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 (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

G

t

{\displaystyle G_{t}}

("

G

{\displaystyle G}

cooled by

t

{\displaystyle t}

") for a game

G

{\displaystyle G}

and a (surreal) number

t

{\displaystyle t}

is defined by

G

t

=

{

{

G

t

L

−

t

∣

G

t

R

+

t

}

for all numbers

t

≤

any number

τ

for which

G

τ

is infinitesimally close to some number

m

,

m

for

t

>

τ

{\displaystyle G_{t}={\begin{cases}\{G_{t}^{L}-t\mid G_{t}^{R}+t\}&{\text{ for all numbers }}t\leq {\text{ any number }}\tau {\text{ for which }}G_{\tau }{\text{ is infinitesimally close to some number }}m{\text{ , }}\\m&{\text{ for }}t>\tau \end{cases}}}

.

The amount

t

{\displaystyle t}

by which

G

{\displaystyle G}

is cooled is known as the temperature; the minimum

τ

{\displaystyle \tau }

for which

G

τ

{\displaystyle G_{\tau }}

is infinitesimally close to

m

{\displaystyle m}

is known as the temperature

t

(

G

)

{\displaystyle t(G)}

of

G

{\displaystyle G}

;

G

{\displaystyle G}

is said to freeze to

G

τ

{\displaystyle G_{\tau }}

;

m

{\displaystyle m}

is the mean value (or simply mean) of

G

{\displaystyle G}

.

Heating is the inverse of cooling and is defined as the "integral"

∫

t

G

=

{

G

if

G

is a number,

{

∫

t

(

G

L

)

+

t

∣

∫

t

(

G

R

)

−

t

}

otherwise.

{\displaystyle \int ^{t}G={\begin{cases}G&{\text{ if }}G{\text{ is a number, }}\\\{\int ^{t}(G^{L})+t\mid \int ^{t}(G^{R})-t\}&{\text{ otherwise. }}\end{cases}}}

Multiplication and overheating

Norton multiplication is an extension of multiplication to a game

G

{\displaystyle G}

and a positive game

U

{\displaystyle U}

(the "unit")

defined by

G

.

U

=

{

G

×

U

(i.e.