The concept of fuzzy measure was introduced by Sugeno in 1974. A notion of signed fuzzy measure is introduced in this paper, and its elementary properties are briefly discussed. An analogue of Hahn decomposition theorem is established under the null-null-additive condition. A version of the Jordan decomposition theorem is proved under the null-additive condition.
Fuzzy Shapley values are developed based on classical Shapley values and used to allocate profit among partners in virtual enterprises (VE). Axioms of the classical Shapley value are extended to Shapley values with fuzzy payoffs by using fuzzy sets theory. Fuzzy Shapley function is defined based on these extended axioms. From the viewpoint the allocation for each partner should be a crisp value rather a fuzzy membership function at the end of cooperation, a crisp allocation scheme based on fuzzy Shapley values is proposed.
