A large proportion of the Bayesian mechanism design literature is restricted to the family of regular distributions $\mathbb{F}_{\tt reg}$ [Mye81] or the family of monotone hazard rate (MHR) distributions $\mathbb{F}_{\tt MHR}$ [BMP63], which overshadows this beautiful and well-developed theory. We (re-)introduce two generalizations, the family of quasi-regular distributions $\mathbb{F}_{\tt Q-reg}$ and the family of quasi-MHR distributions $\mathbb{F}_{\tt Q-MHR}$. All four families together form the following hierarchy: $\mathbb{F}_{\tt MHR} \subsetneq (\mathbb{F}_{\tt reg} \cap \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$ and $\mathbb{F}_{\tt Q-MHR} \subsetneq (\mathbb{F}_{\tt reg} \cup \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$. 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 $\mathbb{F}_{\tt reg}$ and $\mathbb{F}_{\tt MHR}$. Third but foremost, numerous results [BK96, HR09a, CD15, DRY15, HR14, AHN+19, JLTX20, JLQ+19b, FLR19, GHZ19b, JLX23, LM24] established before for regular/MHR distributions now can be generalized, with or even without quantitative losses.

翻译：贝叶斯机制设计文献中很大一部分局限于正则分布族 $\mathbb{F}_{\tt reg}$ [Mye81] 或单调风险率分布族 $\mathbb{F}_{\tt MHR}$ [BMP63]，这在一定程度上遮蔽了这一优美且发展完善的理论。我们（重新）引入两个推广：拟正则分布族 $\mathbb{F}_{\tt Q-reg}$ 与拟单调风险率分布族 $\mathbb{F}_{\tt Q-MHR}$。所有四个分布族共同构成如下层级关系：$\mathbb{F}_{\tt MHR} \subsetneq (\mathbb{F}_{\tt reg} \cap \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$ 且 $\mathbb{F}_{\tt Q-MHR} \subsetneq (\mathbb{F}_{\tt reg} \cup \mathbb{F}_{\tt Q-MHR}) \subsetneq \mathbb{F}_{\tt Q-reg}$。我们提出的新分布族具有多方面的重要意义。首先，其定义条件是对正则性/单调风险率条件（即虚拟价值函数和/或风险率函数的单调性）的直接松弛，这反映了经济直觉。其次，它们满足关于顺序统计量的自然数学性质，而这些性质在原分布族 $\mathbb{F}_{\tt reg}$ 和 $\mathbb{F}_{\tt MHR}$ 中均被违反。第三也是最重要的一点，先前针对正则/单调风险率分布建立的众多结果 [BK96, HR09a, CD15, DRY15, HR14, AHN+19, JLTX20, JLQ+19b, FLR19, GHZ19b, JLX23, LM24] 现在可以被推广，甚至可以在没有定量损失的情况下实现。