Editing Rock paper scissors (section)

=== Additional weapons ===
Generalized rock-paper-scissors games where the players have a choice of more than three weapons have been studied.<ref>{{cite arXiv|last=Akin|first=Ethan|eprint=1806.11241|title=Rock, Paper, Scissors, Etc—Topics in the Theory of Regular Tournaments|year=2018 |class=math.DS}}</ref> Any variation of rock paper scissors is an [[Orientation (graph theory)|oriented graph]], where the [[Vertex (graph theory)|nodes]] represent the symbols (weapons) choosable by the players, and an edge from A to B means that A defeats B. Each oriented graph is a potentially playable rock paper scissors game. According to theoretical calculations, the number of distinguishable (i.e. not [[Graph isomorphism|isomorphic]]) oriented graphs grows with the number of weapons = 3, 4, 5, ... as follows:<ref>{{citation
 | last1 = Harary | first1 = Frank | author1-link = Frank Harary | last2 = Palmer | first2 = Edgar M. | contribution = Formula 5.4.13 | location = New York | mr = 0357214 | page = 133 | publisher = Academic Press | title = Graphical Enumeration | year = 1973}}.</ref><ref name="rpsfwgame"/>
: 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … {{OEIS|A001174}}.

The French game ''pierre, papier, ciseaux, puits'' (stone, paper, scissors, [[water well|well]]) is unbalanced; both the stone and scissors fall in the well and lose to it, while paper covers both stone and well. This means two "weapons", well and paper, can defeat two moves, while the other two weapons each defeat only one of the other three choices. The stone has no advantage to well, so optimal strategy is to play each of the other objects (paper, scissors and well) one-third of the time.<ref name=Umbhauer>{{cite book|title=Game Theory and Exercises|last=Umbhauer|first=Gisèle|isbn=978-1-3173-6299-9|page=340|publisher=Routledge|date=2016}}</ref>

