Axiomatizations of Signed Discrete Choquet Integrals

Abstract

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.

Discrete Integrals and Axiomatically Defined Functionals

“…As an example, we mention the axiomatization of a Choquet type integral based on a signed capacity given in [39].…”

Discrete Integrals Based on Comonotonic Modularity

“…We also recall some axiomatizations of these function classes. For general background, see [3,8,9] A capacity on X = {1, . .…”

“…More generally, signed Choquet integrals, which need not be nondecreasing in their arguments, and the Lovász extensions of pseudo-Boolean functions, which need not vanish at the origin, are natural extensions of the Choquet integrals and have been thoroughly investigated in aggregation theory. For recent references, see, e.g., [3,8].…”