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Decomposing Wrong-Consensus Agreement in LLM Self-Consistency: A GPT-4.1 Case Study

Lizhuo Zhang, Mengmeng Tang, Chenfeng Long, Xiaoyong Tang, Xiang Luo

arXiv:2608.18795v1cs.CLcs.AI

TL;DR

Majority voting can produce high agreement on wrong answers and even reduce accuracy on hard questions. This paper decomposes wrong-consensus agreement into mechanical per-case preference and an unexplained residual, finding that preference largely accounts for agreement on GPQA-Diamond but not on open-domain AIME. The results show that agreement is graded evidence of correctness, not certification.

  • Problem

    Majority voting can yield high agreement on wrong answers and reduce accuracy on hard questions, challenging the assumption that more agreement means greater reliability.

  • Method

    The paper uses a leak-free, difficulty-matched counterfactual to decompose wrong-consensus agreement into mechanical per-case preference and a preference-unexplained residual.

  • Results

    81–93% of agreement is mechanically explained on GPQA-Diamond versus 59–78% on AIME, where a 1.56–2.80 Γ-unit residual survives.

  • Takeaways & Limitations

    High agreement is graded evidence of correctness, never certification, and per-case preference does not fully explain wrong-consensus agreement on open-domain AIME.

  • Takeaways & Limitations

    The decomposition relies on an i.i.d. sampling assumption and cannot separate positive within-case sampling correlation from the preference-unexplained residual.

Abstract

from arXiv · show

Majority voting over multiple LLM samples is widely used to raise answer accuracy, yet its gain varies erratically: on hard questions it can even backfire. This paper gives a quantitative account of this failure. A pluralistic agreement index Gamma is defined as the expected fraction of the samples of a wrong run that agree with the consensus, normalized by a reference scale d=(1-p)/(C-1), and is decomposed into a mechanical component (what a vote delivers given only a per-case answer preference) and a preference-unexplained residual. The mechanical null is difficulty-matched and leak-free: each case is resimulated at its own accuracy and option preference, estimated from the case's other runs, so no run predicts its own agreement. On GPT-4.1 the decomposition shows benchmark-associated direction (an observational ordering over n=4 cells per benchmark, not a significance claim). On multiple-choice GPQA-Diamond, the per-case answer preference explains 81-93% of the held-out test-run agreement index: the shared-bias-dominates account over-claims here, because a wrong but attractive option the whole cohort latches onto is captured by the per-case preference channel (whether that preference is induced by shared training bias is not identified). On open-domain AIME, the mechanical preference explains only 59-78% (21-29% if shrunk to pure noise), and a preference-unexplained residual of 1.56-2.80 Gamma units survives, which a run-level preference-heterogeneity reference more than absorbs (1.4-2.1). A self-consistency backfire on hard questions is reproduced (binned voting gap down to -0.09, coupled CI [-0.12,-0.07]), and the highest-agreement bin reaches an accuracy of only 0.42-0.83, a 1.2-3.6x lift over base rate: agreement is graded evidence, not certification. No new voting method is proposed; code and evidence are committed and reproducible.

1 Introduction

This paper explains wrong-consensus agreement through a hierarchy of leak-free counterfactual nulls, separating generic plurality mechanics from model-specific correlated error. Its descriptive contribution is a quantitative decomposition rather than a new voting method or a new backfire phenomenon.

  • Motivation: Agreement can remain high when answers are wrong, and majority voting can reduce accuracy on hard questions relative to a single sample.These motivate analyzing wrong-consensus agreement rather than treating agreement as certification.
  • Problem: Existing work lacks a counterfactual decomposition distinguishing generic plurality effects from model-specific correlated error in LLM self-consistency.The paper addresses this gap with a hierarchy of counterfactual nulls for an agreement index.
  • Contribution: The paper defines a pluralistic agreement index Γ and compares real per-run data with progressively richer counterfactual nulls.Each null grants the data strictly more mechanical structure than the preceding null.
  • Results: ϕ ∈[0.806, 0.927] on multiple-choice GPQA-Diamond, with a small preference-unexplained residual.ϕ is the fraction of observed wrong-consensus agreement reproduced by a fixed-preference i.i.d. counterfactual, using preference and accuracy estimated from other runs only.
  • Scope: The paper makes a descriptive understanding claim: it proposes no new voting method and claims no new backfire phenomenon.Its claim is the quantitative counterfactual hierarchy for Γ.

