# Temperature (CGT)

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' then is 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.