21. A rectangular array of me numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is the minimum of its row and the maximum of its column. For instance, the array 1

3 2 0 -2 6

.5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described above and suppose that there are two individuals

—A and B—that are playing the following game: A is to choose one of the numbers 1, 2,..., n and B one of the numbers 1, 2,..., m. These choices are announced simultaneously, and if A chose i and B chose j, then A receives from B the amount specified by the number in the ith row, jth column of the array. Now suppose that the array contains a saddlepoint—say the number x in the rth row and column k—call this number x. Now if player A chooses row r, then that player can guarantee herself a win at least x (since x is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x (since x is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x, and as B has a way of playing that guarantees he will lose no more than x, it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x.

If the m numbers in the rectangular array described above are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?