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A Homeomorphism Theorem for the Universal Type Space with the Uniform Topology

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  • Martin Hellwig

    (Max Planck Institute for Research on Collective Goods)

Abstract

The paper shows that the Mertens-Zamir (1985) reconciliation of belief hierarchy and type space models of incomplete information is robust to the requirement that the topology on belief hierarchies reflect the continuity properties of strategic behaviour, taking account of the fact that beliefs of arbitrarily high orders in agents.belief hierarchies can have a signi.cant impact on strategic behaviour. When endowed with one of the .ner topologies proposed by Fudenberg et al. (2006) and Chen et al. (2010, 2017), the space of belief hierarchies is still homeomorphic to the space of probability measures (beliefs) over exogenous data and other agents.belief hierarchies. The canonical mapping from nonredundant abstract type spaces with continuous belief functions to the space of belief hierarchies is an embedding if the range of belief functions has the topology of convergence in total variation.

Suggested Citation

  • Martin Hellwig, 2016. "A Homeomorphism Theorem for the Universal Type Space with the Uniform Topology," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2016_17, Max Planck Institute for Research on Collective Goods, revised Jul 2022.
    • Handle: RePEc:mpg:wpaper:2016_17
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    File URL: http://www.coll.mpg.de/pdf_dat/2016_17online.pdf
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    References listed on IDEAS

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    1. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
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    3. 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.
    4. MERTENS, Jean-François & ZAMIR, Shmuel, 1985. "Formulation of Bayesian analysis for games with incomplete information," LIDAM Reprints CORE 608, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-391, June.
    6. Heifetz, Aviad, 1993. "The Bayesian Formulation of Incomplete Information--The Non-compact Case," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(4), pages 329-338.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    incomplete information; universal type space; uniform weak topology; uniform strategic topology; homeomorphism theorem;
    All these keywords.

    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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