Probability/ Information Theory

Robin Burke

GAM 224

Fall 2005

Outline

AdminErratumDesign groups"Rules" paper

UncertaintyProbability

Information theorySignal and noise

Erratum

The rules of "8 & out" are not isomorphic to "Thunderstorm" Mr. Allen was correct

Consider all players draw no ones for seven turns each player should be 1 turn from being out but each player should have 7*(k-1) points none of them can go out next turn

Moral it is hard to find a confusing alternative to

counting

Design teams

9 people not on teams Please get teamed up! Next milestone Monday

Rules paper

Due 10/10Analysis paper #1: "Rules"You should be playing your game and

taking notes Note

you cannot use lab machines to do word processing

laptops are OK

Important points

Thesis "great game" is not a thesis This is a thesis

• "Inertial navigation, fixed firing direction and accurate collision detection in Asteroids create an environment in which ship orientation is highly coupled, generating emergent forms of gameplay."

No thesis = paper will not be graded Documentation

game itself, book, lectures other sources if used Missing or inadequate documentation = paper will not

be graded

Rules paper 2

Schemas Emergence Uncertainty Information Theory Information Systems Cybernetics Game Theory Conflict

Do not use more than one

Rules paper 3

Outlinessuggestions

Focusdo not catalog every rule, every game

objectidentify those items that contribute to

your argumentdepth over breadth

Rules paper 4

Turn inhardcopy in class 10/10

Turn in to turnitin.comon 10/10

Late policy½ grade per daysubmit by email

Turnitin.com

Class id1353024

Passwordkatamari

Uncertainty

Many games are probabilisticroll the diceshuffle the cards

Some games are notChessCheckersDots and Boxes

Certainty vs uncertainty

Certainty the condition when the outcome of an action

is known completely in advance. Some games operate this way

Chess Dots and Boxes

But even then uncertainty about who will win otherwise what is the point?

Strategically-interesting deterministic games are hard to design

Probability

Probability is the study of chance outcomesoriginated in the study of games

Basic ideaa random variablea quantity whose value is unknown

until it is "sampled"

Random variable 2

We characterize a random variable not by its value but by its "distribution" the set of all values that it might take and the percentage of times that it will take

on that value distribution sums to 1

• since there must be some outcome Probability

the fraction of times that an outcome occurs

Single Die

Random variable# of spots on the side facing up

Distribution1...6each value 1/6 of the time

Idealization

Single die

Random variableodd or even number of dots

Distributionodd or even50%

Distributions are not always uniform

Two dice

Random variable sum of the two die values

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

Non-uniform not the same as picking a random # between 2-12 dice games use this fact

Computing probabilities

Simplest to count outcomes Dice poker

roll five die keep best k, roll 5-k becomes your "hand"

Suppose you roll two 1s what are the outcomes when your roll the

other 3 again to improve your hand?

Outcomes

Each possible combination of outcomes of 3 rolls6 x 6 x 6 = 216 possible outcomes

QuestionsProbability of 3 of kind

or better?Probability of 4 of a

kind or better?

Die #1

Die #2

Die #31, 1, 1

6, 6, 1

6, 6, 6

Basic probability theory

Repeated trials addbut not in a simple wayprobability of a coin flip = heads?probability of heads in 3 flips?

Outcome sequences multiplyprobability of 3 flips all heads?

Relevance for Games

Many game actions are probabilistic even if their effects are deterministic player may not be able to exercise the

controls perfectly every time Asking somebody

to do the same uncertain task over increases the overall chance of success

to succeed on several uncertain tasks in a row decreases (a lot) the overall chance of success

Role of Chance

Chance can enter into the game in various ways Chance generation of resources

dealing cards for a game of Bridge rolling dice for a turn in Backgammon

Chance of success of an action an attack on an RPG opponent may have a

probability of succeeding Chance degree of success

the attack may do a variable degree of damage Chance due to physical limitations

the difficulty of the hand-eye coordination needed to perform an action

Role of Chance 2

Chance changes the players' choices player must consider what is likely to happen

• rather than knowing what will happen

Chance allows the designer more latitude the game can be made harder or easier by

adjusting probabilities Chance preserves outcome uncertainty

with reduced strategic input example: Thunderstorm

Random Number Generation Easy in physical games

rely on physical shuffling or perturbation basic uncertainty built into the environment

• "entropy" Not at all simple for the computer

no uncertainty in computer operations must rely on algorithms that produce "unpredictable"

sequences of numbers• an even and uncorrelated probability distribution

sometime variation in user input is used to inject noise into the algorithm

Randomness also very important for encryption

Psychology

People are lousy probabilistic reasoners Reasoning errors

Most people would say that the odds of rolling a 1 with two die = 1/6 + 1/6 = 1/3

We overvalue low probability events of high risk or reward Example: Otherwise rational people buy

lottery tickets We assume success is more likely after

repeated failure Example: "Gotta keep betting. I'm due."

