Theory and Decision

, Volume 71, Issue 3, pp 297–324 | Cite as

A representation of preferences by the Choquet integral with respect to a 2-additive capacity

  • Brice Mayag
  • Michel Grabisch
  • Christophe Labreuche
Article

Abstract

In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions allowing to represent an ordinal preferential information provided by the decision maker by a Choquet integral w.r.t a 2-additive capacity. We provide also a characterization of this type of preferential information by a belief function which can be viewed as a capacity. These characterizations are based on three axioms, namely strict cycle-free preferences and some monotonicity conditions called MOPI and 2-MOPI.

Keywords

Capacity Möbius transform Choquet integral Multiple criteria decision analysis k-monotone function Belief function 

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References

  1. Bana e Costa C. A., Nunes da Silva F., Vansnick J.-C. (2001) Conflict dissolution in the public sector: A case-study. European Journal of Operational Research 130: 388–401CrossRefGoogle Scholar
  2. Bana e Costa C. A., Correa E. C., De Corte J.-M., Vansnick J.-C. (2002a) Facilitating bid evaluation in public call for tenders: A socio-technical approach. Omega 30: 227–242CrossRefGoogle Scholar
  3. Bana e Costa C. A., De Corte J.-M., Vansnick J.-C. (2002b) On the mathematical foundations of MACBETH. Multiple Criteria Decision Analysis: State of the Art Surveys 78: 409–437Google Scholar
  4. Berrah L., Clivillé V. (2007) Towards an aggregation performance measurement system model in a supply chain context. Computers in Industry 58(7): 709–719CrossRefGoogle Scholar
  5. Chateauneuf A. (1994) Modeling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research 52: 3–20CrossRefGoogle Scholar
  6. Chateauneuf A. (1995) Ellsberg paradox intuition and Choquet expected utility. In: Coletti G., Dubois D., Scozzafava R. (eds) Mathematical models for handling partial knowledge in artificial intelligence. Plenum Press, New York, pp 1–20Google Scholar
  7. Chateauneuf A., Jaffray J. Y. (1989) Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences 17: 263–283CrossRefGoogle Scholar
  8. Chateauneuf A., Kast R., Lapied A. (2004) Conditioning capacities and Choquet integrals: The role of comonotony. Theory and Decision 51(2–4): 367–386Google Scholar
  9. Chateauneuf A., Grabisch M., Rico A. (2008) Modeling attitudes towards uncertainty through the use of Sugeno integral. Journal of Mathematical Economics 44: 1084–1099CrossRefGoogle Scholar
  10. Choquet G. (1953) Theory of capacities. Annales de l’Institut Fourier 5: 131–295Google Scholar
  11. Clivillé V., Berrah L., Mauris G. (2007) Quantitative expression and aggregation of performance measurements based on the MACBETH multi-criteria method. International Journal of Production economics 105: 171–189CrossRefGoogle Scholar
  12. Ellsberg D. (1961) Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics 75: 643–669CrossRefGoogle Scholar
  13. Gajdos T. (2002) Measuring inequalities without linearity in envy: Choquet integrals for symmetric capacities. Journal of Economic Theory 106: 190–220CrossRefGoogle Scholar
  14. Grabisch M. (1996) The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research 89: 445–456CrossRefGoogle Scholar
  15. Grabisch M. (1997) k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92: 167–189CrossRefGoogle Scholar
  16. Grabisch M., Labreuche C. (2004) Fuzzy measures and integrals in MCDA. In: Figueira J., Greco S., Ehrgott M. (eds) Multiple criteria decision analysis. Kluwer Academic Publishers, Dordrecht, pp 563–608Google Scholar
  17. Grabisch M., Labreuche Ch. (2008) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6: 1–44CrossRefGoogle Scholar
  18. Grabisch M., Duchêne J., Lino F., Perny P. (2002) Subjective evaluation of discomfort in sitting position. Fuzzy Optimization and Decision Making 1(3): 287–312CrossRefGoogle Scholar
  19. Grabisch M., Labreuche Ch., Vansnick J.-C. (2003) On the extension of pseudo-Boolean functions for the aggregation of interacting bipolar criteria. European Journal of Operational Research 148: 28–47CrossRefGoogle Scholar
