### Article

## Ограниченное с-ядро в играх с ограниченной кооперацией

A game with {\em{restricted cooperation}} is a triple (N,v,\Omega), where N is a finite set of players,

\Omega is a non-empty collection oft feasible coalitions , and v a characteristic function defined on \Omega.

U.Faigle (1989) obtained necessary and sufficient conditions for the non-emptiness of the core for games with restricted cooperation. Unlike the classical TU games the cores for games with restricted cooperation may be unbounded. Recently Grabisch and Sudh\"olter (2012) studied the core for games whose collections of feasible coalitions has a hierarchical structure generated by a partial order relation of players.For this class of games they proposed a new concept -- the bounded core -- whose definition can be extended to the general class of games with restricted cooperation as the union of all bounded faces of the core. For this class of games the bounded core can be empty even the core is not empty.

An axiomatization of the bounded core for the whole class of games with restricted cooperation is given with the help of axioms efficiency, boundedness, bilateral consistency, a weakening of converse consistency, and ordinality. Another axiomatization of the core is given for the subclass of games with non-empty cores that are bounded. The characterizing axioms are non-emptiness, covariance, boundedness, consistency, the reconfirmation property, superadditivity, and continuity.

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.

A cooperative finite-stage dynamic *n*-person prisoner's dilemma is considered. The time-consistent subset of the core is proposed. The Shapley value for the stochastic model of the *n*-person prisoner's dilemma is calculated in explicit form.

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

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.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.

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.