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Monotone equimeasurable rearrangements with non-additive probabilities

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  • Ghossoub, Mario

Abstract

In the classical theory of monotone equimeasurable rearrangements of functions, “equimeasurability” (i.e. the fact the two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knighitan uncertainty, or ambiguity is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to situations of ambiguity. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of non-additive probabilities, or capacities that satisfy a property that I call strong nonatomicity. The latter is a strengthening of the notion of nonatomicity, and these two properties coincide for additive measures and for submodular (i.e. concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.

Suggested Citation

  • Ghossoub, Mario, 2011. "Monotone equimeasurable rearrangements with non-additive probabilities," MPRA Paper 37629, University Library of Munich, Germany, revised 23 Mar 2012.
  • Handle: RePEc:pra:mprapa:37629
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Massimiliano Amarante & Mario Ghossoub & Edmund Phelps, 2012. "Contracting for Innovation under Knightian Uncertainty," Cahiers de recherche 18-2012, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    2. Ghossoub, Mario, 2010. "Belief heterogeneity in the Arrow-Borch-Raviv insurance model," MPRA Paper 37630, University Library of Munich, Germany, revised 22 Mar 2012.
    3. Amarante, M & Ghossoub, M & Phelps, E, 2013. "Innovation, Entrepreneurship and Knightian Uncertainty," Working Papers 12241, Imperial College, London, Imperial College Business School.

    More about this item

    Keywords

    Ambiguity; Capacity; Non-Additive Probability; Choquet Integral; Monotone Equimeasurable Rearrangement; Production under Uncertainty;

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • D89 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Other
    • D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity

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