Repeated Sales with Multiple Strategic Buyers

Abstract

In a market with repeated sales of a single item to a single buyer, prior work has established the existence of a zero revenue perfect Bayesian equilibrium in the absence of a commitment device for the seller. This counter-intuitive outcome is the result of strategic purchasing decisions, where the buyer worries that the seller will update future prices in response to past purchasing behavior. We first show that in fact almost any revenue can be achieved in equilibrium, but the zero revenue equilibrium uniquely survives natural refinements. This establishes that single buyer markets without commitment are subject to market failure. However, our main result shows that this market failure depends crucially on the assumption of a single buyer. If there are multiple buyers, the seller can approximate the revenue that is possible with commitment. We construct an intuitive equilibrium for multiple buyers that survives our refinements, in which the seller learns from past purchasing behavior and obtains a constant factor of the per-round Myerson optimal revenue. The seller's pricing policy has a natural explore-exploit structure, where the seller starts with low prices that gradually ascend to learn buyers' values, and in later rounds exploits the surviving high-valued buyers. The result resembles an ascending-price auction, implemented over time. This relates to the intuition from the Coase conjecture in the durable goods literature [Coase 1972] which states that in the absence of commitment, one should expect the VCG outcome (which, for multiple buyers, yields non-trivial revenue for the seller). We further explore this relationship to the Coase conjecture by considering a setting with unlimited supply of goods each round. The Coasian intuition would suggest that the seller makes no revenue in this case, since the VCG outcome gives each item away for a trivial price. However, we show that this intuition does not hold for our setting with non-durable goods. As in the single-item setting, when the seller is constrained to posting a single, anonymous price to all buyers, there exist equilibria for which the seller's revenue is within a constant factor of the Myerson optimal revenue. Finally, we consider the importance of our restriction to anonymous prices. We show that if the seller is permitted to offer different prices to each agent then the Coasian intuition from the single-item setting binds once more: the seller is no longer able to extract nontrivial revenue from any equilibrium with sufficiently natural structure. In other words, the restriction of the seller to an anonymous price was crucial in deriving nontrivial revenue with unlimited supply. Intuitively, an anonymous price mitigates the ability of the seller to use the information an individual buyer leaks with each purchasing decision. Consequently, buyers are more willing to make nontrivial purchasing decisions, which in turn allows the seller to learn.