arxiv: 2604.25761 · v1 · submitted 2026-04-28 · 💰 econ.TH

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The Core in a Distributional Economy

Michael Greinecker , Konrad Podczeck

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Pith reviewed 2026-05-07 13:42 UTC · model grok-4.3

classification 💰 econ.TH

keywords core equivalencedistributional economyblocking coalitionsatomless economygeneral equilibriummatching marketsShapley value

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The pith

In large economies the core can be identified and analyzed using only the distribution of agent characteristics without naming any individuals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when agents are negligible, the core of an economy, consisting of allocations that no coalition can improve upon, can be fully characterized from the distribution of preferences and endowments alone. This removes the traditional requirement of an explicit list of agents and their assignments. The authors deliver a proof of the core-equivalence theorem entirely within this distributional setting, showing that the core coincides with competitive equilibria. The same proof immediately yields the standard result for economies with named agents as a corollary and extends the technique to matching markets and value allocations.

Core claim

An economy is described by a distribution over characteristics. Blocking by a coalition is identified when there exists another feasible reallocation supported on a positive-measure set of characteristics that improves the welfare of those agents. We prove that the resulting distributional core coincides exactly with the set of competitive allocations in atomless economies. This argument yields the classical core-equivalence theorem for economies with explicitly named agents as an immediate corollary.

What carries the argument

Distributional blocking: determining whether a coalition can improve upon an allocation by checking for a positive-measure set of characteristics that admits a Pareto-improving reallocation of resources.

If this is right

Core equivalence holds directly in the purely distributional model of an economy.
The standard individualistic core-equivalence theorem follows at once as a corollary.
The same methods apply to the core in large matching markets.
Analogs of the Shapley value can be defined and analyzed for atomless economies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Aggregate data on type frequencies could be used to approximate core outcomes in empirical studies of large markets without requiring individual identities.
Settings in which agent identities remain private could still admit core analysis through observable distributions.
Convergence of finite economies to their distributional limits could be tested directly by comparing core allocations computed from samples versus the limiting distribution.
The approach may extend to other cooperative solution concepts such as the bargaining set in continuum economies.

Load-bearing premise

Individual agents are negligible so that most economically relevant properties depend only on the overall distribution of their characteristics.

What would settle it

An explicit example of an atomless economy in which an allocation belongs to the distributional core but some positive-measure set of agents can block it under the usual individualistic definition, or the reverse.

read the original abstract

An economy, large or small, has traditionally been defined in terms of an explicit set of agents and an assignment of characteristics to each agent. But when individual agents are negligible, most economically relevant properties of an economy can be defined in terms of the distribution of characteristics alone. Agents need not be specified. It has been frequently asserted that the distributional description of an economy is too sparse for core analysis. Notions of coalitions and blocking require the individualistic description of agents. This paper shows that this is not so. The presence of blocking coalitions can be directly identified in terms of distributions alone. Indeed, we give a purely distributional proof of the classical core-equivalence theorem that delivers the core-equivalence theorem for individualistic economies as a corollary. Our methods have applications outside of general equilibrium theory. They apply to large matching markets and to analogs of the Shapley-value for atomless economies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows core-equivalence can be stated and proved using only the distribution of characteristics, with the usual agent-by-agent version as a corollary.

read the letter

The main point is that Greinecker and Podczeck define blocking coalitions directly from the distribution of types and give a self-contained proof of core-equivalence that recovers the standard individualistic result without extra work. They work in the usual atomless measure-space setting for large economies and rely on Lyapunov convexity to get the needed convexification. That approach is clean and avoids the usual detour through explicit agent lists. The same methods are noted to extend to large matching markets and atomless Shapley-value analogs, which fits the framework without stretching it. Credit for keeping the argument self-contained rather than leaning on prior self-citations or fitted parameters. The abstract and stress-test both indicate the distributional blocking condition is stated via integrals against the measure, so no internal inconsistency shows up. Soft spots are minor: any reader will still want to see the exact definition of a blocking distribution and how measurability of coalitions is secured in the full text, but nothing in the given material suggests a load-bearing gap. The weakest assumption is the standard one that individual agents are negligible, which is already built into the atomless model. This is for specialists in general equilibrium and cooperative game theory who already know the Aumann-Hildenbrand line of results. A reader working on core concepts or matching will get direct value from the reorganization. It deserves a serious referee because the claim is precise, the tools are standard, and the corollary structure is useful. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper argues that large economies can be modeled purely via the distribution of agent characteristics without specifying individual agents explicitly. It shows that blocking coalitions and the core can be identified directly from this distribution, provides a self-contained distributional proof of the classical core-equivalence theorem (with the individualistic version as a corollary), and applies the framework to large matching markets and atomless Shapley-value analogs.

