### A separation theorem for finite families

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The uncertainty of expectations and vagueness of the interests belong to natural components of cooperative situations, in general. Therefore, some kind of formalization of uncertainty and vagueness should be included in realistic models of cooperative behaviour. This paper attempts to contribute to the endeavour of designing a universal model of vagueness in cooperative situations. Namely, some initial auxiliary steps toward the development of such a model are described. We use the concept of fuzzy...

The main purpose of this article is to provide an exact theory of the dynamic programming on a sufficiently general basis. Let $M$ be a compact topological Hausdorff’s space, let ${\tilde{T}}^{M}$ be the set of all continuous transformations of the space $M$ into itself. Suppose such a topology is introduced on ${\tilde{T}}^{M}$ that ${\tilde{T}}^{M}$ is Haousdorff’s space and that the transformation $\phi (x,y)=y\left(x\right)$ of the product $M\otimes {\tilde{T}}^{M}$ into $M$ is continuous with respect to Tichonoff’s topology on $M\otimes {\tilde{T}}^{M}$. Suppose ${\tilde{T}}^{M}$ is a compact subspace of ${\tilde{T}}^{M}$ and $\mathcal{M}=M\otimes {T}^{M}\otimes ...\otimes {T}^{M}\otimes ...$. We define the transformations...

In this note a class of convex polyhedral sets of functions is studied. A set of the considered class is non-emplty if it satisfies certain conditions. Using Theorem 1 of this paper in the case of multi-index transportations problems we obtain necessary conditions for the existence of a feasible solution to this problem.

One of the possible models of fuzzification of non-transferable utility (NTU) coalitional games was extensively treated in [4]. In this paper, we suggest an alternative structure of fuzzification of the NTU games, where for every coalition a fuzzy class of (generally crisp) sets of its admissible pay-off vectors is considered. It is shown that this model of a fuzzy coalitional game can be represented by a fuzzy class of deterministic NTU games, and its basic concepts like the superadditivity or...

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.

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...

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...

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