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.
