{{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 {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 games (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)