Axiomatizations of Lov\'asz extensions of pseudo-Boolean functions

TL;DR

This paper characterizes functions related to aggregation properties like Lovász extensions and Choquet integrals, showing their equivalence under certain conditions and introducing a new median-additivity concept.

Contribution

It provides a complete description of functions satisfying key aggregation properties and introduces horizontal median-additivity, unifying and extending existing classes.

Findings

Properties are equivalent and characterize Lovász extensions.

Regularity conditions lead to functions coinciding with Lovász extensions.

Introduces horizontal median-additivity and describes the symmetric Lovász extension class.

Abstract

Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lov\'asz extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lov\'asz extensions, which includes the discrete symmetric Choquet integrals.