Random Sampling Auctions for Limited Supply

Balcan et al. [3] show that the framework of the random sampling auction of Goldberg et al. [12] gives a generic reduction in the context of unlimited supply profit maximization from mechanism design (e.g., Goldberg et al. [12]) to algorithm design (e.g., Guruswami et al. [14]). One major open question left in [3] is to extend the results to limited supply settings. For the case that the seller has a large but limited supply of each commodity, we answer this question by giving a non-trivial adaptation of the random sampling auction framework to the limited supply setting, and prove bounds analogous to those of [3] for both the objectives of profit maximization and welfare maximization. These results generalize the prior work on random sampling limited supply auctions of Borgs et al. and Abrams [5, 1] which consider the special case where agents have linear valuations and budgets.