A new characterization of minimax identity problem in a two-person zero-sum dynamic game system (Nonlinear Analysis and Convex Analysis)

For any two stochastic spaces X and $Y$ , we would hke to search a real valued function $f$ : $X\cross Yarrow \mathbb{R}$ for $(x,y)\in X\cross Y$ satisfying that whether the minimax identity (theorem) $\inf_{x\in X}\sup_{y\in Y}f(x,y)=\sup_{y\in Y}id_{x\in X}f(x,y)$ holds.This problem established in a two- person zero-sum dynamic game under some conditions is solvable.Keywords.Minimax theorem, upper (lower) valuedfunction, dynamic games, saddle value function.1 Preliminary For any spaces X and $Y$ , a real valued function $f$ on $X\cross Y$ is considered to search conditions in the function $f$ : $X\cross Yarrow \mathbb{R}$ , and conditions in spaces X and $Y$ satisfy the identity $iffi_{x\in X}\sup_{y\in Y}f(x,y)=\sup_{y\in Y}\inf_{x\in X}f(x,y)$ , namely minimax identity or mm- $\max$ theorem.There are three types in minimax theorems described by Ky Fan (cf.[1], Fan).Thus the minimax theorems are multicriteria.In this note, we assume the spaces X and $Y$ are regarded as the strategy spaces of players I and II, respectively in a two person dynamic game, and we would assign a game function $f$ in such a game system, and prove the minimax theorem holds.Research means that one tries to find the conditions such that the objective result holds.The achieved research may be used the technique by