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Interpretation of Multicriteria Decision Making Models with Interacting Criteria

  • Michel GrabischEmail author
  • Christophe Labreuche
Chapter
Part of the Multiple Criteria Decision Making book series (MCDM)

Abstract

We consider general MCDA models with discrete attributes. These models are shown to be equivalent to a multichoice game and we put some emphasis on discrete Generalized Independence Models (GAI), especially those which are 2-additive, that is, limited to terms of at most two attributes. The chapter studies the interpretation of these models. For general MCDA models, we study how to define a meaningful importance index, and propose mainly two kinds on importance indices: the signed and the absolute importance indices. For 2-additive GAI models, we study the issue of the decomposition, which is not unique in general. We show that for a monotone 2-additive GAI model, it is always possible to obtain a decomposition where each term is monotone. This has important consequences on the tractability and interpretability of the model.

References

  1. Bacchus, F., & Grove, A. (1995, July). Graphical models for preference and utility. In Conference on Uncertainty in Artificial Intelligence (UAI), Montreal, Canada (pp. 3–10).Google Scholar
  2. Bigot, D., Fargier, H., Mengin, J., & Zanuttini, B. (2012, August). Using and learning GAI-decompositions for representing ordinal rankings. In European Conference on Artificial Intelligence (ECAI), Montpellier, France.Google Scholar
  3. Boutilier, C., Bacchus, F., & Brafman, R. (2001). UCP-networks: A directed graphical representation of conditional utilities. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI-01), Seattle (pp. 56–64).Google Scholar
  4. Bouyssou, D., & Pirlot, M. (2016). Conjoint measurement tools for MCDM. In S. Greco, M. Ehrgott, & J. Figueira (Eds.), Multiple criteria decision analysis (pp. 97–151). Springer, New York.Google Scholar
  5. Braziunas, D. (2012). Decision-theoretic elicitation of generalized additive utilities. Ph.D. thesis, University of Toronto.Google Scholar
  6. Choquet, G. (1953). Theory of capacities. Annales de l’Institut Fourier, 5, 131–295.CrossRefGoogle Scholar
  7. Fishburn, P. (1967). Interdependence and additivity in multivariate, unidimensional expected utility theory. International Economic Review, 8, 335–342.CrossRefGoogle Scholar
  8. Fisher, R. A., & Mackenzie, W. A. (1923). The manurial response of different potato varieties. Journal of Agricultural Science, 13, 311–320.CrossRefGoogle Scholar
  9. Grabisch, M. (1997). k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, 92, 167–189.Google Scholar
  10. Grabisch, M., & Labreuche, Ch. (2003, September). Capacities on lattices and \(k\)-ary capacities. In 3d International Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2003), Zittau, Germany (pp. 304–307).Google Scholar
  11. Grabisch, M., & Labreuche, Ch. (2008). A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR, 6, 1–44.  https://doi.org/10.1007/s10288-007-0064-2.
  12. Grabisch, M., & Labreuche, Ch. (to appear). Monotone decomposition of 2-additive generalized additive independence models. Mathematical Social Sciences.Google Scholar
  13. Grabisch, M., Marichal, J.-L., Mesiar, R., & Pap, E. (2009). Aggregation functions (Vol. 127). Encyclopedia of mathematics and its applications. Cambridge: Cambridge University Press.Google Scholar
  14. Grabisch, M., Labreuche, Ch., & Ridaoui, M. (submitted).On importance indices in multicriteria decision making. working paper.Google Scholar
  15. Greco, S., Mousseau, V., & Słowinski, R. (2014). Robust ordinal regression for value functions handling interacting criteria. European Journal of Operational Research, 239(3), 711–730.CrossRefGoogle Scholar
  16. Hsiao, C. R., & Raghavan, T. E. S. (1990). Multichoice cooperative games. In B. Dutta (Ed.), Proceedings of the International Conference on Game Theory and Economic Applications, New Delhi, India.Google Scholar
  17. Hsiao, C. R., & Raghavan, T. E. S. (1993). Shapley value for multichoice cooperative games, I. Games and Economic Behavior, 5, 240–256.CrossRefGoogle Scholar
  18. Klijn, F., Slikker, M., & Zarzuelo, J. M. (1999). Characterizations of a multichoice value. International Journal of Game Theory, 28, 521–532.CrossRefGoogle Scholar
  19. Labreuche, C., & Grabisch, M. (to appear) Using multiple reference levels in multi-criteria decision aid: the Generalized-Additive Independence model and the Choquet integral approaches. European Journal of Operational Research.Google Scholar
  20. Peters, H., & Zank, H. (2005). The egalitarian value for multichoice games. Annals of Operations Research, 137, 399–409.CrossRefGoogle Scholar
  21. Ridaoui, M., Grabisch, M., & Labreuche, C. (2017a, July). Axiomatization of an importance index for generalized additive independence models. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2017), Lugano, Switzerland (pp. 340–350). Lecture Notes in Artificial Intelligence. Cham: Springer.Google Scholar
  22. Ridaoui, M., Grabisch, M., & Labreuche, C. (2017b, October). An alternative view of importance indices for multichoice games. In 5th International Conference on Algorithmic Decision Theory, Luxembourg.Google Scholar
  23. Ridaoui, M., Grabisch, M., & Labreuche, Ch. (2018). An interaction index for multichoice games. arXiv:1803.07541.
  24. Rota, G. C. (1964). On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 340–368.Google Scholar
  25. Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games, Vol. II (Vol. 28, pp. 307–317). Annals of mathematics studies. Princeton: Princeton University Press.Google Scholar
  26. Tehrani, A. F., Cheng, W., Dembczynski, K., & Hüllermeier, E. (2012). Learning monotone nonlinear models using the Choquet integral. Machine Learning, 89, 183–211.Google Scholar
  27. van den Nouweland, A., Tijs, S., Potters, J., & Zarzuelo, J. (1995). Cores and related solution concepts for multi-choice games. ZOR - Mathematical Methods of Operations Research, 41, 289–311.CrossRefGoogle Scholar
  28. Von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.). Princeton: Princeton University Press.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Paris I – Panthéon-Sorbonne, Paris School of EconomicsParisFrance
  2. 2.Thales Research and TechnologyPalaiseauFrance

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