IDEAS home Printed from https://ideas.repec.org/p/hal/journl/halshs-03881431.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03881431
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-03881431/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    More about this item

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:halshs-03881431. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.