Game Theory
Conflict
Robin Burke
GAM 224
Fall 2005
Outline
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Admin
Quiz
Game Theory
 Utility theory
 Zero-sum and non-zero sum games
 Decision Trees
 Degenerate strategies
Conflict
 Types of goals
 Systems of conflict
Admin

Due Monday
Rule paper
 Hard copy in class
 turnitin.com
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Assassin starts midnight Friday
pick up player kits from Prof. Wilcox or
 wait until Monday

Ernest Adams
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Talk
 10:30 – 12 noon
 go to talk and write reaction paper
 substitute for any other reaction paper
Game design workshop
 1 – 4 pm
 CTI students only, preference to GAM majors
 work with other students / faculty
 apply his techniques to develop game ideas
 personal feedback from the expert
For workshop
 send email to jtrinkle@cs.depaul.edu
 must attend talk to go to workshop
Quiz

30 minutes
Game Theory
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
A branch of economics
Studies rational choice in a adversarial
environment
Assumptions


rational actors
complete knowledge
• in its classic formulation


known probabilities of outcomes
known utility functions
Utility Theory

Utility theory
a single scale
 value with each outcome


Different actors
may have different utility valuations
 but all have the same scale

Expected Utility

Expected utility
what is the likely outcome
 of a set of outcomes
 each with different utility values


Example

Bet
• $5 if a player rolls 7 or 11, lose $2
otherwise

Should you take this bet?
How to evaluate

Expected Utility

for each outcome
• reward * probability


Meaning


(1/6) * 5 + (1/18) * 5 + (7/9) (-2) = -2/9
If you made this bet 1000 times, you would
probably end up $222 poorer.
Doesn't say anything about how a given trial
will end up

Probability says nothing about the single
case
Game Theory
Examine strategies based on
expected utility
 The idea


a rational player will choose the
strategy with the best expected utility
Example


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Non-probabilistic
Cake slicing
Two players


cutter
chooser
Cutter's
Utility
Choose
bigger
piece
Choose
smaller
piece
Cut cake
evenly
½ - a bit
½ + a bit
Cut
unevenly
Small piece
Big piece
Rationality

Choose
bigger
piece
Choose
smaller
piece
Cut cake
evenly
(-1, +1)
(+1, -1)
Cut
unevenly
(-10, +10)
(+10, -10)
Rationality



Both
Utilities
each player will take highest utility option
taking into account the other player's likely
behavior
In example

if cutter cuts unevenly
• he might like to end up in the lower right
• but the other player would never do that
• -10

if the current cuts evenly,
• he will end up in the upper left
• -1
• this is a stable outcome
• neither player has an incentive to deviate
Zero-sum

Note

for every outcome
• the total utility for all players is zero

Zero-sum game


something gained by one player is lost by
another
zero-sum games are guaranteed to have a
winning strategy
• a correct way to play the game

Makes the game not very interesting to play

to study, maybe
Non-zero sum
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A game in which there are nonsymmetric outcomes


better or worse for both players
Classic example

Prisoner's Dilemma
Hold Out Confess
Hold Out [-1, -1]
[-3, 0]
Confess
[-5, -5]
[0, -3]
Degenerate Strategy

A winning strategy is also called


Because



a degenerate strategy
it means the player doesn't have to think
there is a "right" way to play
Problem


game stops presenting a challenge
players will find degenerate strategies if they
exist
Nash Equilibrium

Sometimes there is a "best" solution


A Nash equilibrium is a strategy



Even when there is no dominant one
in which no player has an incentive to
deviate
because to do so gives the other an
advantage
Creator



John Nash Jr
"A Beautiful Mind"
Nobel Prize 1994
Classic Examples

Car Dealers


Why are they always next to each other?
Why aren't they spaced equally around
town?
• Optimal in the sense of not drawing customers to
the competition

Equilibrium


because to move away from the competitor
is to cede some customers to it
Prisoner's Dilemma

Nash Equilibrium


Because

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Confess
in each situation, the prisoner can improve
his outcome by confessing
Solution



iteration
communication
commitment
Rock-Paper-Scissors
Player 2
Player 1
Rock
Paper
Scissors
Rock
[0,0]
[-1, +1]
[+1, -1]
Paper
[+1, -1]
[0,0]
[-1, +1]
Scissors
[-1, +1]
[+1, -1]
[0,0]
No dominant strategy

Meaning

there is no single preferred option
• for either player

Best strategy
(single iteration)
 choose randomly
 "mixed strategy"

Mixed Strategy
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Important goal in game design
Player should feel
 all of the options are worth using
 none are dominated by any others
Rock-Paper-Scissors dynamic
 is often used to achieve this
Example
 Warcraft II
•
•
•
•
Archers > Knights
Knights > Footmen
Footmen > Archers
must have a mixed army
Mixed Strategy 2
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
Other ways to achieve mixed strategy
Ignorance

If the player can't determine the dominance of a strategy
• a mixed approach will be used
• (but players will eventually figure it out!)

