It's well known that zero-sum games and LP are equivalent. Unfortunately, I can only find documentation on the reduction from LP to a zero-sum game. According to this, reducing a ZSG to a LP is trivial, but nevertheless I am not able to do it. For a matrix $A \in \mathbb{R}^{n \times m}$, a ZSG is defined as:
$$ \text{min } \lambda $$ $$ \text{s.t. } Ax \leq \lambda \vec e$$ $$ x \in \Delta^{m} $$ $$ \lambda \in [-1,1] $$
Where $\vec e$ is the vector of all 1, and $\Delta^m$ is the unitary simplex in $m$ dimension. I can only observe that is strange that $x$ is not appearing in the minimization statement.
The line $x \in \Delta^m$ can be substituted by $x^e_m = 1$. The r.h.s of the inequality can be casted into a vector $\vec \lambda$, but we obtain $n$ different constraints, which we could treat as $$ \vec \lambda \leq e_m$$ and $$ \vec \lambda \geq - e_m$$ And I think we need some linear algebraic way to say that all the entries of $\lambda$ should be equal. Then, the minimization statement can be rephrased as
$$ \text{min } \vec \lambda^ e_m $$
Any ideas?