IDEAS home Printed from https://ideas.repec.org/p/wpa/wuwpga/0504004.html
   My bibliography  Save this paper

A new integral for capacities

Author

Listed:
  • Ehud Lehrer

    (Tel Aviv University)

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.

Suggested Citation

  • Ehud Lehrer, 2005. "A new integral for capacities," Game Theory and Information 0504004, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpga:0504004
    Note: Type of Document - pdf; pages: 17
    as

    Download full text from publisher

    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/game/papers/0504/0504004.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Yaron Azrieli & Ehud Lehrer, 2007. "Extendable Cooperative Games," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 9(6), pages 1069-1078, December.
    2. Sarin, Rakesh K & Wakker, Peter, 1992. "A Simple Axiomatization of Nonadditive Expected Utility," Econometrica, Econometric Society, vol. 60(6), pages 1255-1272, November.
    3. Dow James & Werlang Sergio Ribeiro Da Costa, 1994. "Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction," Journal of Economic Theory, Elsevier, vol. 64(2), pages 305-324, December.
    4. Wakker, Peter, 1989. "Continuous subjective expected utility with non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 18(1), pages 1-27, February.
    5. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    6. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
    7. Chris Starmer, 2000. "Developments in Non-expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk," Journal of Economic Literature, American Economic Association, vol. 38(2), pages 332-382, June.
    8. Weber, Robert J., 1994. "Games in coalitional form," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 36, pages 1285-1303, Elsevier.
    9. Ebbe Groes & Hans JÛrgen Jacobsen & Birgitte Sloth & Torben TranÖs, 1998. "Axiomatic characterizations of the Choquet integral," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 441-448.
    10. Yaron Azrieli & Ehud Lehrer, 2007. "Market Games in Large Economies with a Finite Number of Types," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 31(2), pages 327-342, May.
    11. Dow, James & Werlang, Sergio Ribeiro da Costa, 1992. "Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio," Econometrica, Econometric Society, vol. 60(1), pages 197-204, January.
    12. Johannes Leitner, 2005. "A Short Note On Second‐Order Stochastic Dominance Preserving Coherent Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 649-651, October.
    13. Ehud Kalai & Eitan Zemel, 1982. "Generalized Network Problems Yielding Totally Balanced Games," Operations Research, INFORMS, vol. 30(5), pages 998-1008, October.
    14. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    15. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    16. Yaron Azrieli & Ehud Lehrer, 2004. "On Concavification and Convex Games," Game Theory and Information 0408002, University Library of Munich, Germany.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    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," Post-Print hal-01302377, 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. Roee Teper, 2015. "Subjective Independence and Concave Expected Utility," Working Paper 5865, Department of Economics, University of Pittsburgh.
    15. 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.
    16. Stauber, Ronald, 2019. "A strategic product for belief functions," Games and Economic Behavior, Elsevier, vol. 116(C), pages 38-64.
    17. 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.
    18. Roee Teper, 2014. "Sandwich Games," Working Paper 5863, Department of Economics, University of Pittsburgh.
    19. 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.
    20. Pintér, Miklós, 2022. "How to make ambiguous strategies," Journal of Economic Theory, Elsevier, vol. 202(C).
    21. 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.
    22. 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.
    23. Adam Šeliga, 2022. "Convolution of Decomposition Integrals," Mathematics, MDPI, vol. 10(5), pages 1-9, February.
    24. Flesch, János & Vermeulen, Dries & Zseleva, Anna, 2017. "Zero-sum games with charges," Games and Economic Behavior, Elsevier, vol. 102(C), pages 666-686.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marinacci, Massimo, 2000. "Ambiguous Games," Games and Economic Behavior, Elsevier, vol. 31(2), pages 191-219, May.
    2. Andreas Lehnert & Wayne Passmore, 1999. "Pricing systemic crises: monetary and fiscal policy when savers are uncertain," Finance and Economics Discussion Series 1999-33, Board of Governors of the Federal Reserve System (U.S.).
    3. repec:dau:papers:123456789/7323 is not listed on IDEAS
    4. Ramon Casadesus-Masanell & Peter Klibanoff & Emre Ozdenoren, 1998. "Maximum Expected Utility over Savage Acts with a Set of Priors," Discussion Papers 1218, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Pascal Toquebeuf, 2016. "Choquet expected utility with affine capacities," Theory and Decision, Springer, vol. 81(2), pages 177-187, August.
    6. Casadesus-Masanell, Ramon & Klibanoff, Peter & Ozdenoren, Emre, 2000. "Maxmin Expected Utility over Savage Acts with a Set of Priors," Journal of Economic Theory, Elsevier, vol. 92(1), pages 35-65, May.
    7. Takao Asano, 2004. "Portfolio Inertia and [Epsilon]-Contaminations," ISER Discussion Paper 0610, Institute of Social and Economic Research, Osaka University.
    8. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    9. Dominiak, Adam & Lee, Min Suk, 2017. "Coherent Dempster–Shafer equilibrium and ambiguous signals," Journal of Mathematical Economics, Elsevier, vol. 68(C), pages 42-54.
    10. 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.
    11. Ghirardato, Paolo & Le Breton, Michel, 2000. "Choquet Rationality," Journal of Economic Theory, Elsevier, vol. 90(2), pages 277-285, February.
    12. 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.
    13. Eichberger, Jürgen & Kelsey, David, 2007. "Ambiguity," Sonderforschungsbereich 504 Publications 07-50, Sonderforschungsbereich 504, Universität Mannheim;Sonderforschungsbereich 504, University of Mannheim.
      • Eichberger, Jürgen & Kelsey, David, 2007. "Ambiguity," Papers 07-50, Sonderforschungsbreich 504.
    14. Jürgen Eichberger & David Kelsey & Burkhard C. Schipper, 2009. "Ambiguity and social interaction," Oxford Economic Papers, Oxford University Press, vol. 61(2), pages 355-379, April.
    15. Dominiak, Adam, 2013. "Iterated Choquet expectations: A possibility result," Economics Letters, Elsevier, vol. 120(2), pages 155-159.
    16. Stefan Trautmann & Peter P. Wakker, 2018. "Making the Anscombe-Aumann approach to ambiguity suitable for descriptive applications," Journal of Risk and Uncertainty, Springer, vol. 56(1), pages 83-116, February.
    17. Shaw, W. Douglass & Woodward, Richard T., 2008. "Why environmental and resource economists should care about non-expected utility models," Resource and Energy Economics, Elsevier, vol. 30(1), pages 66-89, January.
    18. repec:awi:wpaper:0448 is not listed on IDEAS
    19. Stauber, Ronald, 2019. "A strategic product for belief functions," Games and Economic Behavior, Elsevier, vol. 116(C), pages 38-64.
    20. Dominiak, Adam & Lefort, Jean-Philippe, 2015. "“Agreeing to disagree” type results under ambiguity," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 119-129.
    21. Boff, Hugo Pedro & Werlang, Sergio Ribeiro da Costa, 1998. "Cournot Competition Under Knightian Uncertainty," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 18(2), November.
    22. Adam Dominiak & Jean-Philippe Lefort, 2021. "Ambiguity and Probabilistic Information," Management Science, INFORMS, vol. 67(7), pages 4310-4326, July.

    More about this item

    Keywords

    new integral; capacity; choquet integral; fuzzy capacity; concavity;
    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wpa:wuwpga:0504004. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: EconWPA (email available below). General contact details of provider: https://econwpa.ub.uni-muenchen.de .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.