### Article

## Пред n-ядра в играх с ограниченной кооперацией

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

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 chapter introduces and analyses the Surplus Distributor-prenucleolus for TU games, a lexicographic value that satisfies core stability, strong aggregate monotonicity and null player out property in the class of balanced games. The solution is characterized in terms of balanced collection of sets and can be easily computed in the class of monotonic games with veto players and in the class of bankruptcy games. The SD-prenucleolus stands out as the only known core solution that satisfies coalitional monotonicity in the class of convex games and in the class of veto balanced games. Further, the SD-prekernel for TU games is introduced and analysed.

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.

We offer a general approach to describing power indices that account for preferences as suggested by F. Aleskerov. We construct two axiomatizations of these indices. Our construction generalizes the Laruelle-Valenciano axioms for Banzhaf (Penrose) and Shapley-Shubik indices. We obtain new sets of axioms for these indices, in particular, sets without the anonymity axiom.

Smoking is a problem, bringing signifi cant social and economic costs to Russiansociety. However, ratifi cation of the World health organization Framework conventionon tobacco control makes it possible to improve Russian legislation accordingto the international standards. So, I describe some measures that should be taken bythe Russian authorities in the nearest future, and I examine their effi ciency. By studyingthe international evidence I analyze the impact of the smoke-free areas, advertisementand sponsorship bans, tax increases, etc. on the prevalence of smoking, cigaretteconsumption and some other indicators. I also investigate the obstacles confrontingthe Russian authorities when they introduce new policy measures and the public attitudetowards these measures. I conclude that there is a number of easy-to-implementanti-smoking activities that need no fi nancial resources but only a political will.

One of the most important indicators of company's success is the increase of its value. The article investigates traditional methods of company's value assessment and the evidence that the application of these methods is incorrect in the new stage of economy. So it is necessary to create a new method of valuation based on the new main sources of company's success that is its intellectual capital.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.