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[[Arrow's impossibility theorem]] states that if there are at least three options for which to vote, then there is no system of elections which will always satisfy certain "fairness" criteria. A proof of the theorem uses a situation analogous to RPS, where a candidate A would beat candidate B in a two-person contest, B would beat C, and C would beat A. The proof of the [[Gibbard-Satterthwaite theorem]], stating that no voting system with certain fairness properties is immune to "tactical voting" or a "spoiler effect", uses a similar idea.
[[Arrow's impossibility theorem]] states that if there are at least three options for which to vote, then there is no system of elections which will always satisfy certain "fairness" criteria. A proof of the theorem uses a situation analogous to RPS, where a candidate A would beat candidate B in a two-person contest, B would beat C, and C would beat A. The proof of the [[Gibbard-Satterthwaite theorem]], stating that no voting system with certain fairness properties is immune to "tactical voting" or a "spoiler effect", uses a similar idea.


==Variations==


Players have developed numerous variations on the game, from five-weapon variants (e.g. "rock-paper-scissors-Spock-lizard"<ref>{{cite web
| title = Rock Paper Scissors Spock Lizard
| url = http://www.samkass.com/theories/RPSSL.html
| author = [[Sam Kass|Kass, Sam]]
| accessdate = 2007-11-09
| date = [[2006-07-03]] }}</ref>) to those of purely theoretical interest having 25 or more weapons<ref>{{cite web
| title = RPS-25
| url = http://www.umop.com/rps25.htm
| author = [[David C. Lovelace|Lovelace, David C.]]
| accessdate = 2007-11-09
| date = [[2006-10-06]] }}</ref><ref>{{cite web
| title = RPS-101
| url = http://www.umop.com/rps101.htm
| author = [[David C. Lovelace|Lovelace, David C.]]
| accessdate = 2007-11-09
| date = [[2006-10-06]] }}</ref>. With an odd number of choices, each beats half the weapons and loses to half the weapons. No even number of weapons can be made balanced, unless some pairs of weapons result in a draw; there will always be some weapons superior to others. These also lose some of the aesthetic elegance of the game, which is otherwise one of the simplest possible games of skill.


An example of an unbalanced<ref>{{cite web
| title = The Myth of Dynamite Exposed
| url = http://www.worldrps.com/article4.html
| author = [[The World RPS Society|World RPS Society]]
| accessdate = 2007-11-09
| date = [[2002]] }})</ref> 4-weapon game adds "[[dynamite]]" as a trump. Dynamite, expressed as the extended index finger or thumb, defeats only rock, but is defeated by either scissors or paper. Given that paper performs better than dynamite (it beats anything dynamite will beat, and ties or beats anything dynamite will tie), it is always better to use paper than to use dynamite. In game theory terms, the strategy of "paper" has weak dominance over "dynamite".


One popular variant adds "fire" and "water" as potential trumps. Fire will beat any of the standard weapons (rock, paper, scissors), but a player may only throw it once in his entire lifetime. Water may be played as many times as one wishes, but loses to any throw except fire. (Those who adopt this trump depend on its notability and good sportsmanship to enforce the once-per-lifetime rule.)<ref>{{cite web
| title = Ro-Sham-Bo (The Official UPA Rules)
| url = http://www.upa.org/juniors/roshambo/rules.html
| author = [[Ultimate Players Association]]
| accessdate = 2007-11-09
| date = [[2004-09-04]] }}</ref>


Yet another variation, found in Indonesia, is composed of an earwig, a human, and an elephant. The earwig is able to climb into the elephant's ear and drive it insane, while the human crushes the earwig and the elephant crushes the human.<ref>{{cite web
| title = Children's Games in Street and Playground
| url = http://www.creekcats.com/pnprice/rochamtrue.html
| author = Iona and Peter Opie
| accessdate = 2008-03-10}}</ref>


Another variation includes Bear Man Gun. Bear beats Man. Man beats gun. Gun beats bear. This is plays by having two competitors to stand back to back, both repeating bear man gun and then turning to face eachother. Bear is symbolized by putting a person's hands above their head. Man is symbolized by putting your hands over your privates. Gun is symbolized by placing your hands together in front of you to make a gun with your pointers.
===Greek variation===
In Greece there is a variation called "rock-pencil-paper-scissors", which adds one more weapon, '''pencil''', which is represented by only having the index finger extended.{{Fact|date=April 2008}}


The rules added to the original ones are:
*Pencil writes on paper; pencil wins
*Rock smashes pencil; rock wins
*Scissors cut pencil; scissors win


Some consider this version inferior due to it's unbalanced nature and has long since been replaced by the western version of the game in Greece today.






































































































































































































































































































































































































































































































































































































































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== Analogies in nature and computing ==
== Analogies in nature and computing ==

Revision as of 14:52, 4 May 2008

For the Catatonia album, see Paper Scissors Stone (album).
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Rock-paper-scissors (also known as jan-ken-pon, rochambeau (sometimes spelled roshambo), and many derived terms[1]), is a popular two-person hand game.

