Settling the Communication Complexity of VCG-based Mechanisms for all Approximation Guarantees

TL;DR

The paper resolves the communication complexity of maximal-in-range (MIR) mechanisms for truthful combinatorial auctions, providing tight upper and lower bounds for both general and submodular valuations under poly( m ) communication constraints. It introduces two central notions, $ extsf{MIR}_{ ext SubMod}(m,k)$ and $ extsf{MIR}_{ ext Gen}(m,k)$, showing that with $2^k$ communication, MIR achieves a $ ext{approximation}$ of $ ext{approximately } rac{m}{k}$ for general valuations and $ ext{approximately } rac{ ext{m}}{ ext{k}}$ for submodular valuations, while for subadditive valuations a $ ilde{O}(rac{m}{ ext{k}})$ bound is obtained. It develops algorithmic mechanisms—one based on multiple partitions to attain $m/k$-approximation and another bucketing-based mechanism for subadditive valuations achieving $ ilde{O}(rac{m}{ ext{k}})$—and demonstrates their optimality in value-query and succinct representation models. The paper also delivers matching lower bounds by showing that achieving even modest MIR approximations forces a rich allocation bank and embedding reductions from SetDisjointness, implying exponential communication requirements; together, these results close gaps and clarify the landscape of deterministic truthful mechanisms under limited communication. The work thus advances understanding of the tradeoffs between incentive compatibility, approximation quality, and communication, with implications for mechanism design in resource allocation systems.

Abstract

We consider truthful combinatorial auctions with items $M = [m]$ for sale to $n$ bidders, where each bidder $i$ has a private monotone valuation $v_i : 2^M \to R_+$. Among truthful mechanisms, maximal-in-range (MIR) mechanisms achieve the best-known approximation guarantees among all poly-communication deterministic truthful mechanisms in all previously-studied settings. Our work settles the communication necessary to achieve any approximation guarantee via an MIR mechanism. Specifically: Let MIRsubmod$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with submodular valuations over $m$ items. Then for all $k = Ω(\log(m))$, MIRsubmod$(m,k) = Ω(\sqrt{m/(k\log(m/k))})$. When $k = Θ(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $Ω(m^{1/3}/\log^{2/3}(m))$ to $Ω(\sqrt{m}/\log(m))$. We also have MIRsubmod$(m,k) = O(\sqrt{m/k})$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = Θ(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(\sqrt{m})$ to $O(\sqrt{m/\log(m)})$. Let also MIRgen$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with general valuations over $m$ items. Then for all $k = Ω(\log(m))$, MIRgen$(m,k) = Ω(m/k)$. When $k = Θ(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $Ω(m/\log^2(m))$ to $Ω(m/\log(m))$. We also have MIRgen$(m,k) = O(m/k)$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = Θ(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(m/\sqrt{\log(m)})$ to $O(m/\log(m))$.

Theorems & Definitions (97)

• Theorem 1.1: VCG Mechanism Vickrey61Clarke71Groves73
• Theorem 1.2: restate=LowerBound,name=
• Theorem 1.3: restate=Subadditive,name=
• Theorem 1.4: restate=General,name=
• Definition 2.1: Shattering
• Definition 3.1: Chunking Mechanism
• Definition 3.2: $r$-Itemizing
• Lemma 3.3
• proof
• Theorem 3.4