Perfect Bayesian Equilibria in Repeated Sales

A special case of Myerson’s classic result describes the revenue-optimal equilibrium when a seller offers a single item to a buyer. We study a repeated sales extension of this model: a seller offers to sell a single fresh copy of an item to the same buyer every day via a posted price. The buyer’s value for the item is unknown to the seller but is drawn initially from a publicly known distribution F and remains the same throughout.
We study this setting where the seller is unable to commit to future prices and find several surprises. First, if the horizon is fixed, previous work showed that an equilibrium exists, and all equilibria yield tiny or constant revenue. This is a far cry from the linearly growing benchmark of getting Myerson optimal revenue each day. Our first result shows that this is because the buyer strategies in these equilibria are necessarily unnatural. We restrict to a natural class of buyer strategies called threshold strategies, and show that threshold equilibria rarely exist. Second, if the seller can commit not to raise prices upon purchase, while still retaining the possibility of lowering prices in future, we show that threshold equilibria are guaranteed to exist for the power law family of distributions. As an example, if F is uniform in [0,1], the seller can extract revenue of order n√ in n rounds as opposed to the constant revenue obtainable when he is unable to make commitments. Finally, we consider the infinite horizon game with partial commitment, where both the seller and the buyer discount the future utility by a factor of 1−δ∈[0,1). When the value distribution is uniform in [0,1], there exists a threshold equilibrium with expected revenue at least 43+22√∼69% of the Myerson optimal revenue benchmark.