Consider the game matrix
\begin{bmatrix} -1 & 1 &-1 \\ -1 & -1 &2 \\ 5 & -1 &-1 \end{bmatrix}
Is there any way to reduce it into 2x3 or 3x2 matrix(I don't see a dominating row or column here)? I know how to use the simplex method to solve it but am trying make it simpler.
after putting the question asides for a while and coming back, I suddenly noticed a special feature in the matrix, Adding 1 to every entry and rearranging the rows, we will get
\begin{bmatrix} 6 & 0 &0 \\ 0 & 2 &0 \\ 0 & 0 &3 \end{bmatrix}
Now, its a diagonal matrix and the maximin strategy is simply normalised $$ (1/6, 1/2, 1/3) $$ which , after rearranging back to the original order, is the maximin strategy of the original game.
The minimax strategy and the value can be easily computed.
I actually don't know why I did not see it at the first glance and why I saw this suddenly. I guess it's the tricky beauty of Maths that we all love :)