When Debate Helps: Proposal Supply and Verification-Aware Readout in Multi-Agent Reasoning

Zihao Zhao; Tunyu Zhang; Haizhou Shi; Yusong Zhao; Xinxi Zhang; Hao Wang

cs.AIOct 3, 2026

When Debate Helps: Proposal Supply and Verification-Aware Readout in Multi-Agent Reasoning

Authors: Zihao Zhao, Tunyu Zhang, Haizhou Shi, Yusong Zhao, Xinxi Zhang, Hao Wang

Organizations: University of Illinois Urbana-Champaign · Rutgers University

Abstract

Multi-agent debate can improve reasoning, yet often fails to beat simple majority voting. We argue that successful debate requires two distinct mechanisms: proposal supply must surface a correct answer, and readout must identify that answer when voting misses it. We formalize the first requirement through recoverable headroom, which measures cases where a correct proposal is available but the majority answer is wrong. For the second, we develop Latent Verification Debate (LVD), an accounting model in which candidate proposals receive answer-specific verification evidence before final generation. Controlled fixed-proposal interventions estimate this latent effect in equivalent peer-support units and show that correct evidence changes answer probabilities and generated decisions while proposal supply remains fixed. To improve proposal supply, we construct societies from neural-thicket agents using labeled and label-free coverage objectives. Across two backbones and matched-budget reasoning benchmarks, coverage-selected societies increase complementary proposal supply and improve aggregate accuracy in repeated stochastic evaluations. Round-level controls further show that interaction provides gains beyond applying the same finalizer directly to the initial proposals. These results identify proposal coverage and truth-sensitive evidence use as complementary conditions for debate to outperform voting. Code is available at https://github.com/Wang-ML-Lab/when-debate-helps.

Figures & tables

Figure 1: Overview of the supply-readout view. Transparent regions denote questions for which individual sampled agents can surface the correct answer; solid regions denote questions recovered by the final readout. Top: Overlapping sampled agents provide redundant proposal coverage, so majority vote keeps a small shared region and can reinforce a repeated wrong answer. Bottom: Diverse agents provide complementary proposal coverage; after debate, LVD uses verification-aware evidence to recover a surfaced correct minority answer.

Figure 2: Motivating observations. (a) Qwen2.5-3B-Instruct shows null-like debate dynamics with little gain over vote, while Qwen3-4B-Instruct shows positive gain across rounds. (b) Under the same neural-thicket supply regime, Debate readout improves over vote , while the proposal-hit upper bound remains higher, indicating recoverable headroom. (c) Final debate lift is positively correlated with initial proposal-hit headroom, suggesting that debate is supply-constrained. Together, these observations motivate a supply-readout account: debate helps when correct minority proposals are surfaced and the readout can recover them through verification-aware dynamics.

Backbone: Qwen3-4B-Instruct
Method	Reasoning			MMLU-Redux					Avg	Avg Ranking
Method	AMC12	MATH500	GPQA	Moral	Chem.	CS	Acct.	Med.	Avg	Avg Ranking
Label-Free Methods
SC@20	91.20	92.33	49.13	30.00	70.00	81.67	75.00	83.33	71.58	2.38
Vanilla Debate	92.69	91.67	52.31	50.00	71.67	80.00	76.67	83.33	74.79	1.81
SC-Greedy Debate (Ours)	92.29	92.67	51.45	63.33	68.33	81.67	73.33	86.67	76.22	1.81

Table 1: Accuracy (%) on Qwen3-4B-Instruct and OLMo-3-7B-Instruct under a unified computation budget. SC and Neural thicket vote rows use vote20 ; debate rows use round-3 (R3) majority vote from matched five-round, four-agent traces. MMLU-Redux is split into five 60-example subjects. Avg is the unweighted mean over the eight displayed columns, and Avg Ranking is the mean within-block method rank. Boldface and underline denote the best and second-best result per column within each method block and model. Gray rows mark our proposed agent pool construction methods.

Backbone: Qwen3-4B-Instruct, AMC12 $\rightarrow$ MMLU-Redux
Method	N	Init Vote	R3 Vote	R5 Vote	Final	Init Hit	R3 Hit	R5 Hit	Lift
SC-Greedy Debate (Ours)	300	72.33	75.67	76.33	76.33	84.00	77.67	76.33	+4.00
Top-4 Debate	300	72.67	75.00	74.67	75.33	84.00	77.67	76.67	+2.67
AC-Greedy Debate (Ours)	300	75.33	77.33	77.00	76.67	83.67	78.00	78.00	+1.33
Backbone: OLMo-3-7B-Instruct, AMC12 $\rightarrow$ MMLU-Redux
SC-Greedy Debate (Ours)	300	55.33	56.33	58.33	58.33	70.00	59.67	59.33	+3.00

Table 2: AMC12-selected seed transfer to MMLU-Redux. Init, R3, and R5 report pre-debate, round-3, and round-5 metrics over the same four agents. Final is the R5 vote-then-judge readout; Lift is Final minus Init Vote. Values are percentages except

N

. Gray rows mark our greedy construction methods.

