Midterm Solutions

Midterm Solutions - ECON
166A
MIDTERM
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Unformatted text preview: ECON
166A
MIDTERM
 SInervo
And
Musacchio
 FALL
2010
 
 85
points
total
 
 1)
On
October
14,
1962,
the
US
confirmed
the
presence
of
Soviet
nuclear
missiles
in
 Cuba.
It
was
the
time
of
the
Cold
War
and
the
US
and
USSR
were
archrivals.
The
 USSR
had
placed
these
nukes
in
Cuba.
The
sequence
of
decisions
facing
the
US
 (Kennedy)
and
USSR
(Kruschev)
leaders
is
as
follows.
The
US
moves
first
and
has
to
 decide
to
Blockade
the
island
of
Cuba
so
as
to
prevent
Soviet
ships
from
reaching
the
 island,
or
perform
an
Air
Strike.
If
Air
Strike
the
game
ends,
if
the
former
Blockade
 option
is
used,
the
USSR
faces
a
decision
of
Retaining
the
missiles
or
Withdraw
the
 missiles.
If
the
USSR
withdraws
the
missiles
then
the
game
ends.
If
the
USSR
Retains
 the
missiles
then
the
US
moves
to
either
Blockade
or
Airstrike.
The
US
most
prefers
 the
option
of
Blockade
and
having
the
Soviets
withdraw
and
least
prefers
the
 situation
where
it
Blockades,
the
Soviets
Retain
and
the
US
then
Blockades.
The
US
 also
slightly
prefers
the
situation
of
Blockade
(US),
Maintain
(USSR)
and
then
 Airstrike
(US)
over
the
initial
move
where
the
US
immediately
orders
an
Air
Strike
 (e.g.,
the
sequence
of
intimidation
is
slightly
preferred
to
the
knee
jerk
reaction
of
 attack).
The
USSR
most
prefers
Blockade,
Maintain,
Blockade
(the
US
loses
face)
and
 least
prefers
Blockade,
Maintain,
Airstrike
(the
USSR
loses
faces
and
suffers
an
 attack
on
its
strategic
partner
Cuba).
Of
the
remaining
two
options
the
USSR
slightly
 prefers
Blockade
then
Withdraw
over
the
immediate
Airstrike
move
on
the
part
of
 the
US.

 
 a) Draw
the
EFG
for
the
US
(Kennedy)
and
USSR
(Kruschev),
and
using
 payoff
least
preferred
=
1
and
payoff
most
preferred
=4
(and
the
 intermediate
payoffs
of
2,
3).
Write
out
the
payoffs
to
the
US
and
USSR
on
 your
EFG.
[5
points]

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b) Solve
the
EFG
by
backwards
induction
(Show
all
the
steps)
and
come
up
 with
the
BI
solution.
[10
points]
 
 
 2) Given the game matrix that follows [30 pts total]: a. Categorize the game as symmetric or asymmetric, and zero sum or non-zero sum 2pts). Asymmetric, b. Solve the pure NE (3 pts). Make sure you list your steps (tell us what you are doing). I labeled the BR for each of the row player, and column player by underlying. The NE is indicated in bold. c. During your solution, if the game reduces in # of strategies, categorize the sub-game (3 pts). The game does not reduce, there are no strictly dominated strategies. Common errors. Many of you applied IDWDS – a no no, ONLY IDSDS (there are no SDS). d. The NE is only weakly dominant (e.g., ties for b and c if you fix player 1 on D) A B C D a 4, 4 3, 4 7, 2 0, 0 b 0, 0 0, 2 0, 0 2, 2 c 5, 2 3, 3 3, 7 2, 2 d 2, 7 7, 3 3, 3 0, 0 b weakly dominant with c 2 cont) Given the game matrix that follows: d. Categorize the game as symmetric or asymmetric, and zero sum or non-zero sum 2pts). Asymmetric, non zero sum e. Solve for all pure NE. (4 pts). Make sure you list your steps (tell us what you are doing). I labeled the BR for each of the row player, and column player by underlying. The 3 NE are indicated in bold. f. During your solution, if the game reduces in strategies categorize the sub-game (3 pts). The game does not reduce, there are no strictly dominated strategies. g. Provide (game-theoretic) justifications for choosing among NE. Define the concept. (3pts) Well W, w could be eliminated by IDWDS because it is only a weakly dominant strategy leaving Z, y and Y, z. Notice that W, w is a NE that is NOT payoff dominant (A strategy profile is payoff dominant if there is no other strategy profile for which each player has strictly higher payoffs. W, w would be Payoff dominant if it was 6, 6, because the 6,6 would collectively beat 5, 1 and 1, 5, but that was not the case). W X Y Z w 3, 3 1, 1 1, 1 3 , -1 x 1, 1 1, 1 3, 3 -1, 1 y 1, 1 3, 3 4, 4 5, 1 z -1, 3 1 , -1 1, 5 -1, -1 Here is a more complete answer that goes into the details of weak dominance: W,
w
is
slightly
less
preferred
for
the
reasons
of
weak
dominance
(over
Z,
y
and
Y,
 z).

