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Uniform topologies on types

  • Xiong, Siyang

    ()

    (Department of Economics, Rice University)

  • Chen, Yi-Chun

    ()

    (Department of Economics, National University of Singapore)

  • di Tillio, Alfredo

    ()

    (Department of Economics, Università Luigi Bocconi)

  • Faingold, Eduardo

    ()

    (Department of Economics, Yale University)

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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Article provided by Econometric Society in its journal Theoretical Economics.

Volume (Year): 5 (2010)
Issue (Month): 3 (September)
Pages:

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Handle: RePEc:the:publsh:462
Contact details of provider: Web page: http://econtheory.org

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  1. Eddie Dekel & Drew Fudenberg, 2006. "Topologies on Type," Discussion Papers 1417, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, vol. 59(1), pages 189-198, February.
  3. Jeffrey C Ely & Marcin Peski, 2008. "Critical Types," Levine's Working Paper Archive 122247000000001935, David K. Levine.
  4. Morris, Stephen & Dekel, Eddie & Fudenberg, Drew, 2007. "Interim Correlated Rationalizability," Scholarly Articles 3196333, Harvard University Department of Economics.
  5. 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.
  6. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
  7. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
  8. Executive Board, 2008. ""Topologies on types": Correction," Theoretical Economics, Econometric Society, vol. 3(2), June.
  9. 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.
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