Revenue-Optimal Deterministic Auctions for Multiple Buyers with Ordinal Preferences over Fixed-Price Items

In this article, we introduce a Bayesian revenue-maximizing mechanism design model where the items have fixed, exogenously given prices. Buyers are unit-demand and have an ordinal ranking over purchasing either one of these items at its given price or purchasing nothing. This model arises naturally from the assortment optimization problem, in that the single-buyer optimization problem over deterministic mechanisms reduces to deciding on an assortment of items to “show.” We study its multi-buyer generalization in the simplest setting of single-winner auctions or, more broadly, any service-constrained environment. Our main result is that if the buyer rankings are drawn independently from Markov chain choice models, then the optimal mechanism is computationally tractable, and structurally a virtual welfare maximizer. We also show that for ranking distributions not induced by Markov chains, the optimal mechanism may not be a virtual welfare maximizer. Finally, we apply our virtual valuation notion for Markov chains, in conjunction with existing prophet inequalities, to improve algorithmic guarantees for online assortment problems.