Zero game

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In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, 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

{ | }.

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

Other games can have values of zero. In fact, all second-player win games have exactly that value, 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.

A zero game is the opposite of the star (game) {0|0}, 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.

The Zero Game is also the title of a novel by Brad Meltzer.