2 Related work

Prior work established self-consistency, agreement-based confidence, and ensemble decompositions, while this paper’s distinct contribution is a leak-free Γ decomposition rather than a new voting method or discovery of backfire.

  • Self-consistency and its limits: Self-consistency began with sampling-based voting, while later audits showed its accuracy gains are not uniform.The paper builds on prior empirical documentation of self-consistency’s limits.
  • Agreement as confidence: Agreement-based confidence spans calibration, consistency hypotheses, Bayes-risk unification, semantic entropy, and disciplined measurement protocols.These literatures motivate the paper’s agreement-statistics reporting conventions.
  • Ensemble decompositions: Ensemble theory separates vote-marginal effects from error covariance, providing the vocabulary for the paper’s wrong-run-conditional agreement index Γemp.The LLM-side sharpened answer marginal is the target of per-case preference estimation.
  • Note on prior series: A related vision diagnostic studies consensus and annotation difficulty, but this paper’s agreement decomposition is self-contained and independent of that companion analysis.The companion diagnostic uses a cross-model consensus control.

3 Method

The method measures agreement among samples from wrong-consensus runs and decomposes it against leak-free, difficulty-matched mechanical counterfactuals. Per-case answer preferences are separated from residual clustering, with identical wrong-consensus conditioning across empirical and simulated runs.

  • Data and setup: For each case, K samples under a fixed prompt produce single-sample accuracy p, option count C, plurality consensus maj, and consensus accuracy.GPQA-Diamond uses C = 4; open-ended tasks use the mean number of distinct answer strings per run.
  • Robustness: ϕ shifts by at most 0.009 in any cell under randomized tie-breaking, indicating that the lowest-class-id tie rule does not materially affect the decomposition.K = 50 is even, so ties are possible; empirical tie rates are 7.5% for AIME and 0.8% for GPQA.
  • Agreement index: Γemp normalizes the average sample self-consistency α of wrong-consensus runs by d=(1−p)/(C−1), measuring how tightly samples adhere to a wrong plurality.The reference scale d is not a chance-correction claim or assumed uniform data model.
  • Mechanical counterfactuals: Two i.i.d. multinomial counterfactuals estimate mechanical agreement: Γiid distributes incorrect votes uniformly, while Γrival preserves leak-free per-case option preferences estimated from other runs.Both references recompute each simulated run’s plurality and apply the same wrong-consensus conditioning as the empirical index.
  • Coverage and decomposition: ϕ=Γrival/Γ(t)emp reports mechanical coverage on the held-out test-run population, while Γemp is also reported and differs from Γ(t)emp by at most 4% in any cell.The held-out population prevents a run from predicting its own agreement.
  • Coverage and decomposition: Γrival,pool is far below Γrival in every cell, including 1.01 versus 4.26 on gpt-4.1 AIME, showing that mechanical agreement is carried by per-case rather than global preference.The interpretation distinguishes preference-explained agreement from a preference-unexplained residual; shared training bias itself is not identified.

4 Data

The study is a secondary re-analysis of public per-run self-consistency data, covering eight model–benchmark–prompt cells with K = 50 answers per run. It analyzes GPT-4.1-family models on GPQA-Diamond and a released 196-case AIME subset under zero-shot and chain-of-thought prompting where reported.

  • Data sources: The dataset contains K = 50 per-run answers, correctness, self-consistency α, single-sample accuracy p, and majority labels for each case and model.Eight model–benchmark–prompt cells are analyzed.
  • Data sources: The analyzed models are gpt-4.1, gpt-4.1-mini, and gpt-4.1-nano.The benchmarks are GPQA-Diamond and Ding’s 196-case AIME subset, spanning problems from 1983–2025.
  • Experimental cells: The analysis covers zero-shot and chain-of-thought prompting where reported.The AIME subset was inherited from the release rather than selected by the authors.
  • Reproducibility: This is a secondary re-analysis: no model is run, and every reported number is reproducible from a committed parquet file.Only per-run rows from Ding’s public data are re-analyzed.

