Amplified Hardness of Approximation for VCG-Based Mechanisms

TL;DR

This paper proves that for certain social welfare maximization problems, any polynomial-time truthful mechanism cannot surpass a 1/2 approximation for two players and 1/k for k players, under specific hardness assumptions.

Contribution

It establishes new hardness of approximation bounds for VCG-based mechanisms, extending to randomized and more powerful truthful mechanisms, for a broad class of valuation functions.

Findings

No polynomial-time truthful mechanism can beat 1/2 approximation for two players.

For k players, the approximation limit improves to 1/k, matching trivial mechanisms.

Even for budgeted additive valuations with an FPTAS, the approximation cannot surpass 1/k.

Abstract

If a two-player social welfare maximization problem does not admit a PTAS, we prove that any maximal-in-range truthful mechanism that runs in polynomial time cannot achieve an approximation factor better than 1/2. Moreover, for the k-player version of the same problem, the hardness of approximation improves to 1/k under the same two-player hardness assumption. (We note that 1/k is achievable by a trivial deterministic maximal-in-range mechanism.) This hardness result encompasses not only deterministic maximal-in-range mechanisms, but also all universally-truthful randomized maximal in range algorithms, as well as a class of strictly more powerful truthful-in-expectation randomized mechanisms recently introduced by Dobzinski and Dughmi. Our result applies to any class of valuation functions that satisfies some minimal closure properties. These properties are satisfied by the valuation functions in all well-studied APX-hard social welfare maximization problems, such as coverage, submodular, and subadditive valuations. We also prove a stronger result for universally-truthful maximal-in-range mechanisms. Namely, even for the class of budgeted additive valuations, which admits an FPTAS, no such mechanism can achieve an approximation factor better than 1/k in polynomial time.