Reducing a Payoff Matrix Using the Z2 Group of Symmetries

To use the Z2 group of symmetries to reduce the payoff matrix, we need to find matrices A and B such that A^2 = B^2 = I (the identity matrix) and AB = BA. We can then apply these matrices to the rows and columns of the payoff matrix to get a reduced matrix.

One possible choice for A and B is:

A = [1 0; 0 -1] B = [0 1; 1 0]

We can check that A^2 = B^2 = I and AB = BA.

Now we apply A and B to the rows and columns of the payoff matrix:

A: [ 0 0 0 0 0 -1 -1 -1 -2 -1 -1 -2 -2 -3] [-1 -1 -2 -2 -3 0 0 -2 -1 -2 -3 -1 -3 -2] [-2 -3 -1 -3 -2 -2 -2 0 0 -3 -2 -2 -1 -1] [-3 -2 2 -1 -1 -1 -1 -1 -1 0 0 0 0 0]

B: [ 0 0 0 0 0 -1 -1 -1 -2 -1 -1 -2 -2 -3] [-1 -2 -1 -3 -2 0 -2 0 0 -3 -2 -2 -1 -1] [-3 -1 -3 -1 2 -2 0 -2 -1 -1 0 0 -3 -2] [-2 -3 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0]

The rows and columns of the reduced matrix are obtained by taking the orbits of each row and column under the action of A and B. We can see that the row orbits are:

{1, 2} {3, 4}

and the column orbits are:

{1, 2, 3, 4} {5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

The reduced matrix is then obtained by taking the minimum value in each orbit:

[ 0 -3 -3 -3 -3 -2 -3] [-3 0 2 0 -2 -1 -2]

This is a 2 ￗ 7 matrix.