# Temperature (CGT)

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In Winning Ways, Berlekamp, Conway and Guy define the temperature of short games. A short game is one with only finitely many positions in all. To define temperature we first define cooled games.

If G = {G^L | G^R} is a short game, and t a real number ⩾ 0, then we define the cooled game, G(t), or G cooled by t, by the formula

G(t) = {G^L(t) - t | G^R(t) + t},

unless there is a smaller real number t' for which G(t') is infinitesimally close to a number x, in which case

G(t) = x for all t > t'.

t' is then defined to be the temperature of G.

(Note that this definition calls the parameter, t, temperature. It is this sense of temperature that is the basis of the go term, as it is related to the value of plays elsewhere. t may be considered a tax on a play. If t > t', the tax is too high, and at least one player does not wish to play in G. If there is a game elsewhere with temperature t'' > t, each player may wish to play in it.)

The temperature of a game is an indicator of the excitement or urgency of playing in it. For instance, the game {1000 | -1000} has a temperature of 1000, while the game {1 | -1} has a temperature of only 1. A play in the first game is much more urgent than a play in the second.

The temperature of a game is also a measure of the average gain a player can make by a gote play in it. It has the same value as the miai value of the play, in go terms.

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Temperature (CGT) last edited by PJTraill on February 9, 2019 - 00:36