Zero game
Encyclopedia
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: { | }.
A zero game is the opposite of the star (game theory) {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
.
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.
with no piles or a Hackenbush
diagram with nothing drawn on it.
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...
, 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: { | }.
A zero game is the opposite of the star (game theory) {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
Brad Meltzer
Brad Meltzer is a bestselling American political thriller novelist, non-fiction writer, TV show creator and award-winning comic book author.-Early life:...
.
Sprague-Grundy value
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
Fuzzy game
In combinatorial game theory, a fuzzy game is a game which is incomparable with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move...
because first player has no winning option.
Examples
Simple examples of zero games include NimNim
Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap....
with no piles or a Hackenbush
Hackenbush
Hackenbush is a two-player mathematical game that may be played on any configuration of colored line segments connected to one another by their endpoints and to the ground...
diagram with nothing drawn on it.