Interpretable Price Bounds Estimation with Shape Constraints in Price Optimization

TL;DR

This work tackles the challenge of producing interpretable price bounds for multi-product price optimization by introducing a two-phase framework: (1) estimate price-margin bounds from historical data using naive rules, data mining, or machine learning; (2) adjust the estimated bounds via a convex quadratic program that enforces shape constraints such as monotonicity, convexity for lower bounds, and concavity for upper bounds. The three estimation methods are NR-based (quantile-driven), DM-based (association-rule driven), and ML-based (one-class SVM). The adjustment step yields stable, operator-friendly bounds, mitigating overfitting while preserving interpretability. Numerical experiments on Recruit Co., Ltd. data demonstrate improved RMSE and bound stability, with the DM-based approach offering a favorable balance between accuracy and interpretability, and a deployment of the DM-based method in a real pricing system showing operational benefits.

Abstract

This study addresses the interpretable estimation of price bounds in the context of price optimization. In recent years, price-optimization methods have become indispensable for maximizing revenue and profits. However, effective application of these methods to real-world pricing operations remains a significant challenge. It is crucial for operators responsible for setting prices to utilize reasonable price bounds that are not only interpretable but also acceptable. Despite this necessity, most studies assume that price bounds are given constant values, and few have explored reasonable determinations of these bounds. Therefore, we propose a comprehensive framework for determining price bounds that includes both the estimation and adjustment of these bounds. Specifically, we first estimate price bounds using three distinct approaches based on historical pricing data. Then, we adjust the estimated price bounds by solving an optimization problem that incorporates shape constraints. This method allows the implementation of price optimization under practical and reasonable price bounds suitable for real-world applications. We report the effectiveness of our proposed method through numerical experiments using historical pricing data from actual services.

Figures (25)

• Figure 1: Product A
• Figure 2: Product B
• Figure 3: Product C
• Figure 5: Based on FCV (narrow)
• Figure 6: Based on FCV (wide)