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## Ковариантные и контравариантные гомоморфизмы игр с отношениями предпочтения

In this paper we consider games with preference relations. The main optimality concept for such games is concept of equilibrium. We introduce a notion of homomorphism for games with preference relations and study a problem concerning connections between equilibrium points of games which are in a homomorphic relation. The main result is finding covariantly and contravariantly complete families of homomorphisms.

In this paper, we consider the following problem - what affects the amount of investment in knowledge when one of the network firms enters another innovation network. The solution of this problem will allow us to understand exactly how innovative companies will behave when deciding whether to enter the innovation network of another country or region, what conditions affect it and how the level of future investments in knowledge can be predicted.

An approach to sensitivity (stability) analysis of nondominated alternatives to changes in the bounds of intervals of value tradeoffs, where the alternatives are selected based on interval data of criteria tradeoffs is proposed. Methods of computations for the analysis of sensitivity of individual nondominated alternatives and the set of such alternatives as a whole are developed.

The paper proposes a list of requirements for a game able to describe individually motivated social interactions: be non-cooperative, able to construct multiple coalitions in an equilibrium and incorporate intra and inter coalition externalities. For this purpose the paper presents a family of non-cooperative games for coalition structure construction with an equilibrium existence theorem for a game in the family. Few examples illustrate the approach. One of the results is that efficiency is not equivalent to cooperation as an allocation in one coalition. Further papers will demonstrate other applications of the approach.

The concept of the inclusion map of game with preference relations into a game with payoff functions is introduced. Necessary and sufficient conditions of the embeddability of game in factor-game are indicated. A necessary condition and also sufficient conditions for the existence of the inclusion of game with preference relations into a game with payoff functions are found.

The ninth issue of annual Collection of articles consists of four sections: “Analysis of actual economic processes”, “Modeling of financial and market mechanisms”, “Dynamic models”, “Discussions, Notes and Letters”. As a whole nine articles are presented

In this paper, we consider the following problem - what affects the Nash equilibrium amount of investment in knowledge when some agents of the complete graph enter another full one. The solution of this problem will allow us to understand exactly how game agents will behave when deciding whether to enter the other net, what conditions and externalities affect it and how the level of future equilibrium amount of investments in knowledge can be predicted.

For n person games with preference relations some types of optimality solutions are introduced. Elementary properties of their solutions are considered. One sufficient condition for nonempty Ca-core is found.

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.