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.

翻译：我们研究一种定价场景，其中每位顾客会基于客户和/或产品特征获得情境化定价。由于通常仅有历史销售数据可用，我们观测到的是顾客在指定价格下是否购买产品，而非其真实估值。此类观测数据受到历史定价策略的影响，为评估未来策略的有效性带来了困难。本文旨在构建可直接基于观测数据评估定价策略的损失函数，而非通过可能存在偏差的中间需求估计阶段。为实现这一目标，我们借鉴了带噪声标签的机器学习思想，将每个观测到的购买决策视为顾客真实估值经过已知概率变换的结果。基于该变换，我们推导出一类无偏损失函数，并在此类函数中识别出最小方差估计量及对需求估计误差具有鲁棒性的估计量。进一步地，我们证明在情境定价中，离线策略评估文献中常用的估计量属于此类损失函数。最后，我们为这些估计量有效发挥作用的场景提供了管理启示。