Valuing Winners: When and How to Correct for Selection Bias in Randomized Experiments

Hangcheng Zhao; Ron Berman; Walter W. Zhang

arxiv: 2605.18887 · v1 · pith:7LIC7X4Nnew · submitted 2026-05-16 · 💰 econ.EM · econ.GN· q-fin.EC· stat.AP

Valuing Winners: When and How to Correct for Selection Bias in Randomized Experiments

Ron Berman , Walter W. Zhang , Hangcheng Zhao This is my paper

Pith reviewed 2026-05-20 14:19 UTC · model grok-4.3

classification 💰 econ.EM econ.GNq-fin.ECstat.AP

keywords winner's curseselection biasrandomized experimentsA/B testingempirical likelihoodbias correctionregretmulti-armed trials

0 comments

The pith

Different methods for correcting winner's curse bias in randomized experiments perform best under different conditions, with no approach dominating all evaluation targets.

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

When decision-makers select the top treatment from a multi-arm randomized experiment, the observed performance of that winner is inflated by statistical noise, creating a winner's curse that leads to overestimating its value and potentially choosing a suboptimal option. The paper separates this into global bias relative to the true best treatment and selective bias relative to the chosen treatment's actual mean, then ties both to the regret of deploying the wrong arm. It evaluates seven concrete targets—mean bias, mean squared error, and confidence-interval coverage for each type of bias, plus mean regret—across simulations that vary effect sizes and arm counts, including data drawn from an online A/B testing platform. Results show that plug-in estimators excel when differences are large, cross-fitting when arms are close, and resampling when differences are moderate, while a new adaptive empirical likelihood method produces asymptotically valid intervals without the tuning problems of resampling.

Core claim

The paper establishes that bias-correction methods for the winner's curse must be chosen to match the manager's objective, because performance rankings shift with effect sizes and across the seven decision-relevant metrics of mean bias, mean squared error, and confidence-interval coverage for global and selective bias plus mean regret. Simulations demonstrate that the plug-in estimator yields the lowest errors when treatment differences are large, cross-fitting performs best when treatments are similar, resampling methods achieve low mean squared error for moderate differences, and the introduced adaptive empirical likelihood procedure supplies asymptotically valid confidence intervals in a

What carries the argument

The distinction between global winner's curse (bias relative to the true best treatment) and selective winner's curse (bias relative to the selected treatment's true mean), which connects selection to regret from choosing a suboptimal arm and defines the seven evaluation targets used to compare correction methods.

If this is right

Decision-makers facing large treatment differences can use simple plug-in estimators without substantial loss in accuracy for valuing the winner.
When treatment arms produce similar outcomes, cross-fitting corrections reduce both global and selective bias more reliably than alternatives.
Resampling-based corrections minimize mean squared error when effect sizes are moderate, supporting more precise regret calculations.
The adaptive empirical likelihood procedure allows construction of valid intervals without choosing tuning parameters that resampling methods require.

Where Pith is reading between the lines

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

Platforms running repeated A/B tests could monitor observed effect sizes in real time and switch correction methods automatically to reduce cumulative regret.
The global-versus-selective distinction may extend to non-randomized selection problems such as ranking products from noisy sales data.
Sequential experiments that add arms over time would likely introduce additional forms of selection bias not covered by the current static framework.

Load-bearing premise

The performance rankings and the asymptotic validity of the adaptive empirical likelihood procedure depend on the specific simulation designs and data-generating processes calibrated to an online A/B testing platform.

What would settle it

A field experiment with known true treatment means in which the adaptive empirical likelihood confidence intervals fail to achieve the nominal coverage rate would refute the claim of asymptotic validity across settings.

Figures

Figures reproduced from arXiv: 2605.18887 by Hangcheng Zhao, Ron Berman, Walter W. Zhang.

