Bi-capacities -- Part II: the Choquet integral

Michel Grabisch (1, 2) , Christophe Labreuche (3)

1 DECISION
2 CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique
3 Thales Research and Technology [Palaiseau]

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.

Keywords: Fuzzy measure; Capacity; Bi-capacity; Choquet integral

Domains: Discrete Mathematics [cs.DM]; Computer Science and Game Theory [cs.GT]

Michel Grabisch, Christophe Labreuche. Bi-capacities -- Part II: the Choquet integral. Fuzzy Sets and Systems, 2005, 151 (2), pp.237-259. ⟨10.1016/j.fss.2004.08.013⟩. ⟨hal-00187191⟩