The game is often used as a selection method in a similar way to coin flipping or drawing straws to randomly select a person for some purpose. However, unlike truly random selections, it can be played with skill if the game extends over many sessions, as a player can often recognize and exploit the non-random behavior of an opponent.

Sportspeople often use the game (both officially and unofficially, in place of a coin toss) to decide on opening plays. Similarly, uncertain calls, or even the whole game in case of rain, may be so decided.[citation needed] It is also often used as a method for creating appropriately non-biased random results in live action role-playing games, as it requires no equipment. It is also used in some online gambling sites as a form of novelty betting.

Terminology

The exact name of the game can vary, with the three components appearing in a different order, or with "stone" in place of "rock". Non-English-speakers often refer to the game by their words for "rock-paper-scissors" (though not necessarily in that order). Localised names for the game include the Japanese janken or jankenpon; the Korean gawi bawi bo (가위 바위 보); the Filipino jack-en-poy or bato-bato-pick (rock-rock-choose!); the Brazilian jó-kên-pô; the Hungarian kő-papír-olló; the Chilean ca-chi-pún; the South African ching-chong-cha (the words used in the count); the Cantonese Chinese 包 剪 揼 – cover-cut-hit; the Mandarin Chinese shi tou, jian dao, bu (石頭 剪刀 布 – scissors-rock-fabric); the Italian morra Cinese; and the Croatian kamen škare papir.

In France and the United States, in particular, the game is often referred to as Rochambeau (ro-sham-bo) in honor of Jean-Baptiste Donatien de Vimeur, comte de Rochambeau, French hero of the American Revolution. Rochambeau was present in York when General Cornwallis surrendered to George Washington. It is believed that Washington, Cornwallis and Rochambeau played RPS to decide who would be the last to leave Cornwallis' tent after the exchange of formalities. At the time, it was considered most honorable to be last. Rochambeau "won" the RPS game and the game has been known as "Rochambeau" ever since.[citation needed]

Cheating

A common timing-based tactic on a throw of paper or scissors is to wait until the last possible moment to dupe an opponent into believing that one may actually be throwing a scissors or rock. This is usually regarded as a foul at competitive RPS matches. In competitive RPS, however, a foul can be called on any throw that is revealed after thrower's arm swings past the perpendicular to his torso. In this case, the referee may then assign a throw of rock, or a "forced rock".

Priming is the number of bounces one does before revealing the throw. Another way to cheat is to prime three times to an opponent's twice, so that one sees his opponent's throw without revealing his own. In such cases the round will usually be replayed.

To prevent timing-based cheating, the players may be asked to hold their throwing hands behind their backs and reveal their already-formed throws after the count of three.

Another form of cheating is simpler and harder to detect. The strategy of claiming that your hand is scissors when the opponent claims it is paper. One would hold out their hand making the paper form when one sees your opponent has scissors one would open two fingers but not closing the others claiming that is simply 'how you make your scissors'. Of course it is simple to solve this, a do-over would make it right.

However, before playing, the two playing must decide the count on which they will throw their hands.

Mathematics

Non-transitivity

RPS is also often used as an example of the mathematical concept of non-transitivity. A transitive relation R is one for which a R b and b R c implies a R c. A reflexive, antisymmetric, and transitive relation on a set is known as a partial ordering, from which notions of "greater" and "less" follow. A game option which is "greater" than another is closer to being optimal, but such a notion does not exist in RPS: The relation used to determine which throws defeat which is non-transitive. Rock defeats scissors, and scissors defeat paper, but rock loses to paper. In fact, RPS could be called "intransitive" because A is greater than B, and B is greater than C, yet A is not greater than C.

Example of a commutative non-associative magma

In mathematics, it can be shown that there exist magmas (groupoids) that are commutative but not associative. One example is given by the operation of RPS.

