Axiomatic structure of k-additive capacities

dc.contributor.authorMiranda Menéndez, Pedro
dc.contributor.authorGrabisch, Michel
dc.contributor.authorGil, Pedro
dc.description.abstractIn this paper we deal with the problem of axiomatizing the preference relations modeled through Choquet integral with respect to a k-additive capacity, i.e. whose Mobius transform vanishes for subsets of more than k elements. Thus, k-additive capacities range from probability measures (k=1) to general capacities (k=n). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general k-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.
dc.description.departmentDepto. de Estadística e Investigación Operativa
dc.description.facultyFac. de Ciencias Matemáticas
dc.identifier.citationBen Porath, E., Gilboa, I., 1994. Linear measures, the Gini index, and the income-equality trade-off. Journal of Economic Theory 2 (64), 443– 467. Berge, C., 1971. Principles of Combinatorics, Volume 72 of Mathematics in Science and Engineering Academic Press, New York, London. Calvo, T., De Baets, B., 1998. Aggregation operators defined by k-order additive/maxitive fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 6 (6), 533– 550. Cao-Van, K., De Baets, B., 2001. A decomposition of k-additive Choquet and k-maxitive Sugeno integrals. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 (9), 127– 143. Chateauneuf, A., 1994. Modelling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research (52), 3– 20. Chateauneuf, A., Jaffray, J.-Y., 1989. Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences 3 (17), 263– 283. Choquet, G., 1953. Theory of capacities. Annales de 1’Institut Fourier (5), 131–295. Dempster, A.P., 1967. Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statististics (38), 325– 339. Denneberg, D., 1994. Non-Additive Measures and Integral. Kluwer Academic Publishers, Dordrecht. Fodor, J., Marichal, J.-L., Roubens, M., 1995. Characterization of the ordered weighted averaging operators. IEEE Transactions on Fuzzy Systems 3 (2), 236– 240. Gajdos, T., 2002. Measuring inequalities without linearity in envy: Choquet integral for symmetric capacities. Journal of Economic Theory 106, 190–200. Grabisch, M., 1995. Pattern classification and feature extraction by fuzzy integral. Third European Congress on Intelligent Techniques and Soft Computing (EUFIT), Aachen, Germany, pp. 1465–1469. Grabisch, M., 1996a. Fuzzy measures and integrals: a survey of applications and recent issues. In: Dubois, D., Prade, H., Yager, R. (Eds.), Fuzzy Sets Methods in Information Engineering: A Guide Tour of Applications. Grabisch, M., 1996b. k-order additive discrete fuzzy measures. Proceedings of Sixth International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU), Granada, Spain, pp. 1345–1350. Grabisch, M., 1997a. Alternative representations of OWA operators. In: Yager, R.R., Kacprzyk, J. (Eds.), The Ordered Weighted Averaging Operators: Theory and Applications. Kluwer Academic Publisher, Dordrecht, pp. 73– 85. Grabisch, M., 1997b. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 2 (92), 167– 189. Grabisch, M., 1997c. On the representation of k-decomposable measures. Proceedings of Seventh International Fuzzy Systems Association World Congress, Prague, Czech Republic. Grabisch, M., 1998. k-Additive measures: recent issues and challenges. Fifth International Conference on Soft Computing and Information/Intelligent Systems, Izuka, Japan, pp. 394–397. Grabisch, M., Ducheˆne, J., Lino, F., Perny, P., 2002. Subjective evaluation of discomfort in sitting position. Fuzzy Optimization and Decision Making 1 (3), 287–312. Grabisch, M., Labreuche, C., 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– 47. Hammer, P.L., Holzman, R., 1992. On approximations of pseudo-Boolean functions. Zeitschrift fu¨ r Operations Research. Mathematical Methods of Operations Research 1 (36), 3 – 21. Hardy, G.H., Littlewood, J.E., Pólya, G., 1952. Inequalities Cambridge University Press, Cambridge, UK. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A., 1971. Foundations of Measurement, vo 1. Additive and Polynomial Representations, Academic Press, New York, London. Labreuche, C.h., Grabisch, M., 2003. The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets and Systems 137, 11– 26. Miranda, P., Grabisch, M., 2000. Characterizing k-additive fuzzy measures. Proceedings of Eighth International Conference of Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU), Madrid, Spain, 1063– 1070. Murofushi, T., Sugeno, M., 1993. Some quantities represented by the Choquet integral. Fuzzy Sets and Systems 2 (56), 229– 235. Rota, G.C., 1964. On the foundations of combinatorial theory: I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (2), 340–368. Schmeidler, D., 1986. Integral representation without additivity. Proceedings of the American Mathematical Society 97 (2), 255– 261. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 517– 587. Shafer, G., 1976. A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ, USA. Sugeno, M., 1974. Theory of fuzzy integrals and its applications. Tokyo Institute of Technology (Ph.D. thesis). Wakker, P., 1989. Additive Representations of Preferences. Kluwer Academic Publishers, Dordrecht. Walley, P., 1991. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, UK. Weymark, J.A., 1981. Generalized Gini inequality indices. Mathematical Social Sciences 4 (1), 409–430. Yager, R.R., 1988. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics 1 (18), 183–190.
dc.journal.titleMathematical Social Sciences
dc.rights.accessRightsrestricted access
dc.subject.ucmAnálisis funcional y teoría de operadores
dc.titleAxiomatic structure of k-additive capacities
dc.typejournal article
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
250.7 KB
Adobe Portable Document Format