  In this paper, 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 ranking 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.
