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Levin and Smith showed this by considering an auction model in which the number
of bidders participating was endogenously determined [7]. They modeled a pool of
potential bidders, and showed that, at equilibrium, each potential bidder has a fixed
probability of actually participating in (or entering) the auction. The number of bidders
participating in any auction was thus described by a binomial distribution. Bajari and
Hortacsu considered a similar model and compared their model to data collected from
eBay auctions selling collectable U.S. coins [1]. They note that in such online auctions,
the pool of potential bidders is extremely large. However, the fact that, in general, only
a small number of bids are observed, suggests that the probability that a potential bid-
ders participates in any individual auction is very low. Thus, they assume that, in such
cases, a Poisson distribution is an appropriate approximation for the binomial proposed
by Leven and Smith. In light of this work, we describe the number of bidders partici-
pating in any auction by a Poisson distribution and thus the probability that
n
bidders
participate is given by:
n
e
−
ν
n
!
P
(
n
)=
ν
(4)
Here the parameter
describes the mean of this distribution and thus represents the
expected or average number of participants in any individual auction. Given this dis-
tribution, we can extend the results described in the previous section and express the
probability of the auction closing at any bid level, in terms of the parameter
ν
,rather
than
n
. To do so, we simply sum the probability given in equation 1 multiplied by the
probability of that number of bidders actually occurring:
ν
∞
n
=0
P
(
n
)
P
n
(
l
i
)
P
ν
(
l
i
)=
(5)
Now substituting equations 1 and 4 into this expression and making use of the identity
∑
n
=0
ν
n
/
n
! =
e
ν
allows us to derive the result:
F
(
l
i
)]
e
v
[
F
(
l
i
+1
)
−
1]
⎧
⎨
⎩
−
e
v
[
F
(
l
i
)
−
1]
F
(
l
i
+1
)
−
F
(
l
i
)
[1
−
i
= 0
P
ν
(
l
i
)=
(6)
[1
−
F
(
l
i
)]
e
v
[
F
(
l
i
+1
)
−
1]
−
e
v
[
F
(
l
i
)
−
1]
F
(
l
i
+1
)
−
F
(
l
i
)
+
e
v
[
F
(
l
i
−
1
)
−
1]
−
e
v
[
F
(
l
i
)
−
1]
F
(
l
i
)
−
F
(
l
i
−
1
)
0
<
i
≤
m
Now finally, as before, we are able to perform a weighted sum over all of the discrete
bid levels to determine the expected revenue of the auctioneer given the uncertainty in
the number of bidders that are participating in any specific auction:
l
i
1
−
F
(
l
i
)
−
l
i
+1
1
−
F
(
l
i
+1
)
m
i
=0
e
ν[
F
(
l
i
+1
)
−
1]
−
e
ν[
F
(
l
i
)
−
1]
F
(
l
i
+1
)
−
F
(
l
i
)
E
ν
=
(7)
This is a key result. It expresses the expected revenue of the auction in terms of the
actual bid levels implemented, the bidders valuation distribution and,
, the mean num-
ber of bidders who participate in each auction. We use this result in the next section to
derive optimal bid levels in spite of the inherent uncertainty in the number of bidders
who will participate in any individual auction.
ν
