Sequential Posted-Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously (i.e., everywhere-supported, continuous, symmetric, and normalized (conditional) distributions that satisfy regularity, the MHR condition, and affiliation). In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that a simple generalization of these mechanisms achieves a better revenue performance; namely, if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Ω(1/max{1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d = 0) to arbitrary dependence (d = n − 1).