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.
In this paper we deal with mathematical modeling of real processes that are based on preference relations in the sense that, for every pair of distinct alternatives, the processes are linked to a value of preference degree of one alternative over the other one. The use of preference relations is usual in decision making, psychology, economics, knowledge acquisition techniques for knowledge-based systems, social choice and many other social sciences. For designing useful mathematical models of such processes, it is very important to adequately represent properties of preference relations. We are mainly interested in the properties of such representations which are usually called reciprocity, consistency and transitivity. In decision making processes, the lack of reciprocity, consistency or transitivity may result in wrong conclusions. That is why it is so important to study the conditions under which these properties are satisfied. However, the perfect consistency or transitivity is difficult to obtain in practice, particularly when evaluating preferences on a set with a large number of alternatives. Under different preference representation structures, the multiplicative and additive preference representations are incorporated in the decision problem by means of a transformation function between multiplicative and additive representations. Some theoretical results on relationships between multiplicative and additive representations of preferences on finite sets are presented and some possibilities of measuring their consistency or transitivity are proposed and discussed. Illustrative numerical examples are provided.
Using players' Shapley-Shubik power indices, Peleg [4] proved that voting by count and account is more egalitarian than voting by account. In this paper, we show that a stronger shift in power takes place when the voting power of players is measured by their Shapley-Shubik indices. Moreover, we prove that analogous power shifts also occur with respect to the absolute Banzhaf and the absolute Johnston power indices.