Optimal design of English auctions with discrete bid levels

In this paper we consider a common form of the English auction that is widely used in online Internet auctions. This discrete bid auction requires that the bidders may only submit bids which meet some predetermined discrete bid levels and, thus, there exists a minimal increment with which a bidder may raise the current price. In contrast, the academic literature of optimal auction design deals almost solely with continuous bid auctions, and, as a result, there is little practical guidance as to how an auctioneer, who is seeking to maximise his revenue, should determine the number and value of these discrete bid levels. Consequently, in current online auctions, a fixed bid increment is commonly implemented, despite this having been shown to be optimal in only limited cases. Given this background, in this paper, our aim is to provide the optimal auction design for an English auction with discrete bid levels. To this end, we derive an expression that relates the expected revenue of the auction, to the actual discrete bid levels implemented, the number of bidders participating, and the distribution from which the bidders draw their private independent valuations. We use this expression to derive numerical and analytical solutions for the optimal bid levels in the general case. To compare these results with previous work, we apply these solutions to an example, where bidders' valuations are drawn from a uniform distribution. In this case, we prove that when there are more than two bidders, a decreasing bid increment is optimal and we show that the optimal reserve price of the auction increases as the number of bidders increases. Finally, we compare the properties of an auction in which optimal bid levels are used, to the standard auction approach which implements a fixed bid increment. In so doing, we show that the optimal bid levels result in improvements in the revenue, duration and allocative efficiency of the auction.