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Bi-capacities -- Part 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 M\" obius 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. Thi is done in a cooperative game theoretic perspective. In summary, all familiar notions used for fuzzy measures are available in this more general framework.
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Contributor : Michel Grabisch <>
Submitted on : Tuesday, November 13, 2007 - 6:09:49 PM
Last modification on : Thursday, July 2, 2020 - 12:48:01 PM
Long-term archiving on: : Monday, April 12, 2010 - 2:06:48 AM


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Michel Grabisch, Christophe Labreuche. Bi-capacities -- Part I: definition, Möbius transform and interaction. Fuzzy Sets and Systems, Elsevier, 2005, 151 (2), pp.211-236. ⟨10.1016/j.fss.2004.08.012⟩. ⟨hal-00187189⟩



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