**Figure 1.** Figure 1: Three definitions of winner’s curse Note: The selected arm is kˆ while the truly best arm is k ∗ . Although the true means ensure µkˆ < µk∗ in the population, the estimated means in the sample may lead to ˆµkˆ > µˆk∗ The lines illustrate the values of W Cglobal, W Cselect and the Regret and are linked in our identity Equation (5). ing confidence intervals, which may lead managers to decide against deployme… view at source ↗

**Figure 2.** Figure 2: Winner’s Curse Bias (Global) Notes: WC Global as a function of sample size. Panels vary by ∆µ (columns) and σ (rows). Methods: (1) standard nonparametric bootstrap; (2) m-out-of-n bootstrap with m = ⌊N γ ⌋ and γ ∈ {0.45, 0.95}; (3) parametric shrinkage bootstrap (ParametricPB); (4) (Fang and Santos, 2019) Hadamard-derivative based bootstrap; (5) sample splitting; (6) cross-fitting; (7) AKM24 hybrid and (8)… view at source ↗

**Figure 4.** Figure 4: Mean Squared Error (Global) Notes: Mean squared error (MSE) for WC Global as a function of sample size N. Panels vary by ∆µ (columns) and σ (rows). 22 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

**Figure 6.** Figure 6: Coverage (Global) Notes: Empirical coverage of α-level two-sided confidence intervals for µk∗ , i.e., Pr (µk∗ ∈ CIα(ˆµkˆ)), as a function of sample size N. Panels vary by ∆µ (columns) and σ (rows). EL denotes empirical likelihood. Mean squared error (MSE) Figures 4 and 5 summarize the MSE trade-off. The ranking of methods depends on whether the problem is a tie/near-tie or a clear-winner regime. In ties 23… view at source ↗

**Figure 8.** Figure 8: Winner’s Curse Bias (Binomial, Fixed Conversion Rate) [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

**Figure 10.** Figure 10: Coverage (Binomial, Fixed Conversion Rate) [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

read the original abstract

Decision-makers often deploy the best-performing treatment from a randomized experiment, creating a winner's curse: selection favors treatments whose observed outcomes are high partly because of statistical noise, so the na\"ive estimate of the winner is upward biased. We distinguish two forms of winner's curse, bias relative to the true best treatment (global) and bias relative to the selected treatment's true mean (selective), and link them to regret from deploying a suboptimal treatment. This framework defines seven decision-relevant evaluation targets: mean bias, mean squared error, and confidence interval coverage for the global and selective winner's curse, and mean regret. We then show that methods that perform well on one target can perform poorly on others, so corrections should be matched to the manager's objective. Across simulations with varying effect sizes, multiple-arm settings, and data calibrated to an online A/B testing platform, no method dominates uniformly: the plug-in estimator performs best when treatment differences are large, cross-fitting performs best when treatments are similar, and resampling methods often achieve low mean squared error for moderate differences. We also introduce an adaptive empirical likelihood procedure that delivers asymptotically valid confidence intervals across settings without the tuning sensitivity of resampling-based methods.

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 useful framework for picking winner's-curse corrections by objective and effect size, plus a new adaptive EL method for CIs, but the rankings rest on one platform's simulation setup.

read the letter

The main thing to know is that no correction for the winner's curse works best in every case. The paper shows you should match the method to whether you care about global bias, selective bias, or regret, and to how large the treatment differences are. Plug-in looks good for big effects, cross-fitting for small ones, and resampling for middling cases. They also add an adaptive empirical likelihood procedure that aims for valid intervals without the tuning issues of resampling methods. That distinction between global and selective bias and the link to seven concrete targets including regret is the clearest new piece. It turns the usual bias discussion into something more directly tied to what a decision-maker loses by picking the wrong arm. The simulations are calibrated to real A/B testing data, which makes the comparisons feel grounded rather than abstract. The patterns they report are consistent enough to be worth paying attention to for anyone running multi-arm experiments. The soft spot is that everything on method rankings and asymptotic coverage depends on the specific effect-size distributions and arm-independence assumptions in those simulations. If real deployments have heavier tails or dependence across arms, the ordering could change and the coverage claims might not hold as cleanly. The abstract does not give the full derivations or extra robustness checks, so those would need to be verified in the full manuscript. This is for applied econometricians and experimenters who deploy winners from randomized tests and want to reduce regret without over-correcting. Readers who already think about matching estimators to objectives will find it practical. The framework is clear and the evidence is targeted, so it deserves a serious referee even if more generalizability checks would help.

