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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11727
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomains: Auction Theory, Mechanism Design → Economics & Finance

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 be concave.

The property controls how marginal revenue changes as a seller expands the probability of sale. Concavity rules out upward kinks that would make an intermediate sale probability worse than a lottery between more extreme pricing policies. In Myerson's optimal auction, a regular bidder can be ranked by an increasing transformation of value, and reserve prices exclude negative virtual values. With irregular distributions, raw virtual values can fall as values rise, so the mechanism uses ironing: intervals of quantiles receive averaged virtual values to restore monotonicity and incentive compatibility.

Regularity is an assumption about a distribution, not a synonym for smoothness, normality, or common occurrence. Uniform, exponential, Gaussian under standard settings, and some power-law distributions can satisfy it, while other well-behaved densities do not. Discrete analogues require adjusted definitions. The abstraction matters because it converts a constrained mechanism-design problem into monotone virtual-surplus maximization and supports approximation results in prior-independent auctions. Any use must state the value domain, continuity assumptions, and quantile convention rather than inferring regularity from a familiar distribution name alone.

Structural Signature¶

Sig role-phrases:

the private-value distribution — a bidder-value law with stated domain, continuity, cumulative distribution, and density assumptions
the virtual-value transform — actual value adjusted by the inverse hazard term under the continuous formula
the monotonicity condition — virtual value nondecreasing as actual value rises
the quantile-price map — posted price associated with a sale probability under a declared quantile convention
the revenue curve — expected posted-price revenue as a function of sale probability
the concavity equivalence — absence of upward kinks in the revenue curve corresponding to regularity
the truthful-allocation consequence — virtual-surplus ranking preserving value order in the optimal mechanism
the reserve-price threshold — exclusion of types with negative virtual value
the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline

What It Is Not¶

Not statistical smoothness or normality. Regularity is monotonicity of virtual value, equivalently concavity of the revenue curve under a stated quantile convention.
Not guaranteed by a familiar density name. Domain, parameters, continuity, and exact distribution determine whether the condition holds.
Not a claim that values are evenly spaced or well behaved informally. The property concerns marginal revenue and incentive-compatible allocation order.
Not automatic for discrete distributions. Atomic value laws require adjusted definitions rather than silently applying a continuous density formula.
Not necessary for optimal auction theory to exist. Irregular distributions are handled through ironing, which restores monotone effective virtual values.
Not concavity of the cumulative distribution function. The relevant curve is posted-price revenue parameterized by sale probability.
Not mechanism performance without assumptions. Approximation and prior-independent results depend on bidder model, independence, value domain, and the particular regularity statement.

Scope of Application¶

Myerson regularity applies in auction and mechanism design when a bidder-value distribution has monotone virtual values, equivalently a suitably concave quantile-revenue curve under the adopted conventions.

Optimal auctions. Monotone virtual surplus allows truthful allocation rules and reserve prices without ironing.
Reserve pricing. The distribution's virtual-value zero helps characterize monopoly reserves under the private-value model.
Ironing diagnostics. Failure of monotonicity identifies intervals that require ironed virtual values rather than abandonment of mechanism design.
Approximation theory. Regular distribution classes support prior-independent or simple-mechanism performance guarantees.
Distribution comparison. Parametric and nonparametric families can be tested for regularity on declared supports.
Empirical estimation. Finite samples require uncertainty, tail assumptions, and care because global monotonicity cannot be certified by a smooth-looking estimate alone.
Applicability boundary. Regular does not mean Gaussian, smooth, or generically well behaved; atoms and mixed distributions need adapted formulas, and economic conclusions remain conditional on independence, symmetry, private values, and strategic assumptions.

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. The sharper auction-design question is whether the estimated value distribution satisfies this monotonicity, how violations are handled, and whether the claimed equivalence is valid under the distributional regularity conditions actually present.

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. The analyst tracks the distribution, density or quantile form, virtual value, and revenue slope. This small signature predicts whether standard optimal-auction machinery applies directly or requires pooling and convexification.

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.

Examples¶

Canonical¶

Let a bidder's private value be uniformly distributed on [0,1]. Then F(v)=v and f(v)=1, so the Myerson virtual value is phi(v)=v-(1-F(v))/f(v)=v-(1-v)=2v-1. It increases monotonically with v, making the distribution regular. The corresponding revenue curve in quantile space is concave, and the virtual value crosses zero at v=½. In the simplest single-bidder sale, that crossing yields a reserve price of one half under the model. The calculation illustrates that regularity is a property of the value distribution's transformed incentives, not the everyday smoothness of its density or a claim that every observed bid is orderly.

