Exact bounds of the Möbius inverse of monotone set functions

Michel Grabisch (1, 2) , Pedro Miranda (3)

1 PSE - Paris School of Economics
2 CES - Centre d'économie de la Sorbonne
3 UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid]

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.

Mots clés: Möbius inverse; monotone set function; interaction

Domaines: Mathématique discrète [cs.DM]; Economies et finances

Michel Grabisch, Pedro Miranda. Exact bounds of the Möbius inverse of monotone set functions. Discrete Applied Mathematics, 2015, 186, pp.7-12. ⟨hal-01136668⟩