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(Prof
Copic) ,
Fa...

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Unformatted text preview: Ec101
(Prof
Copic) ,
Fa ll
2009,
Midterm
1,
Octob er
27.

 You
ha ve
1h
to
ans wer
the
qu estions.
Each 
qu estion
h as
exactly
one 
correct
ans wer.
Each 
 question
has 
an
indication
of
eas y,
med ium,
or
hard.
There
are
2
extra‐credit
qu estions
wh ich 
 are
both
hard er
than
the
rest
‐
th ese
are
indicated
as
b onus
qu estions
‐
I
su ggest
you
first
 answer
th e
oth er
qu estions .
(Figures
1
and
2
are
on 
th e
last
pa ge.)
Sta y
ca lm
and
do
well!
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1.
( easy) 
In 
th e
game
of
the
battle
of
th e
s exes
b elow,
row
p la yer
is
Wife
(pla yer
1) 
and
colu mn
 player
is
Husband
( Pla yer
2) .
 
 Ballet
 Boxing
 Ballet
 4,1
 0,0
 Boxing
 0,0
 1,3
 What
are
all
th e
Nash
equ ilibria 
of
this 
ga me?

 a.
( Ballet,Ba llet) ,
( Boxin g,
Ballet)
 b.
( Ballet,Ba llet) ,
( Boxing,
Boxin g),
(1/2
Ba llet+
1/2
Boxing,
1/2
Ballet+
1/2
Boxin g)
 c.
( Ba llet,Ballet),
( Boxing,
Boxing) ,
(3/4
Ba llet+
1/4
Boxing,
1/5
Ballet+
4/5
Boxin g)
 d.
( Ballet,Ba llet) ,
( Boxing,
Boxin g),
(4/5
Ba llet+
1/5
Boxing,
1/2
Ballet+
1/2
Boxin g)
 
 
 
 2.
( easy) 
Cons id er
th e
ga me
tree
in
Figure
1.
What
is
th e
sub game‐p erfect
Nash
equilibriu m
of
 that



game?
(first
entry
is
th e
s trategy
of
p layer
1;
the
rest
is 
strategy
of
p layer
2)
 





 a.
(L;
U|L,
d|R)
 b.
(L;
D|L,
d|R)
 c.
( R;
U|L,
u|R)
 d.
( R;
D|L,
u|R)
 3.
( easy) 
Is 
th ere
a
NE
of
th e
ga me
in
Figure
1
which
is
NOT
a 
subga me‐perfect
Nash
equ ilibriu m
 of
that
ga me?
 
 a.
(L;
U|L,
d|R) 
‐
b ecaus e
this 
is
th e
b est
that
pla yer
1
can
do.
 b.
(L;
D|L,
d|R)
–
b ecaus e
if
on e
p la ys
R,
th e
threa t
b y
player
2
is
not
cred ible.
 c.
( R;
U|L,
u|R) 

‐
b ecause
pla yer
1
cou ld
d o
b etter
by
playing
L.
 d.
( R;
D|L,
u|R)

‐
b ecaus e
p layer
2
shou ld
optimize
in 
every
s itua tion .

 
 
 4.

(eas y)
In
a
ba ckward 
indu ction
argu men t
the
pla yers

 
 
 a.
Believe
that
th e
other
ma y
b e
crazy.
 b.
Th in k
that
th e
oth er
pla yer
will
outs mart
th em.
 c.
Are
sure
that
the
oth er
pla yer
will
follow
through 
even
wh en
h is
threa ts
are
not
 cred ible.
 d.
Act
on
the
assump tion 
tha t
th e
oth er
p la yer
would 
only
carry
ou t
threa ts
tha t
are
 cred ible.


 

5.

( eas y)
In 
a
prison ers’
d ilemma 
ga me
each 
pla yer
h as
a
dominant
strategy
b ecaus e
 a.
It
is
Nash
equilibriu m
to
snitch 
and
in
every
Nash
eq uilibriu m
p la yers 
pla y
dominant
 strategies .
 
 
 
 b.
Regard less
of
what
th e
oth er
p layer
does,
it
is
b est
for
a
pla yer
to
sn itch .
 c.
If
n eith er
sn itch es 
on
the
oth er,
th ey
ma ximize
th e
s um
of
th eir
utilities .
 d.
On e
pla yer
can 
domina te
th e
oth er
on e.
 6.