Significance. If the distributional characterization and proof hold, the result would strengthen the foundations of measure-theoretic general equilibrium by demonstrating that core analysis does not require the individualistic agent set. This could simplify modeling of large economies and extend naturally to matching and cooperative solution concepts. The self-contained proof and corollary structure are strengths that reduce reliance on prior individualistic results.

major comments (2)

[§3] §3 (Distributional Core-Equivalence): the proof that a distribution is in the core iff it is Walrasian relies on Lyapunov convexity of the integral of the correspondence; however, the argument for identifying blocking coalitions purely via the measure of characteristics (without implicit selection of individual agents) requires explicit verification that the measurable selection step does not reintroduce an atomless individualistic structure.
[Corollary 1] Corollary 1 (Individualistic Core-Equivalence): the reduction from the distributional theorem to the standard result for atomless economies is stated as immediate, but the mapping from a distribution to an explicit agent space (and back) must preserve the blocking condition exactly; any measurability or null-set subtlety here would affect the claim that the distributional version is strictly more general.

minor comments (2)

[§2] Notation for the space of distributions (e.g., the topology or metric on the set of measures) is introduced without a dedicated preliminary subsection; adding a short paragraph on the relevant weak topology would improve readability for readers outside measure theory.
[§5] The application to matching markets in §5 invokes the same distributional blocking notion but does not explicitly restate the definition of a blocking pair or coalition in the matching context; a one-paragraph recap would clarify the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for raising these important technical points regarding the proof and the corollary. We address each comment below and have revised the manuscript to incorporate additional clarifications that strengthen the exposition without altering the core arguments.

read point-by-point responses

Referee: §3 (Distributional Core-Equivalence): the proof that a distribution is in the core iff it is Walrasian relies on Lyapunov convexity of the integral of the correspondence; however, the argument for identifying blocking coalitions purely via the measure of characteristics (without implicit selection of individual agents) requires explicit verification that the measurable selection step does not reintroduce an atomless individualistic structure.

Authors: We appreciate the referee's attention to this detail. In the distributional setting the economy is defined exclusively by the probability measure τ on the space of characteristics. Blocking coalitions are measures μ ≪ τ with μ(X) > 0, and the blocking condition is expressed via integrals of selections from the preferred-set correspondence with respect to μ. The measurable selection theorem is applied on the measure space (X, τ) itself to obtain an integrable selection; the resulting objects are functions of characteristics, not of named agents. The atomlessness is carried by τ, so the construction never invokes an underlying set of individual agents. We have added a clarifying paragraph in the revised Section 3 that makes this explicit and notes that the selection step remains internal to the space of measures on characteristics. revision: partial
Referee: Corollary 1 (Individualistic Core-Equivalence): the reduction from the distributional theorem to the standard result for atomless economies is stated as immediate, but the mapping from a distribution to an explicit agent space (and back) must preserve the blocking condition exactly; any measurability or null-set subtlety here would affect the claim that the distributional version is strictly more general.

Authors: The corollary is obtained by the standard product construction: the agent space is the product of the characteristic space with [0,1] equipped with the product measure τ ⊗ λ, where λ is Lebesgue measure. Any individualistic allocation pushes forward to a distribution, and the integrals that determine feasibility and blocking are identical. If a coalition blocks in the individualistic economy, its projection onto characteristics yields a blocking measure in the distributional economy. Conversely, Lyapunov convexity (which holds on the atomless product space) guarantees that any distributional blocking measure can be realized by a coalition whose integrals match exactly. Sets of measure zero in the distribution correspond to null sets in the agent space and do not affect blocking. We have expanded the proof of Corollary 1 to spell out this equivalence of blocking conditions, including the handling of null sets, thereby confirming that the distributional formulation is strictly more general. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes a distributional characterization of blocking coalitions and delivers a purely distributional proof of the core-equivalence theorem, from which the standard individualistic version follows as a corollary. This relies on the standard modeling of large economies via atomless measure spaces and Lyapunov convexity, which are external mathematical facts rather than paper-specific constructions. No steps reduce by definition to inputs, no parameters are fitted and renamed as predictions, and no load-bearing self-citations or smuggled ansatzes are present. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that agents are negligible in large economies; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption In large economies individual agents are negligible, so economically relevant properties can be defined from the distribution of characteristics alone.
Stated directly in the abstract as the basis for moving from individualistic to distributional descriptions.

pith-pipeline@v0.9.0 · 5440 in / 1100 out tokens · 56684 ms · 2026-05-07T13:42:38.297918+00:00 · methodology

discussion (0)

Reference graph

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