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

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.

A decodability criterion predicts when hidden-state selection beats majority voting in large language models

Combining the answers a large language model (LLM) samples for a question into one decision is a test-time information fusion problem, usually solved by majority voting. Voting is unreliable on difficult questions, where the sampled answers share correlated errors, so the wrong answer can win and drawing more samples makes the decision worse. Selecting a candidate by reading a correctness signal from the model's hidden states is a promising alternative, but its accuracy varies across models and tasks, and no measure indicates when it can be trusted. In this paper, we propose CASE (Correctness-Axis SElection), a dynamic selection combiner that trains a linear gate on the answer-token hidden state and selects the highest-scoring candidate. Its main contribution is decodability, a leakage-free measure of how well the gate ranks a question's correct candidates above its incorrect ones, which predicts whether hidden-state selection will outperform voting. A conventional probe appears accurate only because of question-identity leakage, which vanishes under question-grouped evaluation. On held-out data, decodability predicts the accuracy gain of selection over voting with a Pearson correlation r=0.75 and a decision threshold near AUC=0.60. Across general and medical LLMs, CASE improves over voting by up to 19 points on medium-difficulty questions and 16.8 points on hard questions. Decodability depends on the aligned knowledge a model must recall, not on its scale, and its prediction transfers to an unseen scientific domain within 3.8 points. It thus provides a practical criterion, measurable in advance for a given model and task, for choosing between learned selection and majority voting.

Funnel of Thoughts: Efficient Test-Time Scaling via Early Voting and Rollout Pruning

Large Reasoning Models produce diverse, sometimes inconsistent answers across repeated queries on the same problem, so multi-sample inference is a prerequisite for reliable deployment. Majority voting at k rollouts is the standard solution and the de facto accuracy target for this regime, but it is prohibitively expensive at the scale LRMs require. We introduce Funnel of Thoughts (FoT), an inference-time method that preserves the full 32-trajectory voted accuracy while halving its attention FLOPs, a 28.8% reduction in full-model inference cost. Across 115K reasoning trajectories from six LRMs, we find that unproductive trajectories often reveal themselves through repeated hesitation markers such as "Wait", "Actually", and "perhaps." These trajectories are less likely to reach the correct answer and consume disproportionate attention FLOPs, degenerating into no-answer loops in the worst case. Built on this training-free lexical signal, FoT identifies the vocabulary that captures these pathological patterns and prunes affected trajectories before completion, reducing online generation attention FLOPs by 56.1% and wall time by 37.6% without any additional model inference; the same signal transfers without retuning across held-out architectures and out-of-domain tasks.

CoRE: Consensus Rewards via Equilibrium for Test-Time Reinforcement Learning

On unlabeled test data, reinforcement learning lacks a ground-truth reward; test-time RL methods derive one from the model's own roll-outs, rewarding those that match the majority vote over $N$ sampled answers. That vote discards a correct answer whenever it is a minority and scores every majority-matching roll-out identically. We replace it with \emph{CoRE} (Consensus Rewards via Equilibrium): the $N$ roll-outs form a graph whose edges combine answer agreement, reasoning similarity, and generation confidence, and replicator dynamics extract its dominant set, yielding a refined pseudo-label, a graded per-roll-out reward, and a per-question cohesiveness gate. CoRE strictly generalizes voting: majority voting is recovered as a special case; a block-value analysis gives a sharp threshold for when consensus recovers a correct minority against a larger wrong plurality; and confidence calibration provably lowers that threshold multiplicatively. Across seven backbones and five benchmarks (42 model--benchmark cells, three seeds each), \emph{CoRE} improves the untrained base by $+21.7$ points on average versus $+20.4$ for majority-vote TTRL, wins wherever agreement is contestable with margins over the vote of up to $+7.5$ points, and reaches the voting baseline's plateau accuracy in $54$--$70$\% fewer steps. Consensus, not counting: treating the roll-out group as a graph rather than a ballot box turns a brittle vote into a calibrated, graded, self-supervised reward at no extra roll-out cost.

Refining Over Resampling: Test-Time Self-Correction for LLM Reasoning

Test-time scaling improves LLM reasoning by using additional inference compute, but wider sampling alone can suffer from diminishing returns: new rollouts often repeat existing answer patterns instead of adding useful reasoning diversity. Verifier-based selection offers an alternative, but its performance depends on the calibration of an external reward model. We propose a verifier-free breadth--depth refinement framework that uses test-time compute to both explore and improve candidate solutions. The method samples multiple independent reasoning rollouts, refines each rollout through iterative self-critique and self-correction, and aggregates the refined answers by majority voting. Breadth preserves diverse initial attempts, while depth repairs local reasoning errors before aggregation. Across AIME24, AIME25, AMC, OlympiadBench, and MATH500, our method consistently improves over greedy decoding, majority voting, verifier-based best-of-$N$, beam search, and lookahead decoding across multiple open-weight models. For instance, with Qwen2.5-1.5B, accuracy increases from the strongest verifier-based baseline to $58.0\%$ on MATH500, and from $25.0\%$ to $32.5\%$ on AMC. These results show that test-time compute can be more effective when used to refine sampled trajectories rather than only to sample more candidates or rely on verifier-guided selection.

