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.
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 '10
 Bonnano
 Game Theory, Englishlanguage films, common knowledge, Ann

Click to edit the document details