arxiv: 2604.06050 · v2 · submitted 2026-04-07 · 💰 econ.TH

Recognition: no theorem link

Robust Testing Of the Allais Paradox By Paired Choices vs. Paired Valuations

Federico Echenique , Gerelt Tserenjigmid

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:27 UTC · model grok-4.3

classification 💰 econ.TH

keywords common ratio effectAllais paradoxstochastic choicepaired choicesvaluation testsexpected utilityrobust testingdecision under risk

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The pith

A strong paired choice test for the common ratio effect stays unbiased under stochastic choice and shows the effect remains prevalent in data.

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

The paper challenges recent findings that the common ratio effect vanishes under valuation-based tests and instead shows those tests are biased against detecting violations of expected utility. It proves that a strengthened version of the traditional paired choice test avoids bias across standard stochastic choice models such as logit and probit. Reapplying this robust test to existing experimental datasets reveals that the common ratio effect is still highly prevalent. This matters because the common ratio effect is a central empirical challenge to expected utility theory in decisions under risk, and its persistence affects whether stochastic noise can explain away observed anomalies.

Core claim

McGranaghan et al. show that standard paired choice tests for the common ratio effect are structurally biased when choice is stochastic and propose valuation tests as a robust alternative, finding no systematic evidence for the effect. We argue that valuation tests are inherently biased and lack predictive power under standard expected utility assumptions. In contrast, we advocate for a strong paired choice test, proving it remains robustly unbiased across common models of stochastic choice. Applying this strong test to existing experimental data, we find that the common ratio effect remains highly prevalent.

What carries the argument

The strong paired choice test, a reinforced version of binary choice comparisons between lotteries that is proven unbiased for detecting the common ratio effect across stochastic choice models.

If this is right

Valuation tests will systematically under-detect violations of expected utility because they are biased even when the true model satisfies EU.
The common ratio effect survives as a robust empirical finding once testing procedures account for stochastic choice.
Stochastic choice models alone do not overturn the Allais paradox in the common ratio domain.
Existing experimental data on probability scaling in choices continue to support non-expected-utility behavior when analyzed with the strong test.
Testing protocols for other Allais-type violations should prioritize strong paired choice designs over valuation elicitations.

Where Pith is reading between the lines

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

The result implies that explanations for the Allais paradox must go beyond simple randomness in choice and address systematic preference patterns.
Similar robustness checks could be applied to the common consequence effect or other EU violations to test whether they also survive unbiased methods.
The bias in valuation tests may arise because continuous value reports introduce different noise structures than binary choices, a distinction worth modeling explicitly.
If the strong test is adopted widely, meta-analyses of risk preferences could be updated to down-weight older valuation-based studies.

Load-bearing premise

The common stochastic choice models such as logit and probit cover the relevant range of realistic behavior and that reanalysis of existing datasets does not introduce selection or application biases.

What would settle it

A new experiment that applies the strong paired choice test to fresh subjects and finds no systematic common ratio effect would undermine the claim that the effect remains highly prevalent under unbiased methods.

Figures

Figures reproduced from arXiv: 2604.06050 by Federico Echenique, Gerelt Tserenjigmid.

**Figure 1.** Figure 1: Possible values of (E[mAB], E[mCD]) as a function of γ when y = 30, p = 0.8 and r = 0.4. When γ = 1 we obtain the point (py, py) = (24, 24). As γ → 0 we obtain any vector in R2 ++. Proposition 1. Consider an expected utility agent with a CRRA von-NeumannMorgenstern utility function u(x) = x γ , where γ ∈ (0, 1). Fix y > 0 and p, r ∈ (0, 1). Suppose that p > 1/2. (1) For any (z1, z2) ∈ R2 ++, there exists … view at source ↗

**Figure 2.** Figure 2: Comparison of the EU, CRE, and RCRE regions under paired choice tests. Proposition 6. Consider the model in Equation (1) and let f(x) = u(x). For each of the three assumptions of Proposition 5, there exist ϵp, ϵx, ϵp,x that satisfy the assumption and E[εAB] ̸= E[εCD]. This result and its proof show that even when ϵp, ϵx, ϵp,x are independent and symmetric around zero, we may have E[εAB] ̸= E[εCD]. 4. Det… view at source ↗

