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The Topology of Change: Foundations of Probability with Black Swans

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  • Chichilnisky, Graciela

Abstract

Classic probability theory treats rare events as ‘outliers’ that are often disregarded and underestimated. Yet in a moment of change rare events can become frequent and frequent events rare. We therefore postulate new axioms for probability theory that require a balanced treatment for rare and frequent events, based on what we call ”the topology of change”. The axioms extend the foundation of probability to integrate rare but potentially catastrophic events or black swans: natural hazards, market crashes, catastrophic climate change and major episodes of species extinction. The new results presented in this article include a characterization of a family of purely finitely additive measures that are - somewhat surprisingly - absolutely continuous with respect to the Lebesgue measure. This is a new development from an earlier characterization of all the probabilities measures implied by the new axioms as a combination of purely finitely additive and countably additive measures that was first established in Chichilnisky (2000, 2002, 2008, 2009), and the results are contrasted here to the work of Kolmogorov (1932), De Groot (1970), Arrow (1971)), Dubins and Savage (1963), Savage (1954), Von Neumann Morgernstern (1954), and Hernstein and Milnor (1972).

Suggested Citation

  • Chichilnisky, Graciela, 2017. "The Topology of Change: Foundations of Probability with Black Swans," MPRA Paper 86080, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:86080
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    References listed on IDEAS

    as
    1. Chichilnisky, Graciela, 2000. "An axiomatic approach to choice under uncertainty with catastrophic risks," Resource and Energy Economics, Elsevier, vol. 22(3), pages 221-231, July.
    2. Graciela Chichilnisky, 2011. "The Limits of Econometrics: Nonparametric Estimation in Hilbert Spaces," Chapters, in: Miroslav Verbic (ed.), Advances in Econometrics - Theory and Applications, IntechOpen.
    3. Chichilnisky, Graciela & Wu, Ho-Mou, 2006. "General equilibrium with endogenous uncertainty and default," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 499-524, August.
    4. Graciela Chichilnisky, 1996. "An axiomatic approach to sustainable development," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 13(2), pages 231-257, April.
    5. Chichilnisky, Graciela, 2009. "The topology of fear," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 807-816, December.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Lebesgue Measure; Rare Event; Additive Measure; Frequent Event; Heavy Tail; additive measure; Classic probability; vanishing sequence ; continuity axiom; the topology of change;
    All these keywords.

    JEL classification:

    • D7 - Microeconomics - - Analysis of Collective Decision-Making

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