Cost
 Dominance is reduced
• if the cost to exercise the option is increased
• or cost to acquire it

Rarity

Mixture is required
• if the dominant strategy can only be used periodically or
occasionally

Payoff/Probability Environment

Mixture is required
• if the probabilities or payoffs change throughout the game
Mixed Strategy 3

In a competitive setting



mixed strategy may be called for
even when there is a dominant strategy
Example



Football
third down / short yardage
highest utility option
•
•
•

But

if your opponent assumes this
•


running play
best chance of success
lowest cost of failure
defense adjusts to prevent a run
increasing the payoff of a long pass
But

if you are behind in a close game and time is short
•
•

payoff changes
because an incomplete pass stops the clock
opponent may not defend the run
Degeneracies
Are not always obvious
 May be contingent on game state

Example

Liar's Dice



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roll the dice in a cup
state the "poker hand" you have rolled
stated hand must be higher than the
opponent's previous roll
opponent can either
• accept the roll, and take his turn, or
• say "Liar", and look at the dice

if the description is correct
• opponent pays $1

if the description is a lie
• player pays $1
Lie or Not Lie

Make outcome chart
for next player
 assume the roll is not good enough


Roller


lie or not lie
Next player

accept or doubt
Expectation

Knowledge
the opponent knows more than just
this
 the opponent knows the previous roll
that the player must beat

• probability of lying
Note

The opponent will never lie about a
better roll


Outcome cannot be improved by
doing so
The opponent cannot tell the truth
about a worse roll

Illegal under the rules
Expected Utility
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
What is the expected utility of the doubting strategy?
 P(worse) - P(better)
When P(worse) is greater than 0.5
 doubt
Probabilities
 pair or better: 95%
 2 pair or better: 71%
 3 of a kind or better: 25%
So start to doubt somewhere in the middle of the twopair range
 maybe 4s-over-1s
BUT

There is something we are ignoring
Repeated Interactions
Roll 1
Truth
doubt
Lie
doubt
accept
Win
Lose
Roll 2
doubt
Lie
doubt
Truth
accept
doubt
doubt
Roll 1
Truth
Lie
accept
Roll 2
Decision Tree
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Examines game interactions over time
Each node
 Is a unique game state
Player choices
 create branches
Leaves
 end of game (win/lose)
Important concept for design
 usually at abstract level
 question
• can the player get stuck?

Example
 tic-tac-toe
Future Cost

There is a cost to "accept"



To compare doubting and accepting


I may be incurring some future cost
because I may have to lie and get caught
we have to look at the possible futures of the
game
In any case

the game becomes degenerate
• the player must doubt
Conflict

Somewhat obvious

"artificial conflict" is part of the
definition
Struggle to achieve a goal

Single player vs. single player


Group vs. group


Basketball, Soccer, Battlefield 1942
One against many

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Chess, Boxing, Warcraft II
Tag, Most action-adventure and FPS games
Every man for himself

Marathon, Risk, Mario Kart
Struggle to achieve a goal

Single player vs. computer


Group of single players vs. game


Tetris
Blackjack
Group against the game
Lord of the Rings board game
 Cooperative mode in Star Wars
Battlefront, etc.

Game goal
May not be fixed
 May not be obvious from the game's
premise
 Single game may support multiple
goals


from the player's point of view
Example: Centipede

Single player mode

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Two player mode

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Do well against the game
Beat your personal best
Get on the high score list
Beat your opponent
Get on high score list
Compete against friend to get on the high
score list
Example: Joust

Single player mode


Players compare scores
Two player mode
One player against the other
 Battle mode
 Cooperation mode players vs.
computer

Example: Gauntlet
Single player mode
 Two player mode

Cannot attack each other
 Real time display of scores
 Players compete for “food”
 Allow most needy to eat or be greedy
 Compete for spending money

Competitive vs. cooperative
All games are competitive
 Can they be cooperative at the same
time?

Adherence to the Magic Circle
 Players on a Basketball team
 Conflict within a cooperative
framework

Systems of conflict

Games exhibit conflict at multiple
levels


Local goals contribute to larger goals
Conflicts at different levels work
together to create a system

the interrelatedness of the conflicts
determines if any given conflict seems
meaningful
Example: Legend of Zelda:
The Wind Waker

Largest conflict


Intermediate conflict


evil sorcerer, Gannon
defeat level
Local conflict
solve puzzle
 defeat particular enemy
 master particular skill

Wind Waker, cont'd

Conflict system

each level is crafted to introduce new skills
• which are essential to succeeding in this and later levels

each level features enemies with certain common
characteristics
• essential to defeating the boss

Game logic links conflicts

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

hero must collect pearls to defeat Gannon
pearls are obtained by defeating a boss enemy
boss enemies are found at the innermost room of each
dungeon
exploration of the dungeon requires defeating enemies and
solving puzzles
exploration also requires weapons, tools and power-ups
items found by exploring the dungeon
Conflict system may fail

Game lacks meaningful play


if players do not perceive the links between
conflicts
Example

Kingdom Hearts
• goal is to protect various "worlds"
• travel between worlds is done through a 3-D
space shooting game
• no coherence between this local conflict and the
global one
• it feels tacked on and rather pointless
Cooperative games?

"New Games" Movement

Dragon
• conflict between head and tail
• mediated by players with uncertain stakes

Stand-up
• conflict with gravity / stability

UN Food Force game


mission = help refugees
conflict with terrain, elements, limited
budgets, logistics
Monday

Analysis Case Study

Legend of Zelda: Wind Waker