### Article

## Cовместная аксиоматизция пред n-ядра и решения Дутты-Рэя для выпуклых игр

Most of solutions for cooperative games with transferable utilities (TU) are covariant with respect to positive linear transformations of individual utilities. However, this property does not take into account interpersonal comparisons of players' payoffs. The constrained egalitarian solution defined by Dutta and Ray for the class of convex TU games, being not covariant, served as a pretext to studying non-covariant solutions. In the paper a weakening of covariance is given in such a manner that, together with some other properties, it characterizes only two solutions -- the prenucleolus and the Dutta--Ray solution -- on the class convex TU TU games.

Transport industry in economy had been studied for many years, however, only recently researchers have begun to widely apply concepts of cooperative game theory to optimize costs and profits which are incurred in hauling. Today a wide range of cost/profit allocation methods have become a trend in transport segment, particularly in logistics operations. The most of these methods based on cooperative game theory consider the effect of collaboration (cooperation) which means the integration of companies as a key way to share transportation costs or profits. This study aims to contribute to this area of research by exploring different allocation methods such as the Shapley value, the nucleolus and some other excess based solution concepts of transferable utility game (TU game). In this work we overview existing studies on the subject and consider methodology of cooperative game theory. Further, we calculate numerical example of three shipping companies based on real data. In order to compare profit sharing results we compute the set of allocations and examine the constructive and blocking power of coalitions. The importance and originality of the work are that it explores the new field of application of game theory in logistics which can provide additional insights in this research area

The most of solutions for games with non-transferable utilities (NTU) are NTU extensions of solution concepts defined for games with transferable utilities (TU). For example, there are three NTU versions of the Shapley value due to Aumann(1985), Kalai--Samet(1977), and Maschler--Owen(1992). The Shapley value is {\it standard} for two-person games. An NTU analog of standard solution is called the {\it symmetric proportional solution (SP)} (Kalai 1977), and the most of NTU solutions are SP solutions for two-person games. Another popular TU game solution which is not standard for two-person case is the {\it egalitarian Dutta-Ray solution (Dutta, Ray (1989), Dutta 1990). It was defined for the class of convex TU games and then extended to the class of all TU games (Branzei et al. 2006). . The DR solution for superadditive two-person TU games is the solution of constrained egalitarianism, it chooses the payoff vectors the closest to the diagonal of the space R^N. Its extension to superadditive two-person NTU games and then to n-person bargaining problems is the lexicographically maxmin solution}: for each game/bargaining problem it is the individually rational payoff vector which is maximal w.r.t. the lexmin relation. This solution if positively homogenous, but is not covariant w.r.t. shifts of individual payoffs. In the presentation this solution is extended to the class of NTU non-levelled games which are both ordinal and cardinal convex. Since convex TU games considered in NTU setting are ordinal and cardinal convex, the NTU DR solution is, in fact, an extension of the original TU version to the mentioned class of NTU games. It turns out that in this class the DR solution is single-valued and belongs to the core. A result similar to that of Dutta for TU convex games is obtained: the DR solution for the class of non-levelled ordinal and cardinal convex games is the single solution being the lexicographically maxmin solution for two-person games and consistent in (slightly modified) Peleg's definition (Peleg 1985) of the reduced games.

Cooperative games with a restricted cooperation, defined by an arbitrary collection of feasible coalitions are considered. For this class the Equal Split-Off Set (ESOS)is defined by the same way as for cooperative games with transferable utilities (TU). For the subclass of these games with non-empty cores the Lorenz-maximal solution is also defined by the same way as for TU games. It is shown that if the ESOS of a game with a restricted cooperation intersects with its core, then it is single-valued and Lorenz dominates other vectors from the core, i.e. it coincides with the Lorenz-maximal solution. Cooperative games with coalitional structure for which the collection of feasible coalitions consists of the coalitions of partition, their unions, and subcoalitions of the coalitions of the partition, are investigated more in detail. For these games the convexity property is defined, and for convex games with coalitional structure existence theorems for two egalitarian solutions -- Lorenz maximal and Lorenz-Kamijo maximal -- are proved. Axiomatic characterizations for both these solutions are given.

A cooperative game with restricted cooperation is a triple (N,v,Omega), where N is a finite set of players, Omega is a collection of feasible} coalitions, v:Omega -->R is a characteristic function. The definition implies that if Omega=2^N, then the game (N,v,Omega)=(N,v) is a classical cooperative game with transferable utilities (TU). The class of all games with restricted cooperation with an arbitrary {\it universal} set of players is considered. The prenucleolus for the class is defined in the same way as for classical TU games. Necessary and sufficient conditions on a collection Omega providing existence and singlevaluedness of the prenucleoli for the class into consideration are found. Axiomatic characterizations of the prenucleolus for games with two-type collections Omega generated by coalitional structures

Interval cooperative games are models of cooperative situation where only bounds for payoffs of coalitions are known with certainty. The extension of solutions of classical cooperative games to interval setting highly depends on their monotonicity properties. However. both the prenucleolus and the tau-value are not aggregate monotonic on the class of convex TU games Hokari (2000, 2001). Therefore, interval analogues of these solutions either should be defined by another manner, or perhaps they exist in some other class of interval games. Both approaches are used in the paper: the prenucleolus of a convex interval game is defined by lexicographical minimization of the lexmin relation on the set of joint excess vectors of lower and upper games. On the other hand, the tau-value is shown to satisfy extendability condition on a subclass of convex games -- on the class of totally positive convex games. The interval prenucleolus is determined , and the proof of non-emptiness of the interval \tau-value on the class of interval totally positive games is given.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.