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Game Theory

Game Theory. By Margaret Banker. What is Game Theory?. Branch of Applied Mathematics used is variety of disciplines Attempts to mathematically capture behavior in strategic settings in which an individual's success in making choices depends on the choices of others

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Game Theory

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  1. Game Theory By Margaret Banker

  2. What is Game Theory? • Branch of Applied Mathematics used is variety of disciplines • Attempts to mathematically capture behavior in strategic settings in which an individual's success in making choices depends on the choices of others • Traditional applications attempt to find equilibria in these games • Each player of the game adopts a strategy that they are unlikely to change

  3. What is Game Theory? • Formal description of a Strategic Setting Mathematically Precise and Logically Consistent Interdependency! Your path is determined by your behavior AND the behavior of others.

  4. 5 Elements of a Game • Players • Strategies: What actions are available to the strategic players? • Payoffs assigned to every combination of actions • Information: What you know when you have to make your decision • Objective: What are you trying to do; the goal

  5. Classical Games • Prisoner’s Dilemma • Coordination Games • Pareto Coordination • Battle of the Sexes • Pigs

  6. Prisoner's Dilemma Objective and Main Tensions: Example: • Objective: Maximize your own payoff • “Bad things” (e.g. prison sentence) denoted as negative values • Player 1 will Confess, because no matter what Player 2 does, Player 1 is better off confessing • Player 2 will Confess because no matter what Player 1 does, Player 2 is better off confessing 2 C D 1 C D

  7. Coordination Game Objective and Main Tensions: Example: • Objective: Maximize your own payoff • There is NO dominant Strategy • Need pre-planned coordination • No incentive to deviate 2 A B 1 A B

  8. Pareto Coordination Objective and Main Tensions: Example: • Objective: Maximize your own payoff • There is NO dominant Strategy • But the game IS solvable • Both pick Strategy “A” 2 A B 1 A B

  9. Battle of the Sexes Objective and Main Tensions: Example: • Objective: Maximize your own payoff • Date to movies (M) or Wrestling (W) • Only go on date if both players agree on where to go • Main Tension: Player 1 prefers W to M, while Player 2 prefers M to W • There is NO dominant Strategy • Added conflict of preferences 2 W M 1 W M

  10. Pigs Objective and Main Tensions: Example: • Objective: Maximize your own payoff • Game: 2 Pigs in a pen – Dominant (D) and Submissive (S) • On one side of pen is the button must press to get food • Food comes on the opposite side of the button • At least one of the pigs must press the button, but are then farthest from the food • Which pig will press (P) the button? Which pig will Not Press (N) S P N D P N

  11. Strategies for Solving • Dominance • Iterated Dominance/Rationalization • Efficiency • Efficient • More Efficient

  12. Dominance • A strategy is dominated if there exists another strategy that gives better results no matter what the other player does • EX. Prisoner’s Dilemma 2 A B 1 A B

  13. Iterative Dominance • Process of the iterated removal of strictly dominated strategies • Neither of Player 1’s pure strategies is dominant • Player 2’s Strategy X is dominated by Y • After X is eliminated, Strategy B iteratively Dominates A • Player 2 Strategy Y is dominated by Z 2 X Y Z 1 A B

  14. Iterative Dominance • Every player guesses what 2/3 of the average of all the players’ guesses will be. • Numbers restricted to real numbers between 0 and 100 • Winner is the closest to 2/3 of the average. • Write down your guess!!

  15. Iterative Dominance • No Dominant Strategy – solution found through iterative dominance • Guessing any number that lies above 66.67 is dominated for every player since it cannot possibly be 2/3rds of the average of any guess. These can be eliminated • Once these strategies are eliminated for every player, any guess above 44.45 is weakly dominated for every player since no player will guess above 66.67 and 2/3 of 66.67 is approximately 44.45. • This process will continue until all numbers above 0 have been eliminated.

  16. Efficiency • “More Efficient” • Strategy s is more efficient than Strategy s’ if all players prefer or are indifferent to the outcome of s to s’ • Comparison between 2 strategy profiles • “Efficient” • Strategy s is called “Efficient” if there is no s’ that is more efficient

  17. Efficiency 2 2 A B Pigs Pareto Coordination P D 1 1 A • What are the efficient strategies? P D B What are the efficient strategies? What are the efficient strategies?

  18. Pigs - Revisited Objective and Main Tensions: Example: • Objective: Maximize your own payoff • Game: 2 Pigs in a pen – Dominant (D) and Submissive (S) • On one side of pen is the button must press to get food • Food comes on the opposite side of the button • At least one of the pigs must press the button, but are then farthest from the food • Which pig will press (P) the button? Which pig will Not Press (N) S P N D P N What is the best method for solving this puzzle? Dominance? Iterative Dominance? Efficiency?

  19. Mixed Strategies • Probability Distribution over the pure strategies • If there is no pure dominance, a mixed strategy may dominate a pure strategy. • Then can use Iterative dominance to determine the equilibrium • Mixed Strategy denoted with symbol “σ” • Mixed Strategy of Player 2: σ2

  20. Mixed Strategies – Example 2 L C R Is there a Mixed Strategy for Player 2 (σ2) that dominates Strategy “L” 1 U • Neither Player 1 nor Player 2 has a pure dominant strategy • Calculate utility of: σ2=(0, ½, ½) • The mixed strategy σ2 must dominate one of Player 2’s pure strategies (L, C, or R) no matter what Player 1 does (U, M, or D) M D

  21. Other Applications of Game Theory • Contracts • How to ensure that parties in a contract will not break faith • 3 Main Forms • Expectation Damage Principle – entitled to expectations • Reliance Damage Principle – entitled to original state • Restitution Damage Principle – cannot gain from illegal act

  22. References • Watson, Joel. “Strategy: An Introduction to Game Theory.” University of California, San Diego. WW Norton & Company: New York. 2008. • Sönmez, Tayfun. “Econ 308 Lecture Notes 4.” Boston College, Department of Economics. <http://www2.bc.edu/~sonmezt/>. • Duffy, John. “Introduction to Game Theory: Elements of a Game.” University of Pittsburgh, Department of Economics.<http://www.pitt.edu/~jduffy/econ1200/Lectures.htm >. • GameTheory.net < http://www.gametheory.net>. • “Glossary of Game Theory Terms.” Gametheory.net. <http://www.gametheory.net/dictionary/#I>.

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