Zero game - Definition

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 combinatorial notation of the zero game is

{ | }.

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

Other games can have values of zero, and in fact, all second-player win games have exactly that value, though they may not be the zero game.

For example, Nim with two identical piles (of any size) is not the zero game, but has value 0, since it is unequivocally a second-player winning situation.

A zero game is the opposite of the fuzzy game {0|0}, which is a first-player win since each player can (if it is their turn) move to a zero game, and therefore win.