Optimizing prices in descending clock auctions

A descending (multi-item) clock auction (DCA) is a mechanism for buying items from multiple potential sellers. In the DCA, bidder-specific prices are decremented over the course of the auction. In each round, each bidder might accept or decline his offer price. Accepting means the bidder is willing to sell at that price. Rejecting means the bidder will not sell at that price or a lower price. DCAs have been proposed as the method for procuring spectrum from existing holders in the FCC's imminent incentive auctions so spectrum can be repurposed to higher-value uses. However, the DCA design has lacked a way to determine the prices to offer the bidders in each round. This is a recognized, important, and timely problem. We present, to our knowledge, the first techniques for this. We develop a percentile-based approach which provides a means to naturally reduce the offer prices to the bidders through the bidding rounds. We also develop an optimization model for setting prices so as to minimize expected payment while stochastically satisfying the feasibility constraint. (The DCA has a final adjustment round that obtains feasibility after feasibility has been lost in the final round of the main DCA.) We prove attractive properties of this, such as symmetry and monotonicity. We develop computational methods for solving the model. (We also develop optimization models with recourse, but they are not computationally practical.) We present experiments both on the homogeneous items case and the case of FCC incentive auctions, where we use real interference constraint data to get a fully faithful model of feasibility. An unexpected paradox about DCAs is that sometimes when the number of rounds allowed increases, the final payment increases. We provide an explanation for this.