Many marketing applications, including credit card incentive programs, offer rewards to customers who exceed specific spending thresholds to encourage increased consumption. Quantifying the causal effect of these thresholds on customers is crucial for effective marketing strategy design. Although regression discontinuity design is a standard method for such causal inference tasks, its assumptions can be violated when customers, aware of the thresholds, strategically manipulate their spending to qualify for the rewards. To address this issue, we propose a novel framework for estimating the causal effect under threshold manipulation. The main idea is to model the observed spending distribution as a mixture of two distributions: one representing customers strategically affected by the threshold, and the other representing those unaffected. To fit the mixture model, we adopt a two-step Bayesian approach consisting of modeling non-bunching customers and fitting a mixture model to a sample around the threshold. We show posterior contraction of the resulting posterior distribution of the causal effect under large samples. Furthermore, we extend this framework to a hierarchical Bayesian setting to estimate heterogeneous causal effects across customer subgroups, allowing for stable inference even with small subgroup sample sizes. We demonstrate the effectiveness of our proposed methods through simulation studies and illustrate their practical implications using a real-world marketing dataset.

翻译：在包括信用卡激励计划在内的诸多营销应用中，为鼓励消费增长，常向超过特定消费阈值的客户提供奖励。量化这些阈值对客户的因果效应对于设计有效的营销策略至关重要。尽管断点回归设计是此类因果推断任务的标准方法，但当客户知晓阈值并策略性地操纵其消费以获取奖励时，该方法的假设可能被违反。为解决此问题，我们提出了一种在阈值操纵下估计因果效应的新框架。其核心思想是将观测到的消费分布建模为两种分布的混合：一种代表受阈值策略性影响的客户，另一种代表未受影响的客户。为拟合该混合模型，我们采用了一种两步贝叶斯方法，包括对非聚集客户进行建模以及对阈值附近样本拟合混合模型。我们证明了大样本下所得因果效应后验分布的后验收缩性。此外，我们将此框架扩展至分层贝叶斯设定，以估计跨客户子群的异质性因果效应，即使在子群样本量较小的情况下也能实现稳定的推断。我们通过模拟研究证明了所提方法的有效性，并利用真实营销数据集阐释了其实际意义。