Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which can be reached by using dominance relations.
My knowledge:
A saddle point in a matrix is an entry which is the minimum in its row and maximum in its column. I also know that for dominance relations the $i^{th}$ row dominates the $j^{th}$ row if all the entries in the $i^{th}$ row are $\geq$ the corresponding entries in the $j^{th}$ row.
The the $i^{th}$ column dominates the $j^{th}$ column if all the entries in the $i^{th}$ column are $\leq$ the corresponding entries in the $j^{th}$ column.
I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it relates to games. I learned how to do saddle points and would like to see a proof of this question to expand my knowledge.
Let $C$ be the $2\times n$ payoff matrix. Suppose that entry $(i,j)$ is a saddle point of the matrix $C$. We can relabel the rows so that the saddle point is in the first row, so without loss of generality $i = 1$.
Let $c_1, \ldots, c_n$ be the columns of the matrix $C$. Let's think about what it means graphically for $(1,j)$ to be a saddle point. This means that the $x$-coordinate of $c_j$ is less than or equal to the $x$-coordinate of any other $c_\ell$. Graphically, the vector $c_j$ is westmost among all of the $c_\ell$s. It also means that the $y$-coordinate of $c_j$ is less than or equal to the $x$-coordinate of $c_j$. Graphically, this means that $c_j$ is on or below the line $y=x$.
Now, I claim that any column $c_\ell$ other than $c_j$ that is on or above the line $y=x$ is dominated by $c_j$. Graphically, the reason is that $c_j$ dominates everything above and to the right of it. Since there's nothing to the left of it, $c_j$ dominates everything above it. Since $c_j$ is below the line $y=x$, everything above the line $y=x$ is above it.
The previous paragraph more formally is... First, $(c_j)_y \leq (c_j)_x$ since $c_j$ is on or below the line $y=x$. Also, $(c_j)_x \leq (c_\ell)_x$ by choice of $(1,j)$. Finally, $(c_\ell)_x \leq (c_\ell)_y$ since $c_\ell$ is on or above the line $y=x$.
We are left with columns that are all on or below the line $y=x$ after crossing out the columns dominated by $c_j$. But this means that the $x$-coordinate of each column is larger than the $y$-coordinate. This means that the first row dominates the second row, so we can cross out the second row.
We're left with only the first row and we know already that $C_{1j}$ is the minimum value in this row, so it dominates every other entry. We're left with only entry $(1,j)$ not crossed out.
The $m\times 2$ case is exactly the same.