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

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.