A zero sum matrix game is given: $A=\begin{bmatrix}13&29&8\\18&22&31\\23&22&19\\ \end{bmatrix}$ Reduce that game to a 2 x 2 game using dominance.

Question

I know that a 2 x 2 matrix is as follows:

$A=\begin{bmatrix}18&31\\23&19\\ \end{bmatrix}$

But it is not possible to reduce the original matrix by row or by columns, how is this result achieved?

Thanks in advance for any help.

Answer 1

Try this:

Suppose that in a mixed strategy of the column player he chooses Col 1 with a probability of $p$ and Col 3 with a probability of $1-p$. You can find that there exists a $p$ such that this mixed strategy dominates the pure strategy of Col 2.