In the present work the task of finding Nash equi-librium situation or finding the most preffered pure strategies in finite scalar m x n  bimatrix Γ(A, B)  game is studied. The problem of finding an equilibrium in the  Γ(A, B)  game has a long history, but due to the complexity of the known algorithms and methods, that cause various problems, its study is being continued today. Our  approach is different from these methods. In order to find an equilibrium situation in the pure strategies, we mean making a prediction by each player about the second player's behavior for choosing preferred pure strategy. Therefore we use a sequential strictly and weakly procedure of dominance for comparing of two pure strategies. Only in the case of strictly dominance the line of dominance has no meaning in order to maintain an equilibrium situation in pure strategies. We don’t have such kind of situation in every game. In this kind of game each player can act by different principles to define the most preferred pure strategy, that is based on making a prediction about his partner’s behavior by the player and that means the orientation on guaranteed levels in concrete situations. We mean the orientation in A and B  matrix games average V(A) and V( B^T)  playoffs obtained by the players using maximum optimal mixed strategies. Each player’s decisions are discussed about the usage the preferred pure strategies related to his partner’s actions. By using them he will gain much more, then in an equilibrium situation. All kinds of other actions are also discussed. Acceptable results are used for solution of high ranged  m x n  (m>2, n>2) Bimatrix  Γ(A, B)  games.

In this paper the existence problem of the equilibrium situation in differential antagonistic games with perfect information and lexicographic payoffs or in a  -dimensional vector-payoffs' game where criteria are strictly ranged with preference relation is studied. The players' dinamic is defined by vector differential x=f( t,x ,u ),y=g( t,y ,v )   equations, respectively control functions u( .), v(. ) and ∈〔0.T〕time  interval. This is a game Γ^L(x_0, y₀)=(Γ^1,...Γ^m ) where x_0, y₀ are starting positions in t=0 moment respectively the first and second players'. x(t)and y(t) are trajectories, the players final aim is finding their optimal variants. A lexicographic ε -equilibrium situation is defined in the game and the conditions of its existence are investiga-terd. These conditions are mainly about f and g  functi-ons. The main definitions are introduced and some results are formulated from theory of differential games with scalar payoff functions and independent move-ments, they are the main for getting results for analogic differential games in the case of lexicographic payoffs. Some auxiliary statements correctness are also establi-shed, on its basic it is proved that in Γ^L(x_0, y₀) game for any ε>0  there exists a lexicographic ε-equilibrium situation in pure strategies. 

Stochastic games are discussed as a priva-te class of a general dynamic games. A certain class of lexicographic noncooperative games is studied - lexi-cographic stochastic matrix games . The problem of the existence of Nash equilibrium is studied with two analyses - standard and nonstandard way. Standard means using the same kind of mixed strategies in case of scalar games. In this case in lexi-cographic stochastic matrix game Nash equilibrium may not be existed. Its existence takes place in relevant stochastic affine matrix game to the existence of Nash equilibrium. In game a set of Nash equi-librium is given by means of relevant stochastic affine matrix game's set of equilibrium. The sufficient condi-tions of the existance such affine game is proved. In nonstandard way of analyses we use such mixed stra-tegies, they use components with lexicog-raphic probabilites. In this case the kinds of subsets of a set of equilibrium in game are described.

A system of interpersonal relationship and its modeling in the form of finite noncooperative game is studied in this article by means of payoff functions. In such games for the main principle of optimality Nash’s Equilibrium Situation is acknowledged. The stages of development of Game Theory are analyzed including the modern situation. Two groups – nonsolidary and solidary of different behaviors characterized for the relationship are defined. The strategies of nonsolidary behavior characterized for the strategic relationships of the players are described and the strategies of solidary behavior are connected with negotiations and agreements. Teaching organization is defined as a management of system comprising a teacher (professor) and collective of pupils (students). Each participant of system has its own interest and difference from each other. This situation gives us a ground to consider some aspects of Game Theory model for optimal management of

The present work deals with lexicographic noncooperative (strategic) games in which the set of strategies of the players are metric compact spaces and the vector-functions of winning are continuous on the set of situations. In such games we introduce the definition of a weak nonstrict (determined by usual nonstrict lexicographic inequality) of Nash equilibrium situation in pure strategies. It has been defined the stability of such equilibrium situation and of lexicographic noncooperative game in relation to change of vector-functions of the winning of players, a problem of an equilibrium stable situation and availability of lexicographic noncooperative game has been studied. The conditions of their stability have been brought. The identification of the indicated conditions has been connected with those features of the task of lexicographic maximum that differs from the task of scalar maximum: the set of points of lexicographic maximum in the task of lexicographic maximum of continuous vector-function defined on metric compact is compact. And in the lexicographic noncooperative game the set of equilibrium situations may not be compact. In particular, it is certified that if in lexicographic game there is only one equilibrium situation then it is a stable situation and the relative game is stable.

