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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
Reaction papersGrades available
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
Analysis Case StudyFinal Fantasy Tactics Advance