Optimal Dynamic Auctions

A monopolist seller has multiple units of an indivisible good to sell over a discrete,finite time horizon. Potential buyers with unit demand arrive and depart over thisperiod. Each buyer privately knows her arrival time, her value for a unit and the timeby which she must make a purchase. We study the revenue maximizing Bayes-Nashincentive compatible allocation rule. The allocation rule can be characterized as anindex rule: each buyer can be assigned an index, and the allocation rule allots the goodto a buyer if her index exceeds some threshold. We show that ‘simple’ index policiesare optimal under an increasing hazard rate condition on the distribution of valuations,and, roughly speaking, that less patient buyers have stochastically higher valuationsthan more patient buyers. By simple, we mean that the index of a buyer depends onlyon his own valuation and the distribution of values of buyers with the same arrival dateand patience. The optimal allocation rule can therefore be described recursively, andsolved for by backward induction. When the environment violates these conditions,the optimal allocation rule cannot be calculated recursively, and the index of a buyermay depend on allocation decisions made in the past.