Arrow's impossibility theorem

Arrow's impossibility theorem states that if there are at least three options for which to vote, then there is no system of elections which will always satisfy certain "fairness" criteria. A proof of the theorem uses a situation analogous to RPS, where a candidate A would beat candidate B in a two-person contest, B would beat C, and C would beat A. The proof of the Gibbard-Satterthwaite theorem, stating that no voting system with certain fairness properties is immune to "tactical voting" or a "spoiler effect", uses a similar idea.

Variations

Players have developed numerous variations on the game, from five-weapon variants (e.g. "rock-paper-scissors-Spock-lizard"[2]) to those of purely theoretical interest having 25 or more weapons[3][4]. With an odd number of choices, each beats half the weapons and loses to half the weapons. No even number of weapons can be made balanced, unless some pairs of weapons result in a draw; there will always be some weapons superior to others. These also lose some of the aesthetic elegance of the game, which is otherwise one of the simplest possible games of skill.

An example of an unbalanced[5] 4-weapon game adds "dynamite" as a trump. Dynamite, expressed as the extended index finger or thumb, defeats only rock, but is defeated by either scissors or paper. Given that paper performs better than dynamite (it beats anything dynamite will beat, and ties or beats anything dynamite will tie), it is always better to use paper than to use dynamite. In game theory terms, the strategy of "paper" has weak dominance over "dynamite".

One popular variant adds "fire" and "water" as potential trumps. Fire will beat any of the standard weapons (rock, paper, scissors), but a player may only throw it once in his entire lifetime. Water may be played as many times as one wishes, but loses to any throw except fire. (Those who adopt this trump depend on its notability and good sportsmanship to enforce the once-per-lifetime rule.)[6]

Yet another variation, found in Indonesia, is composed of an earwig, a human, and an elephant. The earwig is able to climb into the elephant's ear and drive it insane, while the human crushes the earwig and the elephant crushes the human.[7]

Another variation includes Bear Man Gun. Bear beats Man. Man beats gun. Gun beats bear. This is plays by having two competitors to stand back to back, both repeating bear man gun and then turning to face eachother. Bear is symbolized by putting a person's hands above their head. Man is symbolized by putting your hands over your privates. Gun is symbolized by placing your hands together in front of you to make a gun with your pointers.

Greek variation

In Greece there is a variation called "rock-pencil-paper-scissors", which adds one more weapon, pencil, which is represented by only having the index finger extended.[citation needed]

The rules added to the original ones are:

  • Pencil writes on paper; pencil wins
  • Rock smashes pencil; rock wins
  • Scissors cut pencil; scissors win

Some consider this version inferior due to it's unbalanced nature and has long since been replaced by the western version of the game in Greece today.

Analogies in nature and computing

Video games

Combat or strategy-based video games often feature RPS-like cycles in their characters' or units' effectiveness against others. These often attempt to emulate cycles in real-world combat (such as where cavalry are effective against archers, archers have an edge over spearmen, and spearmen are strongest against cavalry), or simple elemental trinities such as fire, water and air. Such game mechanics can make a game somewhat self-balancing, by preventing any one simple strategy from dominating gameplay.

Many card-based video games in Japan use the RPS system as their core fighting system, with the winner of each round being able to carry out their designated attack. (A popular game involving the RPS strategy is Pokemon, in which the use is fire, water, grass.)

Mating strategies

The Common Side-blotched Lizard (Uta stansburiana) exhibits a RPS pattern in its different mating strategies.

Biologist Barry Sinervo from the University of California, Santa Cruz has discovered an RPS evolutionary strategy in the mating behaviour of the side-blotched lizard species Uta stansburiana. Males have either orange, blue or yellow throats and each type follows a fixed, heritable mating strategy:[8]

  • Orange-throated males are strongest and do not form strong pair bonds; instead, they fight blue-throated males for their females. Yellow-throated males, however, manage to snatch females away from them for mating.
  • Blue-throated males are middle-sized and form strong pair bonds. While they are outcompeted by orange-throated males, they can defend against yellow-throated ones.
  • Yellow-throated males are smallest, and their coloration mimics females. Under this disguise, they can approach orange-throated males but not the stronger-bonding blue-throated specimens and mate while the orange-throats are engaged in fights.

This can be summarized as "orange beats blue, blue beats yellow, and yellow beats orange", which is similar to the rules of rock-paper-scissors.

