Pie rule

Player 1 plays first move as White

Player 2 Player 1

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

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

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 has one of two options:

1. Letting the move stand, in which case the second player remains the second player and moves immediately, or
2. Switching places, in which case the second player becomes the first-moving player, and the "new" second player then makes their "first" move. (I.e., the game proceeds from the opening move already made, but with roles reversed.)

The pie rule's use 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] The rule can be applied to other games which are otherwise solved for one player, such as Tablut.^[4]

The rule gets its name from the divide and choose method of ensuring fairness in the division of a pie between two people; one person cuts the pie in half, then the other person chooses which half to select to eat. The person cutting the pie, knowing 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 a game which 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.

Use for determining komi in Go

In Go, one player can choose the amount of komi and the other player decides whether to accept that or switch colors with the other player. In the long run, this leads players to choose fair komi amounts because if they choose a komi that is too advantageous, the other player can just choose to play White and take advantage of that high komi.

References