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.

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.

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.