Problems for Week 7
122

Game Theory

Professor Giacomo Bonanno
PRACTICE
PROBLEMS for
WEEK 7
Topic:
information partitions, knowledge
and common knowledge
VERY IMPORTANT
: do
not
look at the answers until you have made a
VERY
serious effort to solve the
problem. If you turn to the answers to get clues or help, you are wasting a chance to test how well you are
prepared for the exams. I will
not
give you more practice problems later on.
1.
Let the set of states be
W = {a, b, c, d, e, f, g, h, k, m, n}
and the information partition of an
individual be:
{{a,b,c}, {d}, {e, f, g, h}, {k,m}, {n}}
.
Consider the following event:
E = {a,b,d,k,n}}
.
(1)
Does the individual know E at state
a
?
(2)
Does the individual know E at state
c
?
(3)
Does the individual know E at state
d
?
(4)
Does the individual know E at state
h
?
(5)
Does the individual know E at state
k
?
(6)
Does the individual know E at state
n
?
(7)
Let
KE
denote the event that the individual knows
E
(that is, the set of states where the
individual knows
E
). What is
KE
?
(8)
Given an event, say F, denote by
¬
F be the complement of F, that is, the set of states
that are
not
in F. What is the event
¬
KE
, that is the event that the individual does not
know
E
? What is the event
K
KE
, that is, the event that the individual knows that she
does not know
E
?
2.
Let the set of states be
W = {a, b, c, d, e, f, g, h}.
There are three individuals with the
following information partitions:
Ann
Bob
Carol
a
b
c
d
e
f
g
h
a
b
c
d
e
f
g
h
a
b
c
d
e
f
g
h
(1)
Consider the event
E = {a,b,c,f,g}
. Find the following events:
K
Ann
E
(that is, the event that Ann knows
E
),
K
Bob
E,
K
Carol
E,
K
Carol
K
Ann
E
(that is, the event that Carol knows that Ann knows
E
),
K
Bob
K
Carol
K
Ann
E,
K
Ann
¬
K
Bob
K
Carol
E
(that is, the event that Ann knows that Bob does
not know that Carol knows
E
).
(2)
What is the common knowledge partition?
is common knowledge.