Fast Revenue Maximization

We study a data-driven problem pricing problem in which a seller offers a price for a single item based on demand observed at a small finite number of historical prices. Our goal is to derive precise evaluation procedures of the value of the historical information gathered by the seller, along with prescriptions for more efficient price experimentation. Our main methodological result is an exact characterization of the maximin ratio (defined as the worst-case revenue garnered by a seller who only relies on past data divided by the optimal revenue achievable with full knowledge of the distribution of values). This result allows to measure the value of any historical data consisting of prices and corresponding conversion rates. We leverage this central reduction to provide new insights about price experimentation. Motivated by practical constraints that impede the seller from changing prices abruptly, we first illustrate our framework by evaluating the value of local information and show that the mere sign of the gradient may sometimes provide significant information to the seller. We then showcase how our framework can be used to run efficient price experiments. On the one hand, we develop a method to select the next price experiment that the seller should use to maximize the information obtained. On the other hand, we demonstrate that our result allows to considerably reduce the price experimentation needed to reach preset revenue guarantees through dynamic pricing algorithms.