Referee Report

2 major / 2 minor

Summary. The paper examines selection bias (winner's curse) when deploying the best-performing arm from a randomized experiment. It distinguishes global bias (relative to the true best treatment) from selective bias (relative to the selected arm's true mean), links both to regret, and defines seven decision-relevant targets: mean bias, MSE, and CI coverage for each bias type plus mean regret. Simulations across effect-size regimes, arm counts, and A/B-platform-calibrated data show that no correction method dominates uniformly; the plug-in estimator excels for large treatment differences, cross-fitting for similar arms, and resampling for moderate differences. The authors introduce an adaptive empirical likelihood procedure claimed to deliver asymptotically valid confidence intervals without the tuning sensitivity of resampling methods.

Significance. If the simulation-based performance rankings and the asymptotic validity of the adaptive EL procedure hold beyond the specific DGPs, the paper supplies actionable guidance for experimenters on matching bias-correction tools to managerial objectives (bias vs. regret vs. coverage) and offers a tuning-robust inference method. The explicit mapping from bias measures to regret and the multi-target evaluation framework are useful contributions to the econometrics of selection in experiments.

major comments (2)

[Simulation section (and abstract)] The central performance-ordering claim (plug-in best for large differences, cross-fitting for similar arms, resampling for moderate) and the asymptotic-validity claim for adaptive EL both rest on simulation designs calibrated to one online A/B platform with particular effect-size distributions and arm-independence assumptions (see abstract and the simulation section). These are not generic; if the true effect-size law or dependence structure differs, both the 'match method to objective' recommendation and the 'valid across settings without tuning' claim can fail while the formal definitions of the seven targets remain intact. Full details of the DGP, including the distribution from which arm effects are drawn and any cross-arm correlation structure, are required to assess external validity.
[Asymptotic procedure / adaptive EL section] The manuscript reports simulation results showing method performance varies by setting and introduces an asymptotic procedure for the adaptive EL, but without full derivations, explicit verification of coverage properties, or sensitivity checks to the independence and effect-size assumptions, the soundness of the asymptotic-validity claim receives only moderate support. A formal proof sketch or at least the key steps establishing asymptotic coverage under the paper's maintained conditions would strengthen the contribution.

minor comments (2)

[Section 2 or 3] Clarify the exact definition of the seven targets in a single table or numbered list early in the paper so that later simulation results can be directly mapped to each target.
[Simulation design subsection] Provide more detail on how the A/B-platform data are calibrated (sample sizes, variance estimates, number of arms) so readers can judge how representative the simulation designs are.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us improve the clarity and robustness of our paper. We address each of the major comments in turn below.

read point-by-point responses

Referee: [Simulation section (and abstract)] The central performance-ordering claim (plug-in best for large differences, cross-fitting for similar arms, resampling for moderate) and the asymptotic-validity claim for adaptive EL both rest on simulation designs calibrated to one online A/B platform with particular effect-size distributions and arm-independence assumptions (see abstract and the simulation section). These are not generic; if the true effect-size law or dependence structure differs, both the 'match method to objective' recommendation and the 'valid across settings without tuning' claim can fail while the formal definitions of the seven targets remain intact. Full details of the DGP, including the distribution from which arm effects are drawn and any cross-arm correlation structure, are required to assess external validity.

