A Conway game is a two-player game satisfying the following requirements:
1. There are two players, Left and Right (
and 
),
who move alternately.
2. The first player unable to move loses.
3. Both players have complete information about the state of the game.
4. There is no element of chance.
5. Every chain of successive options is finite, even when the players do not alternate (Schleicher and Stoll 2006).
For example, nim is a Conway game, but chess is not (due to the possibility of draws and stalemate). Note that Conway's "game of life" is (somewhat confusingly) not a Conway game.
A Conway game is either:
1. The zero game, denoted as 0 or 
, or
2. An object (an ordered pair) of the form 
, where 
and 
are sets of Conway games.
The elements of 
and 
are called the Left and Right options
respectively, and are the moves available to Left and Right. For example, in the
game 
, if it is 
's move, he may move to 
or 
,
whereas if it is 
's
move, he has no options and loses immediately.
A game in which both players have the same moves in every position is called an impartial game. A game in which players have different options in at least one position is a partisan game. A game with only finitely many positions is a short game. A loopy game permits a return to a previous position and is excluded by the fifth condition (Siegel 2009, p. 93).
Some simple games which occur frequently in the theory have abbreviated names:
1. 
2. 
3. 
for any positive
integer 
4. 
5. 
6. 
A recursive construction generates all short games. Let 
be the set
of game values born on or before day 
, identifying games that are equal. The zeroth day is 
. On the next day, all games 
are formed with 
and 
ranging independently over the subsets of 
, including the empty set.
Thus 
, and the numbers of elements
of 
for 
, 1, ... are 1, 4, 22, 1474, ... (OEIS A065401).
D. Hickerson and R. Li found the count 1474 in 1974 (Fraser et al. 2005,
pp. 6-7).
The following table gives the games 
for the displayed nonempty Left option sets 
(rows) and Right option sets 
(columns), all of which are subsets
of 
.
 |  |  | 0 | 1 | |
| 1 |  |  |  |  |  |
| 0 |  |  |  |  |  |
 | 0 |  |  |  | 0 |
 | 0 |  |  |  | 0 |
 |  |  |  |  |  |
The equivalence classes of short games, using the equality defined below, form an Abelian group with the operations (Schleicher and Stoll 2006):
Here, expressions of the form 
mean the set of all expressions of the form 
with 
in 
.
These game values form a partially ordered set with respect to the comparison operations:
1. 
. If the second player to move in the
game 
can win (
and 
are equal).
2. 
. If the first player to move in the
game 
can win (
and 
are fuzzy).
3. 
. If Left can win the game 
whether he plays first or not (
is greater than 
).
4. 
. If Right can win the game 
whether he plays first or not (
is less than 
).
We have denoted 
by 
. 
will mean either 
or 
, and similarly for 
. For example, we have 
, 
, and 
.
Each 
is a partially
ordered set under 
.
Every short game has a unique canonical form up to identity of its options (Schleicher and Stoll 2006). The canonical form provides a test for equality and is obtained by two types of simplification:
1. Removal of a dominated option: if 
and 
, then 
; and if 
and 
, then 
.
2. Bypassing a reversible option: if 
has a Left option 
with a Right option 
, replace 
by all Left options of 
. Writing 
for their set, this gives 
. Symmetrically, a Right option 
with a Left option 
can be replaced by all Right options of 
, giving 
.
A short game is in canonical form when all its options are in canonical form and it has no dominated or reversible options. Two short games in canonical form are equal iff they have identical sets of Left and Right options.
."
Integers 5, A06, 2005.