## Solving Zero-Sum Games

Our row mins are

and our column maxes are

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

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

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,