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Difference Game
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Difference games are described in Mathematical Go by Berlekamp and Wolfe. They are a part of combinatorial game theory (CGT), in which games, such as Go positions, can be added and subtracted.


To subtract one Go position from another, set up the first position, and in a separate region set up the negative of the second position. The negative of a position is formed by reversing the colors of its stones.

So we can call the negative the 'reversed' position. But please note that ''reversing'' is something else in the theory.

The difference of a game G and itself can be written as G - G. This may be equal to zero (a test of your understanding of the words as CGT talks about them, mostly, taking into account imitative play). It certainly isn't 'nothing'.

The difference game is G + G~, where G~ changes the colour of the stones.

Go is not strictly a combinatorial game because of kos. So difference games involving kos may not behave according to theory.

Difference games can be used to compare plays. Make the difference game of the position after one possible play and the position after another one.

If G and H are distinct options in the starting position, looking at G - H and how you would play it may reveal much about the relationship of the two ways of proceeding.

If the difference game is a win for one player, playing first, and a win or tie for the same player playing second, the play made by that player to set up the difference game is better (except possibly when ko is involved.)

[Diagram]
Diag.: Example 1

Should White play at a or b?

Let's set up a difference game where White plays at a in this position and Black plays at b in the reversed position.


[Diagram]
Diag.: Example 1, Difference Game (i)

When Black plays first the result is jigo.


[Diagram]
Diag.: Example 1, Difference Game (ii)

When White plays first White wins.

So White's play in the difference game (a in the original position) is correct.


Often the choice between alternate moves will depend on the rest of the board. If each player wins the difference game when they play first, that will be the case.

For examples, see Clamp Connection Comparison and Combinatorial Game Theory.


Authors: Bill Spight, Charles Matthews



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This is a copy of the living page "Difference Game" at Sensei's Library.
(C) the Authors, published under the OpenContent License V1.0.