Pricing Experimental Design: Causal Effect, Expected Revenue and Tail Risk

When launching a new product, historical sales data are often not available, leaving price as a crucial experimental instrument for sellers to gauge market response. When designing pricing experiments, there are three fundamental objectives: estimating the causal effect of price (i.e., price elasticity), maximizing the expected revenue through the experiment, and controlling the tail risk that, if not controlled, may lead to significant financial losses. In this paper, we reveal the relationship among such three objectives. Under a linear structural model, we investigate the trade-offs between causal effect estimation and expected revenue maximization, as well as between expected revenue maximization and tail risk control. Furthermore, we propose an optimal pricing experimental design, which can flexibly adapt to different desired levels of trade-offs. Through the optimal design, we also explore the relationship between causal effect estimation and tail risk control. Moreover, we establish an always-valid confidence sequence and a central limit theorem for the inference of the causal effect. Finally, we extend our results and the design to a misspecified setting, where the structural model is not necessarily linear but the seller still runs our design for linear structural models. The results demonstrate the robustness of our design and the wide existence of the relationships among the three objectives. This paper was accepted by George Shanthikumar, data science. Funding: The authors thank the Massachusetts Institute of Technology (MIT)-IBM partnership in Artificial Intelligence and the MIT Data Science Laboratory for support. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.03209 .