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penn101_09s10_game theory

penn101_09s10_game theory - Introduction to Game Theory...

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Introduction to Game Theory
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Game Theory and Games Game theory studies strategic interactions among individuals. A game is any situation where the outcomes are jointly determined by all players. Intrinsic uncertainty: others’ actions Uncertainty as the outcome of careful reasoning Rationality: best action given one’s belief 2
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Examples Some conventional examples: Oligopolies Provision of public goods Negotiations/bargaining Voting Auction ....... 3
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Examples Some less conventional examples: School choice Social norm Crime Abortion Identity Language ....... 4
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What is a Game? A game consists of a set of players a set of actions for each player the payoffs to each player for every possible choice of strategies by the players. A game can be a simultaneous-move game or a dynamic game . 5
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Simultaneous-Move games 6
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Simultaneous-Move Games A game is a simultaneous-move game (normal-form game) if all players make their decisions (simultaneously), without being informed of anyone else’s decision. Simultaneous-move games are represented by a payoff matrix . 7
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Unique Equilibrium in Dominant Strategies The Prisoner’s Dilemma 8
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The Prisoner’s Dilemma Al and Bob are accomplices in a crime. They are arrested and interrogated separately . If they cooperate on denying the crime, then they both get 1 year jail time for a misdemeanor charge. If one defects on the other, then the defector goes home free while the other gets 3 years. If both defect, then they both get 2 years jail time. 9
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The Prisoner’s Dilemma Two players: A l and B ob. Each has two actions, called “ C ooperate” and D efect”. Suppose the payoff from spending n years in prison is –n . The table showing the payoffs to both players for each of the four possible action combinations is the game’s payoff matrix . 10
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The Prisoner’s Dilemma (-1,-1) (-3,0) (0,-3) (-2,-2) C D C D Bob Al Al’s payoff is shown first. Bob’s payoff is shown second. 11
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The Prisoner’s Dilemma (-1,-1) (-3,0) (0,-3) (-2,-2) C D C D Bob Al Question: what plays are we likely to see? 12
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The Prisoner’s Dilemma Suppose Bob cooperates then Al is better off defecting. ( -1 ,-1) (-3,0) ( 0 ,-3) (-2,-2) C D C D Bob Al 13
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The Prisoner’s Dilemma Suppose Bob cooperates then Al is better off defecting. (-1,-1) ( -3 ,0) (0,-3) ( -2 ,-2) C D C D Bob Al 14
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The Prisoner’s Dilemma Same goes for Bob. (-1, -1 ) (-3, 0 ) (0, -3 ) (-2, -2 ) C D C D Bob Al 15
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The Prisoner’s Dilemma (D,D) is a likely play because now both Al and Bob are willing to stick to their part of the play. (-1,-1) ( -3 ,0) (0, -3 ) ( -2 , -2 ) C D C D Bob Al 16
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Nash Equilibrium If neither player can unilaterally improve their own payoff, then we say this is a likely play, and it is called a Nash equilibrium . 17
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The Prisoner’s Dilemma The PD has a unique Nash equilibrium.
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