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Motivated by the dynamic assortment offerings and item pricings occurring ine-commerce, we study a general problem of allocating finite inventories toheterogeneous customers arriving sequentially. We analyze this problem underthe framework of competitive analysis, where the sequence of customers isunknown and does not necessarily follow any pattern. Previous work in thisarea, studying online matching, advertising, and assortment problems, hasfocused on the case where each item can only be sold at a single price,resulting in algorithms which achieve the best-possible competitive ratio of1-1/e.  In this paper, we extend all of these results to allow for items havingmultiple feasible prices. Our algorithms achieve the best-possibleweight-dependent competitive ratios, which depend on the sets of feasibleprices given in advance. Our algorithms are also simple and intuitive; they arebased on constructing a class of universal ``value functions'' which integratethe selection of items and prices offered.  Finally, we test our algorithms on the publicly-available hotel data set ofBodea et al. (2009), where there are multiple items (hotel rooms) each withmultiple prices (fares at which the room could be sold). We find that applyingour algorithms, as a ``hybrid'' with algorithms which attempt to forecast andlearn the future transactions, results in the best performance.

Algorithms for Online Matching, Assortment, and Pricing with Tight Weight-dependent Competitive Ratios