For a general two-player zero-sum game, max_(i<=m) min_(j<=n) a_(i j)<=min_(j<=n) max_(i<=m) a_(i j). If the two are equal, then write max_(i<=m) min_(j<=n) a_(i j) = min_(j<=n) max_(i<=m) a_(i j) congruent 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.
game | payoff matrix | value
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