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Difference game
Path: CGTPath · Prev: Tedomari · Next: ZeroInCGTTerms
Difficulty: Advanced
Keywords: EndGame, Theory
Difference games are a part of combinatorial game theory (CGT), in which games, such as Go positions, can be added and subtracted. They are described in Mathematical Go by Berlekamp and Wolfe. 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.)
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.
Starting from a even position, Black plays at
When Black plays first the result is jigo.
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...
Froese?: Ah, thanks. I think I got it.
Same position, Black to play. Should Black play at a or b?
Which player, if either, stands better?
If White plays first she wins by 1 point.
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: Tedomari · Next: ZeroInCGTTerms This is a copy of the living page "Difference game" at Sensei's Library. ![]() |