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Bi-capacities. I: Definition, Möbius transform and interaction. (English) Zbl 1106.91023
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.

MSC:
91B06 Decision theory
91A12 Cooperative games
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