Disjointness of fuzzy coalitions

The cooperative games with fuzzy coalitions in which some players act in a coalition only with a fraction of their total “power” (endeavor, investments, material, etc.) or in which they can distribute their “power” in more coalitions, are connected with some formal or interpretational problems. Some of these problems can be avoided if we interpret each fuzzy coalition as a fuzzy class of crisp coalitions, as shown in [9, 10, 11]. In this paper, the relation between this model of fuzziness and the original one (in which a fuzzy coalition is a fuzzy set of players) is elucidated, and properties of the model are analyzed and briefly interpreted. The analysis is focused on the concept of disjointness of fuzzy coalitions and its role in the study of superadditivity of games with fuzzy coalitions. In particular, three variants of disjointness are introduced and their consistency is discussed. The derived results may be used for further development of the theory of games with fuzzy coalitions characterized by fuzzy sets of crisp coalitions. They show that the procedure developed in [11] appears to be the most adequate.