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Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices

Ruicheng Ao, Jiashuo Jiang, David Simchi-Levi

TL;DR

The paper tackles dynamic pricing with knapsack-like resource constraints by proposing three algorithms tailored to informational settings: (i) a Boundary Attracted Re-solve Method that achieves logarithmic regret under full information without the non-degeneracy assumption, (ii) an online-learning policy for zero prior information yielding optimal $O(\sqrt{T})$ regret, and (iii) an estimate-then-select re-solve approach that leverages machine-learned informed prices with a known error bound $\epsilon^0$ to bridge the gap between settings. The results show that the boundary-attraction mechanism and fluid-based re-solving can attain near-optimal performance, while online learning and data-informed decisions provide robust improvements when offline data are available or when demand parameters are unknown. Theoretical guarantees are complemented by numerical experiments demonstrating performance across degenerate and non-degenerate cases and illustrating a phase transition as $\epsilon^0$ varies. Collectively, the work advances online resource allocation and dynamic pricing by delivering practical, information-driven strategies with provable regret bounds.

Abstract

We study the dynamic pricing problem with knapsack, addressing the challenge of balancing exploration and exploitation under resource constraints. We introduce three algorithms tailored to different informational settings: a Boundary Attracted Re-solve Method for full information, an online learning algorithm for scenarios with no prior information, and an estimate-then-select re-solve algorithm that leverages machine-learned informed prices with known upper bound of estimation errors. The Boundary Attracted Re-solve Method achieves logarithmic regret without requiring the non-degeneracy condition, while the online learning algorithm attains an optimal $O(\sqrt{T})$ regret. Our estimate-then-select approach bridges the gap between these settings, providing improved regret bounds when reliable offline data is available. Numerical experiments validate the effectiveness and robustness of our algorithms across various scenarios. This work advances the understanding of online resource allocation and dynamic pricing, offering practical solutions adaptable to different informational structures.

Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices

TL;DR

The paper tackles dynamic pricing with knapsack-like resource constraints by proposing three algorithms tailored to informational settings: (i) a Boundary Attracted Re-solve Method that achieves logarithmic regret under full information without the non-degeneracy assumption, (ii) an online-learning policy for zero prior information yielding optimal regret, and (iii) an estimate-then-select re-solve approach that leverages machine-learned informed prices with a known error bound to bridge the gap between settings. The results show that the boundary-attraction mechanism and fluid-based re-solving can attain near-optimal performance, while online learning and data-informed decisions provide robust improvements when offline data are available or when demand parameters are unknown. Theoretical guarantees are complemented by numerical experiments demonstrating performance across degenerate and non-degenerate cases and illustrating a phase transition as varies. Collectively, the work advances online resource allocation and dynamic pricing by delivering practical, information-driven strategies with provable regret bounds.

Abstract

We study the dynamic pricing problem with knapsack, addressing the challenge of balancing exploration and exploitation under resource constraints. We introduce three algorithms tailored to different informational settings: a Boundary Attracted Re-solve Method for full information, an online learning algorithm for scenarios with no prior information, and an estimate-then-select re-solve algorithm that leverages machine-learned informed prices with known upper bound of estimation errors. The Boundary Attracted Re-solve Method achieves logarithmic regret without requiring the non-degeneracy condition, while the online learning algorithm attains an optimal regret. Our estimate-then-select approach bridges the gap between these settings, providing improved regret bounds when reliable offline data is available. Numerical experiments validate the effectiveness and robustness of our algorithms across various scenarios. This work advances the understanding of online resource allocation and dynamic pricing, offering practical solutions adaptable to different informational structures.
Paper Structure (16 sections, 15 theorems, 89 equations, 4 figures, 3 algorithms)

This paper contains 16 sections, 15 theorems, 89 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results and Contributions
  4. Other Related Literature
  5. Algorithm and Logarithmic Regret with Full Information
  6. Algorithm and Regret with No Information
  7. Improvement with Machine-learned Informed Price
  8. Numerical Validations
  9. Conclusions
  10. Proof of Theorem \ref{['thm:full_information']}
  11. Proof of Theorem \ref{['alg:resolve_learn']}
  12. Proof of Proposition \ref{['prop:impossible']}
  13. Proof of Theorem \ref{['thm:incumbent']}
  14. Proof of Proposition \ref{['prop:impossible_incumbent']}
  15. Proof of lemmas
  16. ...and 1 more sections

Key Result

Proposition 1.3

For any online policy $\pi$, we have $Tr^\star\ge \mathcal{R}^T(\pi).$

Figures (4)

  • Figure 1: Regret of our algorithms under different number of time horizon $T$ with full information.
  • Figure 2: Regret of our algorithms under different number of time horizon $T$ with no information ($\log T$-$\log\text{Regret}$ plot).
  • Figure 3: Regret of our algorithms under different number of time horizon $T$ with informed price.
  • Figure 4: Regret of our algorithms under different number of time horizon $T$ and misspecification error $\epsilon^0$.

Theorems & Definitions (16)

  • Proposition 1.3: gallego1994optimal
  • Theorem 2.1
  • Theorem 3.1
  • Lemma 3.2: Prop 4.32, bonnans2013perturbation
  • Lemma 3.3: keskin2014dynamic
  • Proposition 4.1
  • Theorem 4.3
  • Proposition 4.4
  • Definition A.1
  • Lemma A.2: wainwright2019high
  • ...and 6 more