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Robust Price Discrimination

Itai Arieli, Yakov Babichenko, Omer Madmon, Moshe Tennenholtz

TL;DR

This paper analyzes robust price discrimination when the seller's outside option $s$ is unknown to the market designer. It introduces BBM segmentations, where the designer samples a guessed seller value $s_D$ and then uses the optimal segmentation as if $s=s_D$, effectively reducing the problem to a tractable one-dimensional decision that can be analyzed via a zero-sum game. The main result shows the worst-case regret is bounded by $\frac{U^*(0)}{e}$, and this bound is tight in the binary buyer-type case; the authors also extend the framework to restricted seller knowledge and generalized regret, along with experimental evidence that the robust segmentation performs well under realistic distributions. The work connects robust information design with Bayesian persuasion and provides practical guidance for platforms facing seller-value uncertainty, offering both theoretical guarantees and empirical validation. Overall, the paper makes a meaningful advance in designing buyer-surplus-maximizing, information-design policies under seller uncertainty, with implications for online marketplaces and market-shock environments.

Abstract

We consider a model of third-degree price discrimination where the seller's product valuation is unknown to the market designer, who aims to maximize buyer surplus by revealing buyer valuation information. Our main result shows that the regret is bounded by a $\frac{1}{e}$-fraction of the optimal buyer surplus when the seller has zero valuation for the product. This bound is attained by randomly drawing a seller valuation and applying the segmentation of Bergemann et al. (2015) with respect to the drawn valuation. We show that this bound is tight in the case of binary buyer valuation.

Robust Price Discrimination

TL;DR

This paper analyzes robust price discrimination when the seller's outside option is unknown to the market designer. It introduces BBM segmentations, where the designer samples a guessed seller value and then uses the optimal segmentation as if , effectively reducing the problem to a tractable one-dimensional decision that can be analyzed via a zero-sum game. The main result shows the worst-case regret is bounded by , and this bound is tight in the binary buyer-type case; the authors also extend the framework to restricted seller knowledge and generalized regret, along with experimental evidence that the robust segmentation performs well under realistic distributions. The work connects robust information design with Bayesian persuasion and provides practical guidance for platforms facing seller-value uncertainty, offering both theoretical guarantees and empirical validation. Overall, the paper makes a meaningful advance in designing buyer-surplus-maximizing, information-design policies under seller uncertainty, with implications for online marketplaces and market-shock environments.

Abstract

We consider a model of third-degree price discrimination where the seller's product valuation is unknown to the market designer, who aims to maximize buyer surplus by revealing buyer valuation information. Our main result shows that the regret is bounded by a -fraction of the optimal buyer surplus when the seller has zero valuation for the product. This bound is attained by randomly drawing a seller valuation and applying the segmentation of Bergemann et al. (2015) with respect to the drawn valuation. We show that this bound is tight in the case of binary buyer valuation.
Paper Structure (23 sections, 11 theorems, 59 equations, 6 figures)

This paper contains 23 sections, 11 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Techniques
  3. Paper structure
  4. Related Work
  5. Third-degree price discrimination
  6. Robust Bayesian persuasion
  7. Preliminaries: Known Value of the Seller
  8. Robust Price Discrimination
  9. Tightness in the Case of Binary Buyer Type
  10. Improved Regret Beyond the Case of Binary Buyer Type
  11. Experimental Results
  12. Extensions
  13. The Case of Restricted Seller Valuation
  14. The Case of Generalized Regret
  15. Discussion
  16. ...and 8 more sections

Key Result

Lemma 1

For every prior buyer distribution $\mu$ the optimal buyer surplus function $U^*(\cdot)$ is weakly decreasing, absolutely continuous, nonnegative, and differentiable up to a finite number of points. Moreover, there exists $s^* \ge 0$ such that $U^*(\mathop{\mathrm{s}}\limits)$ is constant over $[0,s

Figures (6)

  • Figure 1: The optimal buyer surplus function in the case of binary buyer type.
  • Figure 2: The indirect utility of the market designer $u_{\beta}(p)$ and its concavification, for a given distribution $F_\beta$ over $\mathop{\mathrm{s}}\limits$.
  • Figure 3: The indirect utility of the market designer and its concavification, corresponding to Example \ref{['example2']}.
  • Figure 4: Expected optimal and robust surplus as a function of a shared seller and buyer distribution parameter.
  • Figure 5: Expected difference between optimal and robust surplus, and its theoretical upper bound, as a function of a shared seller and buyer distribution parameter.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Lemma 1
  • Definition 1
  • Remark 1
  • Theorem 1
  • Lemma 2
  • proof : Proof of Theorem \ref{['main_res_theorem']}.
  • Lemma 3
  • Theorem 2
  • proof : Proof of Theorem \ref{['main_res_bintype_theorem']}.
  • Example 1
  • ...and 17 more