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Communication Separations for Truthful Auctions: Breaking the Two-Player Barrier

Shiri Ron, Clayton Thomas, S. Matthew Weinberg, Qianfan Zhang

TL;DR

This work establishes, for the first time, a separation between poly$(m)$-communication truthful and non-truthful approaches in combinatorial auctions with more than two bidders. Focusing on valuations drawn from $\mathsf{SubAdd}\cup\mathsf{SingleM}$ across three bidders, the authors prove that any deterministic truthful mechanism achieving a $0.366$-approximation requires exponential communication in $m$, while a non-truthful extension of Fei09 achieves a $1/2$-approximation with polynomial communication. The core method leverages the taxation complexity framework via menus, augmented by a novel two-layer hard-instance construction inspired by EFNTW19 to close gaps in prior two-bidder reductions and to handle the three-bidder setting. A supplementary analysis for two bidders shows exponential-cost lower bounds under related valuation classes, underscoring the robustness of the approach. The results illuminate the limits of truthful mechanisms and open avenues for extending stronger lower bounds to more bidders or broader valuation classes, with potential implications for mechanism design in practical auctions.

Abstract

We study the communication complexity of truthful combinatorial auctions, and in particular the case where valuations are either subadditive or single-minded, which we denote with $\mathsf{SubAdd}\cup\mathsf{SingleM}$. We show that for three bidders with valuations in $\mathsf{SubAdd}\cup\mathsf{SingleM}$, any deterministic truthful mechanism that achieves at least a $0.366$-approximation requires $\exp(m)$ communication. In contrast, a natural extension of [Fei09] yields a non-truthful $\mathrm{poly}(m)$-communication protocol that achieves a $\frac{1}{2}$-approximation, demonstrating a gap between the power of truthful mechanisms and non-truthful protocols for this problem. Our approach follows the taxation complexity framework laid out in [Dob16b], but applies this framework in a setting not encompassed by the techniques used in past work. In particular, the only successful prior application of this framework uses a reduction to simultaneous protocols which only applies for two bidders [AKSW20], whereas our three-player lower bounds are stronger than what can possibly arise from a two-player construction (since a trivial truthful auction guarantees a $\frac{1}{2}$-approximation for two players).

Communication Separations for Truthful Auctions: Breaking the Two-Player Barrier

TL;DR

This work establishes, for the first time, a separation between poly-communication truthful and non-truthful approaches in combinatorial auctions with more than two bidders. Focusing on valuations drawn from across three bidders, the authors prove that any deterministic truthful mechanism achieving a -approximation requires exponential communication in , while a non-truthful extension of Fei09 achieves a -approximation with polynomial communication. The core method leverages the taxation complexity framework via menus, augmented by a novel two-layer hard-instance construction inspired by EFNTW19 to close gaps in prior two-bidder reductions and to handle the three-bidder setting. A supplementary analysis for two bidders shows exponential-cost lower bounds under related valuation classes, underscoring the robustness of the approach. The results illuminate the limits of truthful mechanisms and open avenues for extending stronger lower bounds to more bidders or broader valuation classes, with potential implications for mechanism design in practical auctions.

Abstract

We study the communication complexity of truthful combinatorial auctions, and in particular the case where valuations are either subadditive or single-minded, which we denote with . We show that for three bidders with valuations in , any deterministic truthful mechanism that achieves at least a -approximation requires communication. In contrast, a natural extension of [Fei09] yields a non-truthful -communication protocol that achieves a -approximation, demonstrating a gap between the power of truthful mechanisms and non-truthful protocols for this problem. Our approach follows the taxation complexity framework laid out in [Dob16b], but applies this framework in a setting not encompassed by the techniques used in past work. In particular, the only successful prior application of this framework uses a reduction to simultaneous protocols which only applies for two bidders [AKSW20], whereas our three-player lower bounds are stronger than what can possibly arise from a two-player construction (since a trivial truthful auction guarantees a -approximation for two players).
Paper Structure (44 sections, 21 theorems, 41 equations, 1 figure)

This paper contains 44 sections, 21 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: Combinatorial Auctions.
  3. Our Contributions.
  4. Brief Overview of Approach.
  5. Roadmap and Conclusions
  6. Related Work
  7. Communication Complexity of Deterministic Truthful Combinatorial Auctions.
  8. Communication Complexity of Randomized Truthful Combinatorial Auctions.
  9. Computational Complexity of Truthful Combinatorial Auctions.
  10. Preliminaries
  11. Combinatorial auctions.
  12. Communication complexity and protocols.
  13. Valuation classes.
  14. Black-box welfare approximation reduction with $O(\log m)$ single minded bidders.
  15. Menus
  16. ...and 29 more sections

Key Result

lemma 2.3

Let $\mathcal{P}$ be a protocol that achieves an approximation $\alpha$ for $n$ bidders with valuations in the class $\mathcal{C}$ with $\operatorname{poly}(m)$ bits. Then, there exists a protocol $\mathcal{P}'$ that achieves an approximation $\alpha$ for $n$ bidders with valuations in the class $\m

Figures (1)

  • Figure 1: Illustration of valuations in \ref{['thrm:valuation-class-first-attempt']}.

Theorems & Definitions (48)

  • definition 2.1: Single-minded valuations
  • definition 2.2: Subadditive valuations
  • lemma 2.3
  • definition 2.4: Menus
  • proposition 2.5: Taxation Principle
  • proposition 2.6: Menu monotonicity D16b
  • theorem 2.7: Follows from D16b
  • proposition 3.1
  • theorem 3.2: EFNTW19
  • proof : (Failed) Proof.
  • ...and 38 more