Sequential Item Pricing for Unlimited Supply

We investigate the extent to which price updates can increase the revenue of a seller with little prior information on demand. We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k< logn days. For the static (k = 1) case, Balcan et al. [2] show that one random item price (the same on each item) yields revenue within a Θ(logm+logn) factor of optimum and this factor is tight. We introduce hereditary maximizers, a novel property regarding buyer valuations that is sufficient for a significant improvement of the approximation factor in the dynamic (k > 1) setting. The hereditary maximizers property limits complementarities among items and is satisfied by any multi-unit or gross substitutes valuation. Our main result is a non-increasing, randomized, schedule of k item prices, the same on each item, with expected revenue within a O( logm+logn k ) factor of optimum for private valuations with hereditary maximizers. This factor is tight (modulo a constant factor): we show that any pricing scheme over k days has a revenue approximation factor of at least logm+logn 3k . We obtain analogous matching lower and upper bounds of Θ( logn k ) if all valuations have the same maximum. We expect the technique we develop to prove our upper bound to be of broader interest; for example, it can be used to significantly improve over the result of [1]. We also initiate the study of revenue maximization given allocative externalities (i.e. influences) between buyers with combinatorial valuations. We provide a rather general model of positive influence of others’ ownership of items on a buyer’s valuation. For affine, submodular externalities and valuations with hereditary maximizers we present an influence-and-exploit [13] marketing strategy based on our algorithm for private valuations. This strategy preserves our approximation factor, despite an affine increase (due to externalities) in the optimum revenue.