14.12: Problem Set 5 Solution
Ruitian Lang
December 3, 2010
1
(a) There are two players, 1 and 2, with
A
1
=
{
U,D
}
and
A
2
=
{
L,R
}
. The type spaces are
T
1
=
{
1
,
1
}
,T
2
=
{
1
,
1
}
, and the beliefs are
p
1
(
t
2

t
1
) =
p
2
(
t
1

t
2
) = 1
/
2 for all
t
1
∈
T
1
and
t
2
∈
T
2
. The
payoﬀ function is as follows:
u
i
(
D,L
;
t
1
,t
2
) =
u
i
(
U,R
;
t
1
,t
2
) = 0 for
i
= 1
,
2 and all
t
1
,t
2
;
u
1
(
U,L
;
t
1
,t
2
) =
2
t
1
,u
1
(
D,R
;
t
1
,t
2
) =
t
1
,u
2
(
U,L
;
t
1
,t
2
) =
t
2
and
u
2
(
D,R
;
t
1
,t
2
) = 2
t
2
. In the notation of the problem,
t
1
is
θ
, and
t
2
is
λ
.
(b) Suppose that in a BNE Player 2 always plays
L
. Then Player 1’s best response is
U
when
θ
= 1 and
D
when
θ
=

1, which must coincide with Player 1’s strategy in the BNE. Now if Player 2 sees that
λ
= 1,
her payoﬀ by playing
L
is
∑
θ
∈
T
1
u
2
(
s
1
(
θ
)
,L
;
θ,λ
= 1)
p
2
(
θ

λ
= 1) = 1
/
2 while her payoﬀ by playing
R
is
∑
θ
∈
T
1
u
2
(
s
1
(
θ
)
,R
;
θ,λ
= 1)
p
2
(
θ

λ
= 1) = 1. Therefore, her best response is
R
, a contradiction.
Similarly, we can see that Player 2 cannot always play
R
in a BNE, since then Player 1 plays
U
when
θ
=

1 and
D
when
θ
= 1 and Player 2’s best response is to play
L
when
λ
=

1.
Therefore, in a pure strategy Nash equilibrium, Player 2 must play diﬀerent strategies for diﬀerent types.
Since
p
1
(
λ

θ
) = 1
/
2 for all
λ
and
θ
,
∑
λ
∈
T
2
u
1
(
a
1
,s
2
(
λ
);
θ,λ
)
p
1
(
λ

θ
) =
1
2
u
1
(
a
1
,L
;
θ,λ
) +
1
2
u
1
(
a
1
,R
;
θ,λ
). In
other words, Player 1 always believes that Player 2 plays
L
with probability 1
/
2. Therefore, Player 1’s best
response is to play
U
when
θ
= 1 and
D
when
θ
=

1. Given this, Player 2’s best response is to play
L
when
λ
=

1 and
R
when
λ
= 1. Therefore, the only possible BNE is that
s
1
(
θ
= 1) =
U,s
1
(
θ
=

1) =
D,s
2
(
λ
= 1) =
R
and
s
2
(
λ
=

1) =
L
. We have also seen that this is indeed a BNE.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 LieWang
 Game Theory, best response, BNE

Click to edit the document details