Causal Inference under Threshold Manipulation: Bayesian Mixture Modeling and Heterogeneous Treatment Effects
This addresses a practical issue in marketing analytics for designing incentive programs, but it is incremental as it builds on existing causal inference methods.
The paper tackles the problem of estimating causal effects in marketing programs with spending thresholds, where customers may strategically manipulate their behavior, by proposing a Bayesian mixture modeling framework that accounts for threshold manipulation and heterogeneous treatment effects, demonstrating effectiveness through simulations and a real-world dataset.
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.