For a general two-player Zero-Sum Game,

If the two are equal, then write

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. Monthly 95, 912-918, 1988.