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

TL;DR

This paper introduces quasi-Lovász extensions, a class of functions generalizing Lovász extensions, with axiomatizations relevant for decision making under uncertainty and involving transformations of variables.

Contribution

It provides new axiomatic characterizations of quasi-Lovász extensions, including symmetric variants, expanding the theoretical framework for these functions.

Findings

Axiomatization of quasi-Lovász extensions using generalized properties.

Characterization of symmetric quasi-Lovász extensions.

Relevance to decision making under uncertainty and utility transformations.

Abstract

We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,...,x_n)=L(\phi(x_1),...,\phi(x_n))$, where $L$ is the Lov\'asz extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\R$ (i.e., the function $L\colon\R^n\to\R$ whose restriction to each simplex of the standard triangulation of $[0,1]^n$ is the unique affine function which agrees with $\psi$ at the vertices of this simplex) and $\phi\colon I\to\R$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations of properties used to characterize the Lov\'asz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lov\'asz extensions, which are compositions of symmetric Lov\'asz extensions with 1-place nondecreasing odd functions.