where
is called the value of the game.
In this case, there exist optimal strategies for the first and second players.

A necessary and sufficient condition for a saddle point to exist is the presence of a payoff
matrix element which is both a minimum of its row and a maximum of its column.
A game may have more than one saddle point, but all must
have the same value.

Dresher, M. "Saddle Points." §1.5 in The
Mathematics of Games of Strategy: Theory and Applications. New York: Dover,
pp. 12-14, 1981.Llewellyn, D. C.; Tovey, C.; and Trick, M.
"Finding Saddlepoints of Two-Person, Zero Sum Games." Amer. Math. Monthly95,
912-918, 1988.