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Randomized Robust Price Optimization

Xinyi Guan, Velibor V. Mišić

TL;DR

This work introduces Randomized Robust Price Optimization (RRPO), a framework that selects a distribution over price vectors to maximize the worst-case revenue across an uncertainty set of demand models. It formalizes DRPO and RRPO, analyzes when randomization yields gains versus is unnecessary, and develops tractable methods for finite-price or finite-uncertainty settings using constraint-generation and two-layer column generation. For linear, semi-log, and log-log demand models, the paper provides exact mixed-integer exponential-cone formulations and demonstrates substantial worst-case revenue gains (up to 92% in real-data experiments) as well as competitive out-of-sample performance. The results show that randomization is often beneficial in realistic multi-product pricing problems, with practical algorithms and data-driven validation supporting the approach’s potential for risk-averse pricing under model misspecification. The work also discusses extensions to constrained price distributions and discretization-based approximations, highlighting avenues for integrating RRPO with operational pricing decisions in retail.

Abstract

The robust multi-product pricing problem is to determine the prices of a collection of products so as to maximize the worst-case revenue, where the worst case is taken over an uncertainty set of demand models that the firm expects could be realized in practice. A tacit assumption in this approach is that the pricing decision is a deterministic decision: the prices of the products are fixed and do not vary. In this paper, we consider a randomized approach to robust pricing, where a decision maker specifies a distribution over potential price vectors so as to maximize its worst-case revenue over an uncertainty set of demand models. We formally define this problem - the randomized robust price optimization problem - and analyze when a randomized price scheme performs as well as a deterministic scheme versus when it yields a benefit. We also propose solution methods for obtaining an optimal randomization scheme over a discrete set of candidate price vectors and show how these methods are applicable for common demand models, such as the linear, semi-log and log-log demand models. We numerically compare the randomized and deterministic approaches on a variety of synthetic and real problem instances; on instances derived from a real grocery retail scanner dataset, we show that the improvement in worst-case revenue can be as high as 92%. Using the same grocery retail scanner dataset, we also show that the randomized approach can produce price prescriptions that achieve higher out-of-sample revenue than the nominal and deterministic robust approaches.

Randomized Robust Price Optimization

TL;DR

This work introduces Randomized Robust Price Optimization (RRPO), a framework that selects a distribution over price vectors to maximize the worst-case revenue across an uncertainty set of demand models. It formalizes DRPO and RRPO, analyzes when randomization yields gains versus is unnecessary, and develops tractable methods for finite-price or finite-uncertainty settings using constraint-generation and two-layer column generation. For linear, semi-log, and log-log demand models, the paper provides exact mixed-integer exponential-cone formulations and demonstrates substantial worst-case revenue gains (up to 92% in real-data experiments) as well as competitive out-of-sample performance. The results show that randomization is often beneficial in realistic multi-product pricing problems, with practical algorithms and data-driven validation supporting the approach’s potential for risk-averse pricing under model misspecification. The work also discusses extensions to constrained price distributions and discretization-based approximations, highlighting avenues for integrating RRPO with operational pricing decisions in retail.

Abstract

The robust multi-product pricing problem is to determine the prices of a collection of products so as to maximize the worst-case revenue, where the worst case is taken over an uncertainty set of demand models that the firm expects could be realized in practice. A tacit assumption in this approach is that the pricing decision is a deterministic decision: the prices of the products are fixed and do not vary. In this paper, we consider a randomized approach to robust pricing, where a decision maker specifies a distribution over potential price vectors so as to maximize its worst-case revenue over an uncertainty set of demand models. We formally define this problem - the randomized robust price optimization problem - and analyze when a randomized price scheme performs as well as a deterministic scheme versus when it yields a benefit. We also propose solution methods for obtaining an optimal randomization scheme over a discrete set of candidate price vectors and show how these methods are applicable for common demand models, such as the linear, semi-log and log-log demand models. We numerically compare the randomized and deterministic approaches on a variety of synthetic and real problem instances; on instances derived from a real grocery retail scanner dataset, we show that the improvement in worst-case revenue can be as high as 92%. Using the same grocery retail scanner dataset, we also show that the randomized approach can produce price prescriptions that achieve higher out-of-sample revenue than the nominal and deterministic robust approaches.
Paper Structure (66 sections, 17 theorems, 153 equations, 4 figures, 18 tables, 5 algorithms)

This paper contains 66 sections, 17 theorems, 153 equations, 4 figures, 18 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem definition
  4. Nominal price optimization problem
  5. Deterministic robust price optimization problem
  6. Randomized robust price optimization problem
  7. Benefits of randomization
  8. Concave revenue function uncertainty sets
  9. Quasiconcavity in $\mathbf{p}$ and quasiconvexity in $\mathbf{u}$
  10. Finite price set $\mathcal{P}$
  11. Solution algorithm for finite price set $\mathcal{P}$, convex uncertainty set $\mathcal{U}$
  12. General solution approach
  13. Linear demand model
  14. Semi-log demand model
  15. Log-log demand model
  16. ...and 51 more sections

Key Result

Theorem 1

Suppose that $\mathcal{P}$ is a convex set and that $\mathcal{R}$ is such that every $R \in \mathcal{R}$ is a concave function of $\mathbf{p}$. Then the RPO problem is randomization-proof, that is, $Z^*_{\mathrm{RR}} = Z^*_{\mathrm{DR}}$.

Figures (4)

  • Figure EC.1: Plot of $\mathrm{RI}$ versus $\delta$ for semi-log demand models.
  • Figure EC.2: Plot of $\mathrm{RI}$ versus $\Gamma$ for semi-log demand models.
  • Figure EC.3: Plot of $\mathrm{RI}$ versus $\delta$ for log-log demand models.
  • Figure EC.4: Plot of $\mathrm{RI}$ versus $\Gamma$ for log-log demand models.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Proposition 1
  • Theorem 4
  • Proposition EC.1
  • Definition EC.1
  • Theorem EC.1: Theorem 2 of calafiore2005uncertain
  • ...and 21 more