Beyond Regularity: Simple versus Optimal Mechanisms, Revisited
Abstract
A large proportion of the Bayesian mechanism design literature is restricted to the family of regular distributions F reg [Mye81] or the family of monotone hazard rate (MHR) distributions F MHR [BMP63], which overshadows this beautiful and well-developed theory. We (re-)introduce two generalizations, the family of quasi-regular distributions F Q-reg and the family of quasi-MHR distributions F Q-MHR. All four families together form the following hierarchy: F MHR ⊂neq (F reg F Q-MHR) ⊂neq F Q-reg and F Q-MHR ⊂neq (F reg F Q-MHR) ⊂neq F 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 F reg and F 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.
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