Surprisingly-Popular Search: Peer-Prediction Scoring to Break Confident-but-Wrong Consensus

Research perspective: cross-domain structural transfer, transferring a structure from peer prediction in mechanism design/crowdsourcing. What reviewers would remember is a claim that challenges consensus: to judge whether consensus is wrong, examine the "actual minus predicted" gap rather than the agreement rate. Even if the method fails, a clean diagnostic result (whether single-model meta-prediction carries information beyond sample frequencies) would have citation value. But this is a fallback, not the main selling point, and the boundary must be faced honestly: if the result is negative, the paper can only be a diagnostic or negative-results paper.

Majority voting and other agreement-based label-free value signals can confidently produce a wrong consensus when multiple reasoning chains share the same misconception. Peer-prediction theory (Prelec's Surprisingly Popular and Bayesian Truth Serum) indicates that the informative quantity is not the raw agreement rate, but the gap between actual and predicted agreement. We propose SP-Search. Sample N chains per problem and cluster their final answers to obtain empirical frequencies f(a). Then ask the same model, under naive/expert personas, to predict "other independent solvers' answer distributions," averaging the predictions to obtain p(a). Select using Dirichlet-shrunk s(a)=log f(a)−log p(a): argmax f(a)·exp(λ s(a)). The same score can serve as the value of prefix branch nodes for pruning and expansion. Problems with "high f but negative s" are flagged as correlated-error cases, and extra budget is routed to diversified prefix sampling. The key risk is that a single model's i.i.d. samples lack the heterogeneous private signals required by SP theory, so s(a) may simply be meta-prediction calibration error plus noise. The first stage is therefore a go/no-go diagnostic without cross-model tuning: on problems with a wrong plurality, is the proportion for which s(correct answer)>s(wrong plurality) significantly above chance? Proceed with the full method only if this test passes.

PRMs require expensive step-level supervision, are brittle under OOD conditions, and can be reward-hacked by intensive search. Agreement-based label-free signals implicitly assume that "the majority is usually correct," whereas the closest papers repeatedly show that majority voting locks in the wrong modal answer on difficult problems and that LLM errors are correlated across models. A label-free test that distinguishes "correct consensus" from "correlated errors" is currently missing. If such a test exists, it could (a) correct some plurality-wrong problems at the selection stage and (b) serve as a risk signal that shifts budget from easy problems to difficult problems being misled by wrong consensus. The area chair also notes that the recoverable plurality-wrong subset is itself small, limiting the upper bound on practical gains.

The field assumes by default that agreement increases monotonically with correctness. SP theory originates in human populations with heterogeneous respondents, and no one has applied it directly to rollouts from the same model. However, experts' prior belief points the other way: single-model meta-prediction is likely merely to restate its own sampling distribution, giving s(a)≈0 plus noise. Thus "this can work" is not obvious, but "this cannot work" has not been proved either. This is precisely why a cheap diagnostic should come first.

1. Answer clustering and empirical frequencies: sample N chains per problem (for example, 32); use answer equivalence for mathematics, test-output signatures for code, and answer options for GPQA to obtain f(a).
2. Meta-prediction: prompt the same model K times (for example, 8 times) to "estimate other independent solvers' answer distributions," comparing three conditions: no persona, naive, and expert. Average the predictions to obtain p(a). Use prefix caching to control cost.
3. SP scoring and shrinkage: s(a)=log f(a)−log p(a), with Dirichlet-prior shrinkage for rare answers (on the plurality-wrong subset, this step may pull the scores back toward majority voting and must be tested separately).
4. Final selection: argmax f(a)·exp(λ s(a)). Choose λ using single-model signals only (for example, a small labeled development set, or a fixed default with reported sensitivity), without cross-model agreement; otherwise, it becomes confounded with a cross-model jury signal.
5. Prefix-node value: at a branch node, sample completions conditioned on that prefix to obtain an answer distribution, and compare it with meta-prediction conditioned on the same prefix; use the resulting surprise as a value for pruning/expansion.
6. Correlated-error detector: flag problems with high f and negative s, and route extra budget to diversified prefix sampling.
7. Sources of heterogeneity (added in response to the fatal issue): compare the effects of personas, different prefixes, partial information (withholding parts of the problem statement), and different model families as meta-predictors, aligning these as far as possible with the formal conditions of SP theory.