Mapped back: Uniform [0,1] is the private-value distribution, phi the virtual-value transform, and increasing 2v-1 the monotonicity condition. Concavity is the revenue curve equivalence, and v=½ is the reserve-price threshold supporting the truthful-allocation consequence.

Applied / In Practice¶

An auction designer estimates a value distribution from historical, strategically generated bids only after stating a behavioral model. The estimated virtual-value curve is checked for monotonicity and uncertainty. If it is nonmonotone, blindly applying the regular-case allocation can violate the required monotonic structure; ironing or another irregular-case treatment is needed. Reserve recommendations are stress-tested against sampling error, bidder asymmetry, correlation, and shifts in the market. The regularity condition simplifies the theoretical design, but it neither validates the estimated private values nor guarantees fairness, efficiency, or robust revenue in deployment.

Mapped back: Estimated values supply the private-value distribution, transformed into the virtual-value transform and quantile-price map. Monotonicity tests the regularity condition; violations invoke the irregular-case repair, while any recommended reserve remains the reserve-price threshold under explicit empirical assumptions.

Structural Tensions¶

T1 — Identity versus admissible variation. Myerson-regular distribution must remain recognizable across legitimate variants. Admissible variation is bounded by this condition: Monotone virtual surplus allows truthful allocation rules and reserve prices without ironing. The stable element is expressed by this invariant: 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. Treating every surface change as a new abstraction fragments the identity, while allowing a change to the constitutive relation produces a false positive.

Diagnostic: After the proposed variation, can an analyst still establish this invariant: 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?

T2 — Recognition versus proxy. The domain needs observable or inferential evidence for Myerson-regular distribution, but the evidence is not automatically the identity. The working recognition rule is: the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline. A familiar indicator can occur without the defining relation, and the relation can persist when a customary detector is unavailable.

Diagnostic: Does the evidence establish the defining claim—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—or only a correlated sign?

T3 — Definition versus operational judgment. A compact definition aids reuse, whereas actual classification in auction theory can require expert decisions about boundary conditions, measurements, conventions, or exceptions. The property controls how marginal revenue changes as a seller expands the probability of sale. The definition must constrain those judgments without pretending that every admissible case can be recognized from a label alone.

Diagnostic: Which observation would make a competent practitioner reject the classification under the stated definition?

T4 — Scope versus overextension. Myerson-regular distribution has a genuine habitat in which monotone virtual surplus allows truthful allocation rules and reserve prices without ironing. Yet Regular does not mean Gaussian, smooth, or generically well behaved; atoms and mixed distributions need adapted formulas, and economic conclusions remain conditional on independence, symmetry, private values, and strategic assumptions. A useful application map therefore has to be broad enough to cover recurring practice and narrow enough to exclude merely topical or metaphorical occurrences.

Diagnostic: Can the claimed application fill the same carrier and relation roles, or has only the name traveled?

T5 — Transfer versus domain accent. Knowledge about Myerson-regular distribution can travel within its home domain, and some structural lessons may travel farther. 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. What transfers must be separated from the specialist vocabulary, warrant, and closure conditions that remain anchored in auction theory.

Diagnostic: Is the receiving case a literal instance of Myerson-regular distribution, a co-instance of Probability Distribution, or only an analogy?

T6 — Autonomy versus reduction. Myerson-regular distribution is a strict specialization of Probability Distribution, but the edge does not erase the domain differentia. The broader node supplies only the necessary structural relation; auction theory supplies the carrier, warrant, boundary, and exception conditions expressed by this identity: 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. The entry is over-split if those conditions add no discriminating work and under-specified if the parent alone is used for cases that require them.

Diagnostic: Can a domain expert use the added conditions to distinguish Myerson-regular distribution from another case that equally instantiates Probability Distribution?

Structural–Framed Character¶

Myerson-regular distribution is mixed: structurally specifiable but materially dependent on its disciplinary frame. Its structural side consists of the carrier the private-value distribution — a bidder-value law with stated domain, continuity, cumulative distribution, and density assumptions and the constitutive relation 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. Its framed side comes from auction theory, which fixes what the terms denote, what counts as evidence, and when a qualification or exception defeats the classification.

Across the principal tests, the entry is not merely a free-floating pattern. Evaluative weight: the identity can be stated descriptively even when its use has practical or normative consequences. Practice dependence: the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline. Institutional stabilization: disciplinary conventions may stabilize the name and test without necessarily creating every underlying event or relation. Vocabulary portability: the invariant is 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. Import versus recognition: an outside case qualifies literally only if the same typed roles and collapse condition are available; otherwise the comparison is analogical.

