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.
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-00187189
Contributor : Michel Grabisch <>
Submitted on : Tuesday, November 13, 2007 - 6:09:49 PM
Last modification on : Thursday, March 21, 2019 - 2:49:08 PM
Long-term archiving on : Monday, April 12, 2010 - 2:06:48 AM

Files

fss04I.pdf
Files produced by the author(s)

Identifiers

Citation

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⟩

Share

Metrics

Record views

298

Files downloads

184