Models: open-source long-CoT models such as Qwen3-8B and QwQ-32B, with 8×A100 and vLLM. The primary metric is overall accuracy at equal token budgets; plurality-wrong stratification is for analysis only. Additional meta-prediction tokens must count toward the total budget, and baselines receive the same quantity of additional samples. Pool plurality-wrong problems across datasets and seeds and report bootstrap confidence intervals. Stage 0 is a go/no-go study: use only a single model, with no cross-model tuning.

• datasets: MATH-500 (hard-problem subset); AIME 2024/2025 (very few plurality-wrong problems, used only in pooled analysis); OlympiadBench; GPQA-Diamond (small answer space, so meta-prediction may trivially match f); LiveCodeBench (answer=test-output signature); ProcessBench (step-level AUROC/F1)
• baselines: Majority vote; DeepConf / self-certainty-weighted voting; Qwen2.5-Math-PRM-7B best-of-N and beam search; Math-Shepherd best-of-N; self-consistency using the entire equivalent additional-token budget to increase the sample count; a token-matched cross-model jury baseline (added in response to the red team and area chair); PoLR-style prefix consensus (added); the answer distribution from meta-prediction alone (B alone)
• metrics: Accuracy at equal total token budgets; accuracy on the plurality-wrong subset (oracle-stratified, pooled, and with CI); P[s(correct)>s(wrong plurality)] and the correlation between meta-prediction and f; AUROC of correlated-error flags; ProcessBench step AUROC/F1; reward-hacking curves as N varies; the proportion of plurality-right problems flipped to a wrong answer
• ablations: Meta-prediction without a persona; meta-prediction from a different model family; removal of Dirichlet shrinkage; answer-level-only vs prefix-level node value; random personas vs expert/naive personas; fixed λ vs a labeled development set vs cross-model tuning (labeled separately); different model sizes (testing whether the signal disappears in stronger models)
• expected: Original hypothesis: recover 10–25% of plurality-wrong problems on hard-problem sets, achieve an overall +2–4 percentage points on AIME/OlympiadBench at equal token budgets, and obtain flag AUROC>0.7. My judgment: these numbers are optimistic. The area chair notes that single-model meta-prediction is likely merely to restate f, that the signal is weaker in stronger models, and that the AIME subset is too small. A more realistic expectation is a weak signal or one concentrated on a small number of knowledge-asymmetry problems, with an overall gain of 0–1 percentage points and difficulty outperforming additional sampling at equal token budgets.
• falsification criteria: Terminate the method direction if any of the following holds: (1) on the plurality-wrong subset, SP scoring provides less than a 5% absolute improvement over majority voting with additional samples at equal token budgets; (2) the correlation between meta-prediction and empirical frequencies exceeds 0.95 (no information asymmetry); (3) the CI for pooled P[s(correct)>s(wrong plurality)] contains 0.5. If only (3) fails, the work could shift to publication as a diagnostic/negative result.

1. Days 1–2: for Qwen3-8B and QwQ-32B, sample and cache 32 chains per problem on the MATH-500 hard-problem subset and AIME; establish the answer-clustering/equivalence pipeline and label the oracle strata for correct vs wrong plurality.
2. Days 3–4: collect 8 meta-predictions per problem under 3 persona conditions (none/naive/expert), first checking whether the output format can be parsed as a distribution.
3. Day 5: compute the correlation between meta-prediction and f offline. If it is >0.95, stop immediately and write up the result as a diagnostic conclusion.
4. Days 6–7: compute P[s(correct)>s(wrong plurality)] on the plurality-wrong subset (single model, without tuning λ), add OlympiadBench/GPQA to the pooled analysis, and compute bootstrap CI.
5. Days 8–9: add majority voting with additional samples at equal token budgets and a token-matched cross-model jury baseline (the cached second model can be used), and compare flip rates: the targets are to correct ≥10% of plurality-wrong problems and incorrectly flip <2% of plurality-right problems.
6. Day 10: conduct a go/no-go review; if it passes, design prefix-node-value and routing experiments. The budget is approximately 150 GPU-hours.

