A rectangular array of $m n$ 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
$$
\begin{array}{rr}
1 & 3 & 2 \\
0 & -2 & 6 \\
.5 & 12 & 3
\end{array}
$$
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 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 $1,2, \ldots, n$ and $B$ one of the numbers $1,2, \ldots, m$. 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, $j$ th column of the array. Now suppose that the array contains a saddlepoint-say the number in the row $r$ and column $k-$ call this number $x_{r k}$. Now if player $A$ chooses row $r$, then that player can guarantee herself a win at least $x_{r k}$ (since $x_{r k}$ 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 $x_{r k}$ (since $x_{r k}$. is the maximum number in the column $k$ ). Hence, as $A$ has a way of playing that guarantees her a win of $x_{r k}$ and as $B$ has a way of playing that guarantees he will lose no more than $x_{r k}$, it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player $A$ is $x_{r k}$.
If the $n m$ 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?