Problem Set 1 Solutions

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Unformatted text preview: 
 Problem
Set
1
Solutions




































Econ
166a
/
CMPS
166a
 Fall
2010
 Barry
Sinervo
and
John
Musacchio
 
 
 1)
Harrington
2.1
 
 
 
 
 2)
Harrington
2.2
 
 
 
 
 
 
 
 
 3)
Harrington
2.5
 
 
 4)
Harrington
2.9
 
 Problem
supp
1:
In
the
sea
otter
kidnap
game,
write
out
the
extensive
form
game
for
 the
male
kidnap
of
pup,
with
ransom
or
no
ransom
paid
by
the
female,
harm/release
 unharmed
by
the
male,
retaliate/no
retaliate
by
the
female,
and
fill
in
the
payoffs
to
 male
and
female
(consider
the
action
of
retaliate/no
retaliate
to
be
costly,
but
has
a
 deterrent
effect
[so
you
have
to
fashion
payoffs
thinking
towards
the
future]).
 Assign
ordinal
payoffs.
You
will
get
zero
if
you
do
not
explain
your
rationale
for
the
 ordinal
payoffs
that
you
assign.
 
 
 M
 
 
 
 Kidnap
 
 No
Kidnap
 
 
 
 
 5
 F
 
 9
 Ransom
 No
 
 
 Ransom
 
 M
 M
 
 
 
 
 Harm
 Release
 Harm
 Release
 
 
 F
 F
 F
 F
 No
 No
 No
 Retaliate
 Retaliate
 No
 
 Retaliate
 Retaliate
 Retaliate
 Retaliate
 Retaliate
 Retaliate
 
 
 6













8























7












9





































3













4




















1









2
 
 2













1























5












6




































4














3




















7










8
 
 
 
 Male
payoffs
top,
female
payoffs
bottom.

 Explanation:

The
female
cares
most
about
getting
her
pup
back
safely.
After
that,
 she
cares
about
not
paying
a
ransom.
Finally,
she
wants
to
maintain
a
reputation
of
 retaliating
whenever
her
pup
is
harmed,
otherwise
she
would
prefer
not
to
take
the
 risk
of
retaliating.
These
preferences
give
rise
to
the
ordinal
payoffs
above.
The
male
 cares
most
about
collecting
a
ransom.
(If
no
ransom
gets
paid
in
the
end,
he’d
rather
 not
have
kidnapped.)

After
that,
he
cares
next
about
not
being
retaliated
upon
if
the
 ransom
is
paid.
If
the
ransom
is
not
paid
he
cares
more
about
maintaining
his
 reputation
as
a
fearsome
kidnapper
and
thus
wants
to
harm
the
pup.
Finally,
he
 prefers
not
to
be
retaliated
upon.
(Other
payoffs
are
justifiable
here).
 
 Supp
Prob:
Right
and
Left
Jawed
fish.

 
 The
key
to
this
problem
is
to
think
baseball.
The
baseball
game
assumes
that
players
 have
come
to
an
equilibrium
in
terms
of
trial
and
error
learning
or
 experience/training.
Thus,
both
games
assume
the
population
is
in
an
equilibrium
 state
where
prey
have
a
belief
system
and
they
expect
something
(as
do
batters
and
 pitchers
depending
handedness).

You
can
include
a
leave
prey
step
if
you
want
(or
 omit
it,
if
you
include
it
the
payoffs
are
modified
below
and
you
insert
another
set
of
 payoffs,
where
the
pred
incurs
some
penalty
for
having
to
search
some
more.

 
 Let’s
write
out
a
game
ignoring
this
step.
In
the
spirit
of
the
baseball
example
we
 wrote
out
in
class,
consider
the
following
game.

In
this
version
of
the
game,
I
will
 assume
a
predator
always
attacks
from
one
side
(its
preferred
side).

