Fudenberg & Stephen Morris, 2005. "Topologies on Types," Harvard Institute of Economic Research Working Papers 2093, Harvard - Institute of Economic Research.
  2. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
  3. Jeffrey C Ely & Marcin Peski, 2008. "Critical Types," Levine's Working Paper Archive 122247000000001935, David K. Levine.
  4. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, vol. 59(1), pages 189-198, February.
  5. Executive Board, 2008. ""Topologies on types": Correction," Theoretical Economics, Econometric Society, vol. 3(2), June.
  6. Atsushi Kajii & Stephen Morris, 1997. "Refinements and Social Order Beliefs: A Unified Survey," Discussion Papers 1197, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  7. Dekel, Eddie & Fudenberg, Drew & Morris, Stephen, 2007. "Interim correlated rationalizability," Theoretical Economics, Econometric Society, vol. 2(1), pages 15-40, March.
  8. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
  9. Jonathan Weinstein & Muhamet Yildiz, 2007. "A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements," Econometrica, Econometric Society, vol. 75(2), pages 365-400, 03.
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