  We study the performance of anonymous posted-price selling mechanisms for a
standard Bayesian auction setting, where $n$ bidders have i.i.d. valuations for
a single item. We show that for the natural class of Monotone Hazard Rate (MHR)
distributions, offering the same, take-it-or-leave-it price to all bidders can
achieve an (asymptotically) optimal revenue. In particular, the approximation
ratio is shown to be $1+O(\ln \ln n/\ln n)$, matched by a tight lower bound for
the case of exponential distributions. This improves upon the previously
best-known upper bound of $e/(e-1)\approx 1.58$ for the slightly more general
class of regular distributions. In the worst case (over $n$), we still show a
global upper bound of $1.35$. We give a simple, closed-form description of our
prices which, interestingly enough, relies only on minimal knowledge of the
prior distribution, namely just the expectation of its second-highest order
statistic.
  Furthermore, we extend our techniques to handle the more general class of
$\lambda$-regular distributions that interpolate between MHR ($\lambda=0$) and
regular ($\lambda=1$). Our anonymous pricing rule now results in an asymptotic
approximation ratio that ranges smoothly, with respect to $\lambda$, from $1$
(MHR distributions) to $e/(e-1)$ (regular distributions). Finally, we
explicitly give a class of continuous distributions that provide matching lower
bounds, for every $\lambda$.
