Online Appendix to: Optimizing Prices in Descending Clock Auctions y

In each round of the descending clock auction, the auctioneer needs to offer each active bidder a price, i.e., to do Step 2.1. of Algorithm 1. Here we show a dynamic programming model that the optimal set of offer prices should solve. Let V (m,S,u, l) be the minimum expected payment that the auctioneer needs to pay to the bidders in a descending clock auction with m rounds, with a set of active bidders S, with upper bounds u and lower bounds l within which the bidders’ valuations lie. Let ξ be a realization of the bidders’ values. For any offer prices p in the first round, the state of the auction by the end of that first round will be as follows.