Solving the Winner Determination Problem in a Divisible-Object Auction

In an auction of a divisible object, bidders' demand functions are often assumed to be nonincreasing, meaning that bidders are willing to pay less or the same price for every additional unit. Under this assumption, the optimal allocation that maximizes the auctioneer's revenue can be found using a greedy-based procedure. This article argues that situations may arise where a bidder may need to express her preferences through a nondecreasing demand function; when such a bidder is present in the auction, the greedy-based procedure does not guarantee the optimal allocation. Thus, this article proposes a mixed integer program that finds the optimal allocation in a divisible-object auction at which bidders submit their bids as arbitrary stepwise demand functions. The practical aspect of the mathematical program is presented by means of a simple yet illustrative example in a treasury bond auction setting. The results of the auctioneer's revenue are reported as a function of the number of bidders with nonincreasing and nondecreasing demand functions.