5 Results

The results show that wrong-consensus agreement is strongly structured by per-case answer preference, especially on GPQA, but leaves a substantial unexplained component on open-domain AIME. Majority voting also backfires on the hardest questions, while agreement provides predictive evidence that remains far from certification.

  • Agreement decomposition: Γemp ∈[3.56, 7.09] in every cell, showing wrong-run samples are strongly attracted to the consensus; ϕ is used for cross-benchmark comparisons.The Γ scale depends on the number of answer choices, so magnitude comparisons across benchmarks are avoided.
  • Benchmark-associated direction: ϕ ∈[0.806, 0.927] on GPQA-Diamond, so per-case preference reproduces most agreement and makes the shared-bias-dominates account over-claim.The preference-unexplained residual is δ = 0.07–0.19 (0.34–0.88 Γ units); preference origin is not identified.
  • Benchmark-associated direction: 59–78% on AIME is mechanically reproduced by preference, leaving 1.56–2.80 Γ units unexplained and indicating residual clustering beyond estimated per-case option preference.The corresponding coverage is ϕ ∈ [0.586, 0.781] and δ ∈[0.22, 0.41].
  • Benchmark-associated direction: 0.19 is the GPQA–AIME mean ϕ difference, with 95% coupled bootstrap CI [0.11, 0.28] and permutation p = 2/70 ≈0.029; this is an ordering, not significance evidence.Boundary-cell CIs overlap, and within-benchmark ϕ is not monotone in single-sample accuracy.
  • Voting backfire: −0.09 is the voting gap for gpt-4.1-nano GPQA-ZS cases with p ≈0.13, with coupled case-level bootstrap CI [−0.12, −0.07].The hardest bins show the most negative gaps, reproducing majority-voting backfire on difficult questions.
  • Winner stability: 0.40 on GPQA and 0.82 on AIME are the overall flip rates between nominally identical K = 50 runs, with AIME inflated by its larger candidate set.The rates indicate substantial plurality-winner instability, especially for open-ended answers.
  • Confidence aggregation: 0.769 vs. 0.770 pooled AUROC shows no appreciable improvement from soft distributional confidence over hard plurality concentration H.The difference stays within 0.02 on every cell, so soft weighting is reported as an understanding result rather than a new voting method.

6 Limitations and conclusion

The paper decomposes wrong-consensus agreement into a mechanical plurality effect and a preference-unexplained residual, with benchmark-associated patterns across GPQA-Diamond and AIME. Its conclusions are limited by correlated temperature draws, while the falsifiable results show per-case preference largely suffices on GPQA but not AIME, where agreement remains graded evidence rather than certification.

  • Limitations: Correlated temperature draws limit identification: under positive exchangeable correlation, δ is an upper bound on the correlation-free residual, but arbitrary correlation leaves direction unidentified.The counterfactuals are therefore benchmark references rather than certified bounds.
  • Conclusion: 0.42–0.83 is the ceiling for high-agreement accuracy, showing that wrong-run samples can share the consensus and agreement is graded evidence rather than certification.The ceiling is reported for the high-agreement regime in Section 5.2.
  • What this design identifies: ≥81% of the GPQA index is reproduced by the correlation-free per-case preference reference, so shared-bias-dominant accounts over-claim there.The preference channel captures attraction to a wrong consensus option, although the origin of that preference is not identified.
  • Falsifiability: ϕ ∈[0.586, 0.781] on AIME leaves a surviving preference-unexplained residual, falsifying a purely per-case preference account of open-domain saturation.A calibrated run-level preference-heterogeneity null more than absorbs this residual.
  • Conclusion: The paper’s data, analysis scripts, and JSON evidence files are committed in full and designed to regenerate every reported number from the raw per-run data.An anonymized copy is intended for venue repository access.

A The pooled-p control fails

The pooled-p reference fails as a counterfactual because averaging accuracy across difficulty levels collapses predicted wrong-consensus rates. Difficulty-matched resimulation instead reproduces the empirical consensus–wrong share.

