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Pie rule

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(Redirected from Swap rule) Rule for balancing a game with an advantage when going first
A chess game using the pie rule, player 2's options in bold
Player 1 plays first move as White

Player 1

a8 white rookb8 white knightc8 white bishopd8 white kinge8 white queenf8 white bishoph8 white rooka7 white pawnb7 white pawnc7 white pawnd7 white pawne7 white pawnf7 white pawng7 white pawnh7 white pawnf6 white knighta2 black pawnb2 black pawnc2 black pawnd2 black pawne2 black pawnf2 black pawng2 black pawnh2 black pawna1 black rookb1 black knightc1 black bishopd1 black kinge1 black queenf1 black bishopg1 black knighth1 black rook

Player 2

 ↙ ↘ 
Player 2
lets move stand 		Player 2
switches places

Player 1

a8 white rookb8 white knightc8 white bishopd8 white kinge8 white queenf8 white bishoph8 white rooka7 white pawnb7 white pawnc7 white pawnd7 white pawne7 white pawnf7 white pawng7 white pawnh7 white pawnf6 white knighta2 black pawnb2 black pawnc2 black pawnd2 black pawne2 black pawnf2 black pawng2 black pawnh2 black pawna1 black rookb1 black knightc1 black bishopd1 black kinge1 black queenf1 black bishopg1 black knighth1 black rook

Player 2

Player 1

a8 black rookb8 black knightc8 black bishopd8 black queene8 black kingf8 black bishopg8 black knighth8 black rooka7 black pawnb7 black pawnc7 black pawnd7 black pawne7 black pawnf7 black pawng7 black pawnh7 black pawnc3 white knighta2 white pawnb2 white pawnc2 white pawnd2 white pawne2 white pawnf2 white pawng2 white pawnh2 white pawna1 white rookc1 white bishopd1 white queene1 white kingf1 white bishopg1 white knighth1 white rook

Player 2

Player 2 to play
as Black, as before 		Player 1 to play again,
now as Black

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.

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. Among modern games, Hex uses this rule. TwixT in tournament play uses a swap rule. 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.

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.

Use for determining komi in Go

In Go, one player can choose the amount of komi. (These are the points given to the second player as compensation for not going first.) The other player then decides whether to accept that or switch colors with the other player. 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

  1. Parker, Henry (1909). Ancient Ceylon: An Account of the Aborigines and of Part of the Early Civilisation. London: Luzac & Co. pp. 601–602. LCCN 81-909073.
  2. Browne, Cameron. Hex Strategy: Making the Right Connections. ISBN 1-56881-117-9.
  3. Mind Sports Olympiad Twixt page. Archived 2018-07-14 at the Wayback Machine.
  4. Schmittberger, R. Wayne (1992). New Rules for Classic Games. John Wiley & Sons Inc. pp. 25–27. ISBN 978-0471536215.
  5. "Pie Rule at Sensei's Library". Impressum bzw. November 18, 2009. Retrieved June 10, 2022.
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