Editing Zero game

{{Short description|Game where both players can't move}}
{{distinguish|Zero-sum game}}

{{about|combinatorial game theory|the novel entitled "The Zero Game"|Brad Meltzer}}
In [[combinatorial game theory]], the '''zero game''' is the game where neither player has any legal options. Therefore, under the [[normal play convention]], 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: { | }.<ref name="conway-p72">{{citation|first=J. H.|last=Conway|authorlink=John Horton Conway|title=On numbers and games|publisher=Academic Press|year=1976|page=72}}.</ref>

A zero game should be contrasted with the [[star (game theory)|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.<ref name="conway-p72"/>

==Examples==
Simple examples of zero games include [[Nim]] with no piles<ref>{{harvtxt|Conway|1976}}, p. 122.</ref> or a [[Hackenbush]] diagram with nothing drawn on it.<ref>{{harvtxt|Conway|1976}}, p. 87.</ref>

==Sprague-Grundy value==
{{main|Sprague–Grundy theorem}}
The [[Sprague–Grundy theorem]] applies to [[impartial game]]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]].<ref>{{harvtxt|Conway|1976}}, p. 124.</ref> All second-player win games have a Sprague–Grundy value of zero, though they may not be the zero game.<ref>{{harvtxt|Conway|1976}}, p. 73.</ref>

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.<ref>{{citation|page=44|first1=Elwyn R.|last1=Berlekamp|author1-link=Elwyn Berlekamp|first2=John H.|last2=Conway|author2-link=John Horton Conway|first3=Richard K.|last3=Guy|author3-link=Richard K. Guy|title=Winning Ways for your mathematical plays, Volume 1: Games in general|publisher=Academic Press|edition=corrected|year=1983}}.</ref>

==References==
{{reflist}}

[[Category:Combinatorial game theory]]
[[Category:0 (number)]]