I've been trying to solve a nash equilibrium for a game. The game is zero sum and symmetric. Unfortunately, it is also quite large. The payoff matrix is (n! , n!) in size. To simplify, I used n=4 (the game is easily solved for n<4) and I've tried cutting out strictly dominated strategies. This is where I am stuck. I suspect that there are additional dominant strategies, but I can't find them. The result is an 11x11 payoff matrix for player A. (Since the game is zero sum, player B has a similar payoff matrix). How do I reduce this game further?
\begin{bmatrix} 0&1&1&1&1&0&1&1&-1&1&-1 \\-1&0&1&1&0&1&0&1&0&1&-1 \\-1&-1&0&-1&1&1&0&1&0&0&0 \\-1&-1&1&0&1&1&1&1&1&-1&0 \\-1&0&-1&-1&0&-1&1&0&0&0&0 \\0&-1&-1&-1&1&0&1&0&1&0&1 \\-1&0&0&-1&-1&-1&0&1&1&1&1 \\-1&-1&-1&-1&0&0&-1&0&1&1&1 \\1&0&0&-1&0&-1&-1&-1&0&1&1 \\-1&-1&0&1&0&0&-1&-1&-1&0&1 \\1&1&0&0&0&-1&-1&-1&-1&-1&0 \end{bmatrix}
If you are curious, the matrix was derived from a game called goofspiel.
According to https://www.math.ucla.edu/~tom/gamesolve.html
An optimal strategy for Player I is: (0.5,0,0,0,0,0.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) An optimal strategy for Player II is: (0.33333,0,0,0.16667,0,0.33333,0,0,0,0,0,0,0,0,0,0,0,0,0.16667,0,0,0,0,0)
...odd. I would think that player 2 and player 1 would have identical mixed strategies.