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Anonymous Pricing in Large Markets

Yaonan Jin, Yingkai Li

Abstract

We study revenue maximization when a seller offers $k$ identical units to ex ante heterogeneous, unit-demand buyers. While anonymous pricing can be $Θ(\log k)$ worse than optimal in general multi-unit environments, we show that this pessimism disappears in large markets, where no single buyer accounts for a non-negligible share of optimal revenue. Under (quasi-)regularity, anonymous pricing achieves a $2+O(1/\sqrt{k})$ approximation to the optimal mechanism; the worst-case ratio is maximized at about $2.47$ when $k=1$ and converges to $2$ as $k$ grows. This indicates that the gains from third-degree price discrimination are mild in large markets.

Anonymous Pricing in Large Markets

Abstract

We study revenue maximization when a seller offers identical units to ex ante heterogeneous, unit-demand buyers. While anonymous pricing can be worse than optimal in general multi-unit environments, we show that this pessimism disappears in large markets, where no single buyer accounts for a non-negligible share of optimal revenue. Under (quasi-)regularity, anonymous pricing achieves a approximation to the optimal mechanism; the worst-case ratio is maximized at about when and converges to as grows. This indicates that the gains from third-degree price discrimination are mild in large markets.
Paper Structure (22 sections, 11 theorems, 50 equations, 1 figure)

This paper contains 22 sections, 11 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results and Techniques
  3. Related Work
  4. Model
  5. Mechanisms
  6. Revenue Curves
  7. Large Markets
  8. Approximation of Anonymous Pricing
  9. Reduction to Triangular Instances
  10. Order Statistics
  11. Lower bound on $D_j(p)$.
  12. Upper bound on $D_j(p)$.
  13. Asymptotic Upper Bounds
  14. Ex-Ante Relaxation.
  15. Asymptotic Upper Bounds.
  16. ...and 7 more sections

Key Result

Theorem 1

For quasi-regular distributions that satisfy the large market assumption, the worst-case approximation ratio of anonymous pricing to the optimal revenue is Moreover, the worst case gap is maximized at $2.47$ when $k=1$.

Figures (1)

  • Figure 1: Decomposition of a revenue curve into triangular ones.

Theorems & Definitions (24)

  • Example 1
  • Definition 1: Regularity
  • Definition 2: Quasi-regular Distributions
  • Definition 3: $\varepsilon$-large Markets
  • Theorem 1
  • Proposition 1: JJLZ22
  • Corollary 1
  • Corollary 2
  • Definition 4: Triangular Distributions
  • Lemma 1
  • ...and 14 more