A Fuzzy-Core Extension of Scarf Theorem and Related Topics

The paper deals with a generalization of Scarf (Scarf, 1967) and Bondareva-Shapley (Bondareva, 1962; Shapley, 1967) theorems on the core of cooperative game to the case of fuzzy domination. The approach proposed is based on the concept of balanced collection of fuzzy coalitions, introduced by the author (Vasil'ev, 2012). This extension of classic notion of balanced collection of standard coalitions makes it possible to present a natural analog of balancedness for so-called fuzzy TU cooperative games. Moreover, it turns out that similar to the standard situation the new balancedness-like assumption is a necessary and sufficient condition for the non-emptyness of the core of fuzzy cooperative game with side payments.