Prof. William H. Sandholm
Department of Economics
University of Wisconsin
March 13, 2007
Midterm Exam Solutions – Economics 713
1.
By the oneshot deviation principle, it is enough to check that there is no profitable
oneshot deviation in any subgame. There are three classes of subgames corresponding
to the three stages specified in
σ
i
.
In stage I, the equilibrium payo
ff
sequence is (3, 3, 3, ... ); the most profitable oneshot
deviation from
T
is to
B
, which generates the payo
ff
sequence (4, 1, 1, 1, ... ). Thus, the
deviation is not profitable if 3
≥
(1

δ
)
·
4
+
δ
·
1, and hence if
δ
≥
1
3
.
In stage II, the equilibrium payo
ff
sequence is (1, 1, 1, ... ); the most profitable oneshot
deviation from
T
is to
B
, which generates the payo
ff
sequence (3, 0, 0, 0, ... ). Thus, the
deviation is not profitable if 1
≥
(1

δ
)
·
3, and hence if
δ
≥
2
3
.
In stage III, there is no profitable oneshot deviation.
Thus, strategy profile
σ
is a subgame perfect equilibrium whenever
δ
≥
2
3
.
2.
(i)
S
i
is the set of functions of the form
s
i
:
T
i
→
A
i
(ii) Assume without loss of generality that
p
(
t
i
)
=
∑
t

i
∈
T

i
p
(
t
i
,
t

i
)
>
0 for all
t
i
∈
T
i
and
i
∈
N
, and let
p
(
t

i

t
i
)
=
p
(
t
i
,
t

i
)
/
p
(
t
i
). Strategy profile
s
*
is a Nash equilibrium if and only
if
(
†
)
t

i
∈
T

i
p
(
t

i

t
i
)
u
i
(
s
*
i
(
t
i
)
,
s
*

i
(
t

i
))
≥
t

i
∈
T

i
p
(
t

i

t
i
)
u
i
(
a
i
,
s
*

i
(
t

i
))
for all
a
i
∈
A
i
,
t
i
∈
T
i
, and
i
∈
N
.
(iii) Multiplying (
†
) through by
p
(
t
i
) shows that it is equivalent to
(
‡
)
t

i
∈
T

i
p
(
t
i
,
t

i
)
u
i
(
s
*
i
(
t
i
)
,
s
*

i
(
t

i
))
≥
t

i
∈
T

i
p
(
t
i
,
t

i
)
u
i
(
a
i
,
s
*

i
(
t

i
))
.
Let
T
*
i
(
a
i
)
=
{
t
i
∈
T
i
:
s
*
i
(
t
i
)
=
a
i
}
be the set of player
i
types who choose action
a
i
under
strategy
s
*
. Then
t
i
∈
T
*
i
(
a
i
)
t

i
∈
T

i
p
(
t
i
,
t

i
)
u
i
(
s
*
i
(
t
i
)
,
s
*

i
(
t

i
))
=
a

i
∈
A

i
ρ
*
(
a
i
,
a

i
)
u
i
(
a
i
,
a

i
)
.