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Polyhedral Clinching Auctions with a Single Sample

Ryosuke Sato

Abstract

We address auctions in two-sided markets with budget constraints on buyers, a fundamental setting also crucial for applications such as display advertising. Our goal is to design efficient mechanisms that satisfy dominant strategy incentive compatibility, individual rationality, and budget balance. To overcome the limitations of impossibility theorems, we assume prior knowledge of sellers' valuations and focus on liquid welfare, an efficiency objective that takes budgets into account. Our contributions are twofold: First, we improve the efficiency guarantees of the polyhedral clinching auction by Hirai and Sato (2022). Second, using the reduction method of D"{u}tting et al. (2021), we extend the mechanism to an efficient single-sample mechanism for budget-constrained auctions, providing the budget extension of their results. Notably, our results hold even under polymatroid constraints and apply to both divisible and indivisible goods.

Polyhedral Clinching Auctions with a Single Sample

Abstract

We address auctions in two-sided markets with budget constraints on buyers, a fundamental setting also crucial for applications such as display advertising. Our goal is to design efficient mechanisms that satisfy dominant strategy incentive compatibility, individual rationality, and budget balance. To overcome the limitations of impossibility theorems, we assume prior knowledge of sellers' valuations and focus on liquid welfare, an efficiency objective that takes budgets into account. Our contributions are twofold: First, we improve the efficiency guarantees of the polyhedral clinching auction by Hirai and Sato (2022). Second, using the reduction method of D"{u}tting et al. (2021), we extend the mechanism to an efficient single-sample mechanism for budget-constrained auctions, providing the budget extension of their results. Notably, our results hold even under polymatroid constraints and apply to both divisible and indivisible goods.
Paper Structure (42 sections, 23 theorems, 79 equations, 3 figures, 1 table, 4 algorithms)

This paper contains 42 sections, 23 theorems, 79 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. How Do We Measure the Efficiency?
  3. Our Contributions
  4. Related Works.
  5. Auctions in Two-sided Markets (without Budgets).
  6. Clinching Auctions with Budgets.
  7. Another Approaches in Budget-constrained Auctions.
  8. Organization of the paper.
  9. Notation.
  10. Our Model
  11. Polyhedral Clinching Auctions with Truthful Sellers
  12. Preprocessing
  13. Mechanism
  14. Efficiency Guarantees for Polyhedral Clinching Auctions
  15. Optimal Liquid Welfare
  16. ...and 27 more sections

Key Result

Theorem 1

The polyhedral clinching auction (for divisible goods) achieves an LW of more than 1/2 of the optimal LW, and an SW of more than the optimal LW.

Figures (3)

  • Figure 1: The resulting network $\mathcal{N}$.
  • Figure 2: Illustration of the chain of tight sets and the dropping of buyers in Algorithm \ref{['HS_PCA']}. The white circles represent buyers in $\{i_1, i_2, \ldots, i_\tau\}$ and the gray shaded circles represent other buyers.
  • Figure 3: The polymatroidal network $\mathcal{N}$.

Theorems & Definitions (49)

  • Theorem : Theorems \ref{['LWofHS']} and \ref{['SW']}, Informal
  • Remark
  • Remark
  • Remark
  • Theorem 3.1: HS2022
  • Remark
  • Theorem 3.2: HS2022
  • Proposition 3.3: GMP2014GMP2015HS2022
  • Proposition 4.1
  • Proposition 4.2
  • ...and 39 more