Kaushik Basu
Spring 2008
Econ 367
Game Theoretic Methods
Problem Set 9
1.
Suppose two individuals play the game, G, described below ten times.
Is there a
subgame perfect equilibrium (SPE) such that in some periods the players play
)
,
(
C
C
?
Explain your answer.
G:
A
B
C
A
1, 1
0, 0
4, 0
B
0, 0
1, 1
0, 0
C
0, 4
0, 0
3, 3
[Here is some food for thought: Note that (C, C) is not a Nash equilibrium of G, and note
also that SPE requires players to play Nash in every subgame.
How then can (C, C) ever
occur in a stage game?]
Answer:
There is no pure strategy SPE that can support (C, C) in any period of the game described
played ten times.
To see this note that the two Nash equilibria, (A, A) and (B, B) have
the same payoffs for both players π
1
(A, A) = π
2
(A, A) = π
1
(B, B) = π
2
(B, B) = 1.
Thus,
no matter what has happened in previous play, no player can be “punished” in the 10
th
round of play.
To see this, think about if player two has deviated in a previous round.
If
she has deviated, she will receive 1 in the final period and if she has not deviated in any
of the previous rounds she will again only receive 1.
There are SPEs if one uses mixed strategies, though.
For example, if the strategy is:
play (C,C) in the first round followed by (B,B) in rounds 2 – 10;
if (C,C) was not played in the first round, then play mixed strategy {(A,1/2), (B,1/2),
(C,0)} in rounds 2 – 10.
The above mixed strategy yields an SPE.