The Stability of Equilibrium Situation in Lexicographic Strategic Games

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 vectorfunctions 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.