Kybernetika 44 no. 3, 416-429, 2008

Disjointness of fuzzy coalitions

Milan Mareš and Milan Vlach


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 by Mareš and Vlach in [9,10,11]. The relation between this model of fuzziness and the original one, where fuzzy coalition is a fuzzy set of players, is shown and the properties of the model are analyzed and briefly interpreted in this paper. The analysis is focused on very elementary properties of fuzzy coalitions and their payments like disjointness, superadditivity and also convexity. Three variants of their modelling are shown and their consistency is investigated. The derived results may be used for further development of the theory of fuzzy coalitions characterized by fuzzy sets of crisp coalitions. They show that the procedure developed in [11] appears to be the most adequate.