(mediu m)
How
man y
Nash
eq.
are
th ere,
pure
or
mixed,
in
th e
prison ers’
d ilemma 
with 
th e
 bombing
option?
(s ee
Figure
2)

 
 
 
 
 
 
 
 7.
( easy) 
Two
countries,
Japan
an d
Norwa y,
simu ltan eously
d ecide
on
s end ing
fish ing
fleets 
on
 the
ocean .
Ea ch
country
can
either
s end
a
large
fleet
of
2
boats ,
in
ord er
to
fish
a 
lot,
or
a
s mall
 fleet
of
only
1
boat,
to
fish 
just
a
bit.
Th e
catch 
is 
that
the
more
they
a ll
fish ,
the
less er
th e
 quality
of
the
catch;
th e
pa yoff
of
Japan 
who
is
a
b it
fu rther
awa y
from
good 
fish ing
grounds
is
 given
b y
u 1(j,n) =4+j‐2n,
and

th e
pa yoff
of
Norwa y
is
u 2(j,n)=6+2n‐3j,
where
j
is 
th e
nu mb er
of
 fishin g
boa ts
s ent
b y
Japan ,
and
n
is 
th e
nu mb er
of
fish ing
boats

sent
b y
Norwa y.
Wha t
will
 happen
in
equ ilibrium?
(drawin g
a
norma l
form
may
h elp 
–
don ’t
ma ke
mista kes
with
computing
 payoffs!)
 a.
0
 b.
1
 c.
2
 d.
3
 e.
4
 
 
 
 
 a.
Th ey
s end
1
boa t
each
because
tha t
is 
th e
b est
th ing
to
do
for
b oth
of
th em.

 b.
Japan
s ends
1
boat
in 
good
faith
that
Norway
will
do
th e
same.
 c.
Th ey
send 
2
boats
each .
 d.
Th ey
a gree
on
d oin g
someth ing
Pareto
optimal.
 8.
( easy) 
What
are
th e
Pareto
op timal
outcomes 
in 
th e
fishin g
game
b etween
Japan
and 
 Norwa y?
 
 
 
 
 a.
Th ey
each 
send
1
boat.
 b.
Th ey
each
s end
2
boats .
 c.

Th ey
s end
a t
most
3
boats 
b etween 
th e
two
of
th em.
 d.
As
long
as 
Japan 
sends
1
boat
th e
ou tcome
is 
Pareto
optima l.
 9.
( easy) 
Ten
golden
coins
can
b e
shared
between
Betsy
and 
Chris
in 
whole
a moun ts.
Wha t
are
 the
Pareto
optimal
outcomes?
Assu me
that
th ey
each 
care
only
ab out
h ow
man y
coins
th ey
 obtain .
Th e
numb ers 
repres ent
(coins
to
Bets y,
coins
to
Chris)
 
 
 
 
 a.
(0,4),
(1,2),
(2,1),
(4,0)
 b.
( x,y),
wh ere
b oth
x>0
and
y>0,
and
x+y=10
and 
x
an d
y
are
both 
in tegers.
 c.
(0,10) ,
(1,9),
( 2,8) ,
(3,7),
( 4,6) ,
(5,5),
(6,4),
(7,3),
(8,2),
(9,1),
(10,0)
 d.
Any
outcome
that
gives 
at
least
5
coins
to
each 
of
th em.
 Qu estions
10‐11.
In
th e
ultimatu m
ga me,
th ere
are
two
pla yers.
Player
1
d ivid es
4
gold en
coins 
 between
th e
two
of
th em
and
h e
propos es
h is
d ivision 
to
Pla yer
2.
Th us,
Pla yer
1
ma y
ta ke
an y
 whole
number
of
coins 
x
for
h ims elf,
0
<=
x
<=
4,
and 
p ropose
4‐x
coins
to
Pla yer
2.
Player
2
can 
 then 
eith er
accept
or
reject
th e
offer.
If
Pla yer
2
rejects
th e
offer,
th en
they
get
0
coins 
ea ch.
 Aga in,
ea ch
of
th e
p layers
only
cares 
about
th e
nu mb er
of
gold en
coins
th ey
get.

 
 10.
( mediu m)
What
are
all
the
subga me‐p erfect
Nash
equilibriu m
ou tcomes
of
this 
ga me?
The
 numb ers
represen t
( x
coins 
to
Player
1,
y
coins
to
Pla yer
2)
 
 
 
 
 
 a.
(3,2)
and
( 2,3)
 b.
(4,0)
and 
(3,1)
 c.
(4,0),
(3,1),
(2,3),
(1,2)
 d.
(3,1)
and 
(2,2)
 e.
(4,0),
(3,1),
(2,2) ,
(1,3),
(0,4)
 11.
(hard) 
Wha t
are
the
Nash 
equilibria
of
th is
ga me?
( hint:
th ink
of
a 
similar
game
with
3
gold en 
 coins
in
wh ich 
both 
pla yers
ta ke
their
a ctions
simu ltan eously,
and
think
of
the
action 
of
Pla yer
2
 as
writin g
on
a
p iece
of
pap er
th e
sma llest
numb er
of
coins
h e
is
willin g
to
accep t;
dra w
th e
 normal
form
of
this
ga me) 

 
 
 
 
 
 a.
(3,1),
(2,2),
(1,3)
 b.
(4,0)
and 
(3,1)
 c.
(4,0),
(3,1),
(2,3),
(1,2)
 d.
(2,2)
 e.
(4,0),
(3,1),
(2,2) ,
(1,3),
(0,4)
 12.
( eas y)
Which
of
th e
following
is 
not
a
Pareto‐optimal
outcome
of
the
ultimatu m
game?
 
