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Theorem 3.
Fo r t h e
j
th (for
j
=1
,...,m
) auction, the expected revenue (denoted
ER
j
)
)is:
s
=
f
n−j
+1
j
m
∀
j
=1
ER
j
=
E
(
V
)
−
E
(
c
|
)
−
E
(
π
wj
)
(11)
In the following section, we use the above equilibrium to show how the expected rev-
enue and the winner's expected profit vary from auction to auction.
4
Revenue and Winner's Profit
In Section 3, we determined equilibrium for the case where the distribution function for
the value (cost) was different for different objects. In this section, our objective is to
show that even if these distribution functions are identical across objects, the expected
revenue is not the same for all objects. We present an example auction scenario which
shows that in accordance with Ashenfelter's empirical result [1], the revenue for our
model can decline in later auctions.
It must be noted that our objective here is not to provide a comprehensive study of
how the expected revenue varies, but only to illustrate (with an example) that there exist
cases where the variation predicted by our theoretical analysis accords with the experi-
mental results of Ashenfelter [1].
Example auction scenario.
This example in intended to show that:
- the revenue declines in later auctions (this result corresponds with Ashenfelter's
empirical results [1]), and
- the winner's expected profit declines in later auctions.
In more detail, the setting for our analysis is as follows. The bidders' values are
identically distributed across objects and so are their costs. The common values of all
m
objects are drawn from a single distribution function. This function (say
Q
:
R
+
→
[0
,
1]) is used to draw the common value of the
j
th (
j
=1
,...,m
) object. Also, there
is a single distribution function (say
G
:
R
+
[0
,
1]) for the cost of the
j
th (
j
=
1
,...,m
) object. As before, each bidder receives a value signal (from
Q
) and a cost
signal (from
G
) for an auction just before that auction begins.
Since there is a single distribution function for all objects, we drop the subscripts
(for the order statistics) in Equations 9 and 11 for profit and revenue and rewrite them
as:
→
E
(
π
wj
)=
E
(
f
n−j
+1
)
E
(
s
n−j
+1
)+
α
j
+1
−
(12)
s
=
f
n−j
+1
)
ER
j
=
E
(
V
)
−
E
(
c
|
−
E
(
π
wj
)
(13)
We determine the expected revenues for the case where the values and costs are dis-
tributed normally. Recall that the normal distribution satisfies the log concavity assump-
tion mentioned in Section 2. Both value and cost signals are distributed normally. The
