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Decomposition-integral: unifying Choquet and the concave integrals

Author

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  • Yaarit Even
  • Ehud Lehrer

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Abstract

This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A prespecified set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determined by the capacity. The decomposition-integral of a random variable is defined as the highest of these values. Thus, different sets of collections induce different decomposition-integrals. It turns out that this decomposition approach unifies well-known integrals, such as Choquet, the concave and Riemann integral. Decomposition-integrals are investigated with respect to a few essential properties that emerge in economic contexts, such as concavity (uncertainty-aversion), monotonicity with respect to stochastic dominance and translation-covariance. The paper characterizes the sets of collections that induce decomposition-integrals, which respect each of these properties. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Yaarit Even & Ehud Lehrer, 2014. "Decomposition-integral: unifying Choquet and the concave integrals," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(1), pages 33-58, May.
  • Handle: RePEc:spr:joecth:v:56:y:2014:i:1:p:33-58
    DOI: 10.1007/s00199-013-0780-0
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    References listed on IDEAS

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    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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    Cited by:

    1. Mayumi Horie, 2016. "Bayesian Updating for Complementarily Additive Beliefs under Ambiguity," KIER Working Papers 935, Kyoto University, Institute of Economic Research.
    2. Xiangyu Qu, 2017. "Separate aggregation of beliefs and values under ambiguity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 503-519, February.
    3. de Castro, Luciano I. & Liu, Zhiwei & Yannelis, Nicholas C., 2017. "Implementation under ambiguity," Games and Economic Behavior, Elsevier, vol. 101(C), pages 20-33.
    4. Hirbod Assa & Sheridon Elliston & Ehud Lehrer, 2016. "Joint games and compatibility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(1), pages 91-113, January.
    5. Angelos Angelopoulos & Leonidas Koutsougeras, 2015. "Value allocation under ambiguity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(1), pages 147-167, May.
    6. Michel Grabisch, 2015. "Fuzzy Measures and Integrals: Recent Developments," Post-Print hal-01302377, HAL.
    7. Michel Grabisch, 2015. "Fuzzy Measures and Integrals: Recent Developments," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477514, HAL.
    8. Hirbod Assa & Sheridon Elliston & Ehud Lehrer, 2016. "Joint games and compatibility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(1), pages 91-113, January.
    9. repec:eee:apmaco:v:361:y:2019:i:c:p:15-21 is not listed on IDEAS
    10. Lehrer, Ehud & Teper, Roee, 2015. "Subjective independence and concave expected utility," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 33-53.

    More about this item

    Keywords

    Capacity; Non-additive probability; Decision making; Decomposition-integral; Concave integral; Choquet integral; C71; D80; D81; D84;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D84 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Expectations; Speculations

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