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## A Distributed Replicator System Corresponding to a Bimatrix Game

Reaction–diffusion type replicator systems are investigated for the case of a bimatrix. An approach proposed earlier for formalizing and analyzing distributed replicator systems with one matrix is applied to asymmetric conflicts. A game theory interpretation of the problem is described and the relation between dynamic properties of systems and their game characteristics is determined. The stability of a spatially homogeneous solution for a distributed system is considered and a theorem on maintaining stability is proved. The results are illustrated with two-dimensional examples in the case of distribution.

This article concerns the problem of predicting the size of company's customer base in case of solving the task of managing its clients. The author purposes a new approach to segment-oriented predicting the size of clients based on adopting the Staroverov's employees moving model. Besides the article includes the limitations of using this model and its modification for each type of relations of the client and the company.

Cooperative game theory instruments application to the corporate finance M&A research issues provide an ability to extend the field considered and conclusions obtained. The paper presents the M&A cooperative games modeling and its empirical implementation to analyze the airline strategic alliance as M&A deal.

The authors investigate behavioural assumptions underlying the normal performance of market economy. It is assumed that a model of man adequate for market economy can be deduced from the ideal-typical properties of the latter. The main components of such model are rationality and morality. Main ethical categories relevant for market economy are analyzed: trust, justice, equality, virtues, freedom as well as their treatment in modern economics. Behavioural properties specifi c for modern Russian economy are discussed.

In this paper, we want to introduce experimental economics to the field of data mining and vice versa. It continues related work on mining deterministic behavior rules of human subjects in data gathered from experiments. Game-theoretic predictions partially fail to work with this data. Equilibria also known as game-theoretic predictions solely succeed with experienced subjects in specific games – conditions, which are rarely given. Contemporary experimental economics offers a number of alternative models apart from game theory. In relevant literature, these models are always biased by philosophical plausibility considerations and are claimed to fit the data. An agnostic data mining approach to the problem is introduced in this paper – the philosophical plausibility considerations follow after the correlations are found. No other biases are regarded apart from determinism. The dataset of the paper “Social Learning in Networks” by Choi et al 2012 is taken for evaluation. As a result, we come up with new findings. As future work, the design of a new infrastructure is discussed.

The collection contains papers accepted for the Fourth International Conference Game Theory and Management (June 28–30, 2010, St. Petersburg University, St. Petersburg, Russia). The presented papers belong to the field of game theory and its applications to management. The volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments.

In work the developed model of adaptive management by the vertically integrated companies based on the system approach supporting the mechanism of an operational management in a uniform cycle of strategic planning, within the limits of faster time is presented. Thus for a finding of optimum values of operating parameters special algorithms of a class of genetic algorithms are used, neural networks the example of the developed system of adaptive management for the vertically-integrated oil company is etc. presented.

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.