I followed this lecture :
https://www3.nd.edu/~apilking/Math10120/Lectures/Topic%2029.pdf
It gives a way to reduce a n x m matrix to a 2×2 matrix where we know how to solve a problem of mixed game strategy. We are searching for an optimal mixed row and column strategies and the value of the zero sum game defined by the given matrix.
However, if no lines nor columns are dominated, we can't reduce the matrix.
I would like to solve (for example this matrix, I ve simply taken an example of the pdf and changed values in order to make incomparable any lines / columns )
\begin{array}{c|rrrr}
& A & C & D \\\hline X & 6 & 7 & 3 \\ Y & 7 & 5 & 2 \\ Z & 8 & 3 & 9
\end{array}
How should I solve this problem ? I'm searching for a general method in order to then solve harder problems.
Best Answer
If there is a 2x2 matrix, you usually find a probability $p$ for player 1 to make player 2 indifferent between his two actions. Then you find probability $q$ for player 2 to make player 1 indifferent between his two actions. This is then a mixed Nash equilibrium, because both are indifferent between any of the two available actions, so they can play both with positive probability.
The problem is the same with a 3x3 matrix, but you will have to solve for 4 rather than 2 probabilities. First, you need to find probabilities $p_1$ and $p_2$ with which player 1 plays his first and second action, respectively (and he will play the third with probability $1-p_1-p_2$), so that player 2 is indifferent between his available actions. Then you need to find probabilities $q_1$ and $q_2$ with which player 2 plays his first two actions (and the third with probability $1-q_1-q_2$), so that player 1 is indifferent between all of his actions.
How to set up the appropriate equations is explained here in an example.