Post-Estimation Adjustments in Data-Driven Decision-Making with Applications in Pricing

Michael Albert, Max Biggs, Ningyuan Chen, Guan Wang

Keywords: predict-then-optimize, data-driven decision making, pricing

TL;DR: We propose a systematic post-estimation adjustment correcting PTO’s asymmetric bias, with a closed form under a simple curvature condition.

Abstract: The predict-then-optimize (PTO) framework is a standard approach in data-driven decision-making, where a decision-maker first estimates an unknown parameter from historical data and then uses this estimate to solve an optimization problem. While widely used for its simplicity and modularity, PTO can lead to suboptimal decisions because the estimation step does not account for the structure of the downstream optimization problem. We study a class of problems where the objective function, evaluated at the PTO decision, is asymmetric with respect to estimation errors. This asymmetry causes the expected outcome to be systematically degraded by noise in the parameter estimate, as the penalty for underestimation differs from that of overestimation. To address this, we develop a data-driven post-estimation adjustment that improves decision quality while preserving the practicality and modularity of PTO. We show that when the objective function satisfies a particular curvature condition, based on the ratio of its third and second derivatives, the adjustment simplifies to a closed-form expression. This condition holds for a broad range of pricing problems, including those with linear, log-linear, and power-law demand models. Under this condition, we establish theoretical guarantees that our adjustment uniformly and asymptotically outperforms standard PTO, and we precisely characterize the resulting improvement. Additionally, we extend our framework to multi-parameter optimization settings. Numerical pricing experiments demonstrate that our method consistently improves revenue, particularly in small-sample regimes where estimation uncertainty is most pronounced. This makes our approach especially well-suited for pricing new products or in settings with limited historical price variation.

Submission Number: 14