In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first.
Impartial games can be analyzed using the Sprague–Grundy theorem.
Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games like ZÈRTZ and Chameleon are also not impartial, since although they are played with shared pieces, the payoffs are not necessarily symmetric for any given position.
A game that is not impartial is called a partisan game.
- E. Berlekamp; J. H. Conway; R. Guy (1982). Winning Ways for your Mathematical Plays. 2 vols. Academic Press.; vol. 1. ISBN 0-12-091101-9.; vol. 2. ISBN 0-12-091102-7.
- E. Berlekamp; J. H. Conway; R. Guy (2001–2004). Winning Ways for your Mathematical Plays. 4 vols. (2nd ed.). A K Peters Ltd.; vol. 1. ISBN 1-56881-130-6.; vol. 2. ISBN 1-56881-142-X.; vol. 3. ISBN 1-56881-143-8.; vol. 4. ISBN 1-56881-144-6.