Game Theory Robin Burke GAM 224 Spring 2004

Outline Admin Game Theory Utility theory Zero-sum and non-zero sum games Decision Trees Degenerate strategies

Admin Due Wed Homework #3 Due Next Week Rule Analysis Reaction papers Grades 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 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 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 idea a 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 bit½ + a bit Cut unevenly Small pieceBig piece

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 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 outcomes better or worse for both players Classic example Prisoner's Dilemma Hold OutConfess 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 RockPaperScissors Player 1Rock[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 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 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 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 accept Win accept doubt TruthLie Lose doubt LieTruth doubt TruthLie 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 valid no matter what the state of the game is

Wednesday Analysis Case Study Final Fantasy Tactics Advance