Game Theory: Reducing the Payoff Matrix with Z2 Symmetry

List of all 14 strategies P2 can employ:

1. Guess 1, 2, 3, 4 in order
2. Guess 4, 3, 2, 1 in order
3. Guess 1, 3, 2, 4 in order
4. Guess 4, 2, 3, 1 in order
5. Guess 2, 1, 4, 3 in order
6. Guess 3, 4, 1, 2 in order
7. Guess randomly
8. Guess 2, 4, 1, 3 in order
9. Guess 3, 1, 4, 2 in order
10. Guess 1, 4, 2, 3 in order
11. Guess 3, 2, 1, 4 in order
12. Guess 2, 3, 4, 1 in order
13. Guess 4, 1, 3, 2 in order
14. Guess 1, 2, 4, 3 in order

Payoff matrix:

| | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | S9 | S10 | S11 | S12 | S13 | S14 | |----|----|----|----|----|----|----|----|----|----|-----|-----|-----|-----|-----| | 1 | 0 | -1 | -2 | -3 | -4 | -4 | -4 | -3 | -2 | -1 | -1 | -1 | -2 | -3 | | 2 | -1 | 0 | -1 | -2 | -3 | -4 | -4 | -3 | -2 | -2 | -1 | -1 | -2 | -3 | | 3 | -2 | -1 | 0 | -1 | -2 | -3 | -4 | -3 | -2 | -2 | -2 | -1 | -1 | -2 | | 4 | -3 | -2 | -1 | 0 | -1 | -2 | -3 | -2 | -1 | -2 | -2 | -2 | -1 | -1 |

To reduce the payoff matrix using the symmetry group Z2 = {e, σ : σ^2 = e}, we need to find the fixed points of the permutation σ = (2 4)(1 3). The fixed points are (1 3)(2 4) and (2 4)(1 3), which means that the strategies that are symmetric under this permutation are S1, S2, S4, S5, S8, S9, and S13. We can then group these strategies together and take their average payoffs to obtain the reduced payoff matrix:

| | S1, S2, S4, S5, S8, S9, S13 | S3 | S6 | S7 | S10 | S11 | S12 | S14 | |-----|--------------------------|----|----|----|-----|-----|-----|-----| | 1/4 | -2 | -2 | -4 | -4 | -1 | -1 | -1 | -3 | | 2/4 | -2 | -1 | -3 | -4 | -2 | -1 | -1 | -3 | | 3/4 | -2 | -1 | -2 | -3 | -2 | -2 | -1 | -2 | | 4/4 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 |

Note that the reduced payoff matrix has 2 rows and 7 columns, which corresponds to the two orbits of the symmetry group. The first row represents the average payoff for the strategies that are symmetric under the identity permutation, and the second row represents the average payoff for the strategies that are symmetric under the permutation σ.