Pricing based on individual customer characteristics is widely used to maximize sellers' revenues. This work studies offline personalized pricing under endogeneity using an instrumental variable approach. Standard instrumental variable methods in causal inference/econometrics either focus on a discrete treatment space or require the exclusion restriction of instruments from having a direct effect on the outcome, which limits their applicability in personalized pricing. In this paper, we propose a new policy learning method for Personalized pRicing using Invalid iNsTrumental variables (PRINT) for continuous treatment that allow direct effects on the outcome. Specifically, relying on the structural models of revenue and price, we establish the identifiability condition of an optimal pricing strategy under endogeneity with the help of invalid instrumental variables. Based on this new identification, which leads to solving conditional moment restrictions with generalized residual functions, we construct an adversarial min-max estimator and learn an optimal pricing strategy. Furthermore, we establish an asymptotic regret bound to find an optimal pricing strategy. Finally, we demonstrate the effectiveness of the proposed method via extensive simulation studies as well as a real data application from an US online auto loan company.

翻译：基于个体客户特征定价广泛应用于最大化卖方收益。本文利用工具变量方法研究内生性下的离线个性化定价问题。因果推断/计量经济学中的标准工具变量方法要么聚焦于离散处理空间，要么要求工具变量对结果无直接效应的排他性约束，这限制了其在个性化定价中的适用性。本文提出一种新的策略学习方法——基于无效工具变量的个性化定价（PRINT），该方法允许连续处理变量且允许工具变量对结果产生直接影响。具体而言，基于收益和价格的结构模型，我们在无效工具变量的辅助下建立了内生性环境下最优定价策略的识别性条件。基于这一新识别结果（该识别可转化为具有广义残差函数的条件矩约束问题），我们构建了对抗性最小-最大估计量并学习最优定价策略。进一步地，我们建立了发现最优定价策略的渐近遗憾界。最后，通过大量仿真研究及来自美国某线上汽车贷款公司的真实数据应用，验证了所提方法的有效性。