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 natural 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. One key aspect of this game is revelation of the buyer’s type through his actions: while the seller might try to learn this value to extract more revenue, the buyer is motivated to hide it to induce lower prices. If the seller is able to commit to future prices, then it is known that the best he can do is extract the Myerson optimal revenue each day. In a more realistic scenario, the seller is unable to commit and must play a perfect Bayesian equilibrium. It is known that not committing to future prices does not help the seller. Thus extracting Myerson optimal revenue each day is a natural upper bound and revenue benchmark in a setting without commitment. We study this setting without commitment and find several surprises. First, if the horizon is fixed, previous work showed that an equilibrium always exists, and all equilibria yield a very low revenue, often times only a constant amount of revenue. This is unintuitive and a far cry from the linearly growing benchmark of obtaining 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, which we call threshold strategies, and show that pure strategy threshold equilibria rarely exist. This offers an explanation for the non-prevalence of bizarre outcomes predicted by previous results. Second, if the seller can commit not to raise prices upon purchase, while still retaining the possibility of lowering prices in future, we recover the natural threshold equilibria by showing that they exist for the power law family of distributions. As an example, if the distribution 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 any 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 4 3+2 √ 2 ∼ 69% of the Myerson optimal revenue benchmark. Under some mild assumptions, this equilibrium is also unique.