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Quantum Communication Complexity of Classical Auctions

Aviad Rubinstein, Zixin Zhou

TL;DR

This work investigates how quantum communication can change the revenue-maximizing auction problem under incentive compatibility. It develops a quantum auction framework that generalizes classical randomized protocols and proves two main threads: (i) for unit-demand and combinatorial valuations, there exist IC quantum protocols achieving near-optimal revenue with exponentially less expected communication than classical methods, albeit sometimes at the cost of requiring exponentially large payments; and (ii) for a two-item additive setting, there are stark separations showing quantum advantages in worst-case communication, including one-way quantum protocols achieving optimal revenue where finite classical or one-way quantum protocols cannot, as well as barely-interactive protocols. The results reveal a rich landscape where quantum and incentive constraints interact in ways that cannot be explained by standard quantum communication limits, and they establish both powerful speed-ups and fundamental limitations (e.g., when payments are bounded or when protocol rounds are finite). Overall, the paper highlights how careful encodings of distributions and spot-check verification enable efficient, incentive-compatible quantum mechanisms, advancing our understanding of quantum-cooperative communication in economic settings. These findings open avenues for exploring quantum-enabled mechanism design beyond traditional classical confines, with implications for secure, efficient auction design under strategic behavior.

Abstract

We study the fundamental, classical mechanism design problem of single-buyer multi-item Bayesian revenue-maximizing auctions under the lens of communication complexity between the buyer and the seller. Specifically, we ask whether using quantum communication can be more efficient than classical communication. We have two sets of results, revealing a surprisingly rich landscape - which looks quite different from both quantum communication in non-strategic parties, and classical communication in mechanism design. We first study the expected communication complexity of approximately optimal auctions. We give quantum auction protocols for buyers with unit-demand or combinatorial valuations that obtain an arbitrarily good approximation of the optimal revenue while running in exponentially more efficient communication compared to classical approximately optimal auctions. However, these auctions come with the caveat that they may require the seller to charge exponentially large payments from a deviating buyer. We show that this caveat is necessary - we give an exponential lower bound on the product of the expected quantum communication and the maximum payment. We then study the worst-case communication complexity of exactly optimal auctions in an extremely simple setting: additive buyer's valuations over two items. We show the following separations: 1. There exists a prior where the optimal classical auction protocol requires infinitely many bits, but a one-way message of 1 qubit and 2 classical bits suffices. 2. There exists a prior where no finite one-way quantum auction protocol can obtain the optimal revenue. However, there is a barely-interactive revenue-optimal quantum auction protocol. 3. There exists a prior where no multi-round quantum auction protocol with a finite bound on communication complexity can obtain the optimal revenue.

Quantum Communication Complexity of Classical Auctions

TL;DR

This work investigates how quantum communication can change the revenue-maximizing auction problem under incentive compatibility. It develops a quantum auction framework that generalizes classical randomized protocols and proves two main threads: (i) for unit-demand and combinatorial valuations, there exist IC quantum protocols achieving near-optimal revenue with exponentially less expected communication than classical methods, albeit sometimes at the cost of requiring exponentially large payments; and (ii) for a two-item additive setting, there are stark separations showing quantum advantages in worst-case communication, including one-way quantum protocols achieving optimal revenue where finite classical or one-way quantum protocols cannot, as well as barely-interactive protocols. The results reveal a rich landscape where quantum and incentive constraints interact in ways that cannot be explained by standard quantum communication limits, and they establish both powerful speed-ups and fundamental limitations (e.g., when payments are bounded or when protocol rounds are finite). Overall, the paper highlights how careful encodings of distributions and spot-check verification enable efficient, incentive-compatible quantum mechanisms, advancing our understanding of quantum-cooperative communication in economic settings. These findings open avenues for exploring quantum-enabled mechanism design beyond traditional classical confines, with implications for secure, efficient auction design under strategic behavior.

Abstract

We study the fundamental, classical mechanism design problem of single-buyer multi-item Bayesian revenue-maximizing auctions under the lens of communication complexity between the buyer and the seller. Specifically, we ask whether using quantum communication can be more efficient than classical communication. We have two sets of results, revealing a surprisingly rich landscape - which looks quite different from both quantum communication in non-strategic parties, and classical communication in mechanism design. We first study the expected communication complexity of approximately optimal auctions. We give quantum auction protocols for buyers with unit-demand or combinatorial valuations that obtain an arbitrarily good approximation of the optimal revenue while running in exponentially more efficient communication compared to classical approximately optimal auctions. However, these auctions come with the caveat that they may require the seller to charge exponentially large payments from a deviating buyer. We show that this caveat is necessary - we give an exponential lower bound on the product of the expected quantum communication and the maximum payment. We then study the worst-case communication complexity of exactly optimal auctions in an extremely simple setting: additive buyer's valuations over two items. We show the following separations: 1. There exists a prior where the optimal classical auction protocol requires infinitely many bits, but a one-way message of 1 qubit and 2 classical bits suffices. 2. There exists a prior where no finite one-way quantum auction protocol can obtain the optimal revenue. However, there is a barely-interactive revenue-optimal quantum auction protocol. 3. There exists a prior where no multi-round quantum auction protocol with a finite bound on communication complexity can obtain the optimal revenue.
Paper Structure (55 sections, 23 theorems, 81 equations, 6 figures)

This paper contains 55 sections, 23 theorems, 81 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. (Un)expected Quantum CC with Combinatorial Valuations
  4. Worst-Case Quantum CC with Two Items
  5. Key Takeaway: Thinking About Quantum and Incentives Together
  6. Additional Related Works
  7. Organization
  8. Preliminaries I: Quantum
  9. Bra-ket notation
  10. Closeness of states
  11. (Non-Strategic) Quantum Communicatiom Protocols
  12. Quantum systems
  13. Initial state of the system
  14. Local histories
  15. Local manipulations: unitary operators and measurements
  16. ...and 40 more sections

Key Result

Theorem 1

Let $\cal{D}$ be a prior over buyer's combinatorial valuations over $n$ items; assume all valuations are in the range Our protocol assumes that we're given some finite upper bound $U$ on valuations, but the communication complexity does not depend on $U$.$[0,U]$. Let $\mathcal{M}$ be any mechanism t

Figures (6)

  • Figure 1: Summary of most surprising aspects of our results from quantum perspective.
  • Figure 2: Summary of most surprising aspects of our results from AGT perspective.
  • Figure 3: The canonical partition
  • Figure 4: Probability density function of $f_2(y)$
  • Figure 5: The canonical partition induced by $s_1$, $s_2$, $P_{\mathrm{crit}}$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Theorem 1: Efficient in-expectation quantum auction protocols
  • Theorem 2: Efficient protocols require large payments - short version
  • Theorem 3: Separating one-way quantum vs classical
  • Theorem 4: Separating interactive quantum vs one-way quantum
  • Theorem 5: Limitations of finite interactive quantum auction protocols
  • Theorem 6: Choi-Jamiołkowski representation of strategies in interactive quantum protocols GutoskiW07
  • Definition 7: Quantum auction protocols
  • Theorem : Theorem \ref{['thm:combinatorial-protocol']} restated
  • Theorem 8: Full version of Theorem \ref{['thm:large-payments-intro']}
  • proof
  • ...and 25 more