Bi-capacities - Part I: definition, Möbius transform and interactionReport as inadecuate

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1 DECISION LIP6 - Laboratoire d-Informatique de Paris 6 2 CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique 3 Thales Research and Technology Palaiseau

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.

Keywords : Interaction index Shapley value Bi-cooperative game Möbius transform Fuzzy measure Capacity Bi-capacity

Author: Michel Grabisch - Christophe Labreuche -



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