How to find linear programming constraints given a 2 player zero sum matrix Given a problem like the following:

A startup want to build talking washing machines spending the least possible. There are three ways of building them: manually, semi-automatically and automatically. The manual production demands 1 minute of qualified work, 40 minutes of non-qualified work and three minutes of assemblage. The work times are 4, 30 and 2 minutos for the semi-automatic method and 8, 20 and 4 minutos for the fully automatic method. A startup has a pool of 4500 minutes of qualified work, 36000 minutos of non-qualified work and 2700 minutos of assembly. The costs of the production are 70, 80 and 85 euros for the manual, semi-automatic and automatic methods. 
There are constraints regarding the number of machines to be produced (999) and the capacity of the factory

I understand how to set up a linear program like this:
minimize $f(x) = 70x_1 + 80x_2 + 85x_3 \ s.t.$
$x_1 + x_2 + x_3 = 999$
$x_1 + 4x_2 + 8x_3 \leq 4500$
$40x_1 + 30x_2 + 20x_3 \leq 36000$
$3x_1 + 2x_2 + 4x_3 \leq 2700$
$x \geq 0$
However, I am confused on how to set up the constraints for a linear program when all you are given is a payout matrix for a 2 player zero sum game. Given the following matrix (row player payouts listed):
\begin{bmatrix}
1 & -1\\ -1 & 1
\end{bmatrix}
I can get to this setup for the row player:
minimize $z \ s.t.$
$x_1 - x_2 = a$
$-x_1 + x_2 = b$
$x_1 + x_2 = 1$
$x_1, x_2 \geq 0$
So $a$ and $b$ are what I'm confused about finding here. I'm wondering how to find these values given this type of input in a general sense, not just for this problem.

Your setup is perhaps confusing because it's not clear what $a,b,z$ represent, and there's nothing relating them to each other or constraining them. As it stands, you could pick any feasible solution (e.g., $(x_1,x_2,a,b) = (1,0,1,-1)$), and then an arbitrarily low value for $z$. This shows that the problem is unbounded, and isn't doing what you want it to do.
Instead try:
$$\text{minimize } z \text{ s.t.}$$
$$\begin{array}{c}
x_1 &- &x_2 &\le &z\\
-x_1 &+ &x_2 &\le &z\\
x_1 &+ &x_2 &= &1\\
\end{array}$$
$$x_1, x_2 \geq 0$$
Now take the instances of $z$ to the left-hand side.
