On the complexity of sequential posted pricing

We study the well-known Sequential Posted Pricing scheme with one item, under the Bayesian setting that the value of each participating agent to the item is drawn from her own value distribution, which is known to the auctioneer as prior information. Each agent comes in to the auction market sequentially, and is offered a take-it-or-leave-it price. The goal of the auctioneer is to maximize her expected revenue. This family of mechanisms has been proved to perform well compared to optimal mechanism under the Bayesian framework in various settings [11], but nothing was previously known on the complexity of computing an optimal sequential posted pricing. In this paper, we show that finding an optimal sequential posted pricing is NP-complete even when the value distributions are of support size three. For the upper bound, we introduce polynomial-time algorithms when the distributions are of support size at most two, or their values are drawn from any identical distributions. As a by-product, we also show the same results hold for order-oblivious posted pricing scheme where after the auctioneer posts the prices, agents come into the auction in an adversarial order. We also study the constrained sequential posted pricing where the auction only runs for a fixed number of τ rounds, and give polynomial-time algorithms when the distributions are of support size at most two. Moreover, we extend our algorithm to cases when the values are decayed with time or the item has several copies. To the best of our knowledge, this is the first result that fully characterizes the computational complexity of sequential posted pricing family.