Authors: We agree that providing full details of the data-generating process (DGP) is essential for evaluating the external validity of our simulation results. In the revised manuscript, we will include a complete description of the DGP, specifying the distribution from which arm effects are drawn and the cross-arm correlation structure (which is assumed to be zero under independence). While our simulations are calibrated to realistic A/B testing scenarios, we acknowledge that the performance rankings may vary under different effect-size distributions or dependence structures. We will add a discussion of these limitations and emphasize that the recommendations are conditional on the settings considered. The multi-target framework itself remains general and independent of specific DGPs. revision: yes
Referee: [Asymptotic procedure / adaptive EL section] The manuscript reports simulation results showing method performance varies by setting and introduces an asymptotic procedure for the adaptive EL, but without full derivations, explicit verification of coverage properties, or sensitivity checks to the independence and effect-size assumptions, the soundness of the asymptotic-validity claim receives only moderate support. A formal proof sketch or at least the key steps establishing asymptotic coverage under the paper's maintained conditions would strengthen the contribution.

Authors: We appreciate this suggestion. The current version relies primarily on simulation evidence for the adaptive empirical likelihood (EL) procedure. In the revision, we will provide a formal proof sketch outlining the key steps for establishing asymptotic coverage under the maintained assumptions of arm independence and the specified effect-size distributions. Additionally, we will include sensitivity analyses to assess robustness to violations of these assumptions. This will strengthen the support for the asymptotic validity claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper links bias measures to regret and defines seven evaluation targets using standard statistical definitions rather than by construction from fitted quantities. Performance rankings across methods are obtained from explicit simulations under varying effect sizes and A/B-platform-calibrated DGPs, while the adaptive empirical likelihood procedure is introduced as an independent statistical contribution with asymptotic validity claims. No load-bearing step reduces to a self-citation chain, a renamed known result, or an input parameter presented as a prediction; the framework remains self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard asymptotic theory for empirical likelihood and on simulation designs whose details are not supplied in the abstract; no free parameters or invented entities are explicitly introduced.

axioms (1)

domain assumption Standard regularity conditions for asymptotic validity of empirical likelihood estimators
Invoked to support the claim that the adaptive procedure delivers valid confidence intervals across settings.

pith-pipeline@v0.9.0 · 5751 in / 1268 out tokens · 49633 ms · 2026-05-20T14:19:20.545653+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We distinguish two forms of winner’s curse, bias relative to the true best treatment (global) and bias relative to the selected treatment’s true mean (selective), and link them to regret... We also introduce an adaptive empirical likelihood procedure that delivers asymptotically valid confidence intervals
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the maximum function is convex... standard approaches that rely on the Delta method will fail at the kink

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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work page doi:10.1287/mksc.2019.1194 2019

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Journal of the American Statistical Association , volume=

Steven G. Self and Kung-Yee Liang , title =. Journal of the American Statistical Association , volume =. 1987 , publisher =. doi:10.1080/01621459.1987.10478472 , URL =

work page doi:10.1080/01621459.1987.10478472 1987

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Megastudies improve the impact of applied behavioural science , author=. Nature , volume=. 2021 , publisher=

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Non-stationary experimental design under linear trends , author=. Advances in Neural Information Processing Systems , volume=

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work page 1982

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Management Science , volume =

Berman, Ron and Van den Bulte, Christophe , title =. Management Science , volume =. 2022 , doi =. https://doi.org/10.1287/mnsc.2021.4207 , abstract =

work page doi:10.1287/mnsc.2021.4207 2022

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, title =

Bechhofer, Robert E. , title =. The Annals of Mathematical Statistics , year =

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work page

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Econometrica , volume =

Gu, Jiaying and Koenker, Roger , title =. Econometrica , volume =. doi:https://doi.org/10.3982/ECTA19304 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA19304 , abstract =

work page doi:10.3982/ecta19304

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PNAS Nexus , volume =

Milkman, Katherine L and Ellis, Sean F and Gromet, Dena M and DeMay, Isabella M and Graci, Heather N and Jung, Youngwoo and Mobarak, Rayyan S and Silvera Zumaran, Ramon A and Simmons, Mia N and Van den Bulte, Christophe and Benartzi, Shlomo and Hilchey, Matthew and Goodyear, Laura and Karlan, Dean and Mazar, Nina and Mochon, Daniel and Shah, Avni M and So...

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