impu1se

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[10/4/04] I'm 11k on KGS, 20k* on IGS where I don't play much. I'm particularly intrigued by the CGTPath information. This page is intended to be a sort of scrapbook of my attempts at understanding various pages on Senseis. I'm not aiming to rewrite anything; but possibly some of these musings may be helpful to someone out there.


CGT Numbers

A game can be represented in the following form as a tree of possible moves:

    G = { B1, B2, B3, etc... | W1, W2, W3, etc... }

B1, B2, B3, etc... are the games that follow after a certain black move has been played. W1, W2, W3, etc... are the games that follow after a certain white move has been made.

[Diagram]

{ | } = 0

There are no possible moves. This game is a number called 0. The last player who moved wins.


[Diagram]

{ 0 | } = 1

Black has one move, a win for black. { { | } | } => { 0 | }. Since this game is greater than the 0 game, it gets called 1. In the same way, { 1 | } = 2, { 2 | } = 3, and so on.


[Diagram]

{ | 0 } = -1

The opposite of a game with white and black switched is the negative of the original game.


[Diagram]

{ 0 | 0 } = *

Whoever gets the last dame gets the last move and wins. This is a fuzzy or infinitesimal game.


[Diagram]

Game with options

This game is { 1, 0 | 0, 1 } = { 1 | 0 } since either player will always choose "a". Also note that a game such as { {2 | 1} | {-3 | -4} } is equivalent to { 1 | -3 } because of the minimax rule and alternating plays.


[Diagram]

{ 1 | 0 } = 1/2

{ 1 | * } => { 1 | 0 } = 1/2.

Two games may be added by allowing players to take alternating turns making a play in one or the other board. Using this process it's easy to see why the { 1 | 0 } game has a value of 1/2.


[Diagram]

Black first = 1

[Diagram]

White first = 1

Either way the result is 1. { 1 | 0 } + { 1 | 0 } = 1. Therefore, { 1 | 0 } = 1/2.


References: ZeroInCGTTerms, CombinatorialGameTheory, Numbers, GoInfinitesimals, SurrealNumbers, DisjunctiveSums



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