Auctions with Dynamic Populations: Efficiency and Revenue Maximization

We examine an environment where goods and privately informed buyers arrive stochastically to a market. A seller in this setting faces a sequential allocation problem with a changing population. We characterize the set of incentive compatible allocation rules and provide a generalized revenue equivalence result. In contrast to a static setting where incentive compatibility implies that higher-valued buyers have a greater likelihood of receiving an object, in this dynamic setting, incentive compatibility implies that higher-valued buyers have a greater likelihood of receiving an object sooner. We also characterize the set of efficient allocation rules and show that a dynamic Vickrey-Clarke-Groves mechanism is efficient and dominant strategy incentive compatible. We then derive an optimal direct mechanism. We show that the revenue-maximizing direct mechanism is a pivot mechanism with a reserve price. Finally, we consider sequential ascending auctions in this setting, both with and without a reserve price. We construct memoryless equilibrium bidding strategies in this indirect mechanism. Bidders reveal their private information in every period, yielding the same outcomes as the direct mechanisms. Thus, the sequential ascending auction is a natural institution for achieving either efficient or optimal outcomes. Interestingly, this is not the case for sequential second-price auctions, as the bids in a second-price auction do not reveal sufficient information to realize either the efficient or optimal allocation.