Dominance of capacities by k-additive belief functions

Miranda Menéndez, Pedro and Grabisch, Michel and Gil, Pedro (2006) Dominance of capacities by k-additive belief functions. European journal of operational research, 175 (2). pp. 912-930. ISSN 0377-2217

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Abstract

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved. We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of k-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditions cannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating k-additive belief functions from which it is possible to derive any other dominant k-additive belief function, and such that the conditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.

Item Type:	Article
Uncontrolled Keywords:	Linear programming; Decision analysis; Capacity; Dominance; k-Additivity; Belief functions
Subjects:	Sciences > Mathematics > Operations research
ID Code:	17067
References:	A. Chateauneuf, J.-Y. Jaffray, Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical Social Sciences 17 (1989) 263–283. G. Choquet, Theory of capacities, Annales de l'Institut Fourier 5 (1953) 131–295. A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statistics 38 (1967) 325–339. D. Denneberg, Non-Additive Measures and Integral, Kluwer Academic, 1994. D. Denneberg, Non-additive measure and integral, basic concepts and their role for applications, in: T. Murofushi, M. Grabisch, M. Sugeno (Eds.), Fuzzy Measures and Integrals. Theory and Applications, Studies in Fuzziness and Soft Computing, vol. 40, Physica-Verlag, 2000, pp. 42–69. D. Dubois, H. Prade, A class of fuzzy measures based on triangular norms, International Journal of General Systems 8 (1982) 43–61. T. Gajdos, Measuring inequalities without linearity in envy: Choquet integral for symmetric capacities, Journal of Economic Theory 106 (2002) 190–200. D. Gale, The Theory of Linear Economic Models, McGraw-Hill, New York, 1960. M. Grabisch, k-order additive discrete fuzzy measures, in: Proceedings of 6th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Spain, 1996, pp. 1345–1350. M. Grabisch, k-order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems 92 (1997) 167–189. M. Grabisch, Upper approximation of non-additive measures by k-additive measures—The case of belief functions, in: Proceedings of 1st International Symposium on Imprecise Probabilities and Their Applications (ISIPTA), Ghent, Belgium, June 1999. M. Grabisch, The interaction and Mo¨bius representations of fuzzy measures on finite spaces, k-additive measures: A survey, in: M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Fuzzy Measures and Integrals. Theory and Applications, Studies in Fuzziness and Soft Computing, vol. 40, Physica-Verlag, 2000, pp. 70–93. M. Grabisch, On lower and upper approximation of fuzzy measures by k-order additive measures, in: B. Bouchon-Meunier, R.R. Yager, L. Zadeh (Eds.), Information, Uncertainty, Fusion, Kluwer Scientific Publisher, 2000, pp. 105–118, Selected papers from IPMU'98. J.-Y. Jaffray, P. Wakker, Decision making with belief functions: Compatibility and incompatibility with the sure-thing principle, Journal of Risk and Uncertainty 7 (1993) 255–271. J.-L. Marichal, k-intolerant capacities and Choquet integrals. European Journal of Operational Research, in press, doi:10.1016/j.ejor.2005.04.015. P. Miranda, M. Grabisch, Optimization issues for fuzzy measures, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 7 (6) (1999) 545–560, Selected papers from IPMU'98.
Deposited On:	13 Nov 2012 10:20
Last Modified:	07 Feb 2014 09:41

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