### Working paper

## The SD-prekernel for TU games

A collection of TU games solutions intermediate between the prekernel and the prenucleolus is considered. All these solutions are Davis-Maschler consistent, symmetric and covariant. Each solution from the collection is parametrized by a positive integer k such that for all games with the number of players not greater than k, the solution for parameter k coincides with the prenucleolus, and for games with more than k players it is maximal, i.e. it satisfies the "k-converse consistency". The properties of solutions are described and their characterization in terms of balancedness is 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

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

We investigate the prenucleolus, the anti-prenucleolus and the SM-nucleolus in glove market games and weighted majority games. This kind of games looks desirable for considering solution concepts taking into account the blocking power of a coalition S with different weights. Analytical formulae for calculating the solutions are presented for glove market game. Influence of the blocking power on players' payoffs is discussed and the examples which demonstrate similarities and differences comparing with other solution concepts are given

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.

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.