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Beyond Regularity: Simple versus Optimal Mechanisms, Revisited

Yiding Feng, Yaonan Jin

Abstract

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.

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 [Mye81] or the family of monotone hazard rate (MHR) distributions [BMP63], 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 the following hierarchy: and . 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 and . 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.

Paper Structure

This paper contains 21 sections, 37 theorems, 146 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Further Discussions
  4. Preliminaries
  5. Distribution Hierarchy and Structural Results
  6. Basic Structural Results
  7. Relations among Distribution Families
  8. Order Statistics of Symmetric Regular/MHR Distributions
  9. Order Statistics of Asymmetric Quasi-Regular/-MHR Distributions
  10. Revenue Approximation by Bayesian Simple Mechanisms
  11. The Single-Item Setting
  12. The Downward-Closed Settings
  13. Revenue Approximation by Prior-Independent Mechanisms
  14. The Single-Item Setting
  15. The Downward-Closed Settings
  16. ...and 6 more sections

Key Result

Proposition 2.3

Figures (6)

  • Figure 1: A Venn diagram of all five families of regular ($\mathbb{F}_{\textnormal{\tt reg}}$), MHR ($\mathbb{F}_{\textnormal{\tt MHR}}$), quasi-regular ($\mathbb{F}_{\textnormal{\tt Q-reg}}$), quasi-MHR ($\mathbb{F}_{\textnormal{\tt Q-MHR}}$), and general ($\mathbb{F}_{\textnormal{\tt gen}}$) distributions, which together form the following hierarchy: $\mathbb{F}_{\textnormal{\tt MHR}} \subsetneq (\mathbb{F}_{\textnormal{\tt reg}} \cap \mathbb{F}_{\textnormal{\tt Q-MHR}}) \subsetneq \mathbb{F}_{\textnormal{\tt reg}},\ \mathbb{F}_{\textnormal{\tt Q-MHR}} \subsetneq (\mathbb{F}_{\textnormal{\tt reg}} \cup \mathbb{F}_{\textnormal{\tt Q-MHR}}) \subsetneq \mathbb{F}_{\textnormal{\tt Q-reg}} \subsetneq \mathbb{F}_{\textnormal{\tt gen}}$.
  • Figure 2: A Hasse diagram of the single-item mechanisms (i.e., an arrow "${M}_{1} \to {M}_{2}$" means the latter ${M}_{2}$ surpasses the former ${M}_{1}$), hence the hierarchy ${\sf BOUP} \leq {\sf SPA}_{\sf BOUR},\ {\sf BOSP} \leq {\sf BOM}$, with only one incomparable pair "${\sf SPA}_{\sf BOUR}$ vs. BOSP".
  • Figure 3: Graphical illustration of equivalent definitions of quasi-regular distributions $\mathbb{F}_{\textnormal{\tt Q-reg}}$ and quasi-MHR distributions $\mathbb{F}_{\textnormal{\tt Q-MHR}}$.
  • Figure 4: Graphical illustration of the analysis for \ref{['thm:order:regular']} and \ref{['thm:order:mhr']}.
  • Figure 5: Revenue curves of symmetric quasi-regular buyers in \ref{['exp:BOM_BOUP:iid', 'exp:BOM_BOUR:iid', 'exp:single_sample:iid:MA']}. The quasi-regularity can be verified by checking Condition \ref{['condition:quasi-regular:revenue curve']} in \ref{['prop:q-regular equivalent definition']}.
  • ...and 1 more figures

Theorems & Definitions (86)

  • Definition 2.1: Revenue Curve and Ironed Revenue Curve BR89
  • Definition 2.2
  • Proposition 2.3: Revenue Equivalence M81
  • Definition 2.4: Bayesian Optimal Mechanism
  • Lemma 2.5: Folklore, see har16
  • Proposition 3.1: Equivalent Definitions of Quasi-Regularity
  • proof
  • Proposition 3.2: Equivalent Definition of Quasi-MHR
  • proof
  • Lemma 3.3
  • ...and 76 more