FinalPracticeSolutions

FinalPracticeSolutions - Econ
166a
 
 7.7
 
...

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Unformatted text preview: Econ
166a
 
 7.7
 
 Fall
2009
 
 Practice
Final
Solutions

 Sinervo/Musacchio
 
 8.1
 
 
 9.10
 
 
 
 
 
 
 
 10.1
 
 
 11.5
 
 
 
 

 Question
6
 a)
A
security
strategy
for
each
player
is
to
play
B.
Playing
B
guarantees
a
player
a
payoff
of
at
least
2.
 
 b)
Best
response
analysis
shows:
 
 A
 B
 A
 4,4
 0,5
 B
 5,0
 2 ,2 
 For
players
it’s
always
better
to
play
B
than
to
play
A,
no
matter
how
the
other
player
plays.

Thus
 both
players
have
playing
B
as
a
dominant
strategy.
 
 c)
An
example
strategy
is
the
following:
 
 “Play
A
until
someone
is
seen
to
have
played
B.
After
that,
play
B
forever.”
 This
particular
example
is
called
a
“grim
trigger”
strategy.
The
important
thing
is
that
a
specification
 of
strategy
ought
to
say
how
a
player
plays
in
all
possible
contingencies
(information
sets)
.
The
 above
strategy
does
that
unambiguously.
 
 d)
The
highest
payoff
is
achieved
when
both
players
play
A
forever.
This
can
be
achieved
by
both
 players
playing
a
“grim
trigger”
strategy.
The
threat
of
retaliation
by
the
other
player
keeps
players
 from
playing
B,
provided
that
players
care
enough
about
future
losses
(in
other
words
are
sufficiently
 patient).
If
a
player
defects
(plays
B)
in
round
t,
then
he
gets
a
bonus
of
1
(compared
to
what
he
 would
have
gotten
had
he
played
A).
However,
in
all
subsequent
rounds
he
suffers
a
net
loss
of
‐2,
 since
his
grim
trigger
opponent
punishes
him
for
the
defection.
The
present
value
of
the
gain
from
 defection
is
less
than
the
loss
if
 1 ∗ δ t ≤ 2δ t +1 + 2δ t + 2 + .... 1 ≤ 2(δ1 + δ 2 + ...) δ 1≤ 2 1− δ 1 δ≥ 3 
 where
δ
is
the
discount
factor.
 
 e)
It
is
possible.
Consider
the
following
pair
of
strategies:
 Player
1:
Play
A
in
odd
rounds
and
B
in
even
as
long
as
opponent
plays
B
in
odd
and
A
in
even.
If
 € opponent
deviates
from
this,
play
B
forever
thereafter.
 
 Player
2:
Play
B
in
odd
rounds
and
A
in
even
as
long
as
opponent
plays
A
in
odd
and
B
in
even.
If
 opponent
deviates
from
this,
play
B
forever
thereafter.
 
 This
pair
of
strategies
gives
each
player
an
average
payoff
of
approximately
2.5
per
round.
If
they
 cheat
by
deviating,
they
get
punished
and
forced
to
accept
a
payoff
of
no
more
than
2
every
round
 thereafter
(since
their
opponent
keeps
playing
B
after
the
cheating
event).
As
long
as
the
discount
 factor
is
sufficiently
high,
any
one‐time
benefit
of
cheating
is
outweighed
by
the
downside
of
being
 punished
in
subsequent
rounds.
 
 This
is
an
example
of
the
folk
theorem
:
that
any
payoff
vector
of
a
repeated
game
that
both
 1. lies
in
the
convex
hull
of
the
payoff
space
of
the
1
shot
game
 2. 
gives
each
player
at
least
as
much
as
they
could
get
through
their
security
strategy
 can
be
achieved
in
a
Nash
equilibrium
by
constructing
threat
strategies. 16.2
 
 

 
 ...
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