Monotone Hazard Rate Distributions Research Articles

Third-degree price discrimination versus uniform pricing

We compare the profit of the optimal third-degree price discrimination policy against a uniform pricing policy. A uniform pricing policy offers the same price to all segments of the market. Our main result establishes that for a broad class of third-degree price discrimination problems with concave profit functions (in the price space) and common support, a uniform price is guaranteed to achieve one half of the optimal monopoly profits. This profit bound holds for any number of segments and prices that the seller might use under third-degree price discrimination. We establish that these conditions are tight and that weakening either common support or concavity can lead to arbitrarily poor profit comparisons even for regular or monotone hazard rate distributions.

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.

Applications of α-Strongly Regular Distributions to Bayesian Auctions

Two classes of distributions that are widely used in the analysis of Bayesian auctions are the monotone hazard rate (MHR) and regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions, the condition is that for values v < v ′ , ϕ ( v ′ ) - ϕ ( v ) ≥ v ′ - v , and for regular distributions, ϕ ( v ′ ) - ϕ ( v ) ≥ 0. Cole and Roughgarden introduced the interpolating class of α-strongly regular distributions (α-SR distributions for short), for which ϕ ( v ′ ) - ϕ ( v ) ≥ α ( v ′ - v ), for 0 ≤ α ≤ 1. In this article, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders’ valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to α-SR distributions. For four of these settings, the auction mechanism requires knowledge of these distributions (in the remaining setting, the distributions are needed only to ensure good bounds on the expected revenue). In these cases, we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.