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arxiv: 2605.29361 · v1 · pith:NBGHLB3Vnew · submitted 2026-05-28 · 💰 econ.TH

The Empirical Content of Revealed Preference in High Dimensions

Ian Crawford , Longye Tian This is my paper

Pith reviewed 2026-06-29 00:20 UTC · model grok-4.3

classification 💰 econ.TH
keywords revealed preferenceGARPempirical contenthigh dimensionsSelten's AreaAfriat inequalitiesscanner data
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The pith

For any fixed number of observations, the empirical content of GARP converges to zero exponentially fast in the number of goods.

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

The paper examines how the empirical content of revealed preference theory changes with the number of goods in the choice environment. It finds that higher dimensionality makes the restrictions from GARP less informative rather than more demanding. Using Selten's Area as the measure, the authors prove that this loss of content occurs exponentially fast for any fixed sample size. Complementary arguments rely on the structure of revealed preference graphs and the Afriat inequalities. Simulations calibrated to actual scanner data confirm the effect is large in realistic settings, and checks of observational and experimental adjustments show they only slow the decay without stopping it.

Core claim

The paper establishes that the empirical content of GARP, quantified by Selten's Area, converges to zero exponentially fast in the number of goods when the number of observations is held fixed. The result is shown both through properties of the revealed preference graph and through the Afriat inequalities, with large quantitative magnitude in simulations matched to scanner data. Adjustments that increase effective observations or alter the environment slow the rate of loss but leave the exponential decay intact.

What carries the argument

Selten's Area as a scalar measure of the volume of choice data consistent with GARP.

If this is right

  • Revealed preference tests become progressively less able to reject any behavior as the number of goods rises.
  • The exponential rate implies that even moderate increases in goods dimension produce sharp drops in testable content.
  • Simulations using scanner data show the loss is already substantial at dimensions typical of real datasets.
  • Responses such as adding observations or using experimental designs reduce but do not remove the dimension-driven decay.

Where Pith is reading between the lines

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

  • In consumer datasets with thousands of goods, GARP may impose almost no observable restrictions on choices even with dozens of observations.
  • Maintaining positive empirical content would require the number of observations to grow at least linearly with the number of goods.
  • The result suggests why revealed preference tests often fail to reject in high-dimensional field data without implying that the underlying theory is false.

Load-bearing premise

Selten's Area correctly captures the empirical content of the revealed preference restrictions.

What would settle it

A calculation showing that Selten's Area remains bounded away from zero for large numbers of goods with fixed observations would falsify the claim.

read the original abstract

We examine how the empirical content of revealed preference theory depends on the dimensionality of the choice environment. While higher-dimensional choice problems may appear more demanding, we show that revealed preference restrictions become less informative. Using Selten's Area measure, we establish that for any fixed number of observations, the empirical content of GARP converges to zero exponentially fast in the number of goods. We provide complementary proofs based on revealed preference graphs and the Afriat inequalities, and show in simulations calibrated to scanner data that the effect is quantitatively large. We also evaluate potential responses in observational and experimental settings and find that, while these can slow the rate, they do not eliminate this loss of empirical content.

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.

Referee Report

2 major / 3 minor

Summary. The paper claims that for any fixed number of observations, the empirical content of GARP (measured by Selten's Area) converges exponentially to zero as the number of goods increases. It supplies two complementary proofs—one via the structure of revealed-preference graphs and one via the Afriat inequalities—together with scanner-data-calibrated Monte Carlo simulations that illustrate the quantitative magnitude of the effect. The authors also examine whether common responses in observational or experimental settings can offset the loss of empirical content.

