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'`` 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.