Bi-capacities Part II: The Choquet integral1

Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a con-text of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models as Cumulative Prospect The-ory (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 rmly rooted in decision theory, notably cooperative game theory. The present second part focuses on the denition of Choquet integral. We give several expressions of it, including an expression w.r.t. the Mobius transform. This permits to give an expression of the Choquet integral for 2-additive bi-capacities w.r.t. the interaction index