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Sample Complexity of Posted Pricing for a Single Item

Billy Jin, Thomas Kesselheim, Will Ma, Sahil Singla

TL;DR

This paper investigates how many samples are needed from buyers' value distributions to find near-optimal posted prices, considering both independent and correlated buyer distributions, and welfare versus revenue maximization.

Abstract

Selling a single item to $n$ self-interested buyers is a fundamental problem in economics, where the two objectives typically considered are welfare maximization and revenue maximization. Since the optimal mechanisms are often impractical and do not work for sequential buyers, posted pricing mechanisms, where fixed prices are set for the item for different buyers, have emerged as a practical and effective alternative. This paper investigates how many samples are needed from buyers' value distributions to find near-optimal posted prices, considering both independent and correlated buyer distributions, and welfare versus revenue maximization. We obtain matching upper and lower bounds (up to logarithmic factors) on the sample complexity for all these settings.

Sample Complexity of Posted Pricing for a Single Item

TL;DR

This paper investigates how many samples are needed from buyers' value distributions to find near-optimal posted prices, considering both independent and correlated buyer distributions, and welfare versus revenue maximization.

Abstract

Selling a single item to self-interested buyers is a fundamental problem in economics, where the two objectives typically considered are welfare maximization and revenue maximization. Since the optimal mechanisms are often impractical and do not work for sequential buyers, posted pricing mechanisms, where fixed prices are set for the item for different buyers, have emerged as a practical and effective alternative. This paper investigates how many samples are needed from buyers' value distributions to find near-optimal posted prices, considering both independent and correlated buyer distributions, and welfare versus revenue maximization. We obtain matching upper and lower bounds (up to logarithmic factors) on the sample complexity for all these settings.
Paper Structure (16 sections, 15 theorems, 41 equations)

This paper contains 16 sections, 15 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Our Contributions to Posted Pricing for Product Distributions
  4. Our Contributions to Posted Pricing for Correlated Distributions
  5. Further Related Work
  6. Product Distributions
  7. Positive Result for Welfare: Proof of \ref{['thm:prodPos']}
  8. Negative Result for Revenue: Proof of \ref{['thm:prodNeg']}
  9. Optimal policy
  10. Bounding a policy's objective by its number of mistakes
  11. Computing the Hellinger distance
  12. Completing the proof of \ref{['thm:prodNeg']}
  13. Correlated Distributions
  14. Positive Result for Welfare and Revenue: Proof of \ref{['thm:corrPos']}
  15. Negative Result for Welfare and Revenue: Proof of \ref{['thm:corrNeg']}
  16. ...and 1 more sections

Key Result

Theorem 1

For product distributions, the sample complexity of welfare maximization is $O(1/\varepsilon^2 \cdot \log^2(1/\delta))$.

Theorems & Definitions (23)

  • Theorem 1: proved in \ref{['sec:prodPos']}
  • Theorem 2: proved in \ref{['sec:prodNeg']}
  • Theorem 3: corollary of \ref{['thm:corrNeg']}
  • Theorem 4: proved in \ref{['sec:corrPos']}
  • Theorem 5: proved in \ref{['sec:corrNeg']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 13 more