Post-estimation Adjustments in Data-driven Decision-making with Applications in Pricing

TL;DR

This paper introduces a post-estimation adjustment method for the predict-then-optimize framework that improves decision quality in pricing problems by accounting for asymmetries in estimation errors, especially in small-sample scenarios.

Contribution

The authors develop a closed-form post-estimation adjustment for PTO that enhances decision outcomes under certain curvature conditions, extending to multi-parameter and biased estimators.

Findings

The adjustment improves revenue in pricing models with linear, log-linear, and power-law demand.

The method outperforms standard PTO, especially with limited data.

Theoretical guarantees show asymptotic and uniform improvements.

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 and settings with biased estimators. Numerical 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.