Myerson-regular distribution¶
A regular distribution in auction theory is a value distribution whose virtual valuation is nondecreasing, equivalently whose quantile-space revenue curve is concave under the standard continuous formulation.
Core Idea¶
A Myerson-regular distribution is a value distribution whose virtual valuation is nondecreasing in the underlying value, or equivalently, whose posted-price revenue curve is concave when parameterized by sale probability. For a continuous cumulative distribution \(F\) with density \(f\), virtual value is \(\phi(v)=v-(1-F(v))/f(v)\). If higher actual values never yield lower virtual values, allocation by virtual surplus respects the ordering needed for a truthful auction. In quantile form, choosing a price that sells with probability \(q\) yields revenue \(R(q)=qF^{-1}(1-q)\) under the standard convention; regularity requires \(R\) to.
Scope of Application¶
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Optimal auctions. Monotone virtual surplus allows truthful allocation rules and reserve prices without ironing.
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Reserve pricing. The distribution's virtual-value zero helps characterize monopoly reserves under the private-value model.
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Ironing diagnostics. Failure of monotonicity identifies intervals that require ironed virtual values rather than abandonment of mechanism design.
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Approximation theory. Regular distribution classes support prior-independent or simple-mechanism performance guarantees.
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Distribution comparison. Parametric and nonparametric families can be tested for regularity on declared supports.
Clarity¶
Regular distribution in mechanism design means Myerson regularity: virtual value is nondecreasing, equivalently the quantile-parameterized revenue curve is concave under the standard assumptions. This is distinct from statistical normality, smoothness, or merely having a density. Naming the property makes clear why virtual-surplus allocation remains monotone and why no ironing is required.
Manages Complexity¶
Myerson regularity reduces a seller's full value distribution to the monotonicity of virtual value or, equivalently under the standard setup, concavity of the quantile revenue curve. When the condition holds, maximizing virtual surplus yields a monotone allocation without separately resolving every price lottery or bidder type. When it fails, nonmonotone intervals are compressed by ironing, creating the irregular branch.
Abstract Reasoning¶
Regularity test. Compute virtual value or the quantile revenue curve and infer regularity from monotonicity or concavity under the applicable assumptions. Mechanism move. When regularity holds, rank by virtual surplus and infer a monotone implementable allocation without ironing. Repair move. When it fails, convexify or iron the nonmonotone region and infer pooled allocation or pricing behavior there. Boundary move. Normality, smooth density, or familiar distribution shape does not establish Myerson regularity; the exact virtual-value condition must be checked, especially for estimated or discrete value distributions.
Knowledge Transfer¶
Within the home domain. Myerson-regular distributions transfer across auction design and mechanism design when the virtual valuation derived from a bidder's value distribution is monotone, supporting simple allocation and ironing-free optimal-auction analysis. Distribution, hazard behavior, virtual value, incentive compatibility, and revenue retain formal roles. Beyond the home domain (C — model condition). The property applies literally to any private-value model satisfying the definition, but not to distributions merely called statistically regular. Its boundary is strong: empirical estimation error, correlation, multidimensional types, irregular tails, and misspecified bidder symmetry can invalidate the conclusion. Regularity does not guarantee fairness, efficiency, or robust practical revenue.
Relationships to Other Abstractions¶
Current abstraction Myerson-regular distribution Domain-specific
Parents (1) — more general patterns this builds on
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Myerson-regular distribution is a kind of Probability Distribution Domain-specific
Myerson-regular distribution is a domain-specific kind of Probability Distribution: A regular distribution in auction theory is a value distribution whose virtual valuation is nondecreasing, equivalently whose quantile-space revenue curve is concave under the standard continuous formulation.
Hierarchy paths (5) — routes to 3 parentless roots
- Myerson-regular distribution → Probability Distribution → Random Variable → Function (Mapping)
- Myerson-regular distribution → Probability Distribution → Probability → Measure → Set and Membership
- Myerson-regular distribution → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Myerson-regular distribution → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Myerson-regular distribution → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Myerson-regular distribution sits in a moderately populated region (57th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Virtual Valuation — 0.88
- Revenue Equivalence Theorem — 0.88
- Amoroso–Robinson Relation — 0.85
- Bundling — 0.85
- Lerner index — 0.84
Computed from structural-signature embeddings · 2026-10-08