minimax theorem,
n. (Game theory) a theorem justifying the exchange of order in taking the minimum and maximum of a saddle function: 
This number, if it exists, is called the value of the associated two-person game. The Sion minimax theorem shows that this minimax exists when X and Y are compact, and F(x, •) is lower semicontinuous and quasi-convex, while F(•, y) is upper semicontinuous and quasi-concave. The most celebrated case is von Neumann's minimax theorem in which X and Y are polyhedra and F is bilinear, corresponding to a payoff matrix.
This number, if it exists, is called the value of the associated two-person game. The Sion minimax theorem shows that this minimax exists when X and Y are compact, and F(x, •) is lower semicontinuous and quasi-convex, while F(•, y) is upper semicontinuous and quasi-concave. The most celebrated case is von Neumann's minimax theorem in which X and Y are polyhedra and F is bilinear, corresponding to a payoff matrix.