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for the auctioneer. To this end, we extended earlier work and derived an expression for
the revenue of the auction under uncertainty in the number of bidders who are participat-
ing in the auction. We used this result to numerically calculate optimal bid levels under
this uncertainty and showed that the value and distribution of these optimal bid levels
are highly dependent on both the mean number of bidders and the bidders' valuation
distribution. Finally, we considered the case in which these environmental parameters
are unknown to the auctioneer, and used Bayesian inference to estimate these param-
eters through observations of the closing price of previous auctions. We showed that
despite the stochastic nature of the auctions, the estimates generated by this algorithm
rapidly converged to the true values. In addition, we showed that by correctly estimating
the true values of these parameters, the auctioneer is able to bid levels that result in an
increase in auction revenue.
Our future work in this area consists of extending the auction model to incorporate
an explicit expression of the auctioneer's costs (rather than the explicit bound on the
maximum number of bid levels that we have presented here). In addition, we intend to
extend the inference method that we have presented here, and in particular, we would
like to use these techniques to perform model identification and selection. Thus, we
would infer the full parameters of several different valuation distributions (using vari-
ational methods to minimise the computational cost of this task) and then infer which
of these distributions best explains the closing prices that were observed (also consid-
ering the effect that an incorrect assumption will have). In so doing, we believe these
techniques will significantly contribute toward our goal of automating the mechanism
design of optimal discrete bid auctions.
Acknowledgments
This research was funded by the DIF-DTC project on Agent-Based Control and the
ARGUS II Defence and Aerospace Research Partnership. This is a collaborative project
involving BAE SYSTEMS, QinetiQ, Rolls-Royce, Oxford University and Southampton
University, and is funded by the industrial partners together with the EPSRC, Ministry
of Defence (MoD) and Department of Trade and Industry (DTI). Sarit Kraus is also
affiliated with the University of Maryland Institute for Advanced Computer Studies
(UMIACS) and this work was part supported by NSF Grant IIS-0208608.
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