Cooling and heating (combinatorial game theory)


In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory,
which was originally devised for cold games in which the winner is the last player to have a legal move.
Overheating was generalised by Elwyn Berlekamp for the analysis of Blockbusting.
Chilling and warming are variants used in the analysis of the endgame of Go.
Cooling and chilling may be thought of as a tax on the player who moves, making them pay for the privilege of doing so,
while heating, warming and overheating are operations that more or less reverse cooling and chilling.

Basic operations: cooling, heating

The cooled game for a game and a number is defined by
The amount by which is cooled is known as the temperature; the minimum for which is infinitesimally close to is known as the temperature of ; is said to freeze to ; is the mean value of.
Heating is the inverse of cooling and is defined as the "integral"

Multiplication and overheating

Norton multiplication is an extension of multiplication to a game and a positive game
defined by
The incentives of a game are defined as.
Overheating is an extension of heating used in Berlekamp's solution of Blockbusting,
where overheated from to is defined for arbitrary games with as
Winning Ways also defines overheating of a game by a positive game, as

Operations for Go: chilling and warming

Chilling is a variant of cooling by used to analyse the Go endgame of Go and is defined by
This is equivalent to cooling by when is an "even elementary Go position in canonical form".
Warming is a special case of overheating, namely, normally written simply as which inverts chilling when is an "even elementary Go position in canonical form".
In this case the previous definition simplifies to the form