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## Evolutionary games with randomly changing payoff matrices

Evolutionary games are used in various elds stretching from economics to biology. Most assume a constant payoff matrix, although some works also consider dynamic payoff matrices. In this article we propose a possibility of switching the system between two regimes with different sets of payoff matrices. Such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A nite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution by applying the Hamilton-Jacobi formalism. Therefore, we present an exactly solvable version of an evolutionary game with annealed noise in the payoff matrix.

Game theory is the mathematical discipline aimed to model various interactions of living organisms in quantitative terms. Game theory, as the universal method for the analysis of social interactions has wide applications in economics, in the theory of control and management, financial mathematics, evolutionary biology, sociology, psychology and politics, in modelling different social processes, in particular, the processes of democratic elections, processes of fair distributions of resource, processes of arms control, etc. The book is designed for all wishing to get acquainted with main ideas and methods of game theory.

The first part gives an elementary but systematic exposition of the main ideas of modern game theory without any special prerequisites in mathematics (secondary school level should be quite sufficient). The second part is devoted mostly to the mathematical methods of the theory.

To stimulate mathematical and scientific imagination and to add charm to the book we illustrate it by carefully selected artistic graphics of a world renowned mathematician and mathematics imaging artist A.T. Fomenko (to whom the authors express their deepest gratitude for allowing to use his works in this text). Though these works were originally designed to illustrate the geometric structure of the Universe, they fit nicely in the circle of ideas dealt with here.

For the second edition the book was essentially revised and updated. When preparing the 1st part of our book we aimed at selecting material reflecting the most fundamental part of the theory that we did not expected to become outdate soon. Therefore the first part remained mostly unchanged (with couple of new insightful examples added). On the other hand, for the second part, apart from correcting typos, lots of new material was prepared. In particular, Chapter 10 was essentially enlarged by more recent developments on game-theoretic treatment of option pricing and two completely new chapters were written: Chapter 12 on games with many agents analysed via the statistical limit of infinite number of players, and Chapter 13 on quantum games. These two topics represent arguably the most bright new trends in the 21 st century game theory. Due to strong physics input, the theory of quantum games still remains separated from the main stream of game theory, and our exposition is possibly the first appearance of quantum games in a mathematics textbook.

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.

We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S|/ + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/. We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

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.