The History of Combinatorial Game Theory
Richard J. Nowakowski
Dalhousie University
[email protected]
January 24, 2009
Abstract
A brief history of the people and the ideas that have contributed to Combinatorial Game Theory.
1
The Newcomer
Games have been recorded throughout history but the systematic application of mathematics to games is a relatively
recent phenomonon.
Gambling games gave rise to studies of probability in the 16th and 17th century.
What has
become known as
Combinatorial Game Theory
or
Combinatorial Game Theory `a la Conway
—this to distinguish it
from other forms of game theory found in economics and biology, for example—is a babe-in-arms in comparison. It
has its roots in the paper [12] written in 1902, but the theory was not ‘codified’ until 1976-1982 with the publications
of
On Numbers and Games
[15] by John H. Conway and
Winning Ways
[10] by Elwyn R. Berlekamp, John H. Conway
and Richard K. Guy.
In the subject of
Impartial
games (essentially the theory as known before 1976), the first MSc thesis appears to
be in 1967 by Jack C. Kenyon [31] and the first PhD by Yaacov Yesha [63] in 1978. Using the full theory is Laura
J. Yedwab’s MSc thesis in 1985 [62] and David Wolfe’s 1991 PhD [60]. (Note: Richard B. Austin’s MSc thesis [3]
contains a little of the
Partizan
theory, but Yedwab’s thesis is purely Partizan theory.)
In the sequel, there are several Mathematical Interludes that give a peek into the mathematics involved in the theory.
Note that the names of games are given in small capitals such as
CHESS
. Also the notion of game is both general, like
CHESS
which refers to a a set of rules, and specific as in a specific
CHESS
position. Whenever I talk mathematically,
as in the Interludes, it is this specific notion that should be invoked.
2
What is Combinatorial Game Theory?
This
Combinatorial Game Theory
has several important features that sets it apart. Primarily, these are games of pure
strategy with no random elements. Specifically:
1. There are
Two Players
who
Alternate Moves
;
2. There are
No Chance Devices
—hence no dice or shuffling of cards;
3. There is
Perfect Information
—all possible moves are known to both players and, if needed, the whole history of
the game as well;
1

4.
Play Ends, Regardless
—even if the players do not alternate moves, the game must reach a conclusion;
5. The
Last Move
determines the winner—
Normal
play: last player to move wins;
Mis`ere
play last player to move
loses!
The players are usually called
Left
and
Right
and the genders are easy to remember —Left for
L
ouise Guy and Right
for
R
ichard Guy who is a important ‘player’ in the development of the subject. More on him later.
Figure 1: Louise and Richard Guy in Banff
Examples of games
1
NOT covered by these rules are:
DOTS
-&-
BOXES
and
GO
, since these are scoring games, the
last person to move is not guaranteed to have either the highest or the lowest score;
CHESS
, since the game can end in
a draw;
BACKGAMMON
, since there is a chance element (dice);
BRIDGE
, the only aspect that this game satisfies is that
it ends.

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