Repeated sales with multiple strategic buyers

Abstract

We study the repeated sale of non-durable goods to persistent, strategic buyers.1 Each buyer has a value for obtaining a good each day, which is drawn from a known prior at the start of the game and does not change. Every day, the seller posts a price, and buyers decide whether or not to buy a good at that day's price. Crucially, the seller is unable to commit to a pricing policy. To predict the outcome of repeated sales in the absence of commitment, we study Perfect Bayesian Equilibrium: both the buyers and the seller are best-responding, no matter the history of play, and beliefs on buyers' values are derived from Bayesian updating. This setting was first studied with a single buyer by (author?) [3]. They observed that there is an equilibrium with zero revenue for the seller. This extreme and counterintuitive equilibrium is driven by a self-fulfilling prophecy: The buyer never accepts any positive price out of fear that doing so will lead the seller to charge very high prices in the future; as a result, the seller infers that only a buyer with very high type would ever accept a positive price, so the seller would indeed charge very high prices in response. In revisiting this setting, (author?) [2] interpret this issue as an unpredictive pathology of the model. To begin our investigation, we offer a more nuanced view: There is a huge multiplicity of equilibria, but only the zero-revenue equilibrium survives several natural refinements that we propose, suggesting that this equilibrium is focal. This negative result is unsatisfying, since the low-revenue equilibrium is not predictive of real- world outcomes. Why don't we see this behavior in practice? One simplifying assumption in the model is the presence of only a single buyer. This allows the seller to exploit the buyer's revealed preference in a very targetted way. In contrast, if the seller continues to sell by posting a single price, but that price will be faced by multiple buyers, then the opportunity for price-discrimination is diminished. We ask: does the presence of multiple buyers change the equilibrium structure? We answer this question in the affermative. We consider a single seller posting a single price to sell one item to n buyers with values drawn independently from an identical prior. We construct an equilibrium which in which the seller attains non-Trivial revenue. In fact, we show that for a wide range of priors, the seller's revenue is a constant fraction of the optimal revenue even with commitment; i.e., of the Myerson revenue-optimal mechanism. Our equilibrium has two other notable properties. First, it survives the refinements that eliminated all but the zero-revenue equilibrium in the single-buyer case. Second, 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 [1] 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). Our analysis of the non-durable setting provides the further assurance, however, that the rate of learning outpaces the discount factor applied to future revenue. 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. These equilibria have a similar explore-exploit structure to the single-item, multi-buyer equilibrium, and survive natural refinements. The high revenue is driven by the ability of the seller to implement supply restriction to increase competition between buyers. 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.