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On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making

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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, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Christophe Labreuche

    (THALES [France], SINCLAIR AI Lab - Saclay Industrial Lab for Artificial Intelligence Research - THALES [France] - TOTAL FINA ELF - EDF - EDF)

Abstract

The Choquet integral w.r.t. a capacity is a versatile tool commonly used in decision making. Its practical identification requires, however, to solve an optimization problem with exponentially many variables and constraints. The introduction of k-additive capacities, through the use of the Möbius transform, permits to reduce the number of variables to a polynomial size, but leaves the number of constraints exponential. When k = 2, the use of vertices of the set of 2-additive capacities permits to solve the problem as the number of vertices is polynomial. When k > 2, this solution is no more applicable as the set of vertices of k-additive capacities is not known. We propose in this paper to use instead the set of vertices which are 0-1 valued. We show that the number of such vertices is polynomial, and we observe that the loss of generality is very small for n = 4, k = 3, and conjecture that this still holds for larger values of n. Also, we study the geometric properties of the convex hull of 0-1 valued k-additive capacities.

Suggested Citation

  • Michel Grabisch & Christophe Labreuche, 2022. "On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making," Post-Print halshs-03881431, HAL.
    • Handle: RePEc:hal:journl:halshs-03881431
    DOI: 10.1016/j.fss.2022.03.018
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03881431
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    Keywords

    capacity k-additive capacity Choquet integral vertices facets; capacity; k-additive capacity; Choquet integral; vertices; facets;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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