Unblocked imputations of fuzzy games I. Existence

In the paper, a generalization of the famous Scarf theorem on the core of NTU cooperative game is established. The generalization considered deals with an extension of classic blocking via ordinary coalitions to the blocking via the so-called fuzzy coalitions. A well-known concept of a balanced family of standard coalitions is extended to the case of an arbitrary set of fuzzy coalitions, thus making it possible to introduce a natural analog of a balanced game for the characteristic function with arbitrary efficiency domain. Applying an appropriate approximation of a fuzzy game by finitely-generated games, together with the seminal combinatorial Scarf lemma, we obtain a rather general existence theorem for unblocked imputations of an $F$-balanced NTU fuzzy cooperative game.