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Exact bounds of the Möbius inverse of monotone set functions

Author

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  • Michel Grabisch

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

  • Pedro Miranda

    (UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid])

Abstract

We give the exact upper and lower bounds of the Möbius inverse of monotone and normalized set functions (a.k.a. normalized capacities) on a finite set of n elements. We find that the absolute value of the bounds tend to 4 n/2 √ πn/2 when n is large. We establish also the exact bounds of the interaction transform and Banzhaf interaction transform, as well as the exact bounds of the Möbius inverse for the subfamilies of k-additive normalized capacities and p-symmetric normalized capacities.

Suggested Citation

  • Michel Grabisch & Pedro Miranda, 2015. "Exact bounds of the Möbius inverse of monotone set functions," Post-Print hal-01136668, HAL.
    • Handle: RePEc:hal:journl:hal-01136668
    Note: View the original document on HAL open archive server: https://hal.science/hal-01136668
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    Cited by:

    1. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 301-326, July.
    2. Arcidiacono, Sally Giuseppe & Corrente, Salvatore & Greco, Salvatore, 2021. "Robust stochastic sorting with interacting criteria hierarchically structured," European Journal of Operational Research, Elsevier, vol. 292(2), pages 735-754.
    3. Silvia Bortot & Ricardo Alberto Marques Pereira & Anastasia Stamatopoulou, 2020. "Shapley and superShapley aggregation emerging from consensus dynamics in the multicriteria Choquet framework," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 583-611, December.

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