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Combinatorial Game Theory
Path: CGTPath   · Prev:   · Next: GameTheoryInterface
    Keywords: Theory

The first book on Combinatorial Game Theory (CGT), On Numbers and Games, by John Conway, was published in the 1970s and has recently been republished. CGT was in part inspired by Go, since many Go end positions (yose) are combinations of independent regions of play.

Therefore CGT comes in naturally from the consideration of play on sub-boards. Each such position is a combinatorial game, which can be added to or subtracted from other such games. This is the idea that Go can be played on several boards at once, with alternating play still in operation: each player is able to play just one stone per turn, on one or other board

Numbers are special cases of combinatorial games: Go players can take that as meaning that some positions are already secure territory.

Bill Spight: In Mathematical Go, Berlekamp and Wolfe show how to apply the concept of number to go positions that still have some play left (hot positions). See Chilling.

By convention, the players are Left (Black) and Right (White). Left's scores are positive, Right's scores are negative; we are playing a zero-sum game with a convention that plus scores are good for Left. Either player, however, may be the one to start in a game, and the notation reflects that (no convention that Black starts, therefore).

A game may be represented in slash notation (see tree representation for more about this):

                    {A, B, C, ... | D, E, F, ...}

To the left of the central vertical bar are the games to which Left (Black) can move, called options or followers. To the right are those to which Right (White) can move.

For example

                    {1 | -1}

represents a one-point gote play. (Remember that scores are games because numbers are games.) Sometimes it is written +/- 1.


In CGT the number 1 means that Left can make a play, but Right cannot (or the equivalent). Go scoring does not obviously translate to CGT numbers, but you can do it. See Mathematical Go by Wolfe and Berlekamp. Interestingly, the form of Go scoring that most straightforwardly translates to CGT numbers is territory scoring with a group tax. See Ancient Chinese Rules and Philosophy.

Anyway, 1 + (-1) = 0: the introduction of addition and equality of games is compatible with the old meanings of '+' and '='. Black can make a play, and then White can make a play, or vice versa. One way to write that game is

                    { -1 | 1 }.

Most of the material originally here has been redistributed:


Evpsych: Is there anybody who has not read the books who understands the page? (i.e. should I try much harder?) Or is there a more elementary page for those of us who have not read the books?

Thanks.

Bill: These pages are tangential to go. They arose because several SL deshis are interested in the math. If you are interested in CGT, I'd say get Winning Ways, recently republished. It's a good introduction.

For a more practical orientation about how CGT applies to go, read Temperature, Difference Game, Miai, Tedomari, Miai Counting, Chilling, and the pages about Infinitesimals.

Evpsych: Thanks, but I'm still interested to know if anybody got an understanding of CGT from SL alone. I have the general idea, but the notation discussions seem to be missing some basic stuff. That could be wrong; SL could be sufficient.

Randall: I haven't read all of the pages on SL yet, but I almost understand what's being said. I suspect there is adequate information if you have a background in game theory -- I knew the concept of minimax and I had heard of CGT before coming across it here. I'm sure you could make it more accessible if you want -- probably by doing no more than asking questions where things don't make sense (or should I say that you can help others make it more accessible).


Original page by Bill Spight; subsequent edits by Charles Matthews.

See also: CGT path



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