Myerson regularity alone often turns out too weak for conducting quantitative analyses in auction and mechanism design. For instance, ratios between revenue and welfare, or sales probabilities can vanish for distributions that fulfill the standard Myerson regularity assumption. Therefore, mild restrictions on Myerson regularity have been proposed. Two separate literatures have emerged on the topic, in economic theory and in computer science. The goal of this paper is therefore twofold. First, we provide an overview of the refinements that have been studied in the different fields and trace them to an earlier literature in statistics and reliability theory. Often, basically the same refinements have been used, yet under different names. Second, we develop new results based on a particularly successful refinement to which we will refer as λ-regularity. It has been employed under the names of ρ-concavity and α-strong-regularity as well. Specifically, we extend a series of quantitative estimates about the single bidder case from algorithmic mechanism design, and analyze the according n-bidder case. We provide new results on various order statistics and on revenue-to-welfare ratios. Finally, we briefly consider applications that go beyond the standard auctions and mechanism design settings such as the measurement of inequality in populations.
The Quantitative View of Myerson Regularity
Nikolaus Schweizer and Nora Szech