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.

game | payoff matrix | value