The reusable remainder is Probability Distribution under a reviewed subsumption relation. That node preserves the necessary cross-domain organization after the auction theory-specific carrier, evidence, and exceptions are removed. Myerson-regular distribution remains autonomous because its recognition and collapse conditions distinguish cases that the parent alone leaves together.

Structural Core vs. Domain Accent¶

What is skeletal. The portable skeleton is a typed carrier organized by a constitutive relation, an invariant, a recognition test, and a collapse condition. Here the carrier is the private-value distribution — a bidder-value law with stated domain, continuity, cumulative distribution, and density assumptions. The decisive relation is 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, which also states the controlling invariant at this level. Stripped of specialist nouns, this organization is represented by Probability Distribution.

What is domain-bound. auction theory supplies the actual objects or agents, admissible transformations, units or conventions, standards of warrant, and named exceptions. In this case, recognition requires evidence for the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline. Admissible variation is bounded by the condition that monotone virtual surplus allows truthful allocation rules and reserve prices without ironing, and the classification collapses when regularity is monotonicity of virtual value, equivalently concavity of the revenue curve under a stated quantile convention. These are constitutive differentia, not illustrative decoration.

Why it remains a domain-specific node. The reviewed DAG relation is subsumption to Probability Distribution. Outside auction theory, the parent captures only the reusable structural remainder. The specialist name remains literal only where the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline can be established under the domain's standards of warrant.

This entry is a kind of Probability Distribution.

Immediate parent — Probability Distribution (subsumption). 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. The parent supplies the necessary broader identity—The complete specification of how probability mass or density is spread over a random variable's possible values — a measure that, once compressed to a named parametric family, encodes shape, moments, tails, and a generative claim about the process producing the data.—while the candidate adds the source-domain carrier, recognition rule, and failure conditions. The defining source account begins: 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.
Nearest catalog surface declined — Virtual Valuation. Its rematch score was 0.281174. Retrieval proximity did not establish synonymy or parentage; the carrier, invariant, and collapse condition remain different.
Related reasoning operations. Evidence, comparison, boundary testing, and representation can support a case without becoming additional DAG parents.

Relationships to Other Abstractions¶

Current abstraction Myerson-regular distribution Domain-specific

Parents (1) — more general patterns this builds on

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.

Neighborhood in Abstraction Space¶

Myerson-regular distribution sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Probability Distribution. This is the reviewed immediate parent or structural prerequisite, not a synonym. Tell: retain Myerson-regular distribution only when the domain-specific relation 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. and its source-domain warrant are established; otherwise route the case to Probability Distribution.
Virtual Valuation. This is the closest catalog retrieval surface, not an accepted synonym or parent. Tell: Ask which entry's carrier, invariant, and collapse test the case actually satisfies; shared vocabulary or a score of 0.739021 is insufficient.
Not statistical smoothness or normality. Regularity is monotonicity of virtual value, equivalently concavity of the revenue curve under a stated quantile convention. Tell: Require the positive recognition condition that the irregular-case repair — ironing intervals to restore monotonicity and incentive compatibility when raw virtual values decline.
Not guaranteed by a familiar density name. Domain, parameters, continuity, and exact distribution determine whether the condition holds. Tell: Replace the familiar surface feature and test whether 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.
A detector, representation, or consequence. A method may reveal Myerson-regular distribution, a notation may describe it, and an outcome may follow from it without any of those being identical to the abstraction. Tell: Would the defining relation remain if the present detector, notation, or downstream effect changed?
A metaphorical transfer. A case outside the home domain may resemble the structure while lacking its native role types and standards of warrant. Tell: If only the general organization survives, route the comparison to Probability Distribution rather than treating it as another Myerson-regular distribution instance.

References¶

Frozen Wikipedia revision: https://en.wikipedia.org/wiki/Regular_distribution_(economics) (revision 1304906615).
DOI: https://doi.org/10.1145/2728732.2728733
DOI: https://doi.org/10.1287/moor.6.1.58
Supporting reference preserved in the packet: http://jasonhartline.com/MDnA/
Supporting reference preserved in the packet: http://www.cs.princeton.edu/courses/archive/spr08/cos444/papers/bulow_klemperer96

The frozen Wikipedia revision is discovery provenance. The cited source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; URL transport failure alone was not treated as substantive contradiction.