  • Control comparison: Table 7 defines wc as the share of runs whose plurality label is wrong, comparing empirical, difficulty-matched i.i.d., and pooled-p predictions.The difficulty-matched reference is denoted wcperq, while the pooled-p reference is denoted wcpooled.
  • GPQA results: ≤0.6% pooled-p predictions on three of four GPQA cells, versus 42–60% observed and 29–46% recovered by the difficulty-matched reference.For 4.1-nano GPQA-ZS, the pooled-p prediction reaches 8.8%.
  • Why the control fails: 0.8 to 3 orders of magnitude is the pooled-p understatement across GPQA cells, because it treats every case as average difficulty under one global accuracy.Pooling over difficulty therefore attributes essentially all correctness to a single global accuracy.

B Second model family (Qwen3.5-9B)

The Qwen3.5-9B check reuses earlier ctx4k sampling under a different protocol on MMLU and MMLU-Pro, while reporting a uniform per-question null rather than the later leak-free rival null.

  • Protocol: Qwen3.5-9B was evaluated on MMLU and MMLU-Pro using earlier ctx4k sampling and a different protocol.The arm predates the leak-free rival null.
  • Benchmarks: MMLU used C = 4, K = 16, and n = 500 questions; MMLU-Pro used C = 10, K = 32, and n = 70.
  • Null model: ρ = Γiid,perq/Γemp was reported as the uniform per-question null for Qwen3.5-9B.The analogous ordering Γrival ≥Γiid was unverified for Qwen3.5-9B, unlike GPT-4.1's eight cells.

C Preference-estimate shrinkage

Preference-estimate shrinkage tests whether mechanical coverage depends on noisy per-case preference estimates. GPQA remains above 0.5 under pure-noise preferences, whereas AIME coverage declines monotonically with shrinkage and differs in option-support scale from Γiid.

  • Shrinkage setup: λ ∈ {1, 0.5, 0} shrinks the per-case estimated preference qi toward uniform over its observed wrong-label support.The estimate ˆqi uses the case’s other runs, making it noisy on open-domain cases.
  • Comparability: 19.5–69.4 labels, the mean rival support across AIME cells, contrasts with C = 9–20 per-run mean distinct answers, limiting direct comparison with Γiid.The rival λ = 0 null samples wrong-label support, whereas Γiid samples C −1 options.
  • Benchmark results: Above 0.5, GPQA mechanical coverage ϕ remains at λ = 0, when the estimated preference is treated as pure noise.Thus the multiple-choice result does not depend on the preference estimate.
  • Benchmark results: Monotonically, AIME ϕ declines with λ within a fixed support, making the residual 1 −ϕ largest at λ = 0.The headline AIME residual is reported at λ = 1.
  • Robustness: 0.03–0.17 per cell is the reduction in ϕ under the conservative zero-numerator convention, while GPQA remains 0.75–0.84 against AIME 0.51–0.71.The benchmark direction survives this alternative draw-coverage treatment.

D Calibrated explanatory null: run-level preference heterogeneity

The calibrated null adds run-level preference heterogeneity by drawing each run’s wrong-answer preference from a Dirichlet distribution around the case preference. Observed dispersion is 2–3× larger on AIME than GPQA, while GPQA’s fitted null coverage remains near the multinomial rival.

  • Calibrated null: Run-level heterogeneity is modeled by drawing each simulated run’s preference q* from Dirichlet(α q-hat_i) before sampling its wrong votes.The α →∞ limit recovers the multinomial rival, and the coverage denominator uses the dispersion-eligible population rather than the test-subset index Γ(t).
  • Observed heterogeneity: 2–3× larger observed plurality-share dispersion on AIME than GPQA indicates greater cross-run variation in attractive wrong answers.This dispersion is presented as an empirical signature of cross-run structure beyond a fixed per-case marginal.
  • GPQA calibration: 0.85–1.01 coverage ϕdm on GPQA sits near the multinomial rival across fitted α values spanning 1.0–50.Two GPQA cells have α values of 1.0–1.5, characterized as moderate heterogeneity.
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