Social Pressure Breaks Majority Voting in LLM Safety Panels

Large language models (LLMs) are increasingly used to detect unsafe content. A common approach is to combine judgments from a panel of models to correct individual mistakes, but this benefit may disappear when every model sees the same misleading context before voting. We study this risk in a controlled two-round experiment. Each model first judges an item alone, then judges it again after six simulated peers either assert the wrong label or abstain. We combine the final judgments by majority vote. Across six open-weight LLMs and six datasets, we find that the wrong-label peer message raises the average reviewer false-alarm rate from 56.5% under silent peers to 87.5%, and majority voting raises the panel false-alarm rate to 100%. Without an asserted label, the same panel outperforms its average member. The effect is strongly asymmetric: reviewers follow pushes toward "unsafe" far more than pushes toward "safe" (about 75% versus 17%), so the panel's false-alarm rate rises sharply while its harmful-miss rate changes little. The proprietary-model probe shows substantial variation across models. These results identify susceptibility to shared social cues as a failure mode of safety panels and provide a simple pre-deployment diagnostic.

Hi-TTRL: Regulating Consensus with Hints for Test-Time Reinforcement Learning

Test-time reinforcement learning (TTRL) improves the reasoning capabilities of large language models without labeled data by updating the policy with pseudo-labels constructed through majority voting. While effective, the reward signal assigned from majority voting is highly sensitive to consensus strength, defined as the frequency of the most common answer within a rollout group. In TTRL, consensus strength plays a dual role: it reflects both the reliability of the pseudo-label and the distribution of advantages. Low consensus can amplify updates from unreliable pseudo-labels through disproportionately large advantages, whereas high consensus reduces reward contrast and ultimately yields vanishing gradients. In this paper, we introduce Hi-TTRL, a test-time reinforcement learning framework that utilizes hints during sampling to regulate rollout consensus strength. Hi-TTRL first estimates consensus strength from a partial rollout group. When the consensus strength falls outside a target interval, it invokes a Markov chain Monte Carlo (MCMC) hint sampler. The sampler targets the power-transformed prefix distribution and uses finite-step approximate sampling to generate rollout prefixes as hints. By tuning the power exponent, Hi-TTRL generates hints with a sharpened or flattened power target, steering rollout consensus strength toward the target interval. Experiments on multiple datasets and backbones show that Hi-TTRL consistently improves over standard TTRL, with ablations and consensus-steering analyses validating the effectiveness of adaptive hint-guided consensus regulation.

Are Diversity Metrics Measuring Diversity? A Capability-Controlled Audit of Majority-Vote Gain in LLM Ensembles

Majority voting over LLMs is widely assumed to benefit from diversity, and diversity measures are used to choose which models to combine. We ask whether five such measures track diversity or mainly re-express capability, auditing them as predictors of majority-vote gain over the best member across 31,900 subsets of 30 LLMs on MMLU-Pro (29 on TruthfulQA) under explicit capability controls. Three findings emerge. First, latent complementarity is ubiquitous: oracle gain is positive in 100% of subsets, yet simple voting beats the strongest member in only 9.98% of all canonical size-3 subsets (18.71% with held-out best selection); the pooled size-2-4 rate is 1.27%, partly reflecting deterministic even-size voting behavior. Second, a joint-correctness proxy (strict diversity) is nearly collinear with one minus mean accuracy (size-3 Spearman rho = +0.991 / +0.988); raw diversity-gain associations are strongly capability-entangled and, with one exception, unstable under control. Third, three linear contingency-table statistics are algebraically non-separable; after capability control, the empirically stable remainder is a modest residual pairwise co-failure association in which more shared error corresponds to lower gain. This direction is robust, but its magnitude is configuration-dependent. Joint rawspace linear regressions treating strict diversity, disagreement, and double-fault as independent predictors are rank-deficient by construction.

When Does Delegation Beat Majority? A Delegation-Based Aggregator for Multi-Sample LLM Inference

Majority voting is the default unsupervised aggregator for multi-sample LLM inference, but it discards two signals: within-group answer entropy and between-group reasoning geometry. We aggregate by delegation instead (Propagational Proxy Voting, PPV): each group of samples keeps weight on its own answer in proportion to its entropy-based confidence (When) and routes the rest to peers by reasoning-embedding similarity (Whom); the stationary distribution of the resulting delegation matrix picks the consensus answer. This requires neither gold labels nor training. On MMLU-Pro with 128 samples per question, delegation beats majority by +1.5 pp overall and +2.24 pp on non-trivial questions (McNemar p ~ 1.0e-14, n = 8,099), overturning wrong majorities whose answer cluster is geometrically incoherent while the correct minority is tight. We then characterize exactly when delegation overturns majority: a two-option model gives a closed-form flip condition on each option's confidence and the weight it routes to the other, with a do-no-harm corollary for near-unanimous questions. The condition calls the realized winner on 96.5% of non-trivial questions, and its predicted mass gap tracks the realized gap at r = 0.97. We did not find any other unsupervised ensemble methods that close the oracle gap.