Reducing a Payoff Matrix Using Z2 Group of Symmetries

To use the Z2 group of symmetries, we need to identify the symmetries of the game. In this case, we can see that there are two players and three strategies for each player, so the game has a total of 9 possible outcomes. We can represent the outcomes as a 3x3 matrix:

| | L | C | R | |---|---|---|---| | T | 0,0 | 0,-1 | -1,-2 | | M | 0,-1 | -1,-2 | -2,-1 | | B | -1,-2 | -2,-1 | -3,-2 |

A symmetry of the game is a permutation of the rows and/or columns of this matrix that preserves the payoff structure. In other words, if we swap two rows or two columns, the payoffs for each player should remain the same.

The Z2 group of symmetries consists of two elements: the identity element e (which does nothing) and the permutation σ that swaps the first two rows and the first two columns. We can apply σ to the matrix by swapping the first two rows and then swapping the first two columns:

| | C | L | R | |---|---|---|---| | M | -1,-2 | 0,-1 | -2,-1 | | T | 0,-1 | 0,0 | -1,-2 | | B | -2,-1 | -1,-2 | -3,-2 |

Now we need to check if this permutation preserves the payoff structure. We can see that the payoffs for player 1 are still in the same order, and the payoffs for player 2 are also in the same order. Therefore, this is a valid symmetry of the game.

To reduce the matrix to a 2x7 one, we can take the average of each pair of symmetric outcomes. For example, the first two rows are symmetric, so we can take their average to get:

| | L | C | R | |---|---|---|---| | T/M | 0,-1/(-1,-2) | 0,-1 | -1.5,-1.5 |

Similarly, the last two rows are symmetric, so we can take their average to get:

| | L | C | R | |---|---|---|---| | B | -1.5,-1.5 | -1.5,-1.5 | -2.5,-2 |

This gives us a 2x7 reduced payoff matrix:

| | L | C | R | T/M | B | |---|---|---|---|---|---| | 1 | 0,-1/(-1,-2) | 0,-1 | -1.5,-1.5 | 0,-1/(-1,-2) | -1.5,-1.5 | | 2 | -1,-2/(-2,-1) | -2,-1 | -2,-1 | -1.5,-1.5 | -2.5,-2 |

Note that we have written the payoffs in the form (player 1, player 2) for each outcome.