IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v25y2000i2p157-178.html
   My bibliography  Save this article

Equivalent Representations of Set Functions

Author

Listed:
  • Michel Grabisch

    () (LIP6, University Pierre & Marie Curie (UPMC), 8, rue du Capitaine Scott, 75015 Paris, France)

  • Jean-Luc Marichal

    () (Department of Management, FEGSS, University of Liège, Boulevard du Rectorat 7---B31, B-4000 Liège, Belgium)

  • Marc Roubens

    () (Institute of Mathematics, University of Liège, Grande Traverse 12---B37, Campus du Sart Tilman, B-4000 Liège, Belgium)

Abstract

This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation.

Suggested Citation

  • Michel Grabisch & Jean-Luc Marichal & Marc Roubens, 2000. "Equivalent Representations of Set Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 157-178, May.
  • Handle: RePEc:inm:ormoor:v:25:y:2000:i:2:p:157-178
    DOI: 10.1287/moor.25.2.157.12225
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/moor.25.2.157.12225
    Download Restriction: no

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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. E. Borgonovo & C. L. Smith, 2011. "A Study of Interactions in the Risk Assessment of Complex Engineering Systems: An Application to Space PSA," Operations Research, INFORMS, vol. 59(6), pages 1461-1476, December.
    3. Kusunoki, Yoshifumi & Tanino, Tetsuzo, 2017. "Investigation on irreducible cost vectors in minimum cost arborescence problems," European Journal of Operational Research, Elsevier, vol. 261(1), pages 214-221.
    4. Ulrich Faigle & Michel Grabisch, 2016. "Bases and linear transforms of TU-games and cooperation systems," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 875-892, November.
    5. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2018. "An axiomatisation of the Banzhaf value and interaction index for multichoices games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01814977, HAL.
    6. Fabien Lange & László Kóczy, 2013. "Power indices expressed in terms of minimal winning coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 281-292, July.
    7. 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.
    8. Ulrich Faigle & Michel Grabisch, 2019. "Least Square Approximations and Linear Values of Cooperative Game," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02381231, HAL.
    9. Stephan Foldes & Peter L. Hammer, 2005. "Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 453-461, May.
    10. Bottero, M. & Ferretti, V. & Figueira, J.R. & Greco, S. & Roy, B., 2018. "On the Choquet multiple criteria preference aggregation model: Theoretical and practical insights from a real-world application," European Journal of Operational Research, Elsevier, vol. 271(1), pages 120-140.
    11. Massimo Marinacci & Luigi Montrucchio, 2005. "Ultramodular Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 311-332, May.
    12. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2018. "An axiomatisation of the Banzhaf value and interaction index for multichoice games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02381119, HAL.
    13. Petter Arnesen & Håkon Tjelmeland, 2015. "Fully Bayesian Binary Markov Random Field Models: Prior Specification and Posterior Simulation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 967-987, December.

    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:inm:ormoor:v:25:y:2000:i:2:p:157-178. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Matthew Walls). General contact details of provider: http://edirc.repec.org/data/inforea.html .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.