Show that, if a 2x3 matrix has a saddle point, then either one row dominates the other, or one column dominates another (or possibly both).

Question

I tried to do it graphically using the saddle point condition:

$ a_{ij^*}\leq a_{i^*j^*} \leq a_{i^*j}$

But I can't seem to figure it out.

Would really appreciate your help.

Answer 1

If the matrix $$\begin{bmatrix}a & b & c \\ d & e & f\end{bmatrix}$$ has a saddle point, by symmetry we may assume it is $a$. So $a \ge d$ but $a \le b$ and $a \le c$.

If $b \le e$, then $d \le e$ as well (since $d \le a \le b \le e$), so the second column dominates the first. Similarly, if $c < f$, then the third column dominates the first.

If neither of these occur, then $b \ge e$ and $c \ge f$, and we already knew $a \ge d$, so the first row dominates the second.