Pie rule

The pie rule, sometimes referred to as the swap rule, is a rule used to balance abstract strategy games where a first-move advantage has been demonstrated. After the first move is made in a game that uses the pie rule, the second player must select one of two options:

1. Letting the move stand. The second player remains the second player and moves immediately.
2. Switching places. The second player becomes the first-moving player with the move already done by the opponent, and the opponent plays the first move of their new color.

A chess game using the pie rule, player 2's options in bold

Player 1 plays first move as White
Player 1 Player 2
↙	↘
Player 2 lets move stand	Player 2 switches places
Player 1 Player 2	Player 1 Player 2
Player 2 to play as Black, as before	Player 1 to play again, now as Black

Depending on the game, there may be two ways to implement switching places.

1. Switching colors means that the players exchange pieces. The player who made the first move becomes the second player and makes the second move on the board. This is demonstrated in the chess diagrams shown here.
2. Switching the first piece can occur in games where the board starts empty and the first move consists of placing one piece. Suppose the colors are white versus black, and black places the first piece. This piece is replaced by a white piece in the corresponding location for white, and the black piece is returned to black's supply. In a game such as Hex or TwixT, the corresponding location is at a cell "reflected" across the nearest (or either) diagonal. In games such as Y, where the board is not directional, the white stone replaces the black stone in the same cell. Players keep their respective color pieces, and play continues with black making the next move. This is effectively the same as switching colors.

The use of pie rule was first reported in 1909 for a game in the Mancala family.^[1] Among modern games, Hex uses this rule.^[2] TwixT in tournament play uses a swap rule.^[3] In Meridians, the first player places 2 stones on the board before the second player chooses the color. The rule can be applied to other games which are otherwise solved for one player, such as Gomoku or Tablut.^[4]

The rule gets its name from the divide and choose method of ensuring fairness in when dividing a pie between two people: one person cuts the pie in half, then the other person chooses which half to eat. The person cutting the pie, knowing that the other person will choose the larger piece, will make as equal a division as possible.

This rule acts as a normalization factor in games where there may be a first-move advantage. In games that cannot end in a draw, such as Hex, the pie rule theoretically gives the second player a win (since one of the players must have a winning strategy after the first move, and the second player can choose to be this player), but the practical result is that the first player will choose a move neither too strong nor too weak, and the second player will have to decide whether switching places is worth the first-move advantage.