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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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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 loss of generality is small, and that the number of such vertices is polynomial. Also, we study the geometric properties of the convex hull of 0-1 valued k-additive capacities

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  • Michel Grabisch & Christophe Labreuche, 2022. "On the convex hull of K-additive 0-1 capacities and its application to model identification in decision making," Documents de travail du Centre d'Economie de la Sorbonne 22003, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:22003
    DOI: 10.1016/j.fss.2022.03.018
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    Keywords

    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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