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Uniform Topologies on Types

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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.

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  • Yi-Chun Chen & Alfredo Di Tillio & Eduardo Faingold & Siyang Xiong, 2009. "Uniform Topologies on Types," Cowles Foundation Discussion Papers 1734, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1734
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d17/d1734.pdf
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    1. 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.
    2. Jeffrey C. Ely & Marcin Pęski, 2011. "Critical Types," Review of Economic Studies, Oxford University Press, vol. 78(3), pages 907-937.
    3. Dekel, Eddie & Fudenberg, Drew & Morris, Stephen, 2006. "Topologies on types," Theoretical Economics, Econometric Society, vol. 1(3), pages 275-309, September.
    4. Adam Brandenburger & Eddie Dekel, 2014. "Hierarchies of Beliefs and Common Knowledge," World Scientific Book Chapters,in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 2, pages 31-41 World Scientific Publishing Co. Pte. Ltd..
    5. Dekel, Eddie & Fudenberg, Drew & Morris, Stephen, 2007. "Interim correlated rationalizability," Theoretical Economics, Econometric Society, vol. 2(1), pages 15-40, March.
    6. 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, March.
    7. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    8. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
    9. Executive Board, 2008. ""Topologies on types": Correction," Theoretical Economics, Econometric Society, vol. 3(2), June.
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    Cited by:

    1. Yi-Chun Chen & Alfredo Di Tillio & Eduardo Faingold & Siyang Xiong, 2012. "The Strategic Impact of Higher-Order Beliefs," Cowles Foundation Discussion Papers 1875, Cowles Foundation for Research in Economics, Yale University.
    2. Heifetz, Aviad & Kets, Willemien, 2018. "Robust multiplicity with a grain of naiveté," Theoretical Economics, Econometric Society, vol. 13(1), January.
    3. Heinsalu, Sander, 2014. "Universal type structures with unawareness," Games and Economic Behavior, Elsevier, vol. 83(C), pages 255-266.
    4. Qin, Cheng-Zhong & Yang, Chun-Lei, 2009. "An Explicit Approach to Modeling Finite-Order Type Spaces and Applications," University of California at Santa Barbara, Economics Working Paper Series qt8hq7j89k, Department of Economics, UC Santa Barbara.
    5. Kets, Willemien, 2011. "Robustness of equilibria in anonymous local games," Journal of Economic Theory, Elsevier, vol. 146(1), pages 300-325, January.
    6. Battigalli Pierpaolo & Di Tillio Alfredo & Grillo Edoardo & Penta Antonio, 2011. "Interactive Epistemology and Solution Concepts for Games with Asymmetric Information," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 11(1), pages 1-40, March.
    7. repec:oup:restud:v:84:y:2017:i:4:p:1424-1471. is not listed on IDEAS
    8. Qin, Cheng-Zhong & Yang, Chun-Lei, 2013. "Finite-order type spaces and applications," Journal of Economic Theory, Elsevier, vol. 148(2), pages 689-719.
    9. Oury, Marion, 2015. "Continuous implementation with local payoff uncertainty," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 656-677.
    10. Strzalecki, Tomasz, 2014. "Depth of reasoning and higher order beliefs," Journal of Economic Behavior & Organization, Elsevier, vol. 108(C), pages 108-122.
    11. Chen, Yi-Chun & Xiong, Siyang, 2013. "The e-mail game phenomenon," Games and Economic Behavior, Elsevier, vol. 80(C), pages 147-156.
    12. Alia Gizatulina & Martin Hellwig, 2015. "The Genericity of the McAfee-Reny Condition for Full Surplus Extraction in Models with a Continuum of Types," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2015_08, Max Planck Institute for Research on Collective Goods.
    13. repec:eee:jetheo:v:170:y:2017:i:c:p:385-416 is not listed on IDEAS

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    Keywords

    Rationalizability; Incomplete information; Higher-order beliefs; Strategic topology; Electronic mail game;

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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