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Complete Conditional Type Structures

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  • Nicodemo De Vito

Abstract

Hierarchies of conditional beliefs (Battigalli and Siniscalchi 1999) play a central role for the epistemic analysis of solution concepts in sequential games. They are modelled by type structures, which allow the analyst to represent the players' hierarchies without specifying an infinite sequence of conditional beliefs. Here, we study type structures that satisfy a "richness" property, called completeness. This property is defined on the type structure alone, without explicit reference to hierarchies of beliefs or other type structures. We provide sufficient conditions under which a complete type structure represents all hierarchies of conditional beliefs. In particular, we present an extension of the main result in Friedenberg (2010) to type structures with conditional beliefs. KEYWORDS: Conditional probability systems, hierarchies of beliefs, type structures, completeness, terminality. JEL: C72, D80

Suggested Citation

  • Nicodemo De Vito, 2023. "Complete Conditional Type Structures," Papers 2305.08940, arXiv.org, revised Dec 2023.
    • Handle: RePEc:arx:papers:2305.08940
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    References listed on IDEAS

    as
    1. Heifetz, Aviad & Samet, Dov, 1999. "Coherent beliefs are not always types," Journal of Mathematical Economics, Elsevier, vol. 32(4), pages 475-488, December.
    2. Battigalli, Pierpaolo & De Vito, Nicodemo, 2021. "Beliefs, plans, and perceived intentions in dynamic games," Journal of Economic Theory, Elsevier, vol. 195(C).
    3. Amanda Friedenberg & Martin Meier, 2011. "On the relationship between hierarchy and type morphisms," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 377-399, April.
    4. 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.
    5. 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..
    6. 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.
    7. Fukuda, Satoshi, 2024. "The existence of universal qualitative belief spaces," Journal of Economic Theory, Elsevier, vol. 216(C).
    8. Amanda Friedenberg & Martin Meier, 2011. "On the relationship between hierarchy and type morphisms," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 46(3), pages 377-399, April.
    9. Pierpaolo Battigalli & Pietro Tebaldi, 2019. "Interactive epistemology in simple dynamic games with a continuum of strategies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(3), pages 737-763, October.
    10. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
    11. Friedenberg, Amanda, 2010. "When do type structures contain all hierarchies of beliefs?," Games and Economic Behavior, Elsevier, vol. 68(1), pages 108-129, January.
    12. 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..
    13. Marciano Siniscalchi, 2022. "Structural Rationality in Dynamic Games," Econometrica, Econometric Society, vol. 90(5), pages 2437-2469, September.
    14. 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).
    15. Amanda Friedenberg & H. Jerome Keisler, 2021. "Iterated dominance revisited," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 72(2), pages 377-421, September.
    16. Amanda Friedenberg & Martin Meier, 2017. "The context of the game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 347-386, February.
    17. Battigalli, Pierpaolo & Siniscalchi, Marciano, 2002. "Strong Belief and Forward Induction Reasoning," Journal of Economic Theory, Elsevier, vol. 106(2), pages 356-391, October.
    18. Dekel, Eddie & Siniscalchi, Marciano, 2015. "Epistemic Game Theory," Handbook of Game Theory with Economic Applications,, Elsevier.
    19. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, March.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    conditional probability systems; hierarchies of beliefs; type structures; completeness; terminality. jel: c72; d80;
    All these keywords.

    JEL classification:

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

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