Rock, Paper, Scissors…Random?

Nancy Pfenning

Dept. of Statistics

University of Pittsburgh

Intro/OutlineRandomness, a cornerstone of probability theory, is a difficult concept to grasp partly because the human brain is ill-equipped to mimic random behavior.

• What does it mean to be “random”?

• Lack of randomness in Rock/Paper/Scissors• Lack of randomness in Rock/Paper/Scissors

• Non-randomness in other examples

• RPS tournament as a model classroom activity

• Video clip of 2010 RPS Tournament

The Meaning of “Random”

Part One

The Meaning of “Random”

“random”How would you (or your students) define the word?

How would I define the word?

Occurring by chance* alone, and not according to a preference or attempted influence.

*So that the laws of probability apply: results are

unknown in the short run unknown in the short run

but predictable in the long run.

Essential for understanding probability distributions!

Two Questions to Think About:

• Can human beings mimic random behavior?

• How do these ideas relate to the game Rock/Paper/Scissors?

“random”How would you define the word?

• made, done, happening, or chosen without method or consciousdecision (OED) (haphazard?)

• governed by or involving equal chances for each item (OED)

• proceeding, made, or occurring without definite aim, reason, or pattern(American Heritage Dictionary)

• lacking a definite plan, purpose (MW) (haphazard?)• lacking a definite plan, purpose (MW) (haphazard?)

• being or relating to a set or to an element of a set each of whose elements has equal probability of occurrence (MW)

• relating to, having, or being elements or events with definite probability of occurrence (MW)

Can humans mimic randomness?

No! The human brain is structured in a way to facilitate the detection and creation of patterns.

It is generally impossible to suppress these tendencies and generate purely random selections without the aid of some sort of random number generator. The latter range anywhere number generator. The latter range anywhere from a coin

to a device that records radioactive decay.

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Randomness and RPS

Part Two

Randomness and RPS

Can Humans Mimic Random Behavior?

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0.4

Pro

bab

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Expected Random

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Rock Paper scissors

Pro

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Move Thrown

Expected Random Probability

Actual Probability

Data from World RPS Society: http://www.worldrps.com/

The Psychology of RPS• "Rock is for Rookies”: Males have a tendency to use Rock

on their opening throw. Rock is perceived as being a "strong" and “forceful“ move.

• (If opponent is more experienced, start with scissors, since he’s likely to start with paper.)

• Inexperienced players avoid runs of 3, perceived as non-• Inexperienced players avoid runs of 3, perceived as non-random. If such an opponent throws 2 scissors in a row,

• make your next throw paper!

• If you played with eyes closed your throws might be morerandom; good for defense but your offense would suffer!

Automatic imitation in a strategic context: players of rock-paper-scissors imitate opponents’ gestures (Proceedings of the Royal Society of Biological Sciences July 2011) showed significant differences in play, blind vs. sighted.

From World RPS Society: http://www.worldrps.com/

Attempting to Mimic

Randomness• The directors of the World

Rock Paper Scissors Society based in Toronto, Canada published “The Official Rock Paper Scissors Strategy Guide”.Strategy Guide”.

• This guide discusses players’ inability to mimic randomness, and the benefits of seeing patterns in their opponents throws.

• Note: Experts recently created an RPS-playing computer, which Walker said he “suckered” easily.

RPS-Playing Computerhttp://www.nytimes.com/interactive/science/rock-paper-scissors.html

RPS-Playing Computer

Basic Strategy of RPS“Players collect, interpret, and ignore information

both prior to and during the match. They must then make their choices. In turn, their opponents analyze these choices. The result is a tangled dance of strategy and counter-strategy…”

• Offense: try to predict opponent’s throw and • Offense: try to predict opponent’s throw and counter it (discover deviations from randomness)

• Defense: try to thwart all attempts at prediction by opponent (Throw at random?)

From World RPS Society: http://www.worldrps.com/

RPS vs Coin-tossing“RPS games are won and lost as a direct result of

how well one plays against the opponent. There

is no hand of fate in RPS. Agreeing to toss a coin

declares a preference to have fate make

decisions rather than to be actively involved in decisions rather than to be actively involved in

the process.”

From The Official RPS Strategy Guide, D. and G. Walker, Simon & Schuster 2004

RPS vs. Chance“Much to the dismay of RPS-playing parents, many school

teachers have used it to illustrate probability. Consider this hypothetical question from a 6th-grade math test:”

Jimmy and Janie are playing a game of RPS. Each can play only one of the three throws. What is the probability that Jimmy will play Rock next? 1/3?probability that Jimmy will play Rock next?

Counter questions: Is Jimmy a good player?

Has he played against Janie before?

What did he (and Janie) throw last?

Is Janie right- or left-handed?

Does Jimmy have a tattoo?

From World RPS Society: http://www.worldrps.com/

1/3?

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Advice from the Pros

(besides trying to psych out your opponent…)

• Keep your throws as varied as possible. Counter the risk of over-thinking throws by memorizing a random number sequence like π.random number sequence like π.

• Develop a repertoire of multiple throws or gambits. This can increase your unpredictability.

