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A Discrete Choquet Integral for Ordered Systems

Listed author(s):
  • Ulrich Faigle

    (Zentrum für Angewandte Informatik [Köln] - Universität zu Köln)

  • Michel Grabisch


    (CES - Centre d'économie de la Sorbonne - UP1 - Université Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lovász' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00563926.

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Date of creation: 2011
Publication status: Published in Fuzzy Sets and Systems, Elsevier, 2011, 168 (1), pp.3-17. 〈10.1016/j.fss.2010.10.003〉
Handle: RePEc:hal:cesptp:halshs-00563926
DOI: 10.1016/j.fss.2010.10.003
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  1. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
  2. Michel Grabisch & Christophe Labreuche, 2008. "Bipolarization of posets and natural interpolation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00274267, HAL.
  3. 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.
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