An Analog of the Bondareva-Shapley Theorem I. The Non-Emptiness of the Core of a Fuzzy Game

This paper deals with a generalization of the famous Bondareva-Shapley theorem [1, 9] on the core of TU cooperative games to the case of fuzzy blocking. The suggested approach is based on the concept of a balanced collection of fuzzy coalitions. Introduced by the author, this extension of the classical balanced collection of standard coalitions yields a natural analog of balancedness for the so-called fuzzy TU cooperative games. As established below, the general balancedness is a necessary and sufficient condition for the non-emptiness of the core of fuzzy TU cooperative games. The non-emptiness criterion of the core is further refined using the classical Helly's theorem on the intersection of convex sets. The S*-representation of a fuzzy game is studied, which simplifies the existence conditions for non-blocking imputations of this game in a series of cases.