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

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 \(k = \Omega(\log(m))\), MIRsubmod\((m,k) = \Omega(\sqrt{m/(k\log(m/k))})\). When \(k = \Theta(\log(m))\), this improves the previous best lower bound for poly-comm. MIR mechanisms from \(\Omega(m^{1/3}/\log^{2/3}(m))\) to \(\Omega(\sqrt{m}/\log(m))\). 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 \(k = \Theta(\log(m))\), 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 \(k = \Omega(\log(m))\), MIRgen\((m,k) = \Omega(m/k)\). When \(k = \Theta(\log(m))\), this improves the previous best lower bound for poly-comm. MIR mechanisms from \(\Omega(m/\log^2(m))\) to \(\Omega(m/\log(m))\). 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 \(k = \Theta(\log(m))\), this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from \(O(m/\sqrt{\log(m)})\) to \(O(m/\log(m))\).