Chapter 6, Problem 6.23TE

A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 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 previously and suppose that there are two individuals— A and B—who are playing the following game: 4 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 wins from B the amount specified by the number in the

ith row, jth column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number . Now if player A chooses row r, then that player can guarantee herself a win of at least (since 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 (since is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of and as B has a way of playing that guarantees he will lose no more than it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?