Bi-capacities. I: Definition, Möbius transform and interaction
type: Item
instance of: scholarly article
title: Bi-capacities. I: Definition, Möbius transform and interaction
zbMATH Open document ID: 1106.91023
author: Michel Grabisch
author: Christophe Labreuche
published in: Fuzzy Sets and Systems
publication date: 2005-04-07T00:00:00Z
review text: Bi-capacities are introduced as extensions of capacities (fuzzy measures) in decision making and cooperative games. The theory of bi-capacities is developed. A bi-capacity is a real valued function \(v\) on \({\mathcal Q}(N)=\{(A,B) \in {\mathcal P}(N)\times{\mathcal P}(N): A \cap B = \emptyset\}\) with \(v(\emptyset,\emptyset)=0\), \(A\subseteq B\) implies \(v(A,\cdot) \leq v(B,\cdot)\) and \(v(\cdot,A) \geq v(\cdot,B)\) as well as \(v(N,\emptyset)=1=-v(\emptyset,N)\). The authors first analyse the structure of the lattices \({\mathcal Q}(N)\) and \({\mathcal Q}^*(N)\). They then introduce the Möbius transform of bi-capacities and prove the form of the Möbius function on \({\mathcal Q}(N)\), before considering several special cases of bi-capacities. Next, derivatives of bi-capacities are defined and expressed in terms of the Möbius transform. Considering bi-capacities as games, expressions for the Shapley value and the interaction index are derived in terms of the Möbius transform.
reviewed by: Matthias Ehrgott
Mathematics Subject Classification ID: 91B06
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 2153577
zbMATH Keywords: Capacity
zbMATH Keywords: Shapley value
MaRDI profile type: Publication
full work available at URL: j.fss.2004.08.012
OpenAlex ID: W2018696312
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DOI: 10.1016/J.FSS.2004.08.012
description: scientific article; zbMATH DE number 2153577