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Probability logic for type spaces

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  • A. Heifetz
  • Ph. Mongin

Abstract

Using a formal propositional language with operators "individual i assigns probability at least a" for countable many a, we devise an axiom system which is sound and complete with respect to the class of type spaces in the sense of Harsanyi (1967-68). A crucial axiom requires that degrees of belief be compatible for any two sets of assertions which are equivalent in a suitably defined natural sense. The completeness proof relies on a theorem of the alternative from convex analysis, and uses the method of filtration by finite sub-languages.
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Suggested Citation

  • A. Heifetz & Ph. Mongin, 1998. "Probability logic for type spaces," THEMA Working Papers 98-25, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    • Handle: RePEc:ema:worpap:98-25
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    References listed on IDEAS

    as
    1. Samet, Dov, 2000. "Quantified Beliefs and Believed Quantities," Journal of Economic Theory, Elsevier, vol. 95(2), pages 169-185, December.
    2. Robert J. Aumann, 1999. "Interactive epistemology II: Probability," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 301-314.
    3. Dov Samet, 1997. "On the Triviality of High-Order Probabilistic Beliefs," Game Theory and Information 9705001, University Library of Munich, Germany.
    4. MONGIN , Philippe, 1993. "A Non-Minimal but Very Weak Axiomatization of Common Belief," LIDAM Discussion Papers CORE 1993046, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Citations

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    Cited by:

    1. Beißner, Patrick & Khan, M. Ali, 2019. "On Hurwicz–Nash equilibria of non-Bayesian games under incomplete information," Games and Economic Behavior, Elsevier, vol. 115(C), pages 470-490.
    2. Dietrich, Franz & Mongin, Philippe, 2010. "The premiss-based approach to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 562-582, March.
    3. Di Tillio, Alfredo & Halpern, Joseph Y. & Samet, Dov, 2014. "Conditional belief types," Games and Economic Behavior, Elsevier, vol. 87(C), pages 253-268.
    4. Shiri Alon & Aviad Heifetz, 2014. "The logic of Knightian games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(2), pages 161-182, October.
    5. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," HEC Research Papers Series 941, HEC Paris.
    6. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.
    7. Pintér, Miklós, 2011. "Common priors for generalized type spaces," MPRA Paper 34118, University Library of Munich, Germany.
    8. Jagau, Stephan & Perea, Andrés, 2022. "Common belief in rationality in psychological games," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    9. Pintér, Miklós, 2010. "The non-existence of a universal topological type space," Journal of Mathematical Economics, Elsevier, vol. 46(2), pages 223-229, March.
    10. Meier, Martin, 2008. "Universal knowledge-belief structures," Games and Economic Behavior, Elsevier, vol. 62(1), pages 53-66, January.
    11. Mikaël Cozic, 2016. "Probabilistic Unawareness," Games, MDPI, vol. 7(4), pages 1-24, November.
    12. Tsakas, Elias, 2014. "Rational belief hierarchies," Journal of Mathematical Economics, Elsevier, vol. 51(C), pages 121-127.
    13. Mikaël Cozic, 2016. "Probabilistic Unawareness," Post-Print hal-01950702, HAL.
    14. Jean Baccelli & Marcus Pivato, 2021. "Philippe Mongin (1950–2020)," Theory and Decision, Springer, vol. 90(1), pages 1-9, February.
    15. Pintér, Miklós & Udvari, Zsolt, 2011. "Generalized type spaces," MPRA Paper 34107, University Library of Munich, Germany.
    16. Philippe Mongin, 2011. "Judgment aggregation," Working Papers hal-00579346, HAL.
    17. Shmuel Zamir, 2008. "Bayesian games: Games with incomplete information," Discussion Paper Series dp486, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    18. MEIER, Martin, 2001. "An infinitary probability logic for type spaces," LIDAM Discussion Papers CORE 2001061, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    19. Feinberg, Yossi, 2005. "Subjective reasoning--dynamic games," Games and Economic Behavior, Elsevier, vol. 52(1), pages 54-93, July.
    20. Marc Fleurbaey, 2020. "Philippe Mongin 1950–2020," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 55(3), pages 399-403, October.
    21. Dekel, Eddie & Siniscalchi, Marciano, 2015. "Epistemic Game Theory," Handbook of Game Theory with Economic Applications,, Elsevier.

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    JEL classification:

    • C49 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Other

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