• The model merely restates its own answer distribution in meta-prediction, making s(a)≈noise (the area chair's highest-ranked risk). → Measure the correlation and P[s(correct)>s(wrong)] in the first week; introduce heterogeneity through personas, partial information, and different prefixes; if ineffective, shift to a diagnostic paper.
• Log-ratio noise for rare answers dominates, and plurality-wrong cases often involve rare answers; Dirichlet shrinkage pulls the scores back toward majority voting. → Increase N, analyze sensitivity to λ and shrinkage strength, and report a sweep over shrinkage strengths; score only the top-k candidate clusters.
• Gains come from the cross-model signal rather than SP (confounding through λ tuning). → Use no cross-model tuning in the primary results; add a jury baseline; treat cross-model meta-prediction as a separate ablation.
• The recoverable subset is small, and AIME results fall within seed variance. → Pool across datasets, use multiple seeds and bootstrap CI, and make overall equal-token accuracy the primary metric.
• Stronger models are better calibrated, so the signal disappears with scale; extra meta-calls cost at least as much as additional sampling. → Experiment across multiple model sizes; count all meta-prediction tokens in the budget and compare against an equivalent quantity of additional samples.
• Others publish first, or the work overlaps with existing LLM SP/higher-order aggregation methods. → Search and verify 'Beyond Majority Voting' and related work immediately; narrow the contribution to prefix-node value and budget routing.

• Q: SP requires respondents with different private signals. Single-model samples lack this heterogeneity, and s(a) measures meta-prediction calibration error rather than truth. A: This objection is largely valid and is the central risk of this idea. We claim no theoretical guarantee. The claim is only that an empirical signal exists if meta-prediction under particular prompts carries knowledge of "common traps" that the sampler lacks. We test this directly through the go/no-go experiment and introduce heterogeneity through personas, partial information, different models, and similar approaches. If the design cannot be formally mapped to SP conditions, the paper should present it as an empirical method.
• Q: Log-ratio noise for rare answers dominates, and the plurality-wrong subset is precisely where rare answers are concentrated. A: We agree that this is a real problem. Mitigations include Dirichlet shrinkage, increasing N and K, and scoring only candidate clusters, but shrinkage may make the method collapse into majority voting. Report a sweep over shrinkage strengths and validate separately on the plurality-wrong subset. Uncontrollable noise is a kill criterion.
• Q: λ is tuned using agreement with another model family, introducing a cross-model consensus signal that is already known to outperform PRMs, without a corresponding baseline. A: This criticism is valid. We will remove cross-model λ selection, use a fixed λ or a small labeled development set for the primary results, and add a token-matched cross-model jury baseline; cross-model tuning will appear only as a separately labeled ablation.
• Q: The novelty claim of "the first use of SP/BTS with LLM meta-prediction" may be false; LLM aggregation work using second-order predictions already exists. A: This is likely valid: the area chair mentions 'Beyond Majority Voting' (2025) and related work. I did not see these in the supplied closest papers and have not verified their full texts. Remove the "first" claim, narrow the contribution to single-model prefix-node value, correlated-error budget routing, and a label-free calibration protocol, and conduct a full literature search before submission.
• Q: AIME has only 30 problems, its plurality-wrong subset is tiny, and a 10–25% recovery rate falls within seed variance. A: Agreed. Results must be pooled across datasets, use multiple seeds, and include bootstrap CI; the primary metric is overall equal-token accuracy rather than percentages on the AIME subset.
• Q: Stronger models are better calibrated, so the signal will disappear, and they have fewer plurality-wrong problems, limiting practical impact. A: This is quite possible. We will test multiple scales and report the trends faithfully. If the signal disappears with scale, the contribution is limited to small/medium models and diagnostic value, which must be stated explicitly in the paper's positioning.

The intersection of test-time compute and peer prediction fits the subject matter of NeurIPS/ICLR, but the area chair's verdict is reject (evidence_plan 2, differentiation 2, realism 2, feasibility 4). If the go/no-go study passes and jury baselines and confidence intervals are added, a main-conference submission could be pursued. If the signal is negative, a workshop or diagnostic/negative-results paper is more realistic. Use the pilot results within two weeks to decide whether to invest several months. The low experiment cost (approximately 150 GPU-hours, 10 days) makes this suitable for an initial trial.