The Great Eight RPS Gambits

• Scissor Sandwich -

• Paper Dolls -

• Tool Box -

Paper, Scissors, Paper

Paper, Scissors, Scissors

Scissors, Scissors, Scissors• Tool Box -

• Avalanche -

• Bureaucrat -

• Crescendo -

• Fistful o’ Dollars -

• Denouement -

Rock, Rock, RockPaper, Paper, Paper

Paper, Scissors, RockRock, Paper, Paper

Rock, Scissors, Paper

Scissors, Scissors, Scissors

Other Activities on Randomness

Part Three

Other Activities on Randomness

Other Activities to Illustrate Inability to Mimic RandomnessInability to Mimic Randomness

Pick 3 States from 51

Pick a Number from 1 to 20

Alabama Louisiana OhioAlaska Maine OklahomaArizona Maryland Oregon

Arkansas Massachusetts PennsylvaniaCalifornia Michigan Rhode IslandColorado Minnesota South CarolinaConnecticut Mississippi South Dakota

Delaware Missouri Tennessee

CHOOSE 3 STATES AT RANDOM FROM 51 Was your sample truly random?

Delaware Missouri TennesseeFlorida Montana TexasGeorgia Nebraska UtahHawaii Nevada VermontIdaho New Hampshire VirginiaIllinois New Jersey WashingtonIndiana New Mexico West VirginiaIowa New York WisconsinKansas North Carolina WyomingKentucky North Dakota Washington, D.C.

Alabama Louisiana OhioAlaska Maine OklahomaArizona Maryland Oregon

Arkansas Massachusetts PennsylvaniaCalifornia Michigan Rhode IslandColorado Minnesota South CarolinaConnecticut Mississippi South Dakota

Delaware Missouri Tennessee

CHOOSE 3 STATES 3 sep. cols. 24%All 3 in 1: 10%

Delaware Missouri TennesseeFlorida Montana TexasGeorgia Nebraska UtahHawaii Nevada VermontIdaho New Hampshire VirginiaIllinois New Jersey WashingtonIndiana New Mexico West VirginiaIowa New York WisconsinKansas North Carolina WyomingKentucky North Dakota Washington, D.C.

Sampling: First Step in Data Production

Each student chooses a whole number at random

from 1 to 20.

Are the selections truly unbiased? A show of

hands may indicate that certain numbers are

©2011 Brooks/Cole,

Cengage Learning

Elementary Statistics: Looking at the Big Picture©2011 Brooks/Cole,

Cengage Learning

Elementary Statistics: Looking at the Big Picture L1.22

hands may indicate that certain numbers are

favored over others…

True or Biased Selection of Numbers 1 to 20

©2011 Brooks/Cole,

Cengage Learning

Elementary Statistics: Looking at the Big Picture©2011 Brooks/Cole,

Cengage Learning

Elementary Statistics: Looking at the Big Picture L1.23

Tournament!

Part Four

(plus discussion of Game Theory)

RPS Tournament!Keeping these strategies in mind, students can compete in their own RPS tournament in the classroom. Because competitors must be paired off successively in one-on-one matches, the optimal class size is a power of 2. A qualifying round can be held to achieve this. For example, if the class size N is more than 16 but fewer than 32, have 2(N-16) students (randomly selected, of course!) compete in the students (randomly selected, of course!) compete in the qualifying round. For this and all subsequent rounds, best out of nine throws is a manageable number as long as the class isn't much larger than 30 students.

The qualifying round eliminates N-16 players, resulting in N-(N-16)=16 players. Pair them off randomly for the next round, then pair off the winners of each, and so on, until the final one-on-one match. Afterwards, the winner can be ``interviewed'' about the strategies he/she employed, and asked if randomness could be said to enter in.

Report on RPS & Randomness as Student Project

Student researches and reports on these ideas, then supervises tournament

• Elementary approach focuses on game• Elementary approach focuses on game

• Social psych approach focuses on psychological explanations for bias

• Neuroscience approach focuses on tendency of brain to favor patterns

• Computer science approach focuses on generation of (pseudo)-random numbers

Rock Paper Scissors Tournament

In the USA Rock Paper Scissors League

(USARPS) Tournament, best of three throws

is a round. Best of 3 rounds is a match: See

http://www.usarps.com/http://www.usarps.com/

On-the-spot tournament among audience

members: best out of 9 throws determines

winner of each round.

Interview the champion about the extent to

which randomness (or lack thereof) played a

role.

Rock/Paper/Scissors In game theory, Nash equilibrium is a solution

concept of a game involving two or more players.

Stated simply, Jack and Jill are in Nash equilibrium if

Jack is making the best decision he can, taking into Jack is making the best decision he can, taking into

account Jill's decision, and Jill is making the best

decision she can, taking into account Jack's decision.

It is known that RPS has a unique mixed Nash

equilibrium in which each player throws R, P, S at

random with equal probabilities 1/3.

Rock/Paper/Scissors/Well!A variation on RPS, played in some countries around the world, includes a fourth option, the well:

Well beats Rock

Well loses to Paper

Well beats ScissorsWell beats Scissors

Well ties with Well

How can two players reach equilibrium playing RPSW?

2010 RPS Championship Video

Part Five

http://www.mtvu.com/shows/spring-break/rock-paper-scissors/2010-rock-paper-

scissors-championship-week-2/

2010 RPS Championship VideoQuickTime™ and a

Apple ProRes 422 (LT) decompressorare needed to see this picture.

Rock/Paper/Scissors/Well!A variation on RPS, played in some countries around the world, includes a fourth option, the well:

Well beats Rock

Well loses to Paper

Well beats ScissorsWell beats Scissors

Well ties with Well

How can two players reach equilibrium playing RPSW?

Thank you!

If you incorporate RPS into your prob/stats curriculum, please let me know---

nancypfenning@gmail.comnancypfenning@gmail.com