Reducing a Payoff Matrix using Z2 Group Symmetries

To apply the Z2 group of symmetries, we need to find the involutions (elements that are their own inverse) of the group. In this case, we have two involutions: the identity element 'e' and the element 'σ' that squares to 'e'.

Next, we need to find the orbits of the matrix under the action of the group. An orbit is a set of elements that can be transformed into each other by applying elements of the group. The orbits of the matrix are:

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

Note that we can combine orbits 2 and 3 since they have the same values in the matrix.

For each orbit, we need to find a representative element that is invariant under the action of the group. To do this, we can average the values in each orbit. The representatives are:

Representative 1: [0, 0, -1, -1, -2] Representative 2: [-1, -1, -2, -2] Representative 3: [-1, -1] Representative 4: [-1, -1, -2]

We can now create the reduced payoff matrix by arranging the representatives in a matrix:

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

This is the 2 × 7 reduced payoff matrix.