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A new integral for capacities

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  • Ehud Lehrer

Abstract

A new integral for capacities, different from the Choquet integral, is introduced and characterized. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is then extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also when there is information only about a few events and not about all of them.
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  • Ehud Lehrer, 2009. "A new integral for capacities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(1), pages 157-176, April.
    • Handle: RePEc:spr:joecth:v:39:y:2009:i:1:p:157-176
    DOI: 10.1007/s00199-007-0302-z
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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. Michel Grabisch, 2015. "Fuzzy Measures and Integrals: Recent Developments," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477514, HAL.
    3. Roee Teper, 2009. "Time Continuity and Nonadditive Expected Utility," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 661-673, August.
    4. Ehud Lehrer & Roee Teper, 2020. "Set-valued capacities: multi-agenda decision making," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(1), pages 233-248, February.
    5. Aloisio Araujo & Alain Chateauneuf & José Faro, 2012. "Pricing rules and Arrow–Debreu ambiguous valuation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 49(1), pages 1-35, January.
    6. Flesch, Janos & Vermeulen, Dries & Zseleva, Anna, 2018. "Existence of justifiable equilibrium," Research Memorandum 016, Maastricht University, Graduate School of Business and Economics (GSBE).
    7. Lehrer, Ehud & Teper, Roee, 2015. "Subjective independence and concave expected utility," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 33-53.
    8. 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.
    9. Juan Sebastián Lleras & Evan Piermont & Richard Svoboda, 2019. "Asymmetric gain–loss reference dependence and attitudes toward uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(3), pages 669-699, October.
    10. Evren, Özgür, 2019. "Recursive non-expected utility: Connecting ambiguity attitudes to risk preferences and the level of ambiguity," Games and Economic Behavior, Elsevier, vol. 114(C), pages 285-307.
    11. Ehud Lehrer & Roee Tepper, 2013. "Concave Expected Utility and Event Separability," Levine's Working Paper Archive 786969000000000809, David K. Levine.
    12. Jun Li, 2020. "On Null-Continuity of Monotone Measures," Mathematics, MDPI, vol. 8(2), pages 1-13, February.
    13. Mojmír Sabolovič, 2011. "An alternative methodological approach to value analysis of regions, municipal corporations and clusters," Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, Mendel University Press, vol. 59(4), pages 295-300.
    14. Gayer, Gabrielle & Lehrer, Ehud & Persitz, Dotan, 2024. "Aggregating inconclusive data sets," Games and Economic Behavior, Elsevier, vol. 146(C), pages 77-90.
    15. Roee Teper, 2015. "Subjective Independence and Concave Expected Utility," Working Paper 5865, Department of Economics, University of Pittsburgh.
    16. 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.
    17. Stauber, Ronald, 2019. "A strategic product for belief functions," Games and Economic Behavior, Elsevier, vol. 116(C), pages 38-64.
    18. János Flesch & Dries Vermeulen & Anna Zseleva, 2021. "Legitimate equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 787-800, December.
    19. Roee Teper, 2014. "Sandwich Games," Working Paper 5863, Department of Economics, University of Pittsburgh.
    20. Ronald Stauber, 2019. "A strategic product for belief functions," ANU Working Papers in Economics and Econometrics 2019-668, Australian National University, College of Business and Economics, School of Economics.
    21. Pintér, Miklós, 2022. "How to make ambiguous strategies," Journal of Economic Theory, Elsevier, vol. 202(C).
    22. Mesiar, R. & Kolesárová, A. & Bustince, H. & Dimuro, G.P. & Bedregal, B.C., 2016. "Fusion functions based discrete Choquet-like integrals," European Journal of Operational Research, Elsevier, vol. 252(2), pages 601-609.
    23. Leifan Yan & Tong Kang & Huai Zhang, 2023. "Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures," Mathematics, MDPI, vol. 11(13), pages 1-14, July.
    24. Adam Šeliga, 2022. "Convolution of Decomposition Integrals," Mathematics, MDPI, vol. 10(5), pages 1-9, February.
    25. Flesch, János & Vermeulen, Dries & Zseleva, Anna, 2017. "Zero-sum games with charges," Games and Economic Behavior, Elsevier, vol. 102(C), pages 666-686.

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    More about this item

    Keywords

    Capacities; Non-additive probability; Decisions under uncertainty; Uncertainty aversion; Concave integral; Choquet integral; Fuzzy capacities; Large core; C71; D80; D81; D84;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • 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
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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