Sample Midterm Solutions

Sample Midterm Solutions - 
 
 1A)
 
 
 1B)
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Unformatted text preview: 
 
 1A)
 
 
 1B)
 
 1C)

 
 
 The
only
pure
strategy
NE
is
(Hide,Take).
We
will
learn
later
in
the
quarter
how
to
 find
Mixed
strategy
equilibria.
 
 1D)

 
 
 
 
 1E)
 
 2)
 
 
 
 
 
 3)
Harrington
2.11
 
 Only
game
(b)
satisfies
perfect
recall.
In
game
(a),
consider
the
 information
set
for
player
1
that
includes
two
nodes.
One
node
is
associated
with
 1
having
chosen
a
and
2
having
chosen
y.
The
other
is
associated
with
1
having
 chosen
b
and
2
having
chosen
x.
At
this
information
set,
1
is
then
unsure
whether
 she
chose
a
or
b.
That
violates
perfect
recall.
As
to
game
(c),
the
information
set
 for
player
1,
which
includes
four
nodes,
captures
the
property
that,
when
1
chooses
 between
actions
c
and
d,
she
doesn’t
know
what
player
2
chose
(which
is
not
in
 violation
of
perfect
recall)
nor
what
she
originally
chose
at
the
initial
node
(which
 is
in
violation
of
perfect
recall).
Game
(b)
satisfies
perfect
recall.
When
1
chooses
 between
actions
c
and
d,
she
cannot
discriminate
between
the
nodes
in
which
play
 was
b
x
and
play
was
a

y,
nor
between
the
nodes
in
which
play
was
b

x
and
 play
was
b

y.
The
former
reflects
1’s
uncertainty
over
2’s
action
but
knowledge
 that
she
originally
chose
a.
The
latter
reflects
1’s
uncertainty
over
2’s
action
but
 knowledge
that
she
originally
chose
b.
 
 4)
Harrington
3.4
 
 a)
For
player
1,
a
is
strictly
dominated
by
b.
Neither
b
nor
c
is
strictly
 dominated.
For
player
2,
z
is
strictly
dominated
by
x.
Player
1
plays
either
b
or
c
 and
player
2
plays
either
x
or
y.
 
 b)
By
the
assumption,
we
can
go
two
rounds
of
the
iterative
deletion
of
 strictly
dominated
strategies
(IDSDS).
After
eliminating
the
strictly
dominated
 strategies,
the
game
matrix
has
the
c
row
and
z
column
deleted.

 
 
 X
 y
 B
 3,1
 2,2
 c
 0,2
 1,2
 
 
 
 
In
the
reduced
game,
b
strictly
dominates
c
for
player
1.
Neither
of
player
2’s
 strategies
is
strictly
dominated.
Thus,
player
1
chooses
b
and
player
2
chooses
either
 x
or
y.
 
 c)
After
the
first
two
rounds,
the
game
is
as
shown
above.
Now
y
strictly
dominates
x
 for
player
2.
Thus,
player
1
chooses
b
and
player
2
chooses
y.
 
 5)
Harrington
3.9
 
 
 6)
Harrington
4.1
 
 
 There
are
four
symmetric
strategy
profiles
and
thus
four
candidates
for
 symmetric
Nash
equilibrium.
Note
that
if
a
person
fails
to
vote
for
the
person
that
 everyone
else
votes
for,
then
no
one
takes
on
the
task.
Thus,
at
a
symmetric
strategy
 profile,
an
individual
player’s
choice
is
always
between
the
person
who
the
others
 are
voting
for
and
no
one.
Consider
the
symmetric
strategy
profile
in
which
all
vote
 for
Boromir.
This
strategy
is
optimal
for
both
Boromir
and
Frodo
as
each
would
 rather
that
Boromir
take
on
the
task
than
that
no
one
do
so.
This
is
clearly
not
a
 Nash
equilibrium,
however,
as
both
Legolas
and
Gimli
would
prefer
to
vote
for
 someone
else,
and
the
result
would
be
that
no
one
takes
the
ring
to
Mordor.
By
a
 similar
argument,
itis
not
a
Nash
equilibrium
for
all
to
vote
for
Gimli,
or
for
all
to
 vote
for
Legolas.
Now
consider
all
voting
for
Frodo.
Since
each
person
prefers
that
 Frodo
do
it
than
that
no
one
do
it,
each
player’s
strategy
is
optimal.
The
unique
 symmetric
Nash
equilibrium
is
then
for
all
to
vote
for
Frodo.
 
 7)
Harrington
4.8
 
 8)
Harrington
5.1
 
 a)
One
class
of
Nash
equilibria
has
19,999
people
requesting
$100
and
the
 other
80,001
people
requesting
$20.
There
are
then
many
Nash
equilibria,
and
they
 differ
only
in
terms
of
who
are
the
lucky
ones
to
receive
$100.
There
is
also
a
class
 of
Nash
equilibria
in
which
at
least
20,001
people
request
$100,
with
the
remainder
 requesting
$20.
They
all
end
up
with
zero.
 
 b)
No
one
submitting
a
request
is
the
unique
Nash
equilibrium.
Note
that
 submitting
a
request
for
$20
is
strictly
dominated
by
not
submitting
a
request.
If
 less
than
20%
request
$100,
then
a
player’s
payoff
is
‐1.95,
and
if
20%
or
more
 request
$100,
then
her
payoff
is
‐21.95.
In
contrast,
the
payoff
is
0
by
not
submitting
 a
request.
Thus,
a
Nash
equilibrium
must
entail
no
one
submitting
a
request
for
 $20.
But
if
no
one
submits
a
request
for
$20,
then,
if
anyone
participates,
they
all
 
 must
submit
a
request
for
$100.
Since
the
fraction
of
requests
for
$100
exceeds
 20%,
all
participants
get
zero,
which
makes
their
payoff
‐21.95
(the
cost
of
the
 magazine).
As
that
is
lower
then
the
payoff
from
not
submitting
a
request,
then,
in
 equilibrium,
it
must
be
the
case
that
no
one
submits
a
request
for
$100
either.
 
 c)
Going
back
to
part
(a),
it
is
no
longer
a
Nash
equilibrium
to
buy
the
 magazine
when
you
don’t
expect
to
get
some
money.
Nash
equilibria
now
entail
 19,999
people
requesting
$100
and
the
other
80,001
people
requesting
$20;
with
 the
former
players
receiving
a
payoff
of
$80.05
and
the
latter
getting
5
cents.
 
 9)
Harrington
5.2

 
 Any
strategy
profile
with
a
person
bidding
a
positive
amount
below
the
highest
bid
 is
not
a
NE,
because
it
results
in
a
negative
payoff,
but
such
a
person
can
get
a
0
 payoff
by
bidding
zero.
 
 Suppose
top
bid
is
b’

 ‐Suppose
only
1
player
bids
b’,
rest
bid
0.
 
 
 If
b’>1,
top
bidder
can
increase
payoff
by
lowering
bid
by
1.
Thus
b’=1
 
 
 

 10)
Harrington
6.3
 11)
Harrington
6.7
 
 
 ...
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