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Econ106G_081_Lecture8

Econ106G_081_Lecture8 - Econ106G Lecture Note 8 Hong Feng 1...

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Econ106G Lecture Note 8 Hong Feng July 22, 2008 1 Punishment Game We have seen that in the prisoner°s dilemma game, ( Defect; Defect ) is the only (dominant strategy) Nash equilibrium, although ( Cooperate; Cooperate ) is a better outcome in terms of Pareto e¢ ciency. The harsh prediction of theory is at odds with casual introspection of real-life interactions. As we have discussed before, one reason may be that players are altruisitc, which actually implies that the payo/s in the PD are misspeci±ed. An alternative way to explain people°s cooperative behavior is to introduce repeated interaction. Consider an extensive form game with two phases: in the ±rst the PD is played, and then after players observe the outcome of this ±rst phase they can decide to sue each other in court (Punish) or to refrain from doing so (Not punish) in a second phase. The payo/ are additive across these two steps of the game. For the PD they are given by 2 C D 1 C 1 ; 1 ° 1 ; 2 D 2 ; ° 1 0 ; 0 and for the second step, they are given by 2 N P 1 N 0 ; 0 ° 12 ; ° 1 P ° 1 ; ° 12 ° 10 ; ° 10 Draw the game tree by yourself. Verify that there are ±ve subgames (includ- ing the whole game), and ±ve information sets for each player. The question is could we ±nd a SPNE such that ( C; C ) is played in the ±rst period. There are two (pure strategy) Nash equilibria in the second stage game, ( N; N ) and ( P; P ) : The idea for supporting ( C; C ) in the ±rst stage is to "re- ward" ±rst round cooperative behavior with the "good equilibrium" ( N; N ) and "punish" ±rst round defective behavior with the "bad equilibrium" ( P; P ) : Consider the following strategy s ° i : 1. In phase one play C 2. In phase two play ± N if outcome of the ±rst phase was ( C; C ) 1
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± P otherwise, i.e if outcome of the ±rst phase was ( C; D ) ; ( D; C ) or ( D; D ) : Let°s verify these strategies constitute a subgame perfect Nash equilibrium. 1. For phase two: Either ( P; P ) or ( N; N ) will be played depending on the outcome of the ±rst phase. They°re both Nash equilibria. 2. For phase one: Given the second phase payo/s, the reduced strategic-form game in phase one is 2 C D 1 C 1 ; 1 ° 11 ; ° 8 D ° 8 ; ° 11 ° 10 ; ° 10 ( C; C ) is the a Nash equilibrium. Hence ( s ° 1 ; s ° 2 ) is a subgame perfect Nash equilibrium. The outcome is ( C; C ) in phase 1 and ( N; N ) in phase 2. ± Notice that for a complete strategy we need to assign actions for all infor- mation sets. ± In this equilibrium, when ( C; D ) was played in stage one, not only player 1 will play P to punish player 2 in the second stage, player 2 will play P as well. The reason is that ( P; P ) is a Nash equilibrium and ( P; N ) is not. Expecting to be punished by player 1, player 2°s best response is to punish back.
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