The proportion of each male type in a population is similar in the long run, but fluctuates widely in the short term. For periods of 4-5 years, one strategy predominates, after which it declines in frequency as the strategy that manages to exploit its weakness increases. This corresponds to the stable pattern of the game in the replicator dynamics where the dynamical system follows closed orbits around the mixed strategy Nash equilibrium[citation needed] (Sinervo & Lively, 1996; Sinervo, 2001; Alonzo & Sinervo, 2001; Sinervo & Clobert, 2003; Sinervo & Zamudio, 2001).

Bacteria also exhibit a rock-paper-scissors dynamic when they engage in antibiotic production. The theory for this finding was demonstrated by computer simulation and in the laboratory by Benjamin Kerr, working at Stanford University with Brendan Bohannan (Nature. 2002 Jul 11;418(6894):171-4.). The antibiotics in question are the bacteriocins - more specifically, colicins produced by Escherichia coli. Biologist Benjamin C. Kirkup, Jr. further demonstrated that the colicins were active as E. coli compete with each other in the intestines of mice, and that the rock-paper-scissors dynamics allowed for the continued competition between antibiotic producing and antibiotic sensitive strains, because antibiotic resistant strains would out-compete the producing strains, providing an environment in which sensitive strains could successfully become established again (Nature. 2004 Mar 25;428(6981):412-4.).

Cultural references

Because of its widespread use, the game has received substantial references in popular culture. Many television series poke fun at particular characters' incompetence at understanding the rules, or show how mischievous characters are often able to "win" the game by inventing new objects which beat all the others. Popular examples involve a "chain saw" or a "giant squid".

Federal case

In 2006, Federal Judge Gregory Presnell from the Middle District of Florida ordered opposing sides in a lengthy court case to settle a trivial (but lengthily debated) point over the appropriate place for a deposition using the game of rock-paper-scissors.[9] The ruling in Avista Management v. Wausau Underwriters stated:

Upon consideration of the Motion – the latest in a series of Gordian knots that the parties have been unable to untangle without enlisting the assistance of the federal courts – it is ORDERED that said Motion is DENIED. Instead, the Court will fashion a new form of alternative dispute resolution, to wit: at 4:00 P.M. on Friday, June 30, 2006, counsel shall convene at a neutral site agreeable to both parties. If counsel cannot agree on a neutral site, they shall meet on the front steps of the Sam M. Gibbons U.S. Courthouse, 801 North Florida Ave., Tampa, Florida 33602. Each lawyer shall be entitled to be accompanied by one paralegal who shall act as an attendant and witness. At that time and location, counsel shall engage in one (1) game of "rock, paper, scissors." The winner of this engagement shall be entitled to select the location for the 30(b)(6) deposition to be held somewhere in Hillsborough County during the period July 11-12, 2006.

— [10]

The public release of this judicial order, requiring counsel to resort to a childish game to resolve a dispute, was widely circulated among area lawyers. It served the function of shaming the respective law firms regarding their litigation conduct.[citation needed]

Auction house RPS match

Large Trees Under the Jas de Bouffan sold for $11,776,000 at Christie's[11].

When Takashi Hashiyama, CEO of a Japanese television equipment manufacturer, decided to auction off the collection of Impressionist paintings owned by his corporation, including works by Cézanne, Picasso and van Gogh, he contacted two leading U.S. auction houses, Christie's International and Sotheby's Holdings, seeking their proposals on how they would bring the collection to the market as well as how they would maximize the profits from the sale. Both firms made elaborate proposals, but neither was persuasive enough to get Hashiyama’s business. Willing to split up the collection into separate auctions, Hashiyama asked the firms to decide between themselves who would get the Cézanne's "Large Trees Under the Jas de Bouffan", worth $12-16 million.

The houses were unable to reach a decision. Hashiyama told the two firms to play RPS, to decide who would get the rights to the auction, explaining that "it probably looks strange to others, but I believe this is the best way to decide between two things which are equally good".

The auction houses had a weekend to come up with a choice of move. Christie's went to the 11-year-old twin daughters of an employee, who suggested "scissors" because "Everybody expects you to choose 'rock'." Sotheby's said that they treated it as a game of chance and had no particular strategy for the game, but went with "paper".[12]

Christie's won the match, with millions of dollars of commission for the auction house.

Tournaments

WRPS sanctioned tournaments

Starting in 2002, the World Rock Paper Scissors Society (WRPS) standardized a set of rules for international play[13] and has overseen annual International World Championships. These open, competitive championships have been widely attended by players from around the world and have attracted widespread international media attention.[14][15][16][17][18] WRPS events are noted for their large cash prizes, elaborate staging, and colourful competitors.[19] In 2004, the championships were broadcast on the U.S. television network Fox Sports Net.

