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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
##### Keywords:
Capacity; Shapley value
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##### References:
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