 
 The
full
solution
involves
IDWDS
if
this
is
possible
(because
no
strategy
is
pareto
 efficient
 or
payoff
dominant).
 
 w x y z W 3, 3 1, 1 1, 1 -1, 3 X 1, 1 1, 1 3, 3 1, -1 Y 1, 1 3, 3 4, 4 1, 5 Z 3, -1 -1, 1 5, 1 -1, -1 
 For
the
moment,
just
focus
on
the
NE
in
this
case
(ignore
all
other
strategies,
 because
these
strategies
cannot
be
supported
by
mutual
BR
criteria
 and
thus,
are
not
in
the
mutually
rationale
strategy
set).

 
 W Y Z 
 From
Harrington,
Chapter
3:
 A
strategy
s'
weakly
dominates
a
strategy
s"
if:
 1)
if
the
payoff
from
s'
is
at
least
as
great
as
that
from
s"
for
ANY
strategies
chosen
 by
the
other
players

AND
 2)
there
are
some
strategies
chosen
by
the
other
players
whereby
the
strategy
s'
is
 strictly
greater
than
s"
 
 from
the
perspective
of
player
1

 Z
weakly
dominates
W
(and
thus
W
is
weakly
dominated
by
Z):
 1)
Z
ties
W
for
player
2
playing
w
AND
 2)
Z
beats
W
for
player
2
playing
y
 
 from
the
perspective
of
player
2

 z
weakly
dominates
w:
 1)
z
ties
w
if
player
1
plays
W
AND
 2)
z
beats
w
if
player
1
plays
Y,

 
 Thus,
W,
w
is
weakly
dominated
by
strategies
in
player
1
and
2,
but
note
that
this
 need
not
be
weakly
dominated
by
a
single
strategy.
 
 Notice
that
in
this
case
both
Y
and
y
are
in
the
NE
set
for
the
players
(e.g.,
5,
1
and
1,
 5
payoffs)
 but
technically
according
to
the
definition
we
use
from
Harrington,
they
need
not
be
 in
the
NE
set
 but
could
be
from
ANY
strategy.
In
this
case
I
pulled
the
weakly
dominant
strategies
 from
these
other

 two
NE.
Thus,
the
2
NE
are
to
be
preferred
over
W,
w
because
the
other
2
NE
are
not
 weakly
dominated.

 
 Thus
of
the
three
NE
the
strategies
of
Player
1:
W
and
Player
2:
w
is
on
a
little
more
 shakier
footing.
 
 This
is
a
question
of
selecting
among
multiple
equilibria,
not
among
the
strategies
 present
in
the
matrix.
 
 One
could
go
even
further
and
look
at
ANY
strategies
played
by
the
other
players,
 But
that
is
beyond
the
scope
of
the
question,
which
was
addressed
directly
at
a
 selection
criteria
for
choosing
among
the
3
NE.
 
 For
completeness,
then
look
at
the
submatrix:
 w 3, 3 1, 1 3, -1 y 1, 1 4, 4 5, 1 z -1, 3 1, 5 -1, -1 
 Y Z y 4, 4 5, 1 z 1, 5 -1, -1 
 Between
these
two
NE,
we
see
that
Z
dominates
Y
if
player
2
plays
y

 But
that
z
dominates
y
if
player
1
plays
Y
(and
vice
versa)
 We
need
to
retain
both
Y
and
Z
and
y
and
z.
 
 A
cautious
player
tends
to
avoid
weakly
dominated
NE
 
 A
clever
student
willalso
notice
that
while
W
is
weakly
dominated
by
Z,
 
 w y W 3, 3 1, 1 Z 3, -1 5, 1 
 In
the
other
case
it
is
z
that
weakly
dominates
w:
 w z W 3, 3 -1, 3 Y 1, 1 1, 5 
 On
this
questions
many
students
confused
weakly
dominated
and
strictly
 dominated.
 Weakly
dominated
is
a
property
of
a
given
strategy
and
another
strategy
in
a
players
 set
 (but
in
this
case
it
is
drawn
from
the
set
of
NE).
 
