1 Rock, Paper, Scissors Random? Nancy Pfenning Dept. of Statistics University of Pittsburgh
2 Intro/Outline Randomness, 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 Non-randomness in other examples RPS tournament as a model classroom activity Video clip of 2010 RPS Tournament
3 Part One The Meaning of Random
4 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 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?
5 random How would you define the word? made, done, happening, or chosen without method or conscious decision (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?) 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)
6 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 from a coin to a device that records radioactive decay. QuickTime and a decompressor are needed to see this picture.
7 Part Two Randomness and RPS
8 0.4 Can Humans Mimic Random Behavior? Proba ability Expected Random Probability Actual Probability 0 Rock Paper scissors Move Thrown Data from World RPS Society:
9 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 nonrandom. 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 more random; good for defense but your offense would suffer! Automatic imitation in a strategic context: players of rockpaper-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:
10 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. 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.
11 RPS-Playing Computer
12 RPS-Playing Computer
13 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 counter it (discover deviations from randomness) Defense: try to thwart all attempts at prediction by opponent (Throw at random?) From World RPS Society:
14 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 the process. From The Official RPS Strategy Guide, D. and G. Walker, Simon & Schuster 2004
15 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? 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? QuickTime and a decompressor are needed to see this picture. From World RPS Society:
16 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.. Develop a repertoire of multiple throws or gambits. This can increase your unpredictability.
17 The Great Eight RPS Gambits Scissor Sandwich - Paper, Scissors, Paper Paper Dolls - Paper, Scissors, Scissors Tool Box - Scissors, Scissors, Scissors Avalanche - Rock, Rock, Rock Bureaucrat - Paper, Paper, Paper Crescendo - Paper, Scissors, Rock Fistful o Dollars - Rock, Paper, Paper Denouement - Rock, Scissors, Paper
18 Part Three Other Activities on Randomness
19 Other Activities to Illustrate Inability to Mimic Randomness Pick 3 States from 51 Pick a Number from 1 to 20
20 CHOOSE 3 STATES AT RANDOM FROM 51 Was your sample truly random? Alabama Louisiana Ohio Alaska Maine Oklahoma Arizona Maryland Oregon Arkansas Massachusetts Pennsylvania California Michigan Rhode Island Colorado Minnesota South Carolina Connecticut Mississippi South Dakota Delaware Missouri Tennessee Florida Montana Texas Georgia Nebraska Utah Hawaii Nevada Vermont Idaho New Hampshire Virginia Illinois New Jersey Washington Indiana New Mexico West Virginia Iowa New York Wisconsin Kansas North Carolina Wyoming Kentucky North Dakota Washington, D.C.
21 CHOOSE 3 STATES 3 sep. cols. 24% All 3 in 1: 10% Alabama Louisiana Ohio Alaska Maine Oklahoma Arizona Maryland Oregon Arkansas Massachusetts Pennsylvania California Michigan Rhode Island Colorado Minnesota South Carolina Connecticut Mississippi South Dakota Delaware Missouri Tennessee Florida Montana Texas Georgia Nebraska Utah Hawaii Nevada Vermont Idaho New Hampshire Virginia Illinois New Jersey Washington Indiana New Mexico West Virginia Iowa New York Wisconsin Kansas North Carolina Wyoming Kentucky North Dakota Washington, D.C.
22 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 favored over others 2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L1.22
23 True or Biased Selection of Numbers 1 to Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L1.23
24 Part Four Tournament! (plus discussion of Game Theory)
25 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 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.
26 Report on RPS & Randomness as Student Project Student researches and reports on these ideas, then supervises tournament 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
27 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 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.
28 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 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.
29 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 Scissors Well ties with Well How can two players reach equilibrium playing RPSW?
30 Part Five 2010 RPS Championship Video
31 2010 RPS Championship Video QuickTime and a Apple ProRes 422 (LT) decompressor are needed to see this picture.
32 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 Scissors Well ties with Well How can two players reach equilibrium playing RPSW?
33 Thank you! If you incorporate RPS into your prob/stats curriculum, please let me know---