  20. Hsee C. K. (1996) The evaluability hypothesis: An explanation for preference reversals between joint and separate evaluations of alternatives. Organizational Behavior and Human Decision Processes 67: 242–257CrossRefGoogle Scholar
  21. Jaffray J.-Y. (1989) Linear utility theory for belief functions. Operations Research Letters 8: 107–112CrossRefGoogle Scholar
  22. Jaffray J.-Y., Wakker P. (1993) Decision making with belief functions: Compatibility and incompatibility with the sure-thing principle. Journal of Risk and Uncertainty 7: 255–271CrossRefGoogle Scholar
  23. Labreuche Ch., Grabisch M. (2003) The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets and Systems 137: 11–26CrossRefGoogle Scholar
  24. Marchant T. (2003) Towards a theory of MCDM: Stepping away from social choice theory. Mathematical Social Sciences 45(3): 343–363CrossRefGoogle Scholar
  25. Mayag, B., Grabisch, M., & Labreuche, C. (2008). A characterization of the 2-additive Choquet integral. In Proceedings of IPMU’08 (CD), Malaga, Spain, June 2008.Google Scholar
  26. Miranda, P., Grabisch, M., & Gil, P. (2002). p-symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems, 10(Suppl.), 105–123.Google Scholar
  27. Miranda P., Grabisch M., Gil P. (2005) Axiomatic structure of k-additive capacities. Mathematical Social Sciences 49: 153–178CrossRefGoogle Scholar
  28. Murofushi, T. & Soneda, S. (1993). Techniques for reading fuzzy measures III: Interaction index. In 9th Fuzzy System Symposium, Sapporo, Japan, May 1993 (pp. 693–696) (in Japanese).Google Scholar
  29. Pap E. (1995) Null-additive set functions. Kluwer Academic, DordrechtGoogle Scholar
  30. Pignon, J. P. & Labreuche, Ch. (2007). A methodological approach for operational and technical experimentation based evaluation of systems of systems architectures. In Int. conference on software & systems engineering and their applications ICSSEA, Paris, France, December 4–6, 2007.Google Scholar
  31. Schmeidler D. (1986) Integral representation without additivity. Proceedings of the American Mathematical Society 97(2): 255–261CrossRefGoogle Scholar
  32. Schmeidler D. (1989) Subjective probability and expected utility without additivity. Econometrica 57(3): 571–587CrossRefGoogle Scholar
  33. Shafer G. (1976) A mathematical theory of evidence. Princeton University Press, PrincetonGoogle Scholar
  34. Shapley L.S. (1953) A value for n-person games. In: Kuhn H.W., Tucker A.W. (eds) Contributions to the theory of games, Vol. II, number 28 in Annals of Mathematics Studies. Princeton University Press, Princeton, pp 307–317Google Scholar
  35. Simon H. (1956) Rational choice and the structure of the environment. Psychological Review 63(2): 129–138CrossRefGoogle Scholar
  36. Slovic P., Finucane M., Peters E., MacGregor D.G. (2002) The affect heuristic. In: Gilovitch T., Griffin D., Kahneman D. (eds) Heuristics and biases: The psychology of intuitive judgment. Cambridge University Press, Cambridge, pp 397–420Google Scholar
  37. Smets Ph. (2005) Decision making in the tbm: The necessity of the pignistic transformation. International Journal of Approximate Reasoning 38(2): 133–147CrossRefGoogle Scholar
  38. Tversky A., Kahneman D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5: 297–323CrossRefGoogle Scholar
  39. Von Neuman J., Morgenstern O. (1994) Theory of games and economic behavior. Princeton University Press, PrincetonGoogle Scholar
  40. Weber S. (1984) \({\bot}\)-decomposable measures and integrals for Archimedean t-conorms \({\bot}\). Journal of Mathematical Analysis and Application 101: 114–138CrossRefGoogle Scholar
  41. Weymark J.A. (1981) Generalized Gini inequality indices. Mathematical Social Sciences 1: 409–430CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Brice Mayag
    • 1
    • 2
  • Michel Grabisch
    • 1
  • Christophe Labreuche
    • 2
  1. 1.Centre d’Economie de la SorbonneUniversity of Paris IParisFrance
  2. 2.Thales R & TPalaiseau cedexFrance

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