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The symmetric and asymmetric Choquet integrals on finite spaces for decision making

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Abstract

In this paper, we give a mathematical analysis of symmetric and asymmetric Choquet integrals in the view of decision making in a finite setting. These integrals present two ways of dealing with negative integrands. The analysis is done with the aid of the Möbius and interaction transforms, this last one having an interesting interpretation in multicriteria decision making (MCDM). The last part of the paper shows the application of these two integrals in MCDM.

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References

  1. A. Chateauneuf and J.Y. Jaffray. Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion. Mathematical Social Sciences, 17:263–283, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Choquet Theory of capacities. Annales de I’Institut Fourier, 5:131–295, 1953.

    MathSciNet  Google Scholar 

  3. A.P. Dempster Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist, 38:325–339, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic, 1994.

  5. D. Denneberg and M. Grabisch. Interaction transform of set functions over a finite set. Information Sciences, 121:149–170, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Dubois, M. Grabisch, F. Modave, and H. Prade. Relating decision under uncertainty and multicriteria decision making models. Int. J. of Intelligent Systems, 15:967–979, 2000.

    Article  MATH  Google Scholar 

  7. D. Dubois and H. Prade. Possibility Theory. Plenum Press, 1988.

  8. D. Dubois and H. Prade. Possibility theory: qualitative and quantitative aspects. In D.M. Gabbay and Ph. Smets, editors, Handbook of Defeasible Reasoning and Uncertainty Management Systems, pages 169–226. Kluwer Academic Publishers, 1998.

  9. M. Grabisch The application of fuzzy integrals in multicriteria decision making. European J. of Operational Research, 89:445–456, 1996.

    Article  MATH  Google Scholar 

  10. M. Grabisch Alternative representations of discrete fuzzy measures for decision making. Int. J. of Uncertainty, Fuzziness, and Knowledge Based Systems, 5:587–607, 1997.

    Article  MathSciNet  Google Scholar 

  11. M. Grabisch k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, 92:167–189, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Grabisch On the representation of k-decomposable measures. In 7th IFSA World Congress, Prague, Czech Republic, June 1997.

    Google Scholar 

  13. M. Grabisch. The interaction and Mobius representations of fuzzy measures on finite spaces, k-additive measures: a survey. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 70–93. Physica Verlag, 2000.

  14. M. Grabisch and Ch. Labreuche. To be symmetric or asymmetric? A dilemna in decision making. In J. Fodor, B. De Baets, and P. Perny, editors, Preferences and Decisions under Incomplete Knowledge, pages 179–194. Physica Verlag, 2000.

  15. M. Grabisch and Ch. Labreuche. The šipoš integral for the aggregation of interacting bipolar criteria. In 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 395–401, Madrid, Spain, July 2000.

  16. M. Grabisch, Ch. Labreuche, and J.C. Vansnick. On the extension of pseudo-Boolean functions for the aggregation of interacting bipolar criteria. European J. of Operational Research, submitted.

  17. M. Grabisch and M. Roubens. Application of the Choquet integral in multicriteria decision making. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 348–374. Physica Verlag, 2000.

  18. D. Kahneman and A. Tversky. Prospect theory: an analysis of decision under risk. Econometrica, 47:263–291, 1979.

    Article  MATH  Google Scholar 

  19. R.L. Keeney and H. Raiffa. Decision with Multiple Objectives. Wiley, New York, 1976.

    Google Scholar 

  20. D.H. Krantz, R.D. Luce, P. Suppes, and A. Tversky. Foundations of measurement, volume 1: Additive and Polynomial Representations. Academic Press, 1971.

  21. J.L. Marichal. Aggregation operators for multicriteria decision aid. PhD thesis, University of Liège, 1998.

  22. T. Murofushi. A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. In 2nd Fuzzy Workshop, pages 39–48, Nagaoka, Japan, October 1992. In Japanese.

  23. T. Murofushi and S. Soneda. Techniques for reading fuzzy measures (III): interaction index. In 9th Fuzzy System Symposium, pages 693–696, Sapporo, Japan, May 1993. In Japanese.

  24. D. Schmeidler Integral representation without additivity. Proc. of the Amer. Math. Soc., 97(2):255–261, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Schmeidler Subjective probability and expected utility without additivity. Econometrica, 57(3):571–587, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Shafer. A Mathematical Theory of Evidence. Princeton Univ. Press, 1976.

  27. L.S. Shapley. A value for n-person games. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pages 307–317. Princeton University Press, 1953.

  28. M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.

  29. M. Sugeno and T. Murofushi. Fuzzy measure theory, volume 3 of Course on fuzzy theory. Nikkan Kōgyō, 1993. In Japanese.

  30. A. Tversky and D. Kahneman. Advances in prospect theory: cumulative representation of uncertainty. J. of Risk and Uncertainty, 1992.

  31. J. šipoš Integral with respect to a pre-measure. Math. Slovaca, 29:141–155, 1979.

    MATH  MathSciNet  Google Scholar 

  32. P. Walley. Coherent lower (and upper) probabilities. Technical Report 22, University of Warvick, Coventry, 1981.

  33. P. Walley Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.

    MATH  Google Scholar 

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Authors and Affiliations

  1. LIP6, University of Paris VI, 4 Place Jussieu, 75252, Paris, France

    Michel Grabisch

  2. Thomson-CSF, Corporate Research Laboratory, Domaine de Corbeville, 91404, Orsay Cedex, France

    Michel Grabisch & Christophe Labreuche

Authors
  1. Michel Grabisch
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  2. Christophe Labreuche
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Corresponding authors

Correspondence to Michel Grabisch or Christophe Labreuche.

Additional information

This paper is based on a preliminary and short version published at the IPMU’2009 Conference [15].

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Grabisch, M., Labreuche, C. The symmetric and asymmetric Choquet integrals on finite spaces for decision making. Statistical Papers 43, 37–52 (2002). https://doi.org/10.1007/s00362-001-0085-4

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  • DOI: https://doi.org/10.1007/s00362-001-0085-4

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