Probability
/ Information Theory
Robin Burke
GAM 224
Fall 2005
Outline

Admin
Erratum
 Design groups
 "Rules" paper


Uncertainty


Probability
Information theory

Signal and noise
Erratum

The rules of "8 & out" are not isomorphic to
"Thunderstorm"


Consider





Mr. Allen was correct
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 10/10
 Analysis 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

Outlines


suggestions
Focus
do not catalog every rule, every game
object
 identify those items that contribute to
your argument
 depth over breadth

Rules paper 4

Turn in


Turn in to turnitin.com


hardcopy in class 10/10
on 10/10
Late policy
½ grade per day
 submit by email

Turnitin.com

Class id


1353024
Password

katamari
Uncertainty

Many games are probabilistic
roll the dice
 shuffle the cards


Some games are not
Chess
 Checkers
 Dots and Boxes

Certainty vs uncertainty

Certainty


Some games operate this way



Chess
Dots and Boxes
But even then



the condition when the outcome of an action
is known completely in advance.
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
outcomes


originated in the study of games
Basic idea
a random variable
 a 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
Distribution
1...6
 each value 1/6 of the time


Idealization
Single die

Random variable


odd or even number of dots
Distribution
odd or even
 50%


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 rolls


6 x 6 x 6 = 216 possible outcomes
6, 6, 6
Questions
Probability of 3 of kind
or better?
 Probability of 4 of a
kind or better?

6, 6, 1
Die #2
Die #3
1, 1, 1
Die #1
Basic probability theory

Repeated trials add
but not in a simple way
 probability of a coin flip = heads?
 probability of heads in 3 flips?


Outcome sequences multiply

probability 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


We overvalue low probability events of high
risk or reward


Most people would say that the odds of
rolling a 1 with two die = 1/6 + 1/6 = 1/3
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 design
Players may take risky long-shots
more often than expected
 Players may expect bad luck to be
reversed

Information theory

Information can be

public
• 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 quantity
measured in bits
 binary choices


If you are listening on a channel for a
yes or no answer


only one bit needs to be conveyed
Example

LOTR
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 uncertainty
the more uncertainty about something
 the 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



Encounter a new monster



10 different weapons
(we'll ignore combinations of weapons)
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