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A foundation for probabilistic beliefs with or without atoms

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  • Mackenzie, Andrew

    (Department of Economics, Maastricht University)

Abstract

We propose two novel axioms for qualitative probability spaces (Bernstein, 1917; de Finetti, 1937; Koopman, 1940; Savage, 1954): (i) unlikely atoms, which requires that there is an event containing no atoms that is at least as likely as its complement; and (ii) third-order atom-swarming, which requires that for each atom, there is a countable pairwise-disjoint collection of less-likely events that can be partitioned into three groups, each with union at least as likely as the given atom. We prove that under monotone continuity (Villegas, 1964; Arrow, 1970), each of these axioms is sufficient to guarantee a unique countably additive probability measure representation, generalizing Villegas (1964) to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.

Suggested Citation

  • Mackenzie, Andrew, 2019. "A foundation for probabilistic beliefs with or without atoms," Theoretical Economics, Econometric Society, vol. 14(2), May.
    • Handle: RePEc:the:publsh:2427
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    References listed on IDEAS

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    Cited by:

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    2. Giulio Principi & Peter P. Wakker & Ruodu Wang, 2023. "Antimonotonicity for Preference Axioms: The Natural Counterpart to Comonotonicity," Papers 2307.08542, arXiv.org.
    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. Baillon, Aurélien & Bleichrodt, Han & Li, Chen & Wakker, Peter P., 2021. "Belief hedges: Measuring ambiguity for all events and all models," Journal of Economic Theory, Elsevier, vol. 198(C).
    5. Giacomo Bonanno & Elias Tsakas, 2017. "Qualitative analysis of common belief of rationality in strategic-form games," Working Papers 175, University of California, Davis, Department of Economics.
    6. Stanca, Lorenzo, 2020. "A simplified approach to subjective expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 151-160.
    7. Andrew Mackenzie, 2021. "On atom-swarming and Luce’s theorem for probabilistic beliefs," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(1), pages 67-74, April.
    8. Giacomo Bonanno & Elias Tsakas, 2017. "Qualitative analysis of common belief of rationality in strategic-form games," Working Papers 181, University of California, Davis, Department of Economics.
    9. Abdellaoui, Mohammed & Wakker, Peter P., 2020. "Savage for dummies and experts," Journal of Economic Theory, Elsevier, vol. 186(C).

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    More about this item

    Keywords

    Beliefs; qualitative probability; unlikely atoms; atom-swarming;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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