[Paper Review] Complement-Free Couples Must Communicate: A Hardness Result for Two-Player Combinatorial Auctions

Tomer Ezra, Michal Feldman+3 more|arXiv (Cornell University)|Nov 24, 2018

Auction Theory and ApplicationsDecision Sciences32 references3 citations

TL;DR

This paper establishes that achieving a better-than-1/2 approximation in combinatorial auctions with two subadditive bidders requires exponentially large communication, proving that trivial protocols (e.g., random allocation or second-price auction on the grand bundle) are optimal for non-trivial approximation beyond 1/2. The key result resolves an open question by showing that any protocol guaranteeing a (1/2 + 6/log₂m)-approximation must use exponential communication in the number of items m.

ABSTRACT

We study the communication complexity of welfare maximization in combinatorial auctions with $m$ items and two subadditive bidders. A $\frac{1}{2}$-approximation can be guaranteed by a trivial randomized protocol with zero communication, or a trivial deterministic protocol with $O(1)$ communication. We show that outperforming these trivial protocols requires exponential communication, settling an open question of [DobzinskiNS10, Feige09]. Specifically, we show that any (randomized) protocol guaranteeing a $(\frac{1}{2}+\frac{6}{\log_2 m})$-approximation requires communication exponential in $m$. This is tight even up to lower-order terms: we further present a $(\frac{1}{2}+\frac{1}{O(\log m)})$-approximation in poly($m$) communication. To derive our results, we introduce a new class of subadditive functions that are "far from" fractionally subadditive functions, and may be of independent interest for future works. Beyond our main result, we consider the spectrum of valuations between fractionally-subadditive and subadditive via the MPH hierarchy. Finally, we discuss the implications of our results towards combinatorial auctions with strategic bidders.

Motivation & Objective

• To resolve the open question of whether non-trivial approximation beyond 1/2 is possible with subexponential communication in two-player subadditive combinatorial auctions.
• To establish tight communication complexity lower bounds for welfare maximization under subadditive valuations when n=2.
• To introduce a new class of subadditive functions that are far from fractionally subadditive, potentially useful for future research.
• To explore the implications of these results for truthful and non-truthful communication-efficient mechanisms in combinatorial auctions.

Proposed method

• The authors construct a hard instance using a probabilistic method to show the existence of subadditive valuation functions that are far from fractionally subadditive.
• They define an ℓ-independent family of sets to ensure that certain coverage properties hold with positive probability, enabling the construction of hard valuation pairs.
• A key technical component is proving that a derived function fℓS(⋅) is subadditive, using contradiction based on subadditivity of σS(⋅) and properties of the complement function.
• The proof uses a union bound over all ℓ-sized subsets and binary vectors to show that an ℓ-independent family exists with positive probability.
• The construction is used to show that any protocol achieving a (1/2 + 6/log₂m)-approximation must communicate exponentially in m.
• The paper further presents a poly(m)-communication protocol achieving a (1/2 + 1/O(log m))-approximation, showing tightness of the lower bound up to lower-order terms.

Experimental results

Research questions

• RQ1 Can a (1/2 + ε)-approximation for two-player subadditive combinatorial auctions be achieved with subexponential communication?
• RQ2 What is the exact communication complexity threshold for improving beyond the 1/2-approximation in this setting?
• RQ3 Are there subadditive valuation functions that are significantly different from fractionally subadditive functions, and can they be used to construct hard instances?
• RQ4 How does the communication complexity of truthful mechanisms compare to non-truthful ones in the two-player case?
• RQ5 What is the relationship between subadditive valuations and the MPH hierarchy in terms of communication complexity?

Key findings

• Any (randomized) protocol guaranteeing a (1/2 + 6/log₂m)-approximation for two subadditive bidders requires communication exponential in m.
• The 1/2-approximation can be trivially achieved with zero or O(1) communication, and no better approximation is possible with subexponential communication.
• A matching upper bound exists: a (1/2 + 1/O(log m))-approximation can be achieved in poly(m) communication.
• The paper introduces a new class of subadditive functions that are far from fractionally subadditive, which may be of independent interest.
• Every monotone subadditive function over m items is MPH-⌈m/2⌉, establishing a connection to the MPH hierarchy.
• The results imply that no communication-efficient truthful mechanism can outperform trivial protocols in this setting, even under honest reporting.