  Motivated by the dynamic assortment offerings and item pricings occurring in
e-commerce, we study a general problem of allocating finite inventories to
heterogeneous customers arriving sequentially. We analyze this problem under
the framework of competitive analysis, where the sequence of customers is
unknown and does not necessarily follow any pattern. Previous work in this
area, studying online matching, advertising, and assortment problems, has
focused on the case where each item can only be sold at a single price,
resulting in algorithms which achieve the best-possible competitive ratio of
1-1/e.
  In this paper, we extend all of these results to allow for items having
multiple feasible prices. Our algorithms achieve the best-possible
weight-dependent competitive ratios, which depend on the sets of feasible
prices given in advance. Our algorithms are also simple and intuitive; they are
based on constructing a class of universal ``value functions'' which integrate
the selection of items and prices offered.
  Finally, we test our algorithms on the publicly-available hotel data set of
Bodea et al. (2009), where there are multiple items (hotel rooms) each with
multiple prices (fares at which the room could be sold). We find that applying
our algorithms, as a ``hybrid'' with algorithms which attempt to forecast and
learn the future transactions, results in the best performance.
