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Simple axioms for countably additive subjective probability

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  • Kopylov, Igor

Abstract

This paper refines Savage's theory of subjective probability for the case of countably additive beliefs. First, I replace his continuity axioms P6 and P7 with a simple modification of Arrow's (1970) Monotone Continuity. Second, I relax Savage's primitives: in my framework, the class of events need not be a [sigma]-algebra, and acts need not have finite or bounded range. By varying the domains of acts and events, I obtain a unique extension of preference that parallels Caratheodory's unique extension of probability measures. Aside from subjective expected utility, I characterize exponential time discounting in a setting with continuous time and an arbitrary consumption set.

Suggested Citation

  • Kopylov, Igor, 2010. "Simple axioms for countably additive subjective probability," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 867-876, September.
    • Handle: RePEc:eee:mateco:v:46:y:2010:i:5:p:867-876
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    References listed on IDEAS

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    1. Peter Wakker, 1993. "Savage's Axioms Usually Imply Violation of Strict Stochastic Dominance," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(2), pages 487-493.
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    Cited by:

    1. Stanca, Lorenzo, 2023. "Robust Bayesian choice," Mathematical Social Sciences, Elsevier, vol. 126(C), pages 94-106.
    2. Kopylov, Igor, 2016. "Canonical utility functions and continuous preference extensions," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 32-37.
    3. Thai Ha-Huy, 2019. "Savage's theorem with atoms," Documents de recherche 19-05, Centre d'Études des Politiques Économiques (EPEE), Université d'Evry Val d'Essonne.
    4. Harvey, Charles M. & Østerdal, Lars Peter, 2012. "Discounting models for outcomes over continuous time," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 284-294.
    5. Kopylov, Igor, 2010. "Unbounded probabilistic sophistication," Mathematical Social Sciences, Elsevier, vol. 60(2), pages 113-118, September.
    6. Lorenzo Bastianello & Vassili Vergopoulos, 2024. "Discounted Subjective Expected Utility in Continuous Time," Papers 2403.15319, arXiv.org.
    7. Webb, Craig S., 2016. "Continuous quasi-hyperbolic discounting," Journal of Mathematical Economics, Elsevier, vol. 64(C), pages 99-106.
    8. Izhakian, Yehuda, 2020. "A theoretical foundation of ambiguity measurement," Journal of Economic Theory, Elsevier, vol. 187(C).
    9. Burkhard C. Schipper, 2024. "Predicting the Unpredictable under Subjective Expected Utility," Papers 2403.01421, arXiv.org, revised Feb 2025.
    10. Stanca, Lorenzo, 2020. "A simplified approach to subjective expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 151-160.
    11. Mackenzie, Andrew, 2019. "A foundation for probabilistic beliefs with or without atoms," Theoretical Economics, Econometric Society, vol. 14(2), May.
    12. Izhakian, Yehuda, 2017. "Expected utility with uncertain probabilities theory," Journal of Mathematical Economics, Elsevier, vol. 69(C), pages 91-103.
    13. Takanori Adachi, 2023. "Hierarchical Structure of Uncertainty," Papers 2311.14219, arXiv.org, revised Sep 2024.
    14. Yehuda Izhakian & David Yermack & Jaime F. Zender, 2022. "Ambiguity and the Tradeoff Theory of Capital Structure," Management Science, INFORMS, vol. 68(6), pages 4090-4111, June.
    15. Pavlo Blavatskyy, 2020. "Expected discounted utility," Theory and Decision, Springer, vol. 88(2), pages 297-313, March.
    16. Marcus Pivato, 2021. "Intertemporal Choice with Continuity Constraints," Mathematics of Operations Research, INFORMS, vol. 46(3), pages 1203-1229, August.
    17. Alon, Shiri & Lehrer, Ehud, 2014. "Subjective multi-prior probability: A representation of a partial likelihood relation," Journal of Economic Theory, Elsevier, vol. 151(C), pages 476-492.
    18. Pivato, Marcus & Vergopoulos, Vassili, 2020. "Subjective expected utility with imperfect perception," Journal of Mathematical Economics, Elsevier, vol. 88(C), pages 104-122.
    19. Manel Baucells & Lin Zhao, 2020. "Everything in Moderation: Foundations and Applications of the Satiation Model," Management Science, INFORMS, vol. 66(12), pages 5701-5719, December.
    20. Han Bleichrodt & Umut Keskin & Kirsten I. M. Rohde & Vitalie Spinu & Peter Wakker, 2015. "Discounted Utility and Present Value—A Close Relation," Operations Research, INFORMS, vol. 63(6), pages 1420-1430, December.
    21. Craig S. Webb, 2019. "Trichotomic discounted utility," Theory and Decision, Springer, vol. 87(3), pages 321-339, October.
    22. Pivato, Marcus & Vergopoulos, Vassili, 2017. "Subjective expected utility representations for Savage preferences on topological spaces," MPRA Paper 77359, University Library of Munich, Germany.
    23. Craig S. Webb, 2026. "Separating Discounting from Changing Utility," Economics Discussion Paper Series 2601, Economics, The University of Manchester.
    24. Hara, Kazuhiro, 2016. "Characterization of stationary preferences in a continuous time framework," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 34-43.
    25. Spinu, Vitalie & Wakker, Peter P., 2013. "Expected utility without continuity: A comment on Delbaen et al. (2011)," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 28-30.

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