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bidders willing to pay the next discrete bid level. If no bidders are willing to pay this new
price, the auction then closes and the item is sold to the current highest bidder.
Now, in order to determine the optimal bid levels that the auctioneer should an-
nounce, an expression for the expected revenue of the auction must be found. Rothkopf
and Harstad considered this problem and identified three mutually exclusive cases that
described the different ways in which the auction could close at any particular bid level
[13]. These cases are shown in figure 1. They then calculated the probability of each
case occurring in a number of limited cases. In our earlier work we have been able to
use the same descriptive cases, but derive a general result for each probability [4]. Thus
we are able to describe the probability of the auction closing at any particular bid level:
⎨
⎩
−
F
(
l
i
)]
F
(
l
i
+1
)
n
−
F
(
l
i
)
n
F
(
l
i
+1
)
−
F
(
l
i
)
[1
i
= 0
P
n
(
l
i
)=
(1)
[1
−
F
(
l
i
)]
F
(
l
i
+1
)
n
−
F
(
l
i
)
n
F
(
l
i
+1
)
−
F
(
l
i
)
+
F
(
l
i
−
1
)
n
−
F
(
l
i
)
n
F
(
l
i
)
−
F
(
l
i
−
1
)
0
<
i
≤
m
Note that the subscript in
P
n
indicates that the expression is in terms of the actual
number of bidders,
n
, who participate in the individual auction, and that we define
F
(
l
m
+1
)=1. Now, the expected revenue of the auctioneer is simply found by summing
over all possible bid levels and weighting each by the revenue that it generates:
m
i
=0
l
i
P
n
(
l
i
)
E
n
=
(2)
Thus, by substituting equation 1 into this expression and performing some simplifica-
tion, we get the result:
l
i
1
−
F
(
l
i
)
−
l
i
+1
1
−
F
(
l
i
+1
)
m
i
=0
F
(
l
i
+1
)
n
−
F
(
l
i
)
n
E
n
=
(3)
F
(
l
i
+1
)
−
F
(
l
i
)
In our previous work we used this result to generate optimal bid levels when the number
of bidders and the bidders valuation distribution are known.
4
Optimising over Uncertainty in the Number of Bidders
Now, we wish to deal with the more general case that the number of bidders participat-
ing in the auction is not known by the auctioneer. To do so, we have to carefully define
what we mean by participation. Thus, a bidder is said to be participating in (or has en-
tered) the auction, if they have generated a valuation for the item being sold, are present
and are prepared to bid. It is this number of bidders (plus their valuation distribution
and the discrete bid levels implemented) that determines the expected revenue of the
auction (as described in equation 3). However, in the English auction considered here,
not all of the bidders who are participating will necessarily submit bids to the auctioneer
(i.e. many will find that the other bidders have raised the price beyond their own private
valuation and thus they have no opportunity to bid). Thus, the auctioneer is not able to
determine the number of bidders who are participating by simply observing the bids.
In addition, in any specific setting, the number of bidders participating in an auction
is unlikely to be fixed but will most likely be described by a probability distribution.
