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Difference game
Path: CGTPath   · Prev: Tedomari   · Next: ZeroInCGTTerms
  Difficulty: Advanced   Keywords: EndGame, Theory

Table of contents Table of diagrams
Example 1
Example 1, Difference Game Setup (1)
Example 1, Difference Game Setup (2)
Example 1, Difference Game (i)
Example 1, Difference Game (ii)
Modified version of Example 1
Modified version of Example 1
Modified version of Example 1
Example 2
Example 2, Difference Game Setup
Example 2, Difference Game (i)
Example 2, Difference Game (ii)

Introduction

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 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.


Definition

To 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 plays

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).


Example 1

[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 Setup (1)

First we mirror the original position to make things even (zero in CGT terms).


[Diagram]
Example 1, Difference Game Setup (2)

Then Black plays at BC (b in the reversed position) and White plays at WC (a in the original position). (They can make those plays in either order.)

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.

[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.



Voodoo?

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

  • Why also mirror the "negative position"? Confuse the reader? :-)

Bill: We are just mirroring the original position to make an even starting position before making the plays to compare.

  • 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.


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.

[Diagram]
Modified version of Example 1

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 W1 on the left hand side'. Possible ways to refute this argument are to say 'white has just as good a move at a on the right hand side' (true) or to say 'black can refute W1 with B2' (false).

[Diagram]
Modified version of Example 1


The second way to argue that the left hand position is not as good for black is to say 'Black has a superb move B1 on the right hand side' Possible ways to refute this might be to argue that white has a good counter W2 to B1 (false, black's right hand position is better), or that black has an even better position after black a on the left hand side than after B1 on the right (also false).

[Diagram]
Modified version of Example 1

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

[Diagram]
Example 2

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


[Diagram]
Example 2, Difference Game Setup

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 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 [ext] http://www.yomiuri.co.jp/igo_e/igomenu.htm

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.



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This is a copy of the living page "Difference game" at Sensei's Library.
(OC) 2004 the Authors, published under the OpenContent License V1.0.