combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' ev ...

, a game is partisan (sometimes partizan) if it is not

impartial Impartiality (also called evenhandedness or fair-mindedness) is a principle of justice holding that decisions should be based on objective criteria, rather than on the basis of bias, prejudice, or preferring the benefit to one person over anothe ...

. That is, some moves are available to one player and not to the other, or the payoffs are not symmetric. Most games are partisan. For example, in

chess Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...

, only one player can move the white pieces. More strongly, when analyzed using combinatorial game theory, many chess positions have values that cannot be expressed as the value of an impartial game, for instance when one side has a number of extra tempos that can be used to put the other side into

zugzwang Zugzwang (; ) is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move will worsen their position. A ...

. Partisan games are more difficult to analyze than

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. In other words, the only difference be ...

s, as the

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 one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented ...

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..

References

{{DEFAULTSORT:Partisan Game Combinatorial game theory