Bi-capacities -- Part II: the Choquet integral

TL;DR

This paper introduces the Choquet integral for bi-capacities, a generalization of fuzzy measures in bipolar decision-making, providing mathematical expressions and tools for analyzing 2-additive cases.

Contribution

It defines the Choquet integral for bi-capacities, including multiple expressions and the interaction index for 2-additive bi-capacities, advancing the theoretical framework.

Findings

Multiple expressions of the Choquet integral for bi-capacities

Formulation of the integral with respect to the Möbius transform

Expression of the integral for 2-additive bi-capacities using the interaction index

Abstract

Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as Cumulative Prospect Theory (CPT). The aim of this paper in two parts is to present the machinery behind bi-capacities, and thus remains on a rather theoretical level, although some parts are firmly rooted in decision theory, notably cooperative game theory. The present second part focuses on the definition of Choquet integral. We give several expressions of it, including an expression w.r.t. the M\"obius transform. This permits to express the Choquet integral for 2-additive bi-capacities w.r.t. the interaction index.