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Conditional Beliefs and Higher-Order Preferences

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  • Lee, Byung Soo

Abstract

In this paper, we establish the Bayesian foundations of type structures in which beliefs are lexicographic probability systems (LPS’s)—such as those used in Brandenburger et al. (2008)—rather than standard probability measures as in Mertens and Zamir (1985). This is a setting which the distinction between preferences hierarchies (Epstein and Wang, 1996) and beliefs hierarchies is meaningful and the former has conceptual advantages. Type structures in which beliefs are conditional probability systems (CPS’s) are found to describe fewer hierarchies than LPS type structures can if a nonredundancy requirement is imposed. The two families of type structures are found to be capable of describing the same set of hierarchies in the absence of such a requirement. The existence of “largest”—a notion closely related to universality—LPS/CPS type structures is also shown. Finally, we find that some coherent hierarchies cannot be types but those hierarchies may be needed to express epistemic conditions for iterated elimination of weakly dominated strategies.

Suggested Citation

  • Lee, Byung Soo, 2013. "Conditional Beliefs and Higher-Order Preferences," MPRA Paper 48551, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:48551
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    File URL: https://mpra.ub.uni-muenchen.de/49738/8/MPRA_paper_49738.pdf
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    References listed on IDEAS

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    1. 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..
    2. Tan, Tommy Chin-Chiu & da Costa Werlang, Sergio Ribeiro, 1988. "The Bayesian foundations of solution concepts of games," Journal of Economic Theory, Elsevier, vol. 45(2), pages 370-391, August.
    3. Epstein, Larry G & Wang, Tan, 1996. ""Beliefs about Beliefs" without Probabilities," Econometrica, Econometric Society, vol. 64(6), pages 1343-1373, November.
    4. Lawrence Blume & Adam Brandenburger & Eddie Dekel, 2014. "Lexicographic Probabilities and Choice Under Uncertainty," World Scientific Book Chapters, in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 6, pages 137-160, World Scientific Publishing Co. Pte. Ltd..
    5. 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).
    6. Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155-179, January.
    7. Adam Brandenburger & Amanda Friedenberg & H. Jerome Keisler, 2014. "Admissibility in Games," World Scientific Book Chapters, in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 7, pages 161-212, World Scientific Publishing Co. Pte. Ltd..
    8. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    9. Battigalli, Pierpaolo & Siniscalchi, Marciano, 1999. "Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games," Journal of Economic Theory, Elsevier, vol. 88(1), pages 188-230, September.
    10. John C. Harsanyi, 1967. "Games with Incomplete Information Played by "Bayesian" Players, I-III Part I. The Basic Model," Management Science, INFORMS, vol. 14(3), pages 159-182, November.
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    Cited by:

    1. Ganguli, Jayant & Heifetz, Aviad & Lee, Byung Soo, 2016. "Universal interactive preferences," Journal of Economic Theory, Elsevier, vol. 162(C), pages 237-260.
    2. Catonini, Emiliano & De Vito, Nicodemo, 2024. "Cautious belief and iterated admissibility," Journal of Mathematical Economics, Elsevier, vol. 110(C).
    3. Petri, Henrik, 2020. "Lexicographic probabilities and robustness," Games and Economic Behavior, Elsevier, vol. 122(C), pages 426-439.
    4. Dekel, Eddie & Siniscalchi, Marciano, 2015. "Epistemic Game Theory," Handbook of Game Theory with Economic Applications,, Elsevier.
    5. Dekel, Eddie & Friedenberg, Amanda & Siniscalchi, Marciano, 2016. "Lexicographic beliefs and assumption," Journal of Economic Theory, Elsevier, vol. 163(C), pages 955-985.
    6. Lee, Byung Soo, 2016. "A space of lexicographic preferences," Journal of Mathematical Economics, Elsevier, vol. 65(C), pages 16-25.

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    More about this item

    Keywords

    Preferences hierarchies; type structure; weakly dominated strategies; epistemic game theory; lexicographic probability system; conditional probability system;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General

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