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Saddle Point (Game)

For a general two-player Zero-Sum Game,

\max_{i\leq m} \min_{j\leq n} a_{ij} \leq \min_{j\leq n}\max_{i\leq m} a_{ij}.

If the two are equal, then write

\max_{i\leq m} \min_{j\leq n} a_{ij} = \min_{j\leq n}\max_{i\leq m} a_{ij}\equiv v,

where $v$ 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.

See also Game, Payoff Matrix, 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.

© 1996-9 Eric W. Weisstein