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Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008

Game Theory. Quantitative Analysis for Decision Making. M.Ghotbi M.Pajhouh Niya. Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008. Outline. Title. What is Game Theory? History of Game Theory Applications of Game Theory Key Elements of a game

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Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008

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  1. Game Theory Quantitative Analysis for Decision Making M.Ghotbi M.Pajhouh Niya Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008

  2. Outline Title • What is Game Theory? • History of Game Theory • Applications of Game Theory • Key Elements of a game • Types of games • Nash Equilibrium (NE) • Pure Strategies & Mixed Strategies • 2players Zero-Sum games DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  3. Title What is Game Theory? • In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose. • The mathematics of human interactions DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  4. Title History of Game Theory • von Neumann wrote a key paper in 1928 • 1944: “Theory of Games and Economic Behavior” by von Neumann and Morgenstern • 1950: Nash invents concept of Nash equilibrium • Game theory booms after this… • 1994: Harsanyi, Nash, and Selten win Nobel Prize in economics for game theory work DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  5. Title Applications of Game Theory • Psychology • Law • Military Strategy • Management • Sports • Game Playing • Mathematics • Computer Science • Biology • Economics • Political Science • International Relations • Philosophy DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  6. Title Key Elements of a game • Players: Who is interacting? • Strategies: What are their options? • Payoffs: What are their incentives? • Information: What do they know? • Rationality: How do they think? DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  7. Title Types of games • Cooperative or non-cooperative • Zero sum and non-zero sum • Simultaneous and sequential • Perfect information and imperfect information • Finite & Infinite Strategies DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  8. Title Pure Strategies • The upper value of the game is equal to the minimum of the maximum values in the columns. • The lower value of the game is equal to the maximum of the minimum values in the rows. DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  9. Title An Example: DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  10. Title Mixed Strategies A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy. DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  11. Title Nash Equilibrium (NE) A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players. DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  12. Title 2-players Zero-Sum games Head Tail Head Tail Penny Matching: • Each of the two players has a penny. • Two players must simultaneously choose whether to show the Head or the Tail. • Both players know the following rules: -If two pennies match (both heads or both tails) then player 2 wins player 1’s penny. -Otherwise, player 1 wins player 2’s penny. DGDG Player 2 Player 1 Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  13. Title Prisoner’s Dilemma • No communication: - Strategies must be undertaken without the full knowledge of what the other players (prisoners) will do. • Players (prisoners) develop dominant strategies but are not necessarily the best one. DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  14. Title Payoff Matrix for Prisoner’s Dilemma Ted Confess Not Confess Confess Bill Not Confess DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  15. Title An Example of Mixed Strategy game DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  16. Title Now let’s Play This Game Pirate Game C B A DGDG D E Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  17. Title Nash’s Equilibrium • This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players. • A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players. • So when each player within a game follows their best strategy, a Nash equilibrium will occur. DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  18. Title Given others’ choices, player i cannot be better-off if she deviates from si* Definition: Nash Equilibrium DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  19. Nash’s Equilibrium cont.: Title Bayesian Nash Equilibrium • The Nash Equilibrium of the imperfect-information game • A Bayesian Equilibrium is a set of strategies such that each player is playing a best response, given a particular set of beliefs about the move by nature. • All players have the same prior beliefs about the probability distribution on nature’s moves. • So for example, all players think the odds of player 1 being of a particular type is p, and the probability of her being the other type is 1-p DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

  20. Title Refrences • Dixit and Nalebuff: Thinking Strategically • Dutta: Strategies and Games: Theory and Practice • www.gametheory.net DGDG Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas. Loremipsumdolorsitamet, consectetueradipiscingelit. Vivamus et magna. Fuscesedsemsed magna suscipitegestas.

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