  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.
