combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the player ...

, the zero game is the game where neither player has any legal options. Therefore, under the

normal play convention A normal play convention in a game is the method of determining the winner that is generally regarded as standard. For example: *Preventing the other player from being able to move *Being the first player to achieve a target *Holding the highest va ...

, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero. The combinatorial notation of the zero game is: .. A zero game should be contrasted with the star game , which is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

Examples

Simple examples of zero games include Nim with no piles or a Hackenbush diagram with nothing drawn on it.

Sprague-Grundy value

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 as ...

applies to

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 bet ...

s (in which each move may be played by either player) and asserts that every such game has an equivalent Sprague–Grundy value, a "nimber", which indicates the number of pieces in an equivalent position in the game of nim. All second-player win games have a Sprague–Grundy value of zero, though they may not be the zero game. For example, normal Nim with two identical piles (of any size) is not the zero game, but has value 0, since it is a second-player winning situation whatever the first player plays. It is not a fuzzy game because first player has no winning option..

References

{{reflist Combinatorial game theory 0 (number)