Applications of α-strongly regular distributions to Bayesian auctions

Two classes of distributions that arewidely 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: forMHR distributions, the condition is that for valuesv < v' φ(v ) - φ(v') ≥ v-v, and for regular distributions, φ(v') - φ(v) ge; 0. Cole and Roughgarden introduced the interpolating class of α-strongly regular distributions (α-SR distributions for short), for which φ(v') - φ(v) ge; α (v-v), for 0 ≤ α ≤ 1. In this article, we investigate five distinct auction settings for which good expected revenue bounds are knownwhen the bidders' valuations are given by MHR distributions. In every case,we showthat 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.