Game TheoryConflict

Robin Burke

GAM 224

Fall 2005

Outline

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 MondayRule paperHard copy in classturnitin.com

Assassin starts midnight Fridaypick up player kits from Prof. Wilcox orwait until Monday

Ernest Adams

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

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 theorya single scalevalue with each outcome

Different actorsmay have different utility valuationsbut all have the same scale

Expected Utility

Expected utilitywhat is the likely outcomeof a set of outcomeseach with different utility values

ExampleBet

• $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 (1/6) * 5 + (1/18) * 5 + (7/9) (-2) = -2/9

Meaning 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 ideaa rational player will choose the

strategy with the best expected utility

Example

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

Rationality 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

Both Utilities

Choose bigger piece

Choose smaller piece

Cut cake evenly

(-1, +1) (+1, -1)

Cut unevenly

(-10, +10) (+10, -10)

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

A game in which there are non-symmetric outcomesbetter or worse for both players

Classic examplePrisoner's Dilemma

Hold Out Confess

Hold Out [-1, -1] [-3, 0]

Confess [0, -3] [-5, -5]

Degenerate Strategy

A winning strategy is also called a degenerate strategy

Because 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 Even when there is no dominant one

A Nash equilibrium is a strategy 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 Confess

Because in each situation, the prisoner can improve

his outcome by confessing Solution

iteration communication commitment

Rock-Paper-Scissors

Player 2

Rock Paper Scissors

Player 1 Rock [0,0] [-1, +1] [+1, -1]

Paper [+1, -1] [0,0] [-1, +1]

Scissors [-1, +1] [+1, -1] [0,0]

No dominant strategy

Meaningthere is no single preferred option

• for either player

Best strategy(single iteration)choose randomly"mixed strategy"

Mixed Strategy

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

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

• running play• best chance of success• lowest cost of failure

But if your opponent assumes this

• 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 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 chartfor next playerassume the roll is not good enough

Rollerlie or not lie

Next playeraccept or doubt

Expectation

Knowledgethe opponent knows more than just

thisthe opponent knows the previous roll

that the player must beat• probability of lying

Note

The opponent will never lie about a better rollOutcome cannot be improved by

doing so The opponent cannot tell the truth

about a worse rollIllegal under the rules

Expected Utility

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 two-pair range maybe 4s-over-1s

BUT

There is something we are ignoring

Repeated Interactions

Roll 1

Roll 2

Roll 1

acceptWin

accept

doubtTruth Lie

Losedoubt

Lie Truth

doubtdoubt

Truth Lie

doubt doubt

accept

Roll 2

Decision Tree

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" I may be incurring some future cost because I may have to lie and get caught

To compare doubting and accepting 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 Chess, Boxing, Warcraft II

Group vs. group Basketball, Soccer, Battlefield 1942

One against many Tag, Most action-adventure and FPS games

Every man for himself Marathon, Risk, Mario Kart

Struggle to achieve a goal

Single player vs. computerTetris

Group of single players vs. gameBlackjack

Group against the gameLord of the Rings board gameCooperative 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

goalsfrom the player's point of view

Example: Centipede

Single player mode Do well against the game Beat your personal best Get on the high score list

Two player mode Beat your opponent Get on high score list

Compete against friend to get on the high score list

Example: Joust

Single player modePlayers compare scores

Two player mode One player against the otherBattle modeCooperation mode players vs.

computer

Example: Gauntlet

Single player mode Two player mode

Cannot attack each otherReal time display of scoresPlayers compete for “food”Allow most needy to eat or be greedyCompete for spending money

Competitive vs. cooperative

All games are competitive Can they be cooperative at the same

time?Adherence to the Magic CirclePlayers on a Basketball teamConflict within a cooperative

framework

Systems of conflict

Games exhibit conflict at multiple levelsLocal goals contribute to larger goals

Conflicts at different levels work together to create a systemthe interrelatedness of the conflicts

determines if any given conflict seems meaningful

Example: Legend of Zelda: The Wind Waker Largest conflict

evil sorcerer, Gannon Intermediate conflict

defeat level Local conflict

solve puzzledefeat particular enemymaster 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 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 StudyLegend of Zelda: Wind Waker