Game Theory

Robin Burke

GAM 224

Spring 2004

Outline

Admin Game Theory

Utility theoryZero-sum and non-zero sum gamesDecision TreesDegenerate strategies

Admin

Due WedHomework #3

Due Next WeekRule Analysis

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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

Any takers?

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

Cutter's

Utility

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 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

• defenses adjust increasing the payoff of a long pass

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 get caught lying

To compare doubting and accepting we have to look at the possible futures of the

game In any case

the game becomes degenerate what is the effect of adding a cost to

"accept"?

Reducing degeneracy

Come up with a rule for reducing degeneracy in this game

Ideally, both options (accept, doubt) would continue to be validno matter what the state of the game

is

Wednesday

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