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Intuitive probability of non-intuitive events

Author

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  • Aase, Knut K.

    (Dept. of Business and Management Science, Norwegian School of Economics)

Abstract

Quantitative probability in the subjective theory is assumed to be finitely additive and defined on all the subsets of an underlying state space. Functions from this space into an Euclidian n-space create a new probability space for each such function. We point out that the associated probability measures, induced by the subjective probability, on these new spaces can not be finitely additive and defined on all the subsets of Euclidian n-space, for n ≥ 3. This is a consequence of the Banach-Tarski paradox. In the paper we show that subjective probability theory, including Savage’s theory of choice, can be reformulated to take this, and similar objections into account. We suggest such a reformulation which, among other things, amounts to adding an axiom to Savage’s seven postulates, and then use a version of Carathéodory’s extension theorem.

Suggested Citation

  • Aase, Knut K., 2023. "Intuitive probability of non-intuitive events," Discussion Papers 2023/15, Norwegian School of Economics, Department of Business and Management Science.
    • Handle: RePEc:hhs:nhhfms:2023_015
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    File URL: https://hdl.handle.net/11250/3093652
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    Keywords

    The Banach-Tarski paradox; the axiom of choice; Savage’s theory of choice; monotone continuity; countable additivity; Carathéodory’s extension theorem; syndicates; contingent claims;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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