lect441-10

# lect441-10 - Advanced Microeconomics Leonardo Felli EC441...

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Advanced Microeconomics Leonardo Felli EC441: Room S.421 Lecture 10: 6 December 2006

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Leonardo Felli Lecture 10: 6 December 2006 Repeated Games This is the best understood class of dynamic games . Players face in each period the same normal form stage game. Players’ payoffs are a weighted average of the payoffs players receive in every stage game. Slide 1
Leonardo Felli Lecture 10: 6 December 2006 Main point of the analysis: players’s overall payoffs depend on the present and the future stage game payoffs , it is possible that the threat of a lower future payoff may induce a player to choose a strategy different from the stage game Nash equilibrium strategies. Famous example: the repeated prisoner dilemma . C D C 1 , 1 - 1 , 2 D 2 , - 1 0 , 0 Slide 2

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Leonardo Felli Lecture 10: 6 December 2006 Per period payoff depends on current action: g i ( a t ) . Players discount the future with a common discount factor δ . Notice that since we are going to compare the equilibrium payoffs for different time horizons we need to re-normalize the payoffs so that they are comparable. The average discounted payoff is therefore: 1 - δ 1 - δ T T t =0 δ t g i ( a t ) Slide 3
Leonardo Felli Lecture 10: 6 December 2006 Assume that the prisoners’ dilemma game is repeated a finite number of times. Nash equilibrium payoffs: (0 , 0) . Nash equilibrium strategies: each player chooses action D independently of the period and the action of the other player in the past. Proof: backward induction. Subgame Perfection prevents any gain from repeated, but finite interaction. Slide 4

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Leonardo Felli Lecture 10: 6 December 2006 Consider a different finitely repeated game . Assume that the following stage game is played twice: L C R T 1 , 1 5 , 0 0 , 0 M 0 , 5 4 , 4 0 , 0 B 0 , 0 0 , 0 3 , 3 Nash equilibria of the stage game: ( T, L ) and ( B, R ) . Assume the game is played twice and consider the following strategies: Slide 5
Leonardo Felli Lecture 10: 6 December 2006 Player 1: play M in the first period; in the second period play B if the observed outcome is ( M, C ) ; in the second period play T if the observed outcome is not ( M, C ) ; Player 2: play C in the first period; in the second period play R if the observed outcome is ( M, C ) ; in the second period play L if the observed outcome is not ( M, C ) ; Slide 6

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Leonardo Felli Lecture 10: 6 December 2006 Result If δ > 1 2 then these strategies are a subgame perfect equilibrium of the game. Proof: Player 1 conforms to the strategies if and only if: 4 + δ 3 5 + δ while player 2 conforms if and only if: 4 + δ 3 5 + δ Both inequalities are satisfied for δ > 1 2 . Slide 7
Leonardo Felli Lecture 10: 6 December 2006 Consider now the the repeated prisoner dilemma in the case: T = + . C D C 1 , 1 - 1 , 2 D 2 , - 1 0 , 0 Property: Both player choosing strategy D in every period is an SPE of the repeated game. Proof: by one deviation principle. Notice that a infinitely repeated game is continuous at infinity.

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