Psychology 2

Why is this? Evolutionary theories

Pure chance events are actually fairly rare outside of games• Usually there is some human action involved• There are ways to avoid being struck by lightning

We tend to look for causation in everything• Evolutionarily useful habit of trying to make sense of the world

Result• superstition

• "lucky hat", etc. We are adapted to treat our observations as a local sample

of the whole environment• but in a media age, that is not valid

• How many stories in the newspaper about lottery losers?

Psychology 3

Fallacies may impact game designPlayers may take risky long-shots

more often than expectedPlayers may expect bad luck to be

reversed

Information theory

Information can bepublic

• board position in chess

private• one's own poker hand

unknown / hidden (to players)• monsters in the next room

Information Theory

There is a relationship between uncertainty and information Information can reduce our uncertainty

Example The cards dealt to a player in "Gin Rummy"

are private knowledge But as players pick up certain discarded

cards from the pile It becomes possible to infer what they are

holding

Information Theory 2

Classical Information Theory Shannon

Information as a quantity how information can a given communication

channel convey?• compare radio vs telegraph, for example

must abstract away from the meaning of the information

• only the signifier is communicated• the signified is up to the receiver

Information Theory 3

Information as a quantitymeasured in bitsbinary choices

If you are listening on a channel for a yes or no answeronly one bit needs to be conveyed

ExampleLOTR

Information Theory 4

If you need a depiction of an individual's appearance many more bits need to be conveyed

Because there are more ways that people can look young / old race eye color / shape hair color / type height dress

A message must be chosen from the vocabulary of signifier options book: "information is a measure of one's freedom of

choice when selecting a message"

Information Theory 5

This is the connection between information and uncertaintythe more uncertainty about somethingthe more possible messages there are

Noise

Noise interrupts a communication channel by changing bits in the original message increases the probability that the wrong

message will be received Redundancy

standard solution for noise• more bits than required, or• multi-channel

Example 1

Gin Rummy Unknown

What cards are in Player X's hand?Many possible answers

• 15 billion (15,820,024,220)• about 34 bits of information

Once I look at my 10 cards• 1.5 billion (1,471,442,973)• about 30 bits

Example 1 cont'd

I have two Kings Player X picks up a discarded King of Hearts

Discards a Queen of Hearts There are two possible states

Player X now has one King <- certain Player X now has two Kings <- very likely

Possible hands 118 million (118,030,185) about 27 bits factor of 10 reduction in the uncertainty 3 bits of information in the message

Example 1 cont'd

Gin Rummy balances privacy of the cards with messages

• discarding cards• picking up known discards

The choice of a discard becomes meaningful because the player knows it will be interpreted as a

message When it isn't your turn

the game play is still important because the messages are being conveyed

interpreting these messages is part of the skill of the player

• trivial to a computer, but not for us

Example 2

Legend of Zelda: Minish Cap Monsters are not all vulnerable to the same

types of weapons 10 different weapons (we'll ignore combinations of weapons)

Encounter a new monster which weapon to use? 4 bits of unknown information

We could try every weapon but we could get killed

Example 2, cont'd

Messages the monster iconography contains messages

• rocks and metal won't be damaged by the sword• flying things are vulnerable to the "Gust Jar"• etc.

the game design varies the pictorial representations of monsters

• knowing that these messages are being conveyed learning to interpret these messages

• is part of the task of the player• once mastered, these conventions make the player

more capable Often sound and appearance combine

a redundant channel for the information

Game Analysis Issues

Be cognizant of the status of different types of information in the game public private unknown

Analyze the types of messages by which information is communicated to the user How does the player learn to interpret these

messages? How are redundant channels used to

communicate?

Monday

No class Meet with your group

Wednesday

Systems of Information Cybernetics

Ch. 17 & 18