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Bipolarization of posets and natural interpolation

Author

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  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Christophe Labreuche

    (Thales Research and Technology [Palaiseau] - THALES [France])

Abstract

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.

Suggested Citation

  • Michel Grabisch & Christophe Labreuche, 2008. "Bipolarization of posets and natural interpolation," Post-Print hal-00274267, HAL.
    • Handle: RePEc:hal:journl:hal-00274267
    DOI: 10.1016/j.jmaa.2008.02.008
    Note: View the original document on HAL open archive server: https://hal.science/hal-00274267
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    Cited by:

    1. Labreuche, Christophe & Grabisch, Michel, 2018. "Using multiple reference levels in Multi-Criteria Decision aid: The Generalized-Additive Independence model and the Choquet integral approaches," European Journal of Operational Research, Elsevier, vol. 267(2), pages 598-611.
    2. GRABISCH, Michel & LABREUCHE, Christophe & RIDAOUI, Mustapha, 2019. "On importance indices in multicriteria decision making," European Journal of Operational Research, Elsevier, vol. 277(1), pages 269-283.
    3. Christophe Labreuche & Michel Grabisch, 2016. "A comparison of the GAI model and the Choquet integral with respect to a k-ary capacity," Documents de travail du Centre d'Economie de la Sorbonne 16004, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Ulrich Faigle & Michel Grabisch, 2011. "A Discrete Choquet Integral for Ordered Systems," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00563926, HAL.

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