Reducing Payoff Matrices with Z2 Symmetry: A Step-by-Step Example

To reduce the payoff matrix using the Z2 group of symmetries, we first need to identify the symmetries of the matrix. In this case, the only symmetry is the transpose of the matrix, which we can denote as 'σ'.

Using this symmetry, we can reduce the matrix by averaging each element with its corresponding element under the symmetry. For example, the first element in the original matrix is 0, and its corresponding element under the symmetry is also 0. So the reduced element is:

(0 + 0)/2 = 0

Similarly, the second element in the original matrix is -1, and its corresponding element under the symmetry is -1. So the reduced element is:

(-1 + -1)/2 = -1

We can apply this process to all the elements in the matrix to obtain the reduced payoff matrix:

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

Note that the reduced matrix has 2 rows and 7 columns, which is the result of applying the symmetry to the original 4 × 14 matrix.