Quasi-Lov́sz Extensions on Bounded Chains

Abstract

We study quasi-Lovász extensions as mappings f:Cn →ℝ defined on a nonempty bounded chain C, and which can be factorized as f(x 1,...,x n) = L(φ(x1),...,φ(xn)), where L is the Lovász extension of a pseudo-Boolean function ψ:{0,1}n → ℝ and φ: C → ℝ is an order-preserving function. We axiomatize these mappings by natural extensions to properties considered in the authors' previous work. Our motivation is rooted in decision making under uncertainty: such quasi-Lovász extensions subsume overall preference functionals associated with discrete Choquet integrals whose variables take values on an ordinal scale C and are transformed by a given utility function φ: C → ℝ. Furthermore, we make some remarks on possible lattice-based variants and bipolar extensions to be considered in an upcoming contribution by the authors. © Springer International Publishing Switzerland 2014.