A large proportion of the Bayesian mechanism design literature is restricted to the family of  **regular** distributions or the family of **monotone hazard rate (MHR)** distributions, which overshadows this beautiful and well-developed theory. We (re-)introduce two generalizations, the family of **quasi-regular** distributions and the family of  **quasi-MHR** distributions. All four families together form a hierarchy.

The significance of our new families is manifold. First, their defining conditions are immediate relaxations of the regularity/MHR conditions (i.e., monotonicity of the virtual value functions and/or the hazard rate functions), which reflect economic intuition. Second, they satisfy natural mathematical properties (about order statistics) that are violated by both original families. Third but foremost, numerous results established before for regular/MHR distributions now can be generalized, with or even without quantitative losses.

Yaonan Jin is a full-time researcher at the Huawei TCS Lab (lead by Pinyan Lu). His research interests encompass Theoretical Computer Science, with an emphasis on Algorithmic Economics. Before joining Huawei, he obtained his PhD from Columbia University in 2023 (advised by Xi Chen and Rocco Servedio). Before that, he obtained his MPhil from Hong Kong University of Science and Technology in 2019 (advised by Qi Qi) and his BEng from Shanghai Jiao Tong University in 2017.