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

Bi-capacities -- Part II: the Choquet integral