Pricing with Samples

In the present paper, we study a fundamental data-driven pricing problem: how should a decision maker (optimally) price based on a finite and limited number of samples from the customers’ value distribution. The decision maker’s objective is to select a general pricing policy with maximum worst-case ratio of revenue compared with an oracle with knowledge of the value distribution, when the latter is only known to belong to some general nonparametric class. We study achievable performance for two central classes: regular and monotone hazard rate (mhr) distributions. We develop a novel unified general approach to quantify the performance of mechanisms. The approach allows us to characterize optimal performance for the fundamental case of a single sample through lower and upper bounds on the maximin ratio, with corresponding near-optimal mechanisms and near-worst-case distributions. Furthermore, by extending this class of mechanisms to the cases in which more samples are available, we leverage our general approach to analyze a novel family of policies leading to new results on achievable performance as the number of samples increases. At a higher level, this work also uncovers insights on the value of samples for pricing purposes. For example, against mhr distributions, a single sample guarantees 64% of the performance an oracle with full knowledge of the distribution would achieve, two samples suffice to ensure 71%, and 10 samples guarantee 80% of such performance.