Optimal Pricing For MHR and $\lambda$-Regular Distributions

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 $\lambda$.