The process of effective interrelation necessary for teaching the subject at higher school has been represented as a noncooperative game between the professor and the students. This process is the functioning of teaching of organizational  system which comprises -pedagogue (professor) and -collective of students. The preference is given to the democratic model of relation - to objective and optimal mutual responsibility of the pedagogue and a student to the rights-obligations imposed on them. Two classes of models of noncooperatve games corresponding to management of  system have been built - games with relations of preferences and the games with utility. The main principle of optimality is the Nash equilibrium, or it is such kind of situation, none of the player it is not profitable the unilateral deviation from it. According to the indicated principle of equilibrium the tasks originated in the process of  system functioning has been solved. According to the solving results students must study systamatically do their tasks and teachers must be responsible objective for their work.

In classical cooperative game theory one of the most important principle is defined by Shapley with three axioms common payoff fair distribution’s Shapley value (or Shapley vector). In the last decade the field of its usage has been spread widely. At this period of time Shapley value is used in network and social systems. Naturally, the question is if it is possible to use Shapley’s classical axiomatics for lexicographic cooperative games. Because of this in the article for dimensional lexicographic cooperative game is introduced Shapley’s axiomatics, as the principle of a fair distribution in the case of dimensional payoff functions, when the criteria are strictly ranking. It has been revealed that axioms discussed by Shapley for classical games are sufficient in lexicographic cooperative games corresponding with the payoffs of distribution. Besides we are having a very interesting case: according to the proved theorem, Shapley’s classical principle simultaneously transforms on the composed scalar games of a lexicographic cooperative game, nevertheless, games could not be superadditive.

Neumann-Morgenstern’s solutions NM (v) as stable solution’s optimal principle is stated in a lexicographic v = (v1, v2 ,..., vm)T cooperative game. The conditions of NM (v) existence are proved for the cases, when: 1. v1 scalar cooperative game’s C-core C (v1) and NM (v1)solutions are equal; 2. Scalar cooperative v1 game’s C-core and NM (v1) solutions are different. In the first case the sufficient conditions are proved in order to say that a C-core C (v) of a lexicographic cooperative v game must not be empty and it should be coincided to NM (v) . In the second case the necessary condition of NM (v) existence is proved. In the case of the existence of NM (v) solutions their forms can be established. Some properties NM (v) of solutions are stated.

A new concept of a mixed strategy is given for m-dimensional lexicographic noncooperative Γ(Γ^0,Γ^1,...,Γ^(m-1)) game when on a set of pure strategies m-dimensional probability distributions are given. In this case each Γ^k(k=0,1,...,m-1) criteria of Γ game corresponds to its probability distributions on sets of pure strategies. Besides, a lexicographic m-dimensional order relation is given on set of -dimensional probability distribution. The given construction is made by the methodology of nonstandard analysis Therefore, the given mixed strategy is called a nonstandard mixed strategy, and a lexicographic game in such strategies is called a nonstandard mixed extension. An equilibrium situation in mixed strategies is defined in Γ game. A nonstandard mixed extension of lexicographic matrix games is studied thoroughly. In such games, necessary and sufficient conditions of the existence of a saddle point are proved. The analyzed examples show that if in a lexicographic matrix game doesn’t exist a saddle point in standard mixed strategies then a saddle point maybe doesn’t exist in nonstandard mixed strategies. If in a lexicographic matrix game doesn’t exist a saddle point in standard mixed strategies then there can be existed a saddle point in nonstandard mixed strategies. Thus, lexicographic games’ nonstandard mixed distribution is a generalization of a standard mixed extension.

In the article dimensional lexicographic noncooperative games are defined for the players' for which there exists a characteristic function . Some main features are proved of function in a lexicographic case. A lexicographic cooperative game is called a couple, whereis a real vector-function on subsets and the following conditions are fulfilled =O; Such cooperative game is denoted by . imputation and its set is defined in game. It is proved that is nonempty and its full characterisation is given. Domination over set is defined. Thus, the main foundations are given, according to this, it is possible to explore the main principles of optimality.

A model of dyadic non-cooperative game Γ(H) is discussed in the paper for the set of one and the same players’ strategies. The players make their choice sitting round the table and have the opportunity to coordinate only the meanings of utilities in every situation. Therefore the players’ payoffs are given by 2×2 matrixes. A notion “the equalized situation” in mixed strategies which is at the same time the equilibrium is introduced. The theorem has been proved, which establishes the conditions of existance of an equalized situation in the given game. In the case of the existence algorithm is constructed. If equalized situation doesn’t exist in the game, then there exists the equilibrium situation in the pure strategies and it is possible to find it by analysis of situations. Γ(H) game’s with bimatrix game in case of two players is given. The players’ conditions of optimal mixed strategies existence in game is written. Relevant examples are solved and Γ(H) game’s application for finite amount of players’ is discussed.