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Axiomatic structure of k-additive capacities

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Author Info

  • Pedro Miranda

    () (Universidad Complutense de Madrid - Universidad Complutense de Madrid)

  • Michel Grabisch

    () (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - CNRS : UMR8095 - Université Paris I - Panthéon-Sorbonne)

  • Pedro Gil

    (Universidad de Oviedo - Universidad de Oviedo)

Abstract

In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a $k$-additive capacity, i.e. whose Möbius 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.

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Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number hal-00188165.

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Date of creation: Mar 2005
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Publication status: Published, Mathematical Social Sciences, 2005, 49, 2, 153-178
Handle: RePEc:hal:cesptp:hal-00188165

Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00188165
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Related research

Keywords: Axiomatic; Capacities; k-Additivity;

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References

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  1. Elchanan Ben Porath & Itzhak Gilboa, 1991. "Linear Measures, the Gini Index and the Income-Equality Tradeoff," Discussion Papers 944, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-87, May.
  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. Weymark, John A., 1981. "Generalized gini inequality indices," Mathematical Social Sciences, Elsevier, vol. 1(4), pages 409-430, August.
  5. Michel Grabisch & Jacques Duchêne & Frédéric Lino & Patrice Perny, 2002. "Subjective Evaluation of Discomfort in Sitting Positions," Post-Print halshs-00273179, HAL.
  6. Christophe Labreuche & Michel Grabisch, 2003. "The Choquet integral for the aggregation of interval scales in multicriteria decision making," Post-Print hal-00272090, HAL.
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Citations

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Cited by:
  1. Brice Mayag & Michel Grabisch & Christophe Labreuche, 2011. "A characterization of the 2-additive Choquet integral through cardinal information," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00625708, HAL.
  2. 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.
  3. Brice Mayag & Michel Grabisch & Christophe Labreuche, 2011. "A characterization of the 2-additive Choquet integral through cardinal information," Post-Print halshs-00625708, HAL.
  4. Brice Mayag & Michel Grabisch & Christophe Labreuche, 2011. "A representation of preferences by the Choquet integral with respect to a 2-additive capacity," Theory and Decision, Springer, vol. 71(3), pages 297-324, September.
  5. Alessio Bonetti & Silvia Bortot & Mario Fedrizzi & Silvio Giove & Ricardo Alberto Marques Pereira & Andrea Molinari, 2011. "Modelling group processes and effort estimation in Project Management using the Choquet integral: an MCDM approach," DISA Working Papers 2011/12, Department of Computer and Management Sciences, University of Trento, Italy, revised Sep 2011.
  6. Paugam, Luc, 2011. "Valorisation et reporting du goodwill : enjeux théoriques et empiriques," Open Access publications from Université Paris-Dauphine urn:hdl:123456789/8007, Université Paris-Dauphine.
  7. Silvia Bortot & Ricardo Alberto Marques Pereira, 2011. "Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process," DISA Working Papers 2011/06, Department of Computer and Management Sciences, University of Trento, Italy, revised 29 Jul 2011.

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