 Notice
that
1,
5
and
5,
1
are
not
weakly
dominated,
but
are
strictly
dominated
(one
 by
the

 other
for
each
player
and
thus
we
are
required
to
retain
them
both).
 
 A
NE
that
is
weakly
dominated
by
any
other
strategy
is
on
shakier
grounds,
as
I
 mentioned
 in
my
lecture.

Why?
 Look
at
the
strategy
W
in
the
payoff
matrix,
which
gives
3
vs
1
if
player
2
plays
w,
y
 respectively.
 In
contrast,
for
the
strategy
Z
the
payoff
matrix
yields
3
vs
5
for
w,
y
respectively.
 Thus,
a
 cautious
player
would
naturally
avoid
the
weakly
dominated
strategy
of
W
because
 Y
gives
 as
high
or
higher
payoffs
regardless
of
what
player
2
plays.
 
 It
is
a
mistake
to
do
IDSDS,
there
are
no
 strictly
dominated
strategies
to
delete.
 
 I
was
not
referring
to
strictly
dominated,
in
this
question,
the
students
were
 required
to
 use
the
concept
of
weak
dominance
to
distinguish
the
W,
w
and
being
slightly
less
 preferred
 to
the
other
pair.
 h. Categorize
the
game
as
symmetric
or
asymmetric
(2pts).
Symmetric
 a. Solve
for
all
pure
NE.
(4
pts).
Make
sure
you
list
your
steps
(tell
us
what
you
 are
doing).
Do
IDSDS
on
P
and
p
and
you
are
left
with
an
RPS,
which
has
 no
NE
(show
your
work
of
course
with
BR
underline,
etc).

 j. During
your
solution,
if
the
game
reduces
in
strategies
categorize
the
sub‐ game
as
symmetric
or
asymmetric
(3
pts).
Symmetric
 k. Based
on
your
analysis
in
j
(e.g.,
of
all
pure
NE),
do
you
expect
there
will
be
a
 mixed
NE?
Justify
your
answer.
[1
point]
 You
would
expect
an
odd
number
of
NE
so
there
should
be
one
mixed
NE
 (Not
required,
but
if
it
is
symmetric
in
payoffs,
you
might
guess
the
mixture
is
1/3,
 1/3,
1/3
–
This
solution
is
on
the
homework).

 
 
 3)
You,
player
A,
are
engaged
in
a
team
class
project
with
your
buddy
player
B.
You
 each
have
to
decide
what
level
of
work
to
put
into
the
project,
and
your
choices
are
 1‐low,
2‐medium,
3‐
high.

Your
work
is
designated
wa
and
that
of
your
buddy
wb.
 The
grade
on
the
project
is
determined
by
the
total
work
W=wa+wb
as
 
 
 2
 3
 4
 5
 6
 Grade
 50
 70
 83
 95
 100
 
 (You
will
notice
that
your
grade
has
diminishing
returns
as
the
amount
of
work
goes
 up.)
Each
player
i’s
payoff
is

Grade
–
10wi
.
[20
points
total]
 
 a) Suppose
you
agree
with
your
buddy
to
work
independently.
You
will
just
 staple
your
two
parts
together
on
the
day
it
is
due,
without
observing
 how
hard
your
partner
worked.
Express
the
game’s
strategic
form
as
a
 bimatrix.

[4
points]
 
 
 1
 2
 3
 1
 40,40
 60,50
 73,53
 2
 50,60
 63,63
 75,65
 3
 53,73
 65,75
 70,70
 M N O P m 90, 90 120, 80 80, 120 70, 120 n 80, 120 90, 90 120, 80 70, 140 o 120, 80 80, 120 90, 90 40, 70 p 120, 70 140, 70 70, 40 70, 70 
 
 b) What
(pure)
Nash
equilibria
can
you
find?
[4
points]
 
 1
 2
 3
 
 
 
 
 
 
 
 
 
 
 1
 40,40
 50,60
 53,73
 2
 60,50
 63,63
 65,75
 3
 73,53
 75,65
 70,70
 The
pure
NE
are
(3,2)
and
(2,3).
 c) What
is
the
equivalent
extensive
form
of
the
game
you
analyzed
in
parts
a
 and
b?
[4
points]
 A
 1
 ! 2 
 B
 2 3
 1
 3
 B
 2 3
 1
 d) Now
 ! 2 B
 3
 1
 ! 
 
 
 40,40

60,50



73,53

50,60


63,63


75,65


53,73


65,75



70,70
 ! 
 
