We study the problem of a seller dynamically pricing d distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution. The goods are not in limited supply, but can only be produced at a limited rate and are costly to produce. The seller observes only the bundle of goods purchased at each day, but nothing else about the buyer's valuation function. Our main result is a dynamic pricing algorithm for optimizing welfare (including the seller's cost of production) that runs in time and a number of rounds that are polynomial in d and the approximation parameter. We are able to do this despite the fact that (i) the price-response function is not continuous, and even its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not observable to the seller. We derive this result as an application of a general technique for optimizing welfare over divisible goods, which is of independent interest. When buyers have strongly concave, Hölder continuous valuation functions over d divisible goods, we give a general polynomial time dynamic pricing technique. We are able to apply this technique to the setting of unit-demand buyers despite the fact that in that setting the goods are not divisible, and the natural fractional relaxation of a unit-demand valuation is not strongly concave. To apply our general technique, we introduce a novel price randomization procedure that has the effect of implicitly inducing buyers to "regularize" their valuations with a strongly concave function. Finally, we also extend our results to a limited-supply setting in which the supply of each good cannot be replenished.
Bibliographical noteFunding Information:
This work was partially supported by NSF Grant No. CNS-1253345, a Sloan Foundation Fellowship, and a DARPA grant. Authors’ addresses: A. Roth, University of Pennsylvania, 3401 Walnut Street, Philadelphia, PA, 19104, USA; email: email@example.com; A. Slivkins, Microsoft Research, 641 6th Ave, New York, NY 10011; email: firstname.lastname@example.org; J. Ullman, Northeastern University, 805 Columbus Ave, Boston, MA 02118, USA; email: email@example.com; Z. S. Wu, University of Minnesota, 200 Union St SE, Minneapolis, MN 55455; email: firstname.lastname@example.org. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from email@example.com. © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. 2167-8375/2020/04-ART6 $15.00 https://doi.org/10.1145/3381527
- Multidimensional dynamic pricing
- convex optimization
- limited supply
- online learning
- revealed preferences