Problemset3
Due Tuesday November 8th before 5pm
1. Consider the infinitelyrepeated pricecompetition game where two firms
make simultaneous choices of their monthly prices. Here the stagegame
is the standard Bertrandmodel of price competition. Each of two firms
chooses a price for the month of
p
t
i
≥
0, and the profits are given by:
π
i
(
p
t
1
, p
2
) = (
p
t
i

c
)
D
i
(
p
t
1
, p
t
2
)
where the demand for firms
i
’s products in a particular month is given as
a function of their own price
p
i
and the competitor’s price
p
j
is:
D
i
(
p
i
, p
j
) =
a

p
i
if
p
i
< p
j
and
p
i
< a
,
1
2
(
a

p
i
)
if
p
i
=
p
j
< a
,
0
if
p
i
> p
j
or min
{
p
i
, p
j
}
> a
.
Both firms care about the streams of revenue with discount factor
1
1+
r
,
where
r >
0 is the interest rate. So given an infinite price history for both
firms
{
(
p
t
1
, p
t
2
)
}
∞
t
=1
, each firm ranks the history via:
Π
i
(
braceleftbig
(
p
t
1
, p
t
2
)
bracerightbig
∞
t
=1
) =
∞
summationdisplay
t
=1
parenleftbigg
1
1 +
r
parenrightbigg
t

1
π
i
(
p
t
1
, p
t
2
)
(a) Provide an expression for the averagediscounted payoff.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Game Theory, infinitelyrepeated pricecompetition game, feasible Nash Equilibria.

Click to edit the document details