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Econ106G Lecture Note 8
Hong Feng
July 22, 2008
1
Punishment 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 reallife 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.
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
(
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
2. For phase one: Given the second phase payo/s, the reduced strategicform
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
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This note was uploaded on 05/26/2009 for the course ECON 106 taught by Professor Cai during the Winter '04 term at UCLA.
 Winter '04
 cai

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