ECON 212 Game Theory
Prof. Andrew Postlewaite
Fall 2010
University of Pennsylvania
Suggested Solution for Problem Set 4
1.
a. There are essentially two states:
G
in which
(
B, B
)
is expected to be played
and
B
in which
(
C, C
)
is expected. Let
V
i
, i
=
G, B
be the expected payoff
in a repeated game each player expects from the prescribed strategy in state
i
(Note that both players get the same expected payoff according to the strategy
profile). Then
V
G
=
(1

δ
)4 +
δV
G
= 4
V
B
=
(1

δ
)(

1) +
δV
G
= 5
δ

1
.
For the given strategy profile to be a SPNE, no player should have an incentive
to deviate. The strategy profile is incentive compatible in state
G
iff
V
G
≥
(1

δ
)5 +
δV
B
(
deviation to
A
)
V
G
≥
(1

δ
)1 +
δV
B
(
deviation to
C
)
Solving these inequalities, we end up with
δ
≥
1
5
.
The strategy profile is
incentive compatible in state
B
iff
V
B
≥
(1

δ
)0 +
δV
B
(
deviation to either
A
or
B
)
⇔
δ
≥
1
5
We conclude that the strategy profile is a SPNE if
δ
≥
1
5
.
b.
(
A, A
)
is the unique strategygame NE. Since this is a finitely repeated game,
always playing
(
A, A
)
is the unique SPNE.
c. Consider the following strategy profile: each player plays
A
in the first stage.
If
(
A, A
)
was played in the first stage, then play
A
. Otherwise, play
C
. Ob
viously, this is not a SPNE because
(
C, C
)
is not a stagegame NE. Incentive
compatibility in both periods are immediate.
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 Spring '12
 gg
 Game Theory, Deviation, SPNE, strategy profile

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