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Difference Game
Path: CGTPath   · Prev: TedomariSolution   · Next: ZeroInCGTTerms
  Difficulty: Advanced   Keywords: EndGame, Theory

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.

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

The difference of a game G and itself can therefore be written as G + (-G) or 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'.

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 (let's call it G) after one possible play and the position (H) 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 (G - H) 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]
Example 1

Should White play at a or b?

(Note about the diagrams. By convention, stones next to unmarked space on the board are alive.)

Let's set up the difference game.


[Diagram]
Example 1, Difference Game

Starting from a even position, Black plays at B1 (b in the reversed position) and White plays at W2 (a in the original position). Is the resulting position better for one player? If so, their original play is better. Let's compare results when each player goes first.


[Diagram]
Example 1, Difference Game (i)

When Black plays first the result is jigo.


[Diagram]
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.



Froese?: Is it just me? Sounds like some kind of voodoo practice...

  • Why also mirror the "negative position"? Confuse the reader? :-)
  • In "Example 1 Game (i)": Isn't it White's turn? Why is Black playing first?
Bill: In the comparison of positions we let each player play first. To compare different plays for one player we compare positions resulting from the different plays. We want to know which position is better, if any, and that does not depend upon whose move it is.
  • In "Example 1 Game (ii)": Why does White win? White has 17 points on the left side and Black 18 points on the right. B+1.
Bill: I have redone the diagrams to make that clearer. The stones framing the position are alive.

Froese?: Ah, thanks. I think I got it.


[Diagram]
Example 2

Same position, Black to play. Should Black play at a or b?


[Diagram]
Example 2, Difference Game

Which player, if either, stands better?


[Diagram]
Example 2, Difference Game (i)

If White plays first she wins by 1 point.


[Diagram]
Example 2, Difference Game (ii)

If Black plays first the result is even.

So the difference game favors White, and Black's correct play in the original position is at b.


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, Combinatorial Game Theory, Throw In or Not.


Authors: Bill Spight, Charles Matthews



Path: CGTPath   · Prev: TedomariSolution   · Next: ZeroInCGTTerms
This is a copy of the living page "Difference Game" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.