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In both cases, we use 10 bid levels (i.e.
m
= 10), as this makes clear the differences
between the two cases. Note that whilst changing the number of bid levels does affect
their value, it does not affect the general form of the distribution seen in the plot.
Now, Rothkopf and Harstad showed that in the case where there were two bidders,
the optimal discrete bid level distribution for the uniform distribution is a fixed bid
increment with evenly spaced bid levels. In addition, for an exponential distribution,
the optimal bid levels with two bidders is an increasing bid increment with bid levels
becoming more widely spaced as the auction progresses. Our results show that in the
general case, where there is uncertainty over the number of bidders that are partici-
pating, the distribution of the optimal discrete bid levels is complex. For the uniform
distribution there is a decreasing bid increment whereby the discrete bid levels become
closer together as the auction progresses. While, for the exponential distribution, the bid
increment initially decreases, reaches a minimum size and then subsequently increases.
We also see that as the number of bidders increases, the value of
l
0
increases.
Rothkopf and Harstad fixed the values of the first and last bid levels at the extremes
of the valuation distribution. However, we make no such restriction and thus the values
of
l
0
and
l
m
are optimised at the same time as the other bid levels. Now, since
l
0
is
equivalent to the reserve price of the auction (i.e. the item will not sell if there are no
bidders willing to pay at least
l
0
) the results indicate that, in contrast to the literature of
optimal auctions with continuous bid increments, the optimal reserve price of an auc-
tion with discrete bid levels is dependent on the mean number of bidders. In general,
we see that when the number of bid levels is large, or the mean number of bidders is
small, the value of
l
0
tends toward the continuous result (for the uniform distribution,
this is
x
∗
= max(
x
,
x
/
2), and for the exponential distribution it is
x
∗
= 1
/
[10]).
Intuitively we can understand these effects by the fact that given a fixed number of
bid levels, we should position them closer together in areas where they are most likely to
differentiate the bidders with the highest valuations. Thus, for the uniform distribution,
the bid levels become closer together nearer to the upper limit of the distribution. Whilst
in the exponential distribution, they become closer together where we expect to find the
bidder with the second highest valuation. This result suggests that it may be possible
to describe the optimal bid levels in terms of the distribution of the expected second
highest valuation. However, it has not proved possible to describe the revenue of the
discrete bid auction in these terms, so at the moment, this shortcut is not available to us.
α
5
Estimating Auction Parameters
In the previous sections, we showed that the optimal discrete bid levels, and hence the
revenue of the auctioneer, are dependent on the number of bidders that participate and
their valuation distribution. Now, when the values of the parameters that characterise
these distributions are not known, we must estimate their value through observations
of previous auctions. Since, in this paper we have derived an expression for the prob-
ability of the auction closing at any particular bid level (given these parameter values)
it is natural to use Bayesian inference to perform this task. That is, having observed an
auction closing at a certain bid level, we calculate our belief that a particular set of pa-
rameter values gave rise to this event. This method contrasts with statistical maximum
