In
combinatorial game theory, cooling, heating, and overheating are operations on
hot game __NOTOC__
In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move.
By contrast, a cold game is one where each player can only worsen their position by making ...
s to make them more amenable to the traditional methods of the theory,
which was originally devised for
cold game __NOTOC__
In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move.
By contrast, a cold game is one where each player can only worsen their position by making ...
s in which the winner is the last player to have a legal move.
Overheating was generalised by
Elwyn Berlekamp
Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10.1 ...
for the analysis of
Blockbusting
Blockbusting was a business practice in the United States in which real estate agents and building developers convinced white residents in a particular area to sell their property at below-market prices. This was achieved by fearmongering the ho ...
.
Chilling (or unheating) 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
("
cooled by
") for a game
and a
(surreal) 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'' (or simply ''mean'') of
.
Heating is the inverse of cooling and is defined as the "''
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
''"
::
Multiplication and overheating
Norton multiplication is an extension of
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
to a game
and a positive game
(the "unit")
defined by
::
The incentives
of a game
are defined as
.
Overheating is an extension of heating used in Berlekamp's
solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
of
Blockbusting
Blockbusting was a business practice in the United States in which real estate agents and building developers convinced white residents in a particular area to sell their property at below-market prices. This was achieved by fearmongering the ho ...
,
where
''overheated from''
''to''
is defined for arbitrary games
with
as
::
''
Winning Ways'' also defines overheating of a game
by a positive game
, as
::
: Note that in this definition numbers are not treated differently from arbitrary games.
: Note that the "lower bound" 0 distinguishes this from the previous definition by Berlekamp
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
[Berlekamp & Wolfe (1994), pp. 52–55]
::
References
Combinatorial game theory
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