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Dominance of capacities by k-additive belief functions

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  • Miranda, Pedro
  • Grabisch, Michel
  • Gil, Pedro

Abstract

In this paper we deal with the set of $k$-additive belieffunctions dominating a given capacity. We follow the lineintroduced 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case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid.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 conditionscannot 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 theconditions 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.
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  • Miranda, Pedro & Grabisch, Michel & Gil, Pedro, 2006. "Dominance of capacities by k-additive belief functions," European Journal of Operational Research, Elsevier, vol. 175(2), pages 912-930, December.
  • Handle: RePEc:eee:ejores:v:175:y:2006:i:2:p:912-930
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    References listed on IDEAS

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    1. Miranda, P. & Grabisch, M. & Gil, P., 2005. "Axiomatic structure of k-additive capacities," Mathematical Social Sciences, Elsevier, vol. 49(2), pages 153-178, March.
    2. Porath Elchanan Ben & Gilboa Itzhak, 1994. "Linear Measures, the Gini Index, and The Income-Equality Trade-off," Journal of Economic Theory, Elsevier, vol. 64(2), pages 443-467, December.
    3. Gajdos, Thibault, 2002. "Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities," Journal of Economic Theory, Elsevier, vol. 106(1), pages 190-200, September.
    4. Thibault Gajdos, 2002. "Measuring Inequalities without Linearity in Envy Through Choquet Integral with Symmetric Capacities," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00085888, HAL.
    5. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    6. Michel Grabisch & Christophe Labreuche, 2016. "Fuzzy Measures and Integrals in MCDA," International Series in Operations Research & Management Science, in: Salvatore Greco & Matthias Ehrgott & José Rui Figueira (ed.), Multiple Criteria Decision Analysis, edition 2, chapter 0, pages 553-603, Springer.
    7. Jaffray, Jean-Yves & Wakker, Peter, 1993. "Decision Making with Belief Functions: Compatibility and Incompatibility with the Sure-Thing Principle," Journal of Risk and Uncertainty, Springer, vol. 7(3), pages 255-271, December.
    8. Pedro Miranda & Michel Grabisch & Pedro Gil, 2002. "p-symmetric fuzzy measures," Post-Print hal-00273960, HAL.
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    Cited by:

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