Lexicographic Strategic Games' Nonstandard Analysis

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.