  We study the following fundamental data-driven pricing problem. How
can/should a decision-maker price its product based on observations at a single
historical price? The decision-maker optimizes over (potentially randomized)
pricing policies to maximize the worst-case ratio of the revenue it can garner
compared to an oracle with full knowledge of the distribution of values when
the latter is only assumed to belong to a broad non-parametric set. In
particular, our framework applies to the widely used regular and monotone
non-decreasing hazard rate (mhr) classes of distributions. For settings where
the seller knows the exact probability of sale associated with one historical
price or only a confidence interval for it, we fully characterize optimal
performance and near-optimal pricing algorithms that adjust to the information
at hand. As examples, against mhr distributions, we show that it is possible to
guarantee $85\%$ of oracle performance if one knows that half of the customers
have bought at the historical price, and if only $1\%$ of the customers bought,
it still possible to guarantee $51\%$ of oracle performance. The framework we
develop leads to new insights on the value of information for pricing, as well
as the value of randomization. In addition, it is general and allows to
characterize optimal deterministic mechanisms and incorporate uncertainty in
the probability of sale.
