# Stone, Paper, Scissors

From Wikipedia

Called by a variety of names such as Roshambo, Jaken, JanKenPo, and Rock-paper-scissors, the game of Stone-paper-scissors is a whimsical game most often played by children. It is often used in a similar way to coin flipping or drawing straws to randomly select a person for some purpose, though unlike truly random selections it can be played with skill if the game extends over many sessions, because one can often recognize and exploit the non-random behavior of an opponent. It is also often used as an example of the mathematical concept of non-transitivity.

## Game play

Two players each make a fist. They count together "1..2..3..Go". On "Go" each simultaneously changes their fist into one of three "weapons":

• Paper: all fingers extended, palm facing downwards.
• Scissors: forefinger and middle finger extended and separated into a "V" shape.
• Stone (or "rock"): a clenched fist.

The objective is to defeat the opponent by selecting a weapon which defeats their choice under the following rules:

1. Paper covers Stone (paper wins)
2. Scissors cut Paper (scissors win)
3. Stone blunts Scissors (stone wins)

If players choose the same weapon, the game is a tie and is played over.

## Strategies

Strategy between human players obviously involves using psychology to attempt to predict or influence opponent behavior. It is considered acceptable to use deceptive speech ("I'm going to play a rock") to influence your opponent.

Mathematically optimal play (according to game theory) is a simple matter of selecting randomly, and so the game may be considered trivial in that sense when played in a way that eliminates psychology, as with a computer. But "optimal" in this sense means only "incapable of being defeated more than expected by chance", while it does not imply that the random strategy is best at taking advantage of a suboptimal opponent. In fact, if the opponent is human or a non-random program, it is almost certain that he plays suboptimally and that a modified strategy can exploit that weakness. This is easily demonstrated by Roshambot, a computer program that handily defeats most human players (as does its author Perry Friedman, who won an \$800 competition against seven opponents including former world poker champion Phil Hellmuth in August 2001). University of Alberta Ph.D. student and poker player Darse Billings organizes a computer Roshambo competition to explore these possibilities, and their application to computer game play in other fields (notably poker, in which exploiting an opponent's non-random behavior is an important part of strategy).

## Cheating

One of the first tricks learned by a Roshambo novice is to hold back a throw of paper until the last possible moment to dupe an opponent into believing that you may actually be throwing a stone. This allows you the extra few milliseconds for fine-tuning your approach and delivery. Both paper and scissors have this ability, however unless you are employing a "double-back" strategy, cloaking a paper throw is likely to draw an instinctive paper from your opponent.

The opening ritual before the actual throws are made ("1..2..3..Go!"), called "priming", is intended to get both players in sync so as to ensure simultaneous delivery of throws. This can be used to an advantage when two players are meeting for the first time, since it is often unclear as to what the priming speed will be. The tendency is to default to the priming speed of the faster player. This allows the faster priming player the luxury of dictating the flow of play and causes their opponent to dedicate more energy to "catching the prime" rather than concentrating on delivering an effective throw.

## Variations

Players often add other "weapons" to the game on a ad-hoc basis, but it is very likely that this will result in an unbalanced game. In particular, four (or any even number) of weapons cannot be made balanced; there will always be some weapons that will be superior to others. It also loses some of the asthetic simplicity of the game, which is otherwise one of the simplest possible games of skill. For example, "dynamite", expressed as the extended index finger, defeats only stone, but is defeated by either scissors or paper.

## Odd or Even

A similar game, for which much of the same thoughts apply is Odd or Even: Player A gets to select odd or even. Then both players act as above, only this time the "weapons" are just "one" (a fist with outstretched thumb) or "two" (a fist with outstretched thumb and forefinger). The values signified by the players are added, player A winning with a correct prediction about the result.

With a choice between two values (it does not matter that they are 1 and 2, only that they are not both odd or even) the game is balanced, and A has no benefit from making the call. But would you allow three (or any odd number) values to choose from, either odd or even would be a more probably outcome with both players acting randomly. (That is because n choices make n2 possible outcomes. Squares of even numbers are even, squares of odd numbers odd.)