Exact bounds of the M{\"o}bius inverse of monotone set functions

Abstract

We give the exact upper and lower bounds of the M{\"o}bius 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 $\sqrt$ $\pi$n/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 M{\"o}bius inverse for the subfamilies of k-additive normalized capacities and p-symmetric normalized capacities.