This article skims the surface of the vast subject of combinatorial games. It often makes reference to the foundational books Winning Ways for Your Mathematical Plays, abbreviated WW [Berlekamp et al. 1982], and On Numbers and Games, abbreviated ONAG [Conway 1976]. Other references that should be consulted are [Praenkel 1980; Guy 1983; Guy 1991], and the other articles in this volume. See also Fraenkel's master bibliography on pages 493-537.

1. What We Mean by a Combinatorial Game

Our games are unlike those of “classical” game theory, that find application in economics, management, and military strategy. Our games usually, though perhaps not quite always, satisfy the following conditions:

• 1. There are just two players, often called Left and Right. There can be no question of coalitions.

• 2. There are several, usually finitely many, positions, and often a particular starting position.

• 3. There are clearly defined rules that specify the two sets of moves that Left and Right can make from a given position to its options.

• 4. Left and Right move alternately, in the game as a whole.

• 5. In the normal play convention a player unable to move loses.

• 6. The rules are such that play will always come to an end because some player will be unable to move. This is called the ending condition. There are no games that are drawn by repetition of moves.

• 7. Both players know what is going on; there is complete information. There is no occasion for bluffing.

• 8. There are no chance moves: no dealing of cards; no rolling of dice.