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Difference game
Path: CGTPath · Prev: Tedomari · Next: ZeroInCGTTerms
Difficulty: Advanced Keywords: EndGame, Theory
IntroductionDifference 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 form a difference game of two games, subtract one game from the other. Difference games can help to decide which of a pair of plays is better in a go position. Set up a difference game of the resulting positions and see if one side has an advantage. DefinitionTo subtract one Go position from another, set up the first position, and in a separate, independent region (or on another board) 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.[1001] 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'.[1002] Go is not strictly a combinatorial game because of kos. So difference games involving kos may not behave according to theory. Also, it may be right to make the play that the difference game says is wrong because it produces more or bigger ko threats for you or fewer or smaller threats for your opponent. Comparing playsDifference 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). 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.
First we mirror the original position to make things even (zero in CGT terms).
Then Black plays at Is the resulting position better for one player? If so, their original play is better. To find out, let's compare results when each player goes first.
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. Voodoo?Froese?: Is it just me? Sounds like some kind of voodoo practice...
Froese?: Ah, thanks. I think I got it. Tom: Playing the difference game is not as much a voodoo practise as it might at first appear. Borrowing from the example above, but swapping the colours on the right.
Suppose we wish to test whether the left hand position is (under all ko free circumstances) at least as good for black as the right hand position. There are two possible ways in which one could try to argue that this was not the case.
Firstly one could say 'But white has a superb move
The second way to argue that the left hand position is not as good for black is to say 'Black has a superb move
In this example, we see that it is false that blacks position in on the left hand side is as good as that on the right hand side.
The clever thing about the difference game is that each step of this argument corresponds to a move in the difference game! Although, once explained, the difference game is not voodoo, I do think that it is a very clever idea.
Example 2
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 other examples, see Picky Endgame Discussion, Clamp Connection Comparison, Canonical Form, Corridor Infinitesimals, More Infinitesimals, Playing Infinitesimals, Tinies and Minies, Reversible, Throw In or Not, and Practical Endgame Test 5/Difference Games. ilanpi I think that the principle underlying this technique can be summarised as follows: To decide if G > H, check to see whether G - H > 0. Actually, I think that this is the very definition of G > H, so this method is canonical, in that sense. Authors: Bill Spight, Charles Matthews [1001] Bill: Charles, this is not correct, as stated. Do you have a revision? (I think we can delete it.) Or how's this? The difference game, G - H, is G + H~, where H~ reverses the colour of the stones of H. ilanpi I don't think you need to introduce the "~" operator which is exactly the same as the "-" operator on single games (as opposed to the "-" as binary operation). [1002] ilanpi: The answer to this is exactly the strategy used in How anyone can get a 9d rating on go servers without cheating too much. If, in that strategy, both games finish (neither one of your opponents resigns), then the sum of your territorial advantage in these two games will be zero (you will be ahead by N points in one game and behind by N points in the other), and in this sense, the CGT sum of these games is zero. Since the games are identical, but with colors reversed, one is the negative of the other, so this is exactly what is meant by the CGT equation G-G = 0.
ilanpi: The current article in the Daily Yomiuri by
Rob van Zeijst is essentially about the limitations of
this model. See Bill: Well, he shows a position that is worth *, not 0. <shrug> But he also shows an endgame problem with at least one mistake in the "solution." ;-) ilanpi: OK, thanks. The limitation I was talking about is to put the G and -G on the same board. You must be careful! Bill: Yes, you must be careful. Path: CGTPath · Prev: Tedomari · Next: ZeroInCGTTerms This is a copy of the living page "Difference game" at Sensei's Library. ![]() |