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A Discrete Choquet Integral for Ordered Systems

Author

Listed:
  • Ulrich Faigle

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

  • Michel Grabisch

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

Abstract

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.

Suggested Citation

  • Ulrich Faigle & Michel Grabisch, 2011. "A Discrete Choquet Integral for Ordered Systems," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00563926, HAL.
  • Handle: RePEc:hal:cesptp:halshs-00563926
    DOI: 10.1016/j.fss.2010.10.003
    Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00563926
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    File URL: https://halshs.archives-ouvertes.fr/halshs-00563926/document
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    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    2. Mustapha Ridaoui & Michel Grabisch, 2016. "Choquet integral calculus on a continuous support and its applications," Operations Research and Decisions, Wroclaw University of Technology, Institute of Organization and Management, vol. 1, pages 73-93.
    3. Ulrich Faigle & Michel Grabisch & Andres Jiménez-Losada & Manuel Ordóñez, 2014. "Games on concept lattices: Shapley value and core," Documents de travail du Centre d'Economie de la Sorbonne 14070, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Michel Grabisch, 2015. "Fuzzy Measures and Integrals: Recent Developments," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477514, HAL.
    5. repec:hal:cesptp:hal-00803233 is not listed on IDEAS
    6. repec:spr:annopr:v:244:y:2016:i:2:d:10.1007_s10479-012-1166-6 is not listed on IDEAS
    7. Ulrich Faigle & Michel Grabisch & Andres Jiménez-Losada & Manuel Ordóñez, 2014. "Games on concept lattices: Shapley value and core," Documents de travail du Centre d'Economie de la Sorbonne 14070, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.

    More about this item

    Keywords

    Choquet integral; belief function; measurability; set systems; Monge algorithm; supermodularity;

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