  What fraction of the potential social surplus in an environment can be
extracted by a revenue-maximizing monopolist? We investigate this problem in
Bayesian single-parameter environments with independent private values. The
precise answer to the question obviously depends on the particulars of the
environment: the feasibility constraint and the distributions from which the
bidders' private values are sampled. Rather than solving the problem in
particular special cases, our work aims to provide universal lower bounds on
the revenue-to-welfare ratio that hold under the most general hypotheses that
allow for non-trivial such bounds.
  Our results can be summarized as follows. For general feasibility
constraints, the revenue-to-welfare ratio is at least a constant times the
inverse-square-root of the number of agents, and this is tight up to constant
factors. For downward-closed feasibility constraints, the revenue-to-welfare
ratio is bounded below by a constant. Both results require the bidders'
distributions to satisfy hypotheses somewhat stronger than regularity; we show
that the latter result cannot avoid this requirement.
