Approximation Schemes for Sequential Posted Pricing in Multi-unit Auctions

We design algorithms for computing approximately revenuemaximizing sequential posted-pricing mechanisms (SPM) in K-unit auctions, in a standard Bayesian model. A seller has K copies of an item to sell, and there are n buyers, each interested in only one copy, and has some value for the item. The seller posts a price for each buyer, using Bayesian information about buyers' valuations, who arrive in a sequence. An SPM specifies the ordering of buyers and the posted prices, and may be adaptive or non-adaptive in its behavior. The goal is to design SPM in polynomial time to maximize expected revenue. We compare against the expected revenue of optimal SPM, and provide a polynomial time approximation scheme (PTAS) for both nonadaptive and adaptive SPMs. This is achieved by two algorithms: an efficient algorithm that gives a (1 - 1/√2πK)-approximation (and hence a PTAS for sufficiently large K), and another that is a PTAS for constant K. The first algorithm yields a non-adaptive SPM that yields its approximation guarantees against an optimal adaptive SPM - this implies that the adaptivity gap in SPMs vanishes as K becomes larger.