{| class="wikitable"
|+ "pierre, papier, ciseaux, puits";<br> + means that the row player "beats", − means "is beaten" and O means tie
! {{diagonal split header|<br />Row<br />player|Opponent}}
! style=min-width:5em | stone
! style=min-width:5em | paper
! style=min-width:5em | scissors
! style=min-width:5em | well
! nowrap | Optimal strategy for<br />the row player<ref name=Umbhauer/>
|- style="text-align: center;"
! scope=row | stone
          || O     || −     || +        || −    || <math>0</math>
|- style="text-align: center;"
! scope=row | paper
           || +     || O     || −        || +    || <math>\frac{1}{3}</math>
|- style="text-align: center;"
! scope=row | scissors
           || −     || +     || O        || −    || <math>\frac{1}{3}</math>
|- style="text-align: center;"
! scope=row | well
           || +     || −     || +        || O    || <math>\frac{1}{3}</math>
|}
Variants in which the number of moves is an odd number and each move defeats exactly half of the other moves while being defeated by the other half are typically considered. Variations with up to 101 different moves have been published.<ref>{{cite web | title = RPSx | url = http://www.umop.com/rps.htm | access-date = 2011-06-14 | archive-date = 2018-10-24 | archive-url = https://web.archive.org/web/20181024085608/http://www.umop.com/rps.htm | url-status = live }}</ref> Adding new gestures has the effect of reducing the odds of a tie, while increasing the complexity of the game. The probability of a tie in an odd-number-of-weapons game can be calculated based on the number of weapons ''n'' as 1/''n'', so the probability of a tie is 1/3 in standard rock paper scissors, but 1/5 in a version that offered five moves instead of three.<ref>{{cite web | title= Rock Paper Scissors Graphs | url= http://www.kreativekorp.com/miscpages/rps/ | author-first= Rebecca G. | author-last= Bettencourt | author-link= Rebecca G. Bettencourt | access-date= 2017-04-06 | archive-date= 2017-04-07 | archive-url= https://web.archive.org/web/20170407055323/http://www.kreativekorp.com/miscpages/rps/ | url-status= live }}</ref>
[[File:Pierre ciseaux feuille lézard spock aligned.svg|thumb|Rock paper scissors lizard Spock gestures]]
One popular five-weapon expansion is "{{visible anchor|rock paper scissors Spock lizard}}", invented by Sam Kass and Karen Bryla,<ref>{{cite web | title = Original Rock-Paper-Scissors-Spock-Lizard Page | url = http://www.samkass.com/theories/RPSSL.html | author = Sam Kass | access-date = 2009-03-11 | archive-date = 2010-11-13 | archive-url = https://web.archive.org/web/20101113135208/http://www.samkass.com/theories/RPSSL.html | url-status = live }}</ref> which adds "[[Spock]]" and "lizard" to the standard three choices. "Spock" is signified with the ''Star Trek'' [[Vulcan salute]], while "lizard" is shown by forming the hand into a sock-puppet-like mouth. Spock smashes scissors and vaporizes rock; he is poisoned by lizard and disproved by paper. Lizard poisons Spock and eats paper; it is crushed by rock and decapitated by scissors. This variant was mentioned in a 2005 article in ''[[The Times]]'' of London<ref name="TimesSpockLizard">{{cite news |url=https://www.thetimes.com/travel/destinations/asia-travel/japan/and-paper-scissors-5pz9q0xkxct |title=... and paper scissors |date=11 June 2005 |access-date=2009-06-09 |archive-url=https://web.archive.org/web/20100531231332/http://www.timesonline.co.uk/tol/comment/leading_article/article1080425.ece |archive-date=31 May 2010 |url-status=live |newspaper=[[The Times]] }}{{subscription required}}</ref> and was later the subject of an [[List of The Big Bang Theory episodes#ep25|episode]] of the American sitcom ''[[The Big Bang Theory]]'' in 2008 (as ''rock-paper-scissors-lizard-Spock'').<ref>{{cite web |last=Lorre |first=Chuck |author-link=Chuck Lorre |title=The Big Bang Theory Video&nbsp;— Rock, Paper, Scissors, Lizard, Spock&nbsp;— CBS.com |url=http://www.cbs.com/shows/big_bang_theory/video/926519421/the-big-bang-theory-rock-paper-scissors-lizard-spock |publisher=[[CBS]] |format=video |access-date=4 September 2012 |archive-url=https://web.archive.org/web/20180730080428/https://www.cbs.com/shows/big_bang_theory/video/926519421/the-big-bang-theory-rock-paper-scissors-lizard-spock/ |archive-date=30 July 2018 |url-status=dead }}</ref>
{| class="wikitable"
|+ Rock-Paper-Scissors-Spock-Lizard;<br> + means that the row player "beats", − means "is beaten" and O means tie
! {{diagonal split header|<br />Row<br />player|Opponent}}
! style=min-width:5em | rock
! style=min-width:5em | paper
! style=min-width:5em | scissors
! style=min-width:5em | Spock
! style=min-width:5em | lizard
! nowrap | Optimal strategy<br />for row player
|- style="text-align: center;"
! scope=row | rock
           || O     || −     || +        || −    || +      || <math>\frac{1}{5}</math>
|- style="text-align: center;"
! scope=row | paper
           || +     || O     || −        || +    || −      || <math>\frac{1}{5}</math>
|- style="text-align: center;"
! scope=row | scissors
           || −     || +     || O        || −    || +      || <math>\frac{1}{5}</math>
|- style="text-align: center;"
! scope=row | Spock
           || +     || −     || +        || O    || −      || <math>\frac{1}{5}</math>
|- style="text-align: center;"
! scope=row | lizard
           || −     || +     || −        || +    || O      || <math>\frac{1}{5}</math>
|}
[[File:Rpsfw game.svg|300px|thumb|right|Oriented graph of Rock-Paper-Scissors-Fire-Water]]
A [[Game theory|game-theoretic analysis]] showed that 4 variants of 582 possible variations using 5 different weapons have non-trivial [[Strategy (game theory)#Mixed strategy|mixed strategy equilibria]].<ref name="rpsfwgame">{{cite journal |last1=Tagiew |first1=Rustam |title=Towards a Framework for Management of Strategic Interaction |journal=Proceedings of the International Conference on Agents and Artificial Intelligence |date=2009 |issue=ICAART |pages=587–590 |url=https://www.researchgate.net/publication/221539739 |access-date=2020-01-27 |archive-date=2020-07-28 |archive-url=https://web.archive.org/web/20200728170217/https://www.researchgate.net/publication/221539739_Towards_a_Framework_for_Management_of_Strategic_Interaction |url-status=live }}</ref> The most representative game of these 4 is "rock, paper, scissors, fire, water". Rock beats scissors, paper beats rock, scissors beats paper, fire beats everything except water, and water is beaten by everything except it beats fire. The perfect game-theoretic strategy is to use rock, paper, and scissors <math>\frac{1}{9}</math> of the time and <math>\frac{1}{3}</math> of the time for fire and water. Nevertheless, experiments show that people underuse water and overuse rock, paper, and scissors in this game.<ref>{{cite book |last1=Tagiew |first1=Rustam |title=AI*IA 2009: Emergent Perspectives in Artificial Intelligence |chapter=Hypotheses about Typical General Human Strategic Behavior in a Concrete Case |series=Lecture Notes in Computer Science |date=2009 |volume=5883 |pages=476–485 |publisher=Springer |location=Berlin, Heidelberg |doi=10.1007/978-3-642-10291-2_48 |bibcode=2009LNCS.5883..476T |isbn=978-3-642-10290-5  | issn=0302-9743}}</ref>
{| class="wikitable"
|+ Rock-Paper-Scissors-Fire-Water;<br> + means that the row player "beats", − means "is beaten" and O means tie
! {{diagonal split header|<br />Row<br />player|Opponent}}
! style=min-width:5em | rock
! style=min-width:5em | paper
! style=min-width:5em | scissors
! style=min-width:5em | fire
! style=min-width:5em | water
! nowrap | Optimal strategy<br />for row player<ref name="rpsfwgame"/>
|- style="text-align: center;"
! scope=row | rock
           || O     || −     || +        || −    || +      || <math>\frac{1}{9}</math>
|- style="text-align: center;"
! scope=row | paper
           || +     || O     || −        || −    || +      || <math>\frac{1}{9}</math>
|- style="text-align: center;"
! scope=row | scissors
           || −     || +     || O        || −    || +      || <math>\frac{1}{9}</math>
|- style="text-align: center;"
! scope=row | fire
           || +     || +     || +        || O    || −      || <math>\frac{1}{3}</math>
|- style="text-align: center;"
! scope=row | water
           || −     || −     || −        || +    || O      || <math>\frac{1}{3}</math>
|}