 
 
 
 a.
(1,3)
 b.
(3,0)
 c.
(2,1)
 d.
(2,2)
 13.
( eas y)
George
is
tryin g
to
assess 
th e
valu e
of
a
firm
that
genera tes 
a
profit
of
$1
on
th e
27th
 day
of
every
month 
(toda y
is
th e
day
on
wh ich
th e
firm
will
gen era te
$1) .
George
is
certain 
tha t
 the
firm
will
last
precisely
237
months ,
and
h is
monthly
d iscoun t
factor
is
d =2/3.
Which
of
th e
 following
is
th e
b est
approximation
to
th e
NPV
of
th is
firm
to
George?
(h int:
th in k
abou t
th e
firm
 lastin g
infin itely
lon g,
if
you
don ’t
rememb er
how
to
d o
that,
try
computing
th e
NPV 
for
some
 numb er
of
mon ths,
e.g.,
3,4,5…).
 
 
 
 
 
 
 14.

(med ium) 
Th ere
are
2
firms
comp etin g
in 
a
market
for
cand les.
Th e
produ ction
cost
of
 producin g
candles
$0/cand le
for
each 
of
the
two
firms.
Th ey
only
set
prices
in
whole
dollars
( i.e.,
 0,1,2,..)
and 
th e
monopolist
price
in
th e
market
for
can dles 
is
$20/cand le.
Wh ich 
of
th e
 following
is
NOT
tru e.
 a.
11/4
 b.
3
 c.
5
 d.
7
 
 a.
If
th e
firms 
comp ete
b y
s ettin g
prices
th ey
would 
un dercu t
each 
oth er’s 
price.
 b.
If
th e
firms
collude,
then 
th ere
is
an
equ ilibrium
in
which
th e
price
equa ls
th e
 monopolis t
price.

 c.

If
th e
firms 
comp ete
th en 
th ere
is
an
equ ilibriu m
in
which
both
firms 
set
th e
price
at
 $1/candle.
 d.
If
th e
two
firms
collud e
th ey
will
a gree
on
s ettin g
th e
price
a t
$30/candle
because
 that
is
b est
for
th e
two
of
th em.

 
 
 15.
(b onus)
Alb ert
an d
Bob
are
to
share
a 
pie
of
s ize
1.
Any
shares 
are
possib le
as
lon g
as
th ey
 sum
up
to
a t
most
1.
Th ere
are
2
periods ,
and
th ey
each
discount
th e
future
by
d =1/2.
In 
th e
 first
p eriod,
Albert
prop oses 
a
division 
of
the
pie.
If
Bob
accepts ,
they
get
th eir
propos ed
shares.
 If
h e
rejects ,
they
get
noth ing
in 
th e
first
p eriod
and 
move
on 
to
th e
2nd
period 
in
wh ich 
Bob
 proposes
th e
d ivision.
If
Alb ert
accepts
that,
th en
th ey
get
in
th e
2nd
p eriod 
th e
shares
propos ed
 by
Bob ,
and 
if
he
rejects
then
th ey
both
get
0
in
th e
s econd
p eriod
and
th e
game
ends .
In
the
 subgame
p erfect
NE
of
th is
game
( only
on e
of
the
followin g
is
tru e)
 
 
 
 
 a.
Alb ert
gets
th e
whole
pie
in
th e
first
p eriod
 b.
Bob 
gets 
half
of
th e
pie
in
the
s econd 
period
 c.
Alb ert
propos es
to
Bob 
half
of
the
pie
in
th e
first
p eriod
and 
Bob
accepts
 d.
Alb ert
propos es
to
Bob
two
thirds
of
th e
pie
in
the
first
period 
and
Bob 
rejects
 16.
(b onus)
Suppos e
that
th e
game
b etween
Alb ert
an d
Bob
can 
go
on
for
as
man y
p eriods
as
 necessary
–
in
th e
first
p eriod 
Alb ert
propos es,
in
the
s econd 
Bob,
in 
th e
third 
Alb ert,
and
so
on
 (Alb ert
propos es
in
even 
p eriods ,
and
Bob
in
odd 
on es).
Again
th ey
don’t
enjoy
th e
p ie
un til
th ey
 agree
on
th e
divis ion 
(note
that
this
is
n ot
a 
rep eated 
game
but
just
a 
sequ en tia l
game
that
ma y go
on 
for
a
very
long
time
b efore
it
is 
fin ished).
Wha t
is 
th e
ou tcome
of
a 
subga me‐p erfect
Nash
 equilibriu m
now?
 
 
 
 
 
 a.
Th ey
d ivid e
the
pie
in 
th e
first
p eriod
and 
th ey
each 
get
a
ha lf.
 b.
Th ey
d ivid e
th e
p ie
in 
th e
s econd
p eriod
and
th ey
ea ch
get
a
ha lf.
 c.
Th ey
divide
th e
p ie
in
th e
first
p eriod 
and
Alb ert
gets 
3/5
while
Bob
gets
2/5
 d.
Th ey
d ivid e
th e
p ie
in 
th e
first
p eriod
and 
Alb ert
gets
2/3
while
Bob
gets
1/3
 
 ...
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This note was uploaded on 10/23/2010 for the course STATISTICS 2 taught by Professor Ramirez during the Winter '09 term at USC.

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