 
 
 
 
 
 
 
 
 suppose
you
decide
that
you,
player
A,
will
finish
your
part
and
then
pass
 it
to
B.
B
will
see
how
hard
you
worked,
and
then
decide
how
hard
to
 work
on
their
part.
What
is
the
extensive
form
of
this
game?
[4
points]
 A
 1
 ! 2 
 B
 2 3
 1
 3
 B
 2 3
 1
 ! 2 B
 3
 1
 ! 
 
 
 40,40

60,50



73,53

50,60


63,63


75,65


53,73


65,75



70,70
 e) Can
you
find
a
backwards
induction
solution
to
the
game
of
part
d?

Based
 on
this
analysis
would
you
want
to
be
the
player
who
works
on
their
part
 first
or
last?
[4
points]
 
 Backwards
induction
allows
us
to
eliminate
branches
to
get:
 A
 1
 ! ! 
 
 
 
 
 
 
 
 2 
 B
 3
 3
 B
 2 B
 3
 
 



























73,53




























75,65
















65,75




 ! 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The
next
level
of
BI
gets
us
to:
 A
 ! 2 
 B
 ! 3 75,65
















 
 Thus
the
solution
is
(2,3).
The
first
mover
ends
up
with
the
higher
payoff,
 so
it
is
better
to
be
him.
 
 4)
It
is
2020,
and
industry
consolidation
has
led
to
having
only
3
oil
companies
in
 the
world
–
(B)otswana
Petroleum,
(E)xxelon,
and
(S)helldon.
The
price
per
barrel
 of
oil
is
determined
by
the
total
quantity
Q
produced
according
to
the
demand
 function
 
 
 
 
 p
=
(200
–
Q)
 where
Q
is
in
millions
of
barrels
produced
per
day,
and
p
is
in
dollars.
 
 Suppose
that
it
costs
each
company
$50
for
each
barrel
they
produce.
(We
are
 ignoring
other
factors
like
fixed
costs,
and
differences
in
oil
reserves
the
companies
 have
access
to.)
Thus
the
profit
of
each
company
i
=
b,
e,
or
s
is
 
 
 
 
 Vi
=
(200
­
Q
­
50)qi
 where
Q=qb
+
qe
+
qs
,
and
the
units
of
Vi
are
millions
of
dollars
per
day.
[20
points
 total]
 
 a) What
is
the
best
response
function
of
company
i?
Express
your
answer
in
 terms
of
q­i

,
the
sum
of
the
quantities
the
companies
other
than
i
produce.
 [8pts]
 i’s
payoff
with
respect
to
qi
is
“hill
shaped,”
so
to
find
the
best
response,
 we
take
the
derivative
with
respect
to
qi
and
find
when
it
is
0.
 Vi = (150 − qi − q− i )qi = −qi2 + (150 − q− i )qi dVi Vi
=
(200
­
Q
­
50)qi
 = −2qi + 150 − q− i dqi dVi * 150 − q− i (qi ) = 0 ⇒ q* = i dqi 2 
 b) Find
a
symmetric
NE.
[4pts]
 In
any
symmetric
equilibrium,
all
companies
produce
the
same
amount
 € so
q­I
must
equal
2
times
qi.
Thus
 150 − 2q* * i qi = 2 4 q* = 150 
 i 
 c) What
is
the
profit
of
each
company
in
the
equilibrium
you
find
in
part
b?
 (Leave
your
answer
in
millions
of
dollars
per
day
as
that’s
easiest.)
[4pts]
 €
 Just
plug
the
quantity
found
in
part
b
into
the
payoff
function.
 
 
 (150­37.5­37.5­37.5)37.5
=

1406.25
millions
of
dollars
per
day
 d) The
CEOs
of
the
three
companies
propose
to
merge
into
1
mega‐company
 because
they
claim
it
will
be
better
for
the
environment.
What
is
the
most
 q* = 37.5 i 
 
 
 
 
 
 
 
 
 
 profitable
output
level
of
the
mega‐company?
What
will
be
its
profits
at
this
 production
level?
[4pts]
 
 Now
the
payoff
function
for
the
mega
company
looks
like
 
 
 V=(150­Q)Q
 
 To
optimize,
we
find
the
derivative
w.r.t.
Q
and
set
it
to
0.
 
 
 dV/dQ
=
150
–
2Q

 
 
 Q*
=
75
 
 Plug
this
into
the
payoff
function
to
get
the
profit
 
 
 (150­75)*75
=
5625
millions
of
dollars
per
day
 ...
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This note was uploaded on 03/03/2011 for the course ECON 414 taught by Professor Staff during the Spring '08 term at Maryland.

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