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. Moreover, we describe a simple and computationally tractable pricing algorithm for the seller that achieves this approximation when buyers best-respond.