In combinatorial game theory

Combinatorial game theory

Combinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...

, a game is partisan or partizan if it is not impartial

Impartial game

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...

. That is, some moves are available to one player and not to the other.

Most games are partisan; for example, in chess

Chess

Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...

, only one player can move the white pieces.

Partisan games are more difficult to analyze than impartial game

Impartial game

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...

s, as the Sprague–Grundy theorem

Sprague–Grundy theorem

In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber. The Grundy value or nim-value of an impartial game is then defined as the unique nimber that the game is equivalent to...

does not apply. However, the application of combinatorial game theory to partisan games allows the significance of numbers as games to be seen, in a way that is not possible with impartial games.