We study markets where a set of indivisible items is sold to bidders with unit-demand valuations, subject to a hard budget limit. Without financial constraints and pure quasilinear bidders, this assignment model allows for a simple ascending auction format that maximizes welfare and is incentive-compatible and core-stable. Introducing budget constraints, the ascending auction requires strong additional conditions on the unit-demand preferences to maintain its properties. We show that, without these conditions, we cannot hope for an incentive-compatible and core-stable mechanism. We design an iterative algorithm that depends solely on a trivially verifiable ex-post condition and demand queries, and with appropriate decisions made by an auctioneer, always yields a welfare-maximizing and core-stable outcome. If these conditions do not hold, we cannot hope for incentive-compatibility and computing welfare-maximizing assignments and core-stable prices is hard: Even in the presence of value queries, where bidders reveal their valuations and budgets truthfully, we prove that the problem becomes NP-complete for the assignment market model. The analysis complements complexity results for markets with more complex valuations and shows that even with simple unit-demand bidders the problem becomes intractable. This raises doubts on the efficiency of simple auction designs as they are used in high-stakes markets, where budget constraints typically play a role.
Bio: Martin Bichler is a full professor at the Department of Computer Science of the Technical University of Munich (TUM) and is affiliated with the TUM School of Management. Martin received his MSc degree from the Technical University of Vienna, and his Ph. D. as well as his Habilitation from the Vienna University of Economics and Business. He was a research fellow at UC Berkeley, and a research staff member at the IBM T. J. Watson Research Center, Yorktown Heights, New York. For the fall semester 2023, he joined the Simons Laufer Mathematical Sciences Institute in Berkeley as a research professor. His research primarily focuses on optimization and game theory with applications in market design.