Performance bounds for optimal sales mechanisms beyond the monotone hazard rate condition

In mechanism design, Myerson’s classical regularity assumption is often too weak to imply quantitative results about the performance of sales mechanisms. For example, ratios between revenue and welfare, or sales probabilities may vanish at the boundary of Myerson regularity. Therefore, for quantitative results, many authors have resorted to much stronger assumptions such as the monotone hazard rate condition. This motivates us to explore performance bounds for sales mechanisms that follow from a quantitative version of Myerson regularity, which we call λ-regularity. The parameter λ interpolates from Myerson regularity to the monotone hazard rate condition and beyond. We provide four equivalent definitions of the concept. These rely on a growth condition on the virtual valuations function (known as α-strong regularity), a monotonicity condition on a generalized hazard rate, a ρ-concavity condition on survival functions and a comparison relation in the convex transform order. By highlighting the interplay between these different perspectives, we unify previous work in economics, computer science, applied mathematics and statistics. We demonstrate the usefulness of λ-regularity for quantitative mechanism design by proving various performance bounds for sales mechanisms. In addition, we briefly consider applications beyond auctions and mechanism design such as the measurement of inequality in populations.