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Game Theory

This outline covers the basics of game theory, utility theory, zero-sum and non-zero-sum games, decision trees, degenerate strategies, and Nash equilibrium. It also includes examples from various industries such as car dealerships and the prisoner's dilemma.

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Game Theory

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  1. Game Theory Robin Burke GAM 224 Spring 2004

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

  3. Admin • Due Wed • Homework #3 • Due Next Week • Rule Analysis • Reaction papers • Grades available

  4. 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

  5. Utility Theory • Utility theory • a single scale • value with each outcome • Different actors • may have different utility valuations • but all have the same scale

  6. 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?

  7. 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

  8. Game Theory • Examine strategies based on expected utility • The idea • a rational player will choose the strategy with the best expected utility

  9. Example • Non-probabilistic • Cake slicing • Two players • cutter • chooser

  10. 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

  11. 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

  12. Non-zero sum • A game in which there are non-symmetric outcomes • better or worse for both players • Classic example • Prisoner's Dilemma

  13. 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

  14. 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

  15. 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

  16. Prisoner's Dilemma • Nash Equilibrium • Confess • Because • in each situation, the prisoner can improve his outcome by confessing • Solution • iteration • communication • commitment

  17. Rock-Paper-Scissors

  18. No dominant strategy • Meaning • there is no single preferred option • for either player • Best strategy • (single iteration) • choose randomly • "mixed strategy"

  19. 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

  20. 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

  21. 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

  22. Degeneracies • Are not always obvious • May be contingent on game state

  23. 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

  24. 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

  25. Expectation • Knowledge • the opponent knows more than just this • the opponent knows the previous roll that the player must beat • probability of lying

  26. 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

  27. 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

  28. BUT • There is something we are ignoring

  29. Repeated Interactions Roll 1 Truth Lie doubt doubt accept Lose Win Roll 2 doubt doubt Lie Truth accept doubt doubt Roll 1 Truth Lie accept Roll 2

  30. 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

  31. 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"?

  32. 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

  33. Wednesday • Analysis Case Study • Final Fantasy Tactics Advance

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