Definitions
Nearby Words

# Rock-paper-scissors

Rock-paper-scissors (also known as scissors-paper-stone, jan-ken-pon, rochambeau (sometimes spelled roshambo), ching-chang-wulla, and many derived terms), 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. 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.

## Game play

 - Each of the three basic hand-signs (from left to right: rock, paper and scissors ) beats one of the other two.

The players both count aloud to three, or speak the name of the game (e.g. "Rock! Paper! Scissors!" or "Ro! Cham! Beau!"), each time raising one hand in a fist and swinging it down on the count. On the third count (saying "scissors!" or "Beau!" ), the players change their hands into one of three gestures, which they then "throw" by extending it towards their opponent. A variation on this version (often played in the Northeastern United States) involves a fourth count—"SHOOT"—before players throw their gesture.

• Rock, represented by a clenched fist.
• Paper, represented by an open hand, with the fingers connected.
• Scissors, represented by the index and middle fingers extended and separated.

The objective is to select a gesture which defeats that of the opponent. Gestures are resolved as follows:

• Rock breaks or blunts scissors; rock wins.
• Paper covers rock; paper wins.
• Scissors cut paper; scissors wins.

If both players choose the same gesture, the game is tied and played again. The average number of plays required to decide a winner is 1.5 (see Mathematics below).

In some variations of the game, the winner of each round "uses" the weapon on the opponent's weapon, to demonstrate that they have won. Otherwise the game is settled.

RPS is frequently played in a "best two out of three" match, and tournament players often prepare sequences of three gestures ahead of time.

Jason Simmons, a competitive RPS champion, claims that women tend to start with scissors, while the World RPS Society states that males have a tendency to lead with rock. At World RPS tournaments, scissors is statistically the least common throw.

During the 17th century, Japanese children played a street game where 2 teams would meet on opposite ends of a field and then send individual runners off towards each other's "Camp". These runners would follow a prescribed course passing several landmarks such as a tree, a rock, a wagon rut or whatever was around; whenever/wherever the two runners would meet up, they would stop and "Challenge" each other to "Rock-Paper-Scissors" or as the Japanese say, "Jan Ken Pon" until one runner won. The two runners would quickly continue running, but with the losing runner racing back to his own "Camp" in disgrace. His only hope for redemption was to get back to his own Camp faster than the runner who beat him at RPS/JKP. If he could pass or "tag" his first landmark before the other team's runner does, then his team was allowed to send a fresh runner out towards their first landmark and intercept the other team's runner and "Challenge" them to RPS/JKP and possibly send that runner back along the route to his own camp. The first team to get all of their runners into the other's camp was declared the winner. This idea has recently been adapted into a board game called "Hand2Hand".

## 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).

## Mathematics

### Expected number of rounds to determine a winner

The expected value for the number of rounds required to decide a winner, given that $p_x$ is the probability of reaching round x, is
$sum_\left\{n=1\right\}^infty n\left(p_n -p_\left\{n+1\right\}\right)$. In each round of RPS there is a 1/3 probability of a tie, so this equals
$sum_\left\{n=1\right\}^infty n\left(\left(tfrac\left\{1\right\}\left\{3\right\}\right)^\left\{n-1\right\} - \left(tfrac\left\{1\right\}\left\{3\right\}\right)^n\right)$. Regrouping (which is permitted since terms converge to zero) then leads to the geometric series
$sum_\left\{n=0\right\}^infty 2n\left(tfrac\left\{1\right\}\left\{3\right\}\right)^n$, which converges to 1.5.

### 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.

## Variations

Players have developed numerous cultural and personal variations on the game, from simply playing the same game with different objects, to expanding into more weapons.

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 four-weapon game adds "dynamite" as a trump. Dynamite, expressed as the extended index finger or thumb, always defeats rock, but is defeated by scissors. The paper-dynamite relationship is disputed; using it as a trump generally implies that "dynamite shreds paper," but there are those who claim that the paper would supposedly smother the wick. Because of this dispute (and the potential unfair advantage that would result), organized rock-paper-scissors contests never use dynamite.

The official roshambo rules of the Ultimate Players Association add "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 good sportsmanship to enforce the once-per-lifetime rule.)

One popular balanced five-weapon expansion adds "Spock" and "lizard" to the standard three. "Spock" is signified with the "live long and prosper" hand gesture, while "lizard" is shown by forming the hand into a sock-puppet-like mouth. Spock crushes scissors and vaporizes rock; he is poisoned by the lizard and disproved by the paper. Lizard poisons Spock and eats paper; it is crushed by the rock and decapitated by the scissors.

### Different weapons

A 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.

The popular Television series That 70's Show features a nuclear war-like version. The cockroach survies the nuclear bomb, the nuclear bomb destroys the foot (or human as it may apply) and the foot/human crushes the cockroach. When the character Fez grew upset over his repeated losses he cries out "It's as if everything is beaten by something else!"

## 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.) 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 an extended RPS strategy is Pokemon, in which attacks have varied effectiveness based on 17 elemental types.)

### 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:

• 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 (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.

#### 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. The ruling in Avista Management v. Wausau Underwriters stated:

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.

#### Auction house RPS match

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".

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 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. WRPS events are noted for their large cash prizes, elaborate staging, and colorful competitors. In 2004, the championships were broadcast on the U.S. television network Fox Sports Net.

### USARPS Tournaments

USA Rock Paper Scissors League (USARPS) is a US-based rock-paper-scissors league. 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.

The \$50,000 2007 USARPS Tournament took place at the Las Vegas Mandalay Bay in May 2007.

In 2008, Sean Sears beat out 300 other contestants and walked out of the Mandalay Bay Hotel & Casino with \$50,000.

The inaugural Budweiser International Rock, Paper, Scissors Federation Championship was held in Beijing, China after the close of the 2008 Summer Olympic Games. Belfast man won the competition.

### National XtremeRPS Competition 2007-2008

The XtremeRPS National Competition 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.

## 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. (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. (HTML abstract)
• Sinervo, Barry & Clobert, Jean (2003): Morphs, Dispersal Behavior, Genetic Similarity, and the Evolution of Cooperation. Science 300(5627): 1949-1951. (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. (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).