# Is A Simplex Method Appropriate for Solving Nash Equilibria?

I've been attempting to teach myself some game theory and in the process, some linear programming.

While muddling through this, I've been attempting to use a variety of simplex methods to attempt to find mixed-strategy Nash equilibria in zero-sum games.

Unfortunately, most of my computations end up with incomplete, or seemingly wrong answers.

Is the simplex method (or two-phase, or dual simplex) appropriate for attempting to find mixed-strategy Nash equilibria, and if so, what is the proper way to set up the starting tableaux?

I've attempting starting with something like this (example is RPS):

\begin{bmatrix}0 & 0 & 0 & -1 & -1 & 0\\1 & 1 & 1 & 0 & 1 & 1\\0.5 & 0 & 1 & 1 & 0 & 0.5\\1 & 0.5 & 0 & 1 & 0 & 0.5\\0 & 1 & 0.5 & 1 & 0 & 0.5\end{bmatrix}

or

\begin{bmatrix}0 & 0 & 0 & -1 & -1 & -1 & -1 & 0\\1 & 1 & 1 & 1 & 0 & 0 & 0 & 1\\0.5 & 0 & 1 & 0 & 1 & 0 &0 & 0.5\\1 & 0.5 & 0 & 0 & 0 & 1 & 0 & 0.5\\0 & 1 & 0.5 & 0 & 0 & 0 & 1 & 0.5\end{bmatrix}

or

\begin{bmatrix}0 & 0 & 0 & 0 & -1 & -1 & -1 & 0\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\1 & 1 & 1 & 1 & 0 & 0 & 0 & 1\\0.5 & 0 & 1 & 0 & 1 & 0 &0 & 0.5\\1 & 0.5 & 0 & 0 & 0 & 1 & 0 & 0.5\\0 & 1 & 0.5 & 0 & 0 & 0 & 1 & 0.5\end{bmatrix}

And this seems simple enough to solve, but anything more complex, or examples with dominated strategies often seem to be unsolveable.