Spring 2008
Econ 367.
GameTheoretic Method
Prelim Exam 2  Solutions
[1 hour.
Total 15 points]
1.
Two players are going to play the Prisoner’s Dilemma described below an
infinitely many times. Each player has a discount factor of 1, such that 0 1 1 < 1.
C
D
C
5, 5
0, 6
D
6, 0
2, 2
(a)
If 1 = 0, describe the subgame perfect equilibrium of this infinitely repeated
game. Briefly explain your answer.
Answer:
1 = 0 means that players don’t value their future at all. So at every period of
the infinitely repeated game, they play the game as if they are playing a one
shot game. (Note that this does NOT mean that it actually is a oneshot game).
In an infinitely repeated game, each subgame is an infinitely repeated game as
well. In this case, where players don’t care about the future at all, each
subgame reduces to the oneshot game. So the only subgame perfect
equilibrium of the game is the strategy “to play D at every period” for both the
players.
(b) Suppose the two players decide to use the ‘trigger strategy,’ that is, each will
begin by playing C and then in any period if they find that nothing but C’s
have been played by both players in the past, they will play C. Under all other
circumstance they will play D.
Calculate the critical value of 1 such that for all values above that the trigger
strategies constitute a Nash equilibrium.
Answer:
Stay: 5
5
5
5
5
…
Deviate: 6
2
2
2
2
…
Therefore PV(stay) = 5/(1  1) and PV(deviate) = 6 + 21/(1  1).
The trigger strategies constitute a Nash equilibrium iff –
PV(stay) ≥ PV(deviate)
i.e., 5/(1  1) ≥ 6 + 21/(1  1)
i.e., 1 ≥ ¼.
Therefore the critical value of 1 such that for all values above that the trigger
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 Spring '08
 BASU
 Game Theory, Nash, Subgame perfect equilibrium, SPE, perfect equilibrium

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