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The relation between degrees of belief and binary beliefs: A general impossibility theorem

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Abstract

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take.

Suggested Citation

  • Franz Dietrich & Christian List, 2019. "The relation between degrees of belief and binary beliefs: A general impossibility theorem," Documents de travail du Centre d'Economie de la Sorbonne 19001, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  • Handle: RePEc:mse:cesdoc:19001
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    References listed on IDEAS

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    1. Franz Dietrich & Christian List, 2018. "From degrees of belief to binary beliefs: Lessons from judgment-aggregation theory," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01744085, HAL.
    2. Franz Dietrich & Christian List, 2007. "Arrow’s theorem in judgment aggregation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(1), pages 19-33, July.
    3. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations with abstentions," Journal of Economic Theory, Elsevier, vol. 145(2), pages 544-561, March.
    4. Nehring, Klaus & Puppe, Clemens, 2010. "Abstract Arrowian aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 467-494, March.
    5. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations," Journal of Economic Theory, Elsevier, vol. 145(2), pages 495-511, March.
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    Cited by:

    1. Franz Dietrich, 2021. "Categorical versus graded beliefs," Documents de travail du Centre d'Economie de la Sorbonne 21032, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.

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    More about this item

    Keywords

    binary beliefs (yes/no); subjective probabilities; construction of binary beliefs from subjective probabilities; impossibility theorem;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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