Bi-capacities—II: the Choquet integral

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 Möbius transform. This permits to express the Choquet integral for 2-additive bi-capacities w.r.t. the interaction index.

A Characterization of the 2-Additive Symmetric Choquet Integral Using Trinary Alternatives

“…The symmetric Choquet integral, also calledŠipoš integral, is a simple way to generalize Choquet integral for bipolar scales [5,8,9]. Given an alternative x := (x 1 , ..., x n ) ∈ X and a 2-additive capacity, the expression of the symmetric Choquet integral is given by:…”

“…Because this aggregation function uses bipolar scales, it can be also defined from a symmetric bi-capacity, see [9] for more details about this concept. We just recall that a function ν : 3 N → R is a bi-capacity on 3 N if it satisfies the following two conditions :…”

“…Bipolar scale is defined as a scale composed of a negative, a positive and a neutral part which respectively allow representing a negative, a positive and a neutral affect towards an option. The Choquet integral has been extended to the bipolar scale based on the notion of capacity or on the notion of bi-capacity [5,8,9]. One of these extensions is the symmetric Choquet integral, also calleď Sipoš integral, defined by a capacity.…”

The core of bicapacities and bipolar games

“…Monotonic bicooperative games satisfying v(N, ∅) = 1 and v(∅, N) = −1 are called bicapacities [8,9]. A specific section is devoted to bicooperative games and bicapacities (Sec.…”

Identification of a 2-Additive Bi-Capacity by Using Mathematical Programming

“…To apply the Choquet integral in the case of bipolar scales, the bipolar Choquet integral (BCI) was introduced in [6] and [7]. In this paper, we particularly focus on BCI which use the concept of a bi-capacity (BC) introduced in [6] and which was further studied in [8,9] and in [10,11]. The BCI typically requires the DM to set 3 n − 1 values where n is the number of attributes.…”