  We study a pricing setting where each customer is offered a contextualized
price based on customer and/or product features. Often only historical sales
data are available, so we observe whether a customer purchased a product at the
price prescribed rather than the customer's true valuation. Such observational
data are influenced by historical pricing policies, which introduce
difficulties in evaluating the effectiveness of future policies. The goal of
this paper is to formulate loss functions that can be used for evaluating
pricing policies directly from observational data, rather than going through an
intermediate demand estimation stage, which may suffer from bias. To achieve
this, we adapt ideas from machine learning with corrupted labels, where we
consider each observed purchase decision as a known probabilistic
transformation of the customer's valuation. From this transformation, we derive
a class of unbiased loss functions. Within this class, we identify minimum
variance estimators and estimators robust to poor demand estimation.
Furthermore, we show that for contextual pricing, estimators popular in the
off-policy evaluation literature fall within this class of loss functions. We
offer managerial insights into scenarios under which these estimators are
effective.
