### Article

## Existence of Equilibrium Strategies in Fuzzy Stochastic Games with Finite Sets of States and Decisions

Noncooperative discounted stochastic n-person games are considered; the payoffs at each step are represented by trapezoidal fuzzy numbers. The existence of stationary Nash equilibrium strategies is proved.

The article substantiates the concept of quantitative assessment of knowledge uncertainty in accident reconstruction tasks based on application of mathematical tools of the fuzzy set theory allowing considering an uncertainty of initial data caused for instance by varying resistance to the motion of investigated objects at the apex stages of the contact–separation processes. The application of the mathematical tools of the fuzzy set theory can substantially expand the potential of applying the methodology to the automobile and technical expertise and provide the enhancement of authenticity and improve the accuracy of making conclusions about the accident reconstruction results.

We study existence of Nash equilibria (NE) in pure stationary strategies in n-person positional games with no moves of chance, with perfect information, and with the mean or total effective cost function.

We construct a NE-free three-person game with positive local costs, thus disproving the conjecture suggested in Boros and Gurvich (2003). Still, the following four problems remain open: Whether NE exist in all *two*-person games with total effective costs such that

(I) all local costs are strictly positive *or* (II) there are no dicycles of the total cost zero?

Whether NE exist in all n-person games with the terminal (transition-free) cost functions, provided all dicycles form a unique outcome c and

(III) assuming that c is worse than any terminal outcome *or* (IV) without this assumption?

For n=3 the case (I) (and hence (II)) is answered in the negative. This is the main result of the present paper. For n=2 the case (IV) (and hence (III)) was answered in the positive earlier.

We briefly survey the above and some other negative and positive results on Nash-solvability in pure stationary strategies for the games under consideration.

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real ϵϵ, let us call a stochastic game ϵϵ-ergodic, if its values from any two initial positions differ by at most ϵϵ. The proposed new algorithm outputs for every ϵ>0ϵ>0 in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an ϵϵ-range, or identifies two initial positions uu and vv and corresponding stationary strategies for the players proving that the game values starting from uu andvv are at least ϵ/24ϵ/24 apart. In particular, the above result shows that if a stochastic game is ϵϵ-ergodic, then there are stationary strategies for the players proving 24ϵ24ϵ-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980) claiming that if a stochastic game is 00-ergodic, then there are ϵϵ-optimal stationary strategies for every ϵ>0ϵ>0. The suggested algorithm extends the approach recently introduced for stochastic games with perfect information, and is based on the classical potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.

The construction of time series linguistic summaries is a topic that draws attention of researchers for many years. The full-fledged software implementation (the pilot web-application) that supports the complete process of linguistic summarization of time series construction is presented in the paper. The program can be used in professional groups for discussions and rapid data analysis. Virtual mobile crash reporting system (MCRS) supplies the test input data used as an example.

We discuss the video classification problem with the matching of feature vectors extracted using deep convolutional neural networks from each frame. We propose the novel recognition method based on representation of each frame as a sequence of fuzzy sets of reference classes whose degrees of membership are defined based on asymptotic distribution of the Kullback–Leibler information divergence and its relation with the maximum likelihood method. In order to increase the classification accuracy, we perform the fuzzy intersection (product triangular norms) of these sets. Experimental study with YTF (YouTube Faces) and IJB-A (IARPA Janus Benchmark A) video datasets and VGGFace, ResFace and LightCNN descriptors shows that the proposed approach allows us to increase the accuracy of recognition by 2–6% compering with the known classification methods.

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.