Optimal design of english auctions with discrete bid levels

This article considers a canonical auction protocol that forms the basis of nearly all current online auctions. Such discrete bid auctions require that the bidders submit bids at predetermined discrete bid levels, and thus, there exists a minimal increment by which the bid price may be raised. In contrast, the academic literature of optimal auction design deals almost solely with continuous bid auctions. As a result, there is little practical guidance as to how an auctioneer, seeking to maximize its revenue, should determine the number and value of these discrete bid levels, and it is this omission that is addressed here. To this end, a model of an ascending price English auction with discrete bid levels is considered. An expression for the expected revenue of this auction is derived and used to determine numerical and analytical solutions for the optimal bid levels in the case of uniform and exponential bidder's valuation distributions. Finally, in order to develop an intuitive understanding of how these optimal bid levels are distributed, the limiting case where the number of discrete bid levels is large is considered, and an analytical expression for their distribution is derived.