World Championship results since 2002

Year Host City Medal Champion Gender Nationality
2002[20] Toronto Gold Peter Lovering Male  Canada
Silver Moe Asem Male  Canada
Bronze Dave Ferris Male  Canada
2003[21] Gold Rob Krueger Male  Canada
Silver Marc Rigaux Male  Canada
Bronze Patrick Merry Male  Canada
2004[22] Gold Lee Rammage Male  Canada
Silver Heather Birrell Female  Canada
Bronze Chris Berggeren Male  United States
2005[23] Gold Andrew Bergel Male  Canada
Silver Stan Long Male  United States
Bronze Stuart Waldman Male  United States
2006[24] Gold Bob Cooper Male  Great Britain
Silver Bryan Bennett Male  United States
Bronze Tom Smith Male  United States
2007[25] Gold Andrea Farina Female  United States
Silver David "Maximus" Arnold Male  United States
Bronze James Doolittle Male  Canada

Tour events

In addition to the International World Championships the WRPS also endorses or sanctions a year-round series of tournaments world wide. "Endorsed" tournaments agree to abide by the WRPS standardized international rules of play and code of conduct, while "Sanctioned" tournaments will net the winner a trip to compete at the International World Championships. Some of the major events of this tour include:

World Series of Roshambo

Professional poker player Phil Gordon hosts the annual World Series of Roshambo during the World Series of Poker every year. In 2006, Annie Duke outlasted a field of 64 players who paid $500 each to enter the tournament. First place prize was $10,000. This is the largest buy-in Roshambo event in the world.

USARPS Tournaments

USA Rock Paper Scissors League (USARPS) is the principal rock-paper-scissors league of the United States. It is sponsored by Bud Light. Matti Leshem is the co-commissioner of the USARPS.

In April 2006, the inaugural USARPS Championship was held in Las Vegas, Nevada. Following months of regional qualifying tournaments held across the US, 257 players were flown to Las Vegas for a single-elimination tournament at the House of Blues where the winner received $50,000. The tournament was shown on the A&E Network on June 12, 2006.

At the first USARPS Championship, Dave "The Drill" McGill defeated Robert "Fast Twitch" Twitchel to win the tournament. In addition, Jason "King of the Morning" Wood won a best-of-500 marathon to earn a brand new car.

The $50,000 2007 USARPS Tournament took place at the Las Vegas Mandalay Bay in May 2007 and was covered by RawVegas.tv and will also be airing on ESPN later this year. The winner was Jamie Langridge from Odessa, Texas and he went home with the Bud Light Trophy and $50,000.

National XtremeRPS Competition 2007-2008

The XtremeRPS National Competition[26] is a US nationwide RPS competition with Preliminary Qualifying contests starting in January 2007 and ending in May 2008, followed by regional finals in June and July 2008. The national finals will be held in Des Moines in August 2008, with a chance to win up to $5,000.

