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In-Depth Information
The remainder of the paper is organised as follows: in section 2 we present related
work and in section 3 we describe our auction model and present the previously de-
rived results for the expected revenue of this auction (in order to make this paper self-
contained). In section 4 we extend this result to the case that the number of bidders
participating in the auction is described by a distribution and we use this new result
to derive optimal discrete bid levels in this case. In section 5 we present our Bayesian
inference algorithm and finally we conclude and discuss future work in section 6.
2
Related Work
The problem of optimal auction design has been studied extensively for the case of
auctions with continuous bid increments [12,10]. In contrast, auctions with discrete bid
levels have received much less attention, and much of the work that does exist is based
on the assumption that there is a fixed bid increment and thus the price of the auction
ascends in fixed size steps [15,3,16]. In contrast, Rothkopf and Harstad considered the
more general question of determining the optimal number and distribution of these bid
levels [13]. They provided a full discussion of how discrete bid levels affect the expected
revenue of the auction and they considered two different distributions for the bidders'
private valuations (uniform and exponential). In the case of the uniform distribution,
they considered two specific instances: (i) two bidders with any number of allowable bid
levels, and (ii) two allowable bid levels with any number of bidders. In the first instance,
evenly spaced bid levels (i.e. a fixed bid increment) was found to be the optimal. In the
second instance, the bid increment was shown to decrease as the auction progressed.
Conversely, for the exponential distribution (again with just two bidders), the optimal
bid increment was shown to increase as the auction progressed.
In previous work, we extended the analysis of Rothkopf and Harstad [13], and, rather
than analyse the ascending price English auction in limited cases, we presented a gen-
eral expression that relates the revenue to the actual bid levels implemented. For a uni-
form valuation distribution we were able to derive analytical results for the optimal bid
levels, and in general, we were able to numerically determine the optimal bid levels
for any bidders' valuation distribution, any number of bid levels and any number of
bidders. In addition, we showed that in general, increasing the number of discrete bid
levels, causes the revenue to approach that of a continuous bid auction.
In this paper, we extend this previous work and address the problem of estimat-
ing the number and valuation distribution of the bidders through observing the closing
price of previous auctions. This problem is similar to that studied in the econometrics
literature, where it has been used to identify the behaviour of bidders in real world auc-
tions [6]. More recently, it has received attention within electronic commerce, with the
goal of determining the reserve price in a repeated procurement auction [2]. Typically,
this work uses statistical maximum likelihood estimators to determine the parameters
that describe the bidders' valuation distribution through observations of their bidding
behaviour. In our case, this task is somewhat different as much of this information is
lost in the discretisation of the bids. Thus, we use the expression that we have already
derived for the revenue of the discrete bid auction, and use Bayesian inference to infer
parameter values through observations of the closing price of previous auctions. This
