A probabilistic communication structure considers the setting with communication restrictions in which each pair of players has a probability to communicate directly. In this paper, we consider a more general framework, called a probabilistic communication structure with fuzzy coalition, that allows any player to have a participation degree to cooperate within a coalition. A maximal product spanning tree, indicating a way of the greatest possibility to communicate among the players, is introduced where the unique path from one player to another is optimal. We present a feasible procedure to find the maximal product spanning trees. Furthermore, for games under this model, a new solution concept in terms of the average tree solution is proposed and axiomatized by defining a restricted game in Choquet integral form.
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.