On the existence of Nash equilibriums for infinite matrix games

The fundamental theorem of game theory states that for every matrix game with finite strategies there exists at least one optimal strategy. It is known that the fundamental theorem of game theory does not hold, in general, in infinite matrix games. In this paper, we provide a characteristic of Nash equilibriums for ∞×∞∞×∞ matrix games and prove an existence theorem of optimal strategies by using the Fan–KKM theorem. We give some applications of these theorems.