Impartial game
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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 Sprague-Grundy theory.
Impartial games include sprouts and nim. Chess, however, is not impartial, since one player can only move white pieces, and the other only black.
A game that is not impartial is called a partisan game.