Winner Determination in Geometrical Combinatorial Auctions

Abstract

We consider auctions of items that can be arranged in rows. Examples of such a setting appear in allocating pieces of land for real estate development, or seats in a theater or stadium. The objective is, given bids on subsets of items, to find a subset of bids that maximizes auction revenue (often referred to as the winner determination problem). We describe a dynamic programming algorithm which, for a k-row problem with connected and gap-free bids, solves the winner determination problem in polynomial time. We study the complexity for bids in a grid, complementing known results in literature. Additionally, we study variants of the geometrical winner determination setting. We provide a NP-hardness proof for the 2-row setting with gap-free bids. Finally, we extend this dynamic programming algorithm to solve the case where bidders submit connected, but not necessarily gap-free bids in a 2-row and a 3-row problem.

Combinatorial auctions: theory, experiments, and practice

“…We use four different item/bidder structures, as shown in Figures 3.3a and 3.3b. The item structures are similar to settings in Kazumori (2010), Scheffel et al (2011), and Vangerven et al (2017b). Note that the structure corresponds to the geometrical setting discussed in Chapter 2, specifically to a 2-or 3-row problem.…”

“…In combinatorial auctions, bidders can place bids on combinations of items, called packages or bundles. Clearly, combinatorial auctions allow bidders to better express their preferences compared to the traditional auction formats, where bidders place bids on individual This chapter is based on Vangerven et al (2017b).…”