Game Saddle Point

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.

REFERENCES:

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.