Optimal Pricing for MHR and λ-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)≈ 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 λ-regular distributions that interpolate between MHR (λ =0) and regular (λ =1). Our anonymous pricing rule now results in an asymptotic approximation ratio that ranges smoothly, with respect to λ, 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 λ.