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 private value for the item is drawn initially from a publicly known distribution $F$ and remains the same throughout. A key aspect of this game is that the seller might try to learn the buyer's private value to extract more revenue, while the buyer is motivated to hide it. We study the Perfect Bayesian Equilibria (PBE) in this setting with varying levels of commitment power to the seller. We find that the seller having the commitment power to not raise prices subsequent to a purchase significantly improves revenue in a PBE.