## Solving Zero-Sum Games

$\begin{bmatrix} 4 & 2 & 5 & 2\\ 2 & 1 & -1 & -20\\ 3 & 2 & 4 & 2\\ -16 & 0 & 16 & 1\\ \end{bmatrix}$

Our row mins are

$\begin{bmatrix} 2\\ -20\\ 2\\ -16\\ \end{bmatrix}$

and our column maxes are

$\begin{bmatrix} 4 & 2 & 16 & 2\\ \end{bmatrix}$

$\begin{pmatrix} \begin{pmatrix} 1\\0\\0\\0 \end{pmatrix}& \begin{pmatrix} 0\\1\\0\\0 \end{pmatrix}\\ \end{pmatrix}$

which corresponds to $$(e_1, e_2)$$, and the others are $$(e_3, e_4)$$, $$(e_3, e_2)$$, and $$(e_3, e_4)$$.

Saddle points arise from optimal pure strategies, also called Pure Nash Equilibria.

Nash Equilibrium: Strategy pairs such that neither player can do better by unilaterally deviating from that strategy.

## Example 2

Now take the example of

$\begin{bmatrix} -2 & 5 & 1 & 0 & -4\\ 3 & -3 & -1 & 3 & 8\\ \end{bmatrix}$

Taking $$\vec{p} = [p, 1-p]$$ and $$A =$$ our matrix, we get

$\vec{p}^\top A = [3-5p, 8p-3, 2p-1, 3-3p, 8-12p]$

Drawing our diagram, we can see that only columns 1 and 3 matter, so we set them equal to each other: $$3-5p = 2p-1$$. Solving, we get $$p = 4/7$$.

Similarly,