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Cooling
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    Keywords: Theory

Cooling is a concept from Combinatorial Game Theory which reduces a hot game in which both players are eager to play to one with lower temperature. This is achieved by having each player pay a certain tax (the amount by which the game is cooled). Or, in other terms, having other more interesting situations on the goban.


BillSpight: Cooling may be defined in terms of play in an environment of plays. This environment is richer than the go board, but is a reasonable approximation of it when there are many plays.


A go example:

[Diagram]

(The outer strings of stones are assumed to be alive)

Here, if Black goes first, he gets three points versus White none. If White connects, the score will be zero points for both players. So the temperature here is three.

Bill: The temperature is 1 1/2.


We can cool this game by requiring that each player gives a certain number of stones to his opponent as prisoner, to pay for the privilege of playing here. If, for instance you need to pay one stone, the result when Black goes first will be 3 vs 1 (because of the tax), and when White goes first it will be 1 for Black and 0 for White. The temperature has been reduced to one.

Bill: If you cool by 1 point the result when Black plays first will be 2. The result when White plays first will be 1. (Neither player will play the dame.)

If the tax were two stones however, none of the players would be interested in playing in this situation: Black would after capturing be just 1 point ahead due to the tax, and White would give Black 2 points by connecting. Clearly, each player is going to wait for the other to play.

The situation becomes interesting when the tax is determined at one and a half stone (half of the temperature, but that rule of thumb doesn't apply in more complex situations).

Bill: The temperature is 1 1/2. The go term for the temperature of a position is its miai value.

The result would then be that Black comes out 3/2 points ahead regardless of whoever plays. Since this is then a gote play (Black stops 1.5 ahead with White to play and vice versa), no play is going to be made here.

Bill: If the tax is 1.5, this cools to *, a fuzzy game in which the player to move wins. Of course, a player will prefer to play in a hotter game, but the tax is not high enough to prevent a play.

If you graph the left and right stops as you increase the cooling factor, you get this diagram:

[Diagram]
A lo-res Thermograph

The direction 'a' to 'g' indicates decreasing value for Black, and 'h' to 'n' indicates a larger tax. (a = 3, d = 1.5, g = 0 and h = 0, j = 1, k = 1.5, n = 3)



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