 Look
 Right
 
 
 To
write
out
one
of
the
EFG
for
the
Fish
we
have
to
assume
a
given
population
state
 exists.
Assume
Right‐jawed
fish
are
prevalent
(at
high
frequency
in
the
population)
 and
thus,
this
will
determine
one
of
the
branches
of
the
nature
move
that
a
prey
 species
is
to
expect.
Given
that
right‐jawed
fish
are
prevalent,
then
prey
species
will
 be
more
likely
to
be
looking
over
their
left
shoulder
(remember
as
I
drew
it
in
 lecture
the
R‐jawed
fish
has
a
jaw
that
hooks
right).
Thus,
this
is
another
nature
 move.
The
EFG
has
two
nature
moves.
Now
consider
the
payoffs.
This
is
just
like
the
 baseball
game.
A
Right‐jawed
fish
will
always
get
highest
payoffs
from
attacking
the
 left
side,
and
the
Left‐jawed
fish
will
always
get
highest
payoffs
from
attacking
from
 the
right
(they
are
adapted
to
do
that).
However,
because
the
fish
are
expecting
 more
attacks
from
the
left
(e.g.,
right‐jawed
prevalent
in
the
population),
then
R‐ jawed
fish
will
have
lower
payoffs
than
L‐jawed
fish
from
their
chosen
sides
of
 attack.
The
same
holds
true
from
the
least
favored
sides
of
attack.
So
the
payoffs
are:
 R‐jaw
/
Prey
look
L
<
L‐jaw
Prey
look
L
<
R‐jaw
/
Prey
look
R
<
L‐jaw/prey
look
L,
 given
R‐jaw
is
prevalent
 or
1<2<3<4
 The
prey
payoffs
are
the
converse
of
this
(just
like
pitching
gets
converse
payoffs
to
 batting!):
 4>3>2>1.
 
 This
assumes
that
even
though
the
prey
look
in
the
right
direction,
when
they
 encounter
the
rare
predator
the
rare
predator
gets
an
advantage.
 
 The
final
part
of
the
puzzle
is
to
assume
the
opposite,
that
left‐jawed
is
prevalent,
 but
we
do
not
have
to
draw
this
out,
it
is
just
the
inequality
above
with
the
given
 states
reversed.
 
 There
are
indeed
other
ways
to
write
out
the
game,
as
long
as
your
logic
is
good.
For
 example,
you
might
assume
that
R‐jaw
fish
can
attack
from
either
the
R
or
the
L
side
 of
the
prey.
In
this
case
the
EFG
is
much
longer,
having
three
sets
of
branches.

This
 assumption
is
like
allow
all
baseball
players
to
switch
hit
though,
so
think
it
through.

 
 Now
this
is
not
required
for
the
homework,
but
just
for
full
comprehension.
 We
could
write
out
a
version
of
the
game
where
we
assume
a
probability
p
that
 predators
are
R‐jawed
and
1‐p
that
predators
are
left
jawed,
and
then
the
 probability
p
that
prey
look
over
their
left
(because
they
are
habituated
to
that
 frequency)
and
1‐p
that
they
look
over
to
the
right
(this
assumes
prey
look
in
direct
 proportion
to
population
attack
rates).
This
is
a
fully
specified
game.

 
 
 
 Supp
Prob
Bad
Guy
vs
Good
Guy.
 
 In
the
Bad
Guy
vs
Good
Guy,
I
assumed

agiven
set
of
pays
and
arrived
at

 Kidnap/pay/release
as
the
BI
solution.
 
 For
the
murderous
“bad”
guy,
I
will
allow
two
possibilities:
Let
us
assume
that
the
 Murderous
Guy
prefers
to
Kill
and
next
get
ransom
money,
but
he
prefers
ransom
 money
and
no
kill
over
any
other
option.
We
get
the
following
EFG:
 
 
 
 Which
reduces
to:
 
 which
reduces
to:
 
 which
reduces
to:
 
 
 
 However,
we
could
assume
that
Guy
really
prefers
murder
above
all
else:
 
 which
reduces
as
before
to
(notice
Vivica’s
payoffs
do
not
change):
 which
reduces
to:
 
 
 Now
in
the
final
step
the
nature
of
the
murderous
guy
changes
the
game:
 
 Recall
that
in
lecture
we
had
a
different
outcome
for
the
good
guy
who
chooses
to
 kidnap/she
paid/and
he
released.
 ...
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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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