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Game Theory Presentation
- 1. Abbas Keramati ( AssistantProfessor) University of Tehran – MBA Fall 2008 Game Theory Quantitative Analysis for Decision Making M.Ghotbi M.Pajhouh Niya
- 2. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Outline 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
- 3. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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
- 4. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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
- 5. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Mathematics Computer Science Biology Economics Political Science International Relations Philosophy Psychology Law Military Strategy Management Sports Game Playing Applications of Game Theory
- 6. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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?
- 7. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Types of games Cooperative or non-cooperative Zero sum and non-zero sum Simultaneous and sequential Perfect information and imperfect information Finite & Infinite Strategies
- 8. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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.
- 9. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG An Example: -12 7 -12 X2 7 6 Y2 6 7 10 Maximum 6 10 X1 Minimum Y1 A B
- 10. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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.
- 11. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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.
- 12. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 2-players Zero-Sum games Player 1 Player 2 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. -1 , 1 1 , -1 1 , -1 -1 , 1 Tail Head Tail Head
- 13. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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.
- 14. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Ted Confess Not Confess Confess Bill Not Confess Payoff Matrix for Prisoner’s Dilemma Both get 3 years 10 years for Bill 1 year for Ted 1 year for Bill 10 years for Ted Both get 5 years
- 15. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG An Example of Mixed Strategy game 0.5 0 A doesn’t bluff 0 B doesn’t believe 1 A bluffs B believes A B
- 16. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Now let’s Play This Game Pirate Game A B D C E
- 17. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG 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.
- 18. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Definition: Nash Equilibrium Given others’ choices, player i cannot be better-off if she deviates from s i *
- 19. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Nash’s Equilibrium cont.: 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
- 20. Title Lorem ipsumdolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. DGDG Dixit and Nalebuff: Thinking Strategically Dutta: Strategies and Games: Theory and Practice www.gametheory.net Refrences