  We compare the expected efficiency of revenue maximizing (or {\em optimal})
mechanisms with that of efficiency maximizing ones. We show that the efficiency
of the revenue maximizing mechanism for selling a single item with k +
log_{e/(e-1)} k + 1 bidders is at least as much as the efficiency of the
efficiency maximizing mechanism with k bidders, when bidder valuations are
drawn i.i.d. from a Monotone Hazard Rate distribution. Surprisingly, we also
show that this bound is tight within a small additive constant of 5.7. In other
words, Theta(log k) extra bidders suffice for the revenue maximizing mechanism
to match the efficiency of the efficiency maximizing mechanism, while o(log k)
do not. This is in contrast to the result of Bulow and Klemperer comparing the
revenue of the two mechanisms, where only one extra bidder suffices. More
precisely, they show that the revenue of the efficiency maximizing mechanism
with k+1 bidders is no less than the revenue of the revenue maximizing
mechanism with k bidders.
  We extend our result for the case of selling t identical items and show that
2.2 log k + t Theta(log log k) extra bidders suffice for the revenue maximizing
mechanism to match the efficiency of the efficiency maximizing mechanism.
  In order to prove our results, we do a classification of Monotone Hazard Rate
(MHR) distributions and identify a family of MHR distributions, such that for
each class in our classification, there is a member of this family that is
pointwise lower than every distribution in that class. This lets us prove
interesting structural theorems about distributions with Monotone Hazard Rate.
