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

TL;DR

This paper precisely determines the maximum and minimum values of the M{"o}bius inverse for monotone normalized set functions, revealing their asymptotic behavior and providing bounds for related transforms and specific subclasses.

Contribution

It introduces exact bounds for the M{"o}bius inverse of monotone normalized set functions and related transforms, including for specific subclasses like k-additive and p-symmetric capacities.

Findings

Bounds tend to 4^{n/2} * sqrt(pi*n/2) as n grows large

Exact bounds established for the interaction and Banzhaf interaction transforms

Bounds derived for k-additive and p-symmetric normalized capacities

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.