**Figure 3.** Figure 3: Implications of [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of the construction in the proof of Proposition 1. Define the random variable X piecewise, conditional on the value of Z and Y , by X =    Z if Z < −d a + bY if Z ≥ −d where a = c−d 2 and b = c+d 2d . This construction is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

read the original abstract

McGranaghan, Nielsen, O'Donoghue, Somerville, and Sprenger [2024] show that standard paired choice tests for the common ratio effect are structurally biased when choice is stochastic, proposing valuation tests as a robust alternative. Using valuation tests, they find no systematic evidence for the common ratio effect, seemingly overturning much of the extant literature. We evaluate this conclusion in light of stochastic choice theory. We argue that valuation tests are inherently biased and lack predictive power under standard expected utility assumptions. In contrast, we advocate for a ``strong'' paired choice test, proving it remains robustly unbiased across common models of stochastic choice. Applying this strong test to existing experimental data, we find that the common ratio effect remains highly prevalent.

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

The paper gives a clean theoretical fix for testing the common ratio effect under stochastic choice and uses it to argue the effect is still common in old data.

read the letter

The main point here is that paired choice tests can be made robust to noise if you use the right strong version, and when you apply that version to existing experiments the common ratio effect shows up strongly again. This directly challenges the McGranaghan et al. 2024 finding that the effect vanishes under valuation tests. The theoretical part is the stronger contribution. They prove the test stays unbiased across standard stochastic models like logit and probit, which addresses the bias concern head-on and gives a concrete alternative to valuations. That robustness result is new and useful on its own. The critique that valuation tests lack power under expected utility plus noise is also straightforward and worth having on the record. The reanalysis is where things get thinner. They report high prevalence after applying the strong test to prior datasets, but those experiments were not designed around the exact pairing structure the test needs. Any subsetting or post-hoc pairing could tilt the sample, and the abstract does not detail how much of that happened or how they checked for it. The theoretical unbiasedness does not automatically protect against that selection issue in legacy data. The paper stays coherent and engages the stochastic choice literature without obvious contradictions. It is aimed at experimental and behavioral economists who run or interpret tests of risk preferences. Readers who care about Allais-type violations or how to handle noisy choice data will find the test itself worth looking at. It deserves a serious referee because the methodological contrast is direct and the robustness claim is checkable. I would send it to peer review, with the data application section getting the closest look.

Referee Report

2 major / 2 minor

Summary. The paper critiques McGranaghan et al. (2024) for concluding that the common ratio effect lacks systematic evidence when using valuation tests, which they proposed as robust to stochastic choice. It argues instead that valuation tests are inherently biased and lack predictive power under standard expected utility assumptions with stochastic choice. The authors advocate a 'strong' paired choice test, prove it is robustly unbiased across common stochastic choice models (e.g., logit, probit), and reapply it to existing experimental data to conclude that the common ratio effect remains highly prevalent.

Significance. If the unbiasedness proof for the strong paired choice test holds and the reanalysis avoids selection biases, the result would restore the common ratio effect as a prevalent phenomenon in the Allais paradox literature, challenging recent valuation-based findings and underscoring the sensitivity of anomaly detection to test design under stochastic choice. The work provides a model-independent theoretical argument and an empirical reanalysis that could shift methodological recommendations in behavioral economics experiments.

major comments (2)

[Empirical reanalysis section] The reanalysis of pre-existing datasets to apply the strong paired choice test: subsetting observations to satisfy the stricter pairing structure (or imputing missing pairs) risks selection on unobservables or altering the effective sample in ways that could inflate prevalence estimates. The manuscript should detail the exact matching procedure, report sample sizes before/after subsetting, and include robustness checks (e.g., comparing to full samples or alternative pairings) to address this load-bearing concern for the empirical claim.
[Theoretical section on strong test] The proof of robustness for the strong paired choice test across stochastic models: while the abstract states it remains unbiased for common models like logit and probit, the derivation must explicitly enumerate the full set of models covered and demonstrate that the test statistic's expectation is zero under each (independent of the common ratio violation). If any realistic stochastic model is omitted, the 'robustly unbiased' claim is weakened.

