Bi-capacities—I: definition, Möbius transform and interaction

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 first part is devoted to the introduction of bi-capacities and the structure on which they are defined. We define the Möbius transform of bi-capacities, by just applying the well-known theory of Möbius functions as established by Rota to the particular case of bi-capacities. Then, we introduce derivatives of bi-capacities, by analogy with what was done for pseudo-Boolean functions (another view of capacities and set functions), and this is the key point to introduce the Shapley value and the interaction index for bi-capacities. This is done in a cooperative game theoretic perspective. In summary, all familiar notions used for fuzzy measures are available in this more general framework.

A value for bi-cooperative games

“…It is an expected value computed from the value defined earlier. The expression we obtain is different from the existing ones [5,7,2]. Section 5 discusses on the relevance of our value on several examples.…”

“…Three different definitions of a value have been proposed for bi-cooperative games [5,7,2]. Beyond the question of the choice of the best suited value among the three previous ones, the interpretation of the notion of solution for a bi-cooperative game is not so obvious.…”

“…This axiom is technical and lacks of clear interpretation. As a result, the solutions defined in [5,7,2] cannot be viewed as a share of the wealth of the players but represents a mean prospect over the three possible choices of all players (i.e. being positive or negative contributor, or do not participate).…”

“…A second proposal was done by the authors in [7], which distinguishes the cases where i is a positive contributor (denoted by φ + ) or a negative contributor (denoted by φ − ):…”

A Survey of Bicooperative Games

“…Throughout this chapter, we will write S ∪ i and S \ i instead of S ∪ {i} and S \ {i} respectively. Grabisch and Labreuche [9] proposed a relation in 3 N given by…”