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l
i
−
1
l
i
l
i
+1
Case 1
Two or more bidders have valuations between [
l
i
,
l
i
+1
) and none have valuations
x
≥
l
i
+1
.
Case 2
One bidder has a valuation
x
≥
l
i
+1
, one or more bidders have valuations in the range
[
l
i
,
l
i
+1
) and the bidder with the highest valuation was the current highest bidder at
l
i
.
Case 3
One bidder has a valuation
x
≥
l
i
, one or more bidders have valuations in the range [
l
i
−
1
,
l
i
),
and the bidder with the highest valuation was not the current highest bidder at
l
i
−
1
.
Fig. 1.
Diagram showing the three cases whereby the auction closes at the bid level
l
i
. In each
case, the circles indicate a bidder's private valuation and the arrow indicates the bid level at which
that bidder was selected as the current highest bidder.
method is attractive, as rather than providing a single parameter estimate at each itera-
tion, it provides a full distribution that describes the auctioneer's belief over the entire
range of possible parameter values. Thus indicating the confidence that the auctioneer
should have in his current estimate [9]. In addition, Bayesian inference tends to be com-
putationally simpler than maximum likelihood methods, since it does not require us to
maximise a function over several dimensions [1].
3
Auction Model and Expected Auction Revenue
In this work we consider a common model of an English auction that was used by
Rothkopf and Harstad [13]. In this model,
n
risk neutral bidders are attempting to buy a
single item from a risk neutral auctioneer. Bidders have independent private valuations,
x
i
, drawn from a common continuous probability density function,
f
(
x
), within the
range [
x
,
x
], and with a cumulative distribution function,
F
(
x
), where with no loss of
generality,
F
(
x
)=0and
F
(
x
)=1. The bidders participate in an ascending price auction,
whereby the bids are restricted to discrete levels which are determined by the auctioneer.
We assume there are
m
+ 1 discrete bid levels, starting at
l
0
and ending at
l
m
(at this
point, we make no constraints on the actual number of these bid levels).
The auction starts with the auctioneer announcing the first discrete bid level (i.e. the
reserve price of the auction) and asks the bidders to indicate their willingness to pay this
amount. In traditional English auction houses, this indication is normally accomplished
by a nod to the auctioneer, while in current online auctions such as
www.onsale.com
it
requires a click of a mouse. If no bidders are willing to pay this amount within a prede-
termined and publically announced interval, the auction closes and the item remains un-
sold. However, if a bid is received, the auction proceeds and the auctioneer again requests