minor comments (2)

[Introduction] Clarify notation for the 'strong' paired choice test versus standard paired choice tests early in the paper to avoid confusion with the McGranaghan et al. terminology.
[Comparison table] Add a table summarizing the key properties (bias, power) of valuation tests, standard paired choice tests, and the proposed strong test under EU and stochastic choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment point by point below, indicating where revisions will be made to improve the manuscript.

read point-by-point responses

Referee: [Empirical reanalysis section] The reanalysis of pre-existing datasets to apply the strong paired choice test: subsetting observations to satisfy the stricter pairing structure (or imputing missing pairs) risks selection on unobservables or altering the effective sample in ways that could inflate prevalence estimates. The manuscript should detail the exact matching procedure, report sample sizes before/after subsetting, and include robustness checks (e.g., comparing to full samples or alternative pairings) to address this load-bearing concern for the empirical claim.

Authors: We agree that the empirical reanalysis requires additional transparency to address potential selection concerns. In the revised manuscript, we will provide a detailed description of the exact matching procedure used to subset observations for the strong paired choice test. We will report sample sizes before and after subsetting for each dataset. We will also add robustness checks, such as comparisons to the full samples and alternative pairing methods, to confirm that the prevalence estimates of the common ratio effect are not driven by the subsetting process. These revisions will directly strengthen the empirical section. revision: yes
Referee: [Theoretical section on strong test] The proof of robustness for the strong paired choice test across stochastic models: while the abstract states it remains unbiased for common models like logit and probit, the derivation must explicitly enumerate the full set of models covered and demonstrate that the test statistic's expectation is zero under each (independent of the common ratio violation). If any realistic stochastic model is omitted, the 'robustly unbiased' claim is weakened.

Authors: We thank the referee for this suggestion to enhance the explicitness of the theoretical proof. The current derivation establishes that the strong paired choice test is unbiased under standard stochastic choice models, including logit and probit. In the revised manuscript, we will explicitly enumerate the full set of models covered (logit, probit, and other common variants such as tremble models) and include step-by-step derivations showing that the expectation of the test statistic is zero under each model, independent of any common ratio violation. This will make the robustness claim more precise. revision: yes

Circularity Check

0 steps flagged

Theoretical unbiasedness proof and external-data reanalysis are independent of each other

full rationale

The paper's core derivation is a mathematical proof that a proposed 'strong' paired-choice test statistic remains unbiased under standard stochastic choice models (logit, probit, etc.). This proof is presented as a first-principles result and does not rely on fitting parameters to the target data or on self-citations for its validity. The subsequent empirical claim—that the common-ratio effect is prevalent—arises from applying the already-proven test to pre-existing experimental datasets collected by other researchers. No equation reduces to its own input by construction, no fitted quantity is relabeled as a prediction, and no load-bearing premise collapses to a self-citation chain. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on standard stochastic choice theory and expected utility as background; it introduces the strong test as a new procedure without additional fitted parameters or new entities.

axioms (2)

domain assumption Common models of stochastic choice (logit, probit, etc.) describe realistic choice behavior.
The proof that the strong test is unbiased is stated to hold across these models.
domain assumption Valuation tasks lack predictive power under standard expected utility assumptions when choice is stochastic.
Central argument against the 2024 valuation approach.

invented entities (1)

Strong paired choice test no independent evidence
purpose: A version of paired choice testing that remains unbiased for the common ratio effect under stochastic choice.
Newly advocated and proven in the paper.

pith-pipeline@v0.9.0 · 5427 in / 1419 out tokens · 76030 ms · 2026-05-10T18:27:23.716211+00:00 · methodology

discussion (0)

Reference graph

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