Notes

  1. ^ "Ready, set ... Roshambo! Contestants vie for $1,000 purse in Rock, Scissors, Paper contest". San Francisco Chronicle. Retrieved 2007-11-20.
  2. ^ Kass, Sam (2006-07-03). "Rock Paper Scissors Spock Lizard". Retrieved 2007-11-09. {{cite web}}: Check date values in: |date= (help)
  3. ^ Lovelace, David C. (2006-10-06). "RPS-25". Retrieved 2007-11-09. {{cite web}}: Check date values in: |date= (help)
  4. ^ Lovelace, David C. (2006-10-06). "RPS-101". Retrieved 2007-11-09. {{cite web}}: Check date values in: |date= (help)
  5. ^ World RPS Society (2002). "The Myth of Dynamite Exposed". Retrieved 2007-11-09. {{cite web}}: Check date values in: |date= (help))
  6. ^ Ultimate Players Association (2004-09-04). "Ro-Sham-Bo (The Official UPA Rules)". Retrieved 2007-11-09. {{cite web}}: Check date values in: |date= (help)
  7. ^ Iona and Peter Opie. "Children's Games in Street and Playground". Retrieved 2008-03-10.
  8. ^ Sinervo, Barry (2001-02-20). "The rock-paper-scissors game and the evolution of alternative male strategies". Retrieved 2006-08-20. {{cite web}}: Check date values in: |date= (help)
  9. ^ "Exasperated judge resorts to child's game". Associated Press. 2006-06-26. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  10. ^ Presnell, Gregory (June 7, 2006). "Order of the court: Avista Management vs. Wausau Underwriters Insurance Co". CNN.com. Retrieved 2006-06-08.
  11. ^ Art/Auctions logo, Impressionist & Modern Art, Christie's, 7PM, May 4, 2005, Sale 1514.
  12. ^ Vogel, Carol (April 29), "Rock, Paper, Payoff: Child's Play Wins Auction House an Art Sale", New York Times {{citation}}: Check date values in: |date= and |year= / |date= mismatch (help)
  13. ^ "Game Basics". World Rock Paper Scissors Society. Retrieved 2006-08-20.
  14. ^ Hruby, Patrick (2004-12-10). "Fists fly in game of strategy". Washington Times. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  15. ^ "2003 World Rock Paper Scissors Championship". All Things Considered. National Public Radio. 2003-10-24. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  16. ^ "Rock, Paper, Scissors A Sport?". CBS News. 2003-10-23. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  17. ^ "Rock Paper Scissors contest being held". Associated Press. 2003-10-27. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  18. ^ Park, Michael Y. (2006-03-20). "Rock, Paper, Scissors, the Sport". Fox News. Retrieved 2006-08-20. {{cite news}}: Check date values in: |date= (help)
  19. ^ "Gallery". World RPS society. 2005-11-13. Retrieved 2006-08-20. {{cite web}}: Check date values in: |date= (help)
  20. ^ "2002 International Rock Paper Scissors Championships Official Results". World RPS society. Retrieved 2006-08-20.
  21. ^ "2003 Championships Official Results". World RPS society. Retrieved 2006-08-20.
  22. ^ "2004 Championships Official Results". World RPS society. Retrieved 2006-08-20.
  23. ^ "2005 Championships Official Results". World RPS society. Retrieved 2006-08-20.
  24. ^ "2006 Championships Official Results". World RPS society. Retrieved 2006-11-16.
  25. ^ "2007 Championships Official Results". World RPS society. Retrieved 2007-10-15.
  26. ^ "XTreme RPS Competition by Showtime Entertainment". Retrieved 2007-01-07.

References

  • Alonzo, Suzanne H. & Sinervo, Barry (2001): Mate choice games, context-dependent good genes, and genetic cycles in the side-blotched lizard, Uta stansburiana. Behavioral Ecology Sociobiology 49(2-3): 176–186. doi:10.1007/s002650000265 (HTML abstract)
  • Culin, Stewart (1895): Korean Games, With Notes on the Corresponding Games at China and Japan. (evidence of nonexistence of rock-paper-scissors in the West)
  • Gomme, Alice Bertha (1894, 1898): The traditional games of England, Scotland, and Ireland, 2 vols. (more evidence of nonexistence of rock-paper-scissors in the West)
  • Opie, Iona & Opie, Peter (1969): Children's Games in Street and Playground Oxford University Press, London. (Details some variants on rock-paper-scissors such as 'Man, Earwig, Elephant' in Indonesia, and presents evidence for the existence of 'finger throwing games' in Egypt as early as 2000 B.C.)
  • Sinervo, Barry (2001): Runaway social games, genetic cycles driven by alternative male and female strategies, and the origin of morphs. Genetica 112-113(1): 417-434. doi:10.1023/A:1013360426789 (HTML abstract)
  • Sinervo, Barry & Clobert, Jean (2003): Morphs, Dispersal Behavior, Genetic Similarity, and the Evolution of Cooperation. Science 300(5627): 1949-1951. doi:10.1126/science.1083109 (HTML abstract) Supporting Online Material
  • Sinervo, Barry & Lively, C. M. (1996): The Rock-Paper-Scissors Game and the evolution of alternative male strategies. Nature 380: 240-243. doi:10.1038/380240a0 (HTML abstract)
  • Sinervo, Barry & Zamudio, K. R. (2001): The Evolution of Alternative Reproductive Strategies: Fitness Differential, Heritability, and Genetic Correlation Between the Sexes. Journal of Heredity 92(2): 198-205. PDF fulltext
  • Sogawa, Tsuneo (2000): Janken. Monthly Sinica 11(5). [Article in Japanese]
  • Walker, Douglas & Walker, Graham (2004): The Official Rock Paper Scissors Strategy Guide. Fireside. (RPS strategy, tips and culture from the World Rock Paper Scissors Society).
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rock_paper_scissors&oldid=210112551"