Uncertainty

Robin BurkeGAM 206

Outlineo Quiz (30 min)o Uncertainty

o Lotso Dice

Quizo 30 minQuiz Answers

Uncertaintyo Many games are

probabilistico roll the diceo shuffle the cards

o Some games are noto Chesso Tic-Tac-Toe

Certainty vs uncertaintyo Certainty

o the condition when the outcome of an action is known completely in advance.

o But even theno uncertainty about who will wino otherwise what is the point?

o Deterministico no chance element involved

o Strategically-interesting deterministic games are hard to designo examples?

Race gameso Race games

o characterized by the goalo get pieces to the end position

first

o Almost always involve chanceo one known exception

oHare and Tortoise (1974)

o Distinguished byo the topology of the tracko the number of pieces

Taxonomyo Cruciform games

o Nyout (Korean)o Pachisio Parcheesio Sorry

o Tables gameso Royal Game of Ur (Mesopotamia)o Game of Twenty (Egypt)o Senet (Egypt)o Liubo (China)o Nard (Persia)

Probabilityo Probability is the study of

chance outcomeso originated in the study of

gamesoBlaise Pascal (1654)

o Basic ideao (modern conception)o a random variableo a quantity whose value is

unknown until it is "sampled"

Random variable 2o We characterize a random

variableo not by its valueo but by its "distribution"o the set of all values that it might

takeo and the percentage of times that it

will take on that valueo distribution sums to 1

o since there must be some outcome

o Probabilityo the fraction of times that an

outcome occurs

Lotso Coin flipso Random variable

o heads?otrue or false

o Distributiono trueo falseo each value ½ of the time

Multiple Lotso More possible outcomes

o 0..4 heads

o Probabilities?o Not Uniform

Possibility Spaceo How many possible outcomes?

o TTTT, HTTT, THTT, HHTT, TTHT, HTHT, THHT, HHHT, TTTH, HTTH, THTH, HHTH, TTHH, HTHH, THHH, HHHH

o 24 = 16o How many of each type?

o 0 H = 1 = 1/16o 1 H = 4 = 1/4o 2 H = 6 = 3/8o 3 H = 4 = 1/4o 4 H = 1 = 1/16

o This is why you get an extra throw on 4 or 5 in Seneto why do you get an extra throw for a

1?

Single dieo Random variable

o odd or even number of dots

o Distributiono odd?o 50%

o The same as binary lots

Single Dieo Random variable

o # of spots on the side facing up

o Distributiono 1...6o each value 1/6 of the time

o Idealization

Two diceo Random variable

o usually we care about the sum of the two die values

o Distributiono 2, 12 = 1/36o 3, 11 = 1/18o 4, 10, = 1/12o 5, 9 = 1/9o 6, 8 = 5/36o 7 = 1/6

o Non-uniformo not the same as picking a random #

between 2-12o dice games use this fact

Computing probabilitieso Simplest to count outcomeso Dice poker

o roll five dieo keep best k, roll 5-ko becomes your "hand"

o Suppose you roll two 1so what are the outcomes when your

roll the other 3 again to improve your hand?

Role of Chanceo Chance can enter into the game in

various wayso Chance generation of resources

o dealing cards for a game of Bridgeo rolling dice for a turn in Backgammon

o Chance of success of an actiono a particular choice in rock-paper-scissors

has a 1/3 chance of winning

o Chance degree of successo in "The Game of Life", a selected card

determines your salary

o Chance due to physical limitationso the difficulty of the hand-eye coordination

needed to perform an action

Role of Chance 2o Chance changes the players'

choiceso player must consider what is likely to

happeno rather than knowing what will happen

o Chance allows the designer more latitudeo the game can be made harder or

easier by adjusting probabilities

o Chance preserves outcome uncertainty o with reduced strategic inputo example: Thunderstorm

Psychologyo People are lousy probabilistic

reasonerso Reasoning errors

o Most people would say that the odds of rolling at least one 1 with two die = 1/6 + 1/6 = 1/3

o (what's the real probability?)o We overvalue low probability

events of high risk or rewardo Example: Otherwise rational

people buy lottery ticketso We assume success is more

likely after repeated failureo Example: "Gotta keep betting. I'm

due."

Psychology 2o Why is this?o Evolutionary theories

o Pure chance events are actually fairly rare outside of games

o Usually there is some human action involvedo There are ways to avoid being struck by

lightningo We tend to look for causation in everything

o Evolutionarily useful habit of trying to make sense of the world

o Resulto superstition

o "lucky hat", etc.o We are adapted to treat our observations

as a local sample of the whole environmento but in a media age, that is not valid

o How many stories in the newspaper about lottery losers?

Analysiso Structure

o Less certainty of outcomeoa good player can have bad luck

o Different strategic choicesoplayers can make risk / return

tradeoffs

o Experienceo Generation / revelation of

chance elementsobecomes an activityomoments of dramatic tension

Wednesdayo New Unito "The Checkered Game of Life"

o USA, 1860

o Readingso 10/11: Context

oChapter from The Enduring Vision"

o 10/16: Game playoChapter from Huck's Raft

o 10/18: Discussiono2nd Chapter from Huck's Raft