Bipolarization of posets and natural interpolation

TL;DR

This paper generalizes the Choquet integral using lattice geometry and introduces a bipolarization mechanism for ordered structures, with applications in multicriteria decision making.

Contribution

It extends the Choquet integral to a broad geometric framework and proposes a novel bipolarization scheme for various ordered structures.

Findings

Generalized Choquet integral via lattice triangulation

Introduced bipolarization mechanism for ordered structures

Applied to multicriteria aggregation with multiple reference levels

Abstract

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.