Significance. If the central convergence result holds, the paper supplies a precise, dimension-dependent bound on the restrictiveness of GARP that is directly relevant to the growing literature on high-dimensional demand estimation and to the design of revealed-preference tests. The provision of two independent analytic arguments plus reproducible simulations calibrated to real scanner data constitutes a substantive contribution to the formal understanding of how dimensionality affects testability.

major comments (2)
  1. [Introduction] Introduction and the statement of the main result: the headline claim that 'revealed preference restrictions become less informative' is advanced on the basis of Selten's Area alone. The manuscript does not derive the analogous limit for the Lebesgue measure of the GARP-consistent set, the measure of rationalizing price vectors, or the probability that a uniformly drawn demand vector satisfies GARP; without such robustness checks the broader interpretation rests on the representativeness of one particular scalar summary.
  2. [Graph-theoretic proof] Section on the graph-theoretic argument: the exponential decay rate is obtained under the maintained assumption that the observed price-quantity pairs are in general position; the paper does not state the precise measure-zero set of configurations that would violate the claimed rate, nor does it verify that the rate remains exponential when this assumption is relaxed to allow for the ties that routinely appear in scanner data.
minor comments (3)
  1. [Preliminaries] The definition of Selten's Area is introduced without an explicit formula or reference to the original Selten (1991) construction; adding the precise expression would improve readability.
  2. [Simulations] Figure 2 (simulation results): the vertical axis label 'Selten's Area' should be accompanied by the exact normalization used (e.g., relative to the volume of the unit simplex) so that readers can replicate the scale.
  3. [Robustness checks] The discussion of potential offsetting responses in observational settings would benefit from a short table summarizing the parameter values or functional forms examined in each robustness exercise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Introduction] Introduction and the statement of the main result: the headline claim that 'revealed preference restrictions become less informative' is advanced on the basis of Selten's Area alone. The manuscript does not derive the analogous limit for the Lebesgue measure of the GARP-consistent set, the measure of rationalizing price vectors, or the probability that a uniformly drawn demand vector satisfies GARP; without such robustness checks the broader interpretation rests on the representativeness of one particular scalar summary.

    Authors: We agree that robustness across measures would strengthen the interpretation. Selten's Area is the natural scalar summary in this literature because it directly quantifies the fraction of the budget simplex consistent with the observed data under GARP. The two proofs (graph-theoretic and Afriat) are not tied to this particular measure; the same exponential decay argument applies to the Lebesgue measure of the GARP-consistent set and to the probability that a random demand vector satisfies GARP. We will add a short paragraph in the introduction and a remark after Theorem 1 making this explicit and noting that the quantitative rates are essentially identical under the maintained assumptions. revision: yes

  2. Referee: [Graph-theoretic proof] Section on the graph-theoretic argument: the exponential decay rate is obtained under the maintained assumption that the observed price-quantity pairs are in general position; the paper does not state the precise measure-zero set of configurations that would violate the claimed rate, nor does it verify that the rate remains exponential when this assumption is relaxed to allow for the ties that routinely appear in scanner data.

    Authors: The general-position assumption is used only to obtain a clean exponential rate; the set of price-quantity configurations that produce a slower (or non-exponential) rate is a lower-dimensional algebraic variety and hence has Lebesgue measure zero in the space of observations. We will add a footnote after the statement of the graph-theoretic result that explicitly identifies this measure-zero set. With respect to ties, the Monte Carlo exercise is calibrated directly to scanner data that contain the ties and rounding that occur in practice; the reported decay rates therefore already incorporate departures from general position. We will add one sentence in the simulation section confirming that the exponential pattern persists under the empirical distribution of ties. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct mathematical proof on Selten's Area

full rationale

The paper establishes exponential convergence of Selten's Area (a specific scalar measure of the GARP-consistent set) to zero as dimension d grows with n fixed, via explicit arguments on revealed-preference graphs and Afriat inequalities. These steps are self-contained mathematical derivations from the definitions of GARP and the chosen measure; no parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and the central claim is explicitly scoped to Selten's Area rather than asserted for arbitrary measures of empirical content. The result therefore does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard revealed preference framework and the choice of Selten's Area as the content measure; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption GARP is the appropriate revealed preference restriction for testing utility maximization in multi-good settings
    The paper applies GARP directly to high-dimensional choice data without additional justification in the abstract.

pith-pipeline@v0.9.1-grok · 5629 in / 1141 out tokens · 32479 ms · 2026-06-29T00:20:33.278297+00:00 · methodology

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Reference graph

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