Page 1 of 5
Department of Economics, University of California, Davis
Ecn 122

Game Theory

Professor Giacomo Bonanno
ANSWERS
TO
PRACTICE
PROBLEMS for
WEEK 7
1.
(1)
No, because the information set containing
a
is {
a,b,c
} which is not contained in
E
=
{
a,b,d,k,n
}.
(2)
No, because the information set containing
c
is {
a,b,c
} which is not contained in
E
=
{
a,b,d,k,n
}.
(3)
Yes, because the information set containing
d
is {
d
} which
is
contained in
E
=
{
a,b,d,k,n
}.
(4)
No, because the information set containing
h
is {
e,f,g,h
} which is not contained in
E
=
{
a,b,d,k,n
}.
(5)
No, because the information set containing
k
is {
k,m
} which is not contained in
E
=
{
a,b,d,k,n
}.
(6)
Yes, because the information set containing
n
is {
n
} which
is
contained in
E
=
{
a,b,d,k,n
}.
(7)
KE
= {
d,n
}.
(8)
¬
KE
= {
a,b,c,e,f,g,h,k,m
}.
K
¬
KE
= {
a,b,c,e,f,g,h,k,m
}.
2.
(1)
K
Ann
E = {a,b,c}
,
K
Bob
E = {a,b,f},
K
Carol
E = {b,c,f,g},
K
Carol
K
Ann
E = K
Carol
{a,b,c} = {b,c},
K
Bob
K
Carol
K
Ann
E = K
Bob
{b,c} =
∅
(this is the
symbol for the empty set)
,
K
Ann
¬
K
Bob
K
Carol
E = K
Ann
¬
K
Bob
{b,c,f,g} = K
Ann
¬
{f}
=
K
Ann
{a,b,c,d,e,g,h} = {a,b,c,d,h}.
(2)
The common knowledge partition is:
a
b
c
d
e
f
g
h
(3)
CKE
=
∅
(4)
If
F
= {
a,b,c,d,e,g
} then
CKF
= {
a,b,c,d
}.
3.
We can describe a state by the number of socks you have taken out and the color of each
sock.
After you have taken out only one sock:
1
B
1
W
(the information set capture the
fact that you cannot see the color of the sock because it is dark)
After you have taken out two socks:
2
B
B
2
B
W
2
W
B
2
W
W
After you have taken out three socks:
3
B
B
B
3
B
B
W
3
B
W
B
3
B
W
W
3
W
B
B
3
W
B
W
3
W
W
B
3
W
W
W
Now in every state there are (at least) two socks of the same color, thus you know that you have