Axiomatizations of Lovasz extensions of pseudo-Boolean functions

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.

💡 Research Summary

The paper investigates three fundamental properties in aggregation theory—horizontal min‑additivity, horizontal max‑additivity, and comonotonic additivity—by interpreting them as relaxed forms of the multivariate Cauchy functional equation. The authors first define each property precisely: horizontal min‑additivity requires that for any vectors x and y, the function value at x can be decomposed into the sum of its values at the componentwise minimum of x and y and at the residual vector; horizontal max‑additivity is the analogous condition using the componentwise maximum; comonotonic additivity demands linearity when the arguments are comonotonic (i.e., share the same ordering of components).

Through a series of functional‑equation arguments, the paper proves that these three conditions are equivalent. The key step is showing that any function satisfying one of the properties must admit a representation of the form
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