Quasi-Lovász Extensions and Their Symmetric Counterparts

Abstract

We introduce the concept of quasi-Lovász extension as being a mapping $f: I^n \rightarrow \textrm{I\!R}$ defined over a nonempty real interval I containing the origin, and which can be factorized as f(x ₁,…,x _n ) = L(ϕ(x ₁),…,ϕ(x _n )), where L is the Lovász extension of a pseudo-Boolean function $\psi:\{0,1\}^n\rightarrow \textrm{I\!R}$ (i.e., the function $L:\textrm{I\!R}^n\rightarrow \textrm{I\!R}$ whose restriction to each simplex of the standard triangulation of [0,1]ⁿ is the unique affine function which agrees with ψ at the vertices of this simplex) and $\varphi\colon I\rightarrow \textrm{I\!R}$ 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-Lovász extensions, we propose generalizations of properties used to characterize the Lovász 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-Lovász extensions, which are compositions of symmetric Lovász extensions with 1-place nondecreasing odd functions.