Figure 3: Label-based five-round dynamics. Solid lines show round-level accuracy before finalization; dashed hollow legend entries marked “Oracle” show proposal-hit coverage. Later rounds can improve vote accuracy while reducing answer diversity, so final gains depend on recovering correct proposals before coverage collapses. The label-free counterpart is in Fig. 4 .

Backbone	Dataset	$\bm{\Delta m_{\mathrm{trace}}}$ [95% CI]	Median $\bm{\widehat{W}}$	Answer-stripped $\bm{T-F}$	$\bm{\Delta}$ Acc at $\bm{d=3}$
Qwen3-4B	GPQA	8.15 [4.96, 11.41]	2.23	4.59	+.14
Qwen3-4B	MMLU-Redux	11.40 [9.02, 13.97]	$>3$	5.50	+.04
OLMo3-7B	GPQA	4.57 [3.65, 5.50]	1.92	2.01	+.56
OLMo3-7B	MMLU-Redux	5.47 [4.52, 6.43]	$>3$	2.10	+.58

Table 3: Controlled fixed-proposal evidence for verification-aware readout. Trace effect is the answer-margin shift from an original correct peer response relative to neutral context;

\widehat{W}

is its median equivalent controlled support over recovery cases.

T-F

is the gold-margin difference between answer-stripped correct and wrong rationales.

\Delta

Acc is correct-support accuracy minus neutral accuracy at

d=3

. All entries use the main readout configuration; secondary configuration controls are reported in the appendix.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Experiment family	Workers per run	Observed wall time and compute
Qwen seed profiling (500 agents per dataset)	6–8 GPUs	3.3–18.4 hours per dataset; approximately 27–110 GPU-hours
OLMo seed profiling (300 retained agents per dataset)	8 GPUs	1.3–6.6 hours per dataset; about 10–53 GPU-hours
Five-round downstream and prompt-control evaluation	1 GPU per job	0.2–4.9 hours per job
Five-run stochastic robustness experiment	1 GPU per job; 120 jobs	156.8 GPU-hours in total
Controlled mechanism, readout, and transfer diagnostics	1 GPU per job	Within the same 0.2–4.9-hour single-job envelope

Appendix

Table 4: Compute used by the reported experiment families. Profiling times vary with dataset size and output length; downstream times vary with the backbone, dataset, and number of debate rounds.

Asset and version	License or terms	Use in this work
Qwen3-4B-Instruct-2507	Apache-2.0	Main 4B backbone; research evaluation only
OLMo-3-7B-Instruct	Apache-2.0; Ai2 Responsible Use Guidelines	Main 7B backbone; research evaluation only
Qwen2.5-3B-Instruct	Qwen Research License	Motivating experiment; non-commercial research evaluation
GPQA	CC BY 4.0	Gated benchmark; examples are not redistributed
MMLU-Redux 2.0	CC BY 4.0	Benchmark evaluation
OlympiadBench	Apache-2.0	Benchmark evaluation

Appendix

Table 5: Licenses and stated terms for existing model and dataset assets.

Backbone	Profiling split	$\sigma$	Seeds	Base ref.	Seeds $>$ base	Mean	Best	Used
Qwen3-4B	MATH500 first200	0.0005	160	90.00	59/160	89.82	92.50	yes
Qwen3-4B	MATH500 first200	0.001	174	90.00	51/174	89.58	92.50	yes
Qwen3-4B	MATH500 first200	0.002	166	90.00	11/166	88.10	91.00	yes
Qwen3-4B	AMC12 train360	0.0005	160	84.17	5/160	82.40	85.83	yes
Qwen3-4B	AMC12 train360	0.001	174	84.17	7/174	81.96	86.11	yes
Qwen3-4B	AMC12 train360	0.002	166	84.17	0/166	78.97	83.33	yes

Appendix

Table 7: Seed-quality diagnostics for the main-table backbones. Values are percentages on the profiling split. Base ref. is the same-split unperturbed pass@1 reference. The Used column marks scales retained for downstream construction.

Backbone	Dataset	$\bm{\Delta m_{\mathrm{trace}}}$ [95% CI]	Median $\bm{\widehat{W}}$	Correct $-$ wrong margin, $\bm{d=1/2/3}$
Qwen3-4B	GPQA	8.15 [4.96, 11.41]	2.23	2.59 / 5.94 / 13.63
Qwen3-4B	MMLU-Redux	11.40 [9.02, 13.97]	$>3$	1.03 / 5.36 / 11.76
OLMo3-7B	GPQA	4.57 [3.65, 5.50]	1.92	3.33 / 5.06 / 5.79
OLMo3-7B	MMLU-Redux	5.47 [4.52, 6.43]	$>3$	3.59 / 5.18 / 5.53

Appendix

Table 8: Original peer-response effects and equivalent conclusion-only support over recovery cases. Trace effects are means with 95% confidence intervals;

\widehat{W}

is the median interpolated equivalent support. The final column reports the correct-support minus wrong-support answer margin under the Verify prompt at each dose.

Backbone / dataset	Help-over-harm asymmetry $\bm{A}$	Correct-over-wrong gap $\bm{T-F}$
Qwen3-4B / GPQA	7.34 [2.14, 12.45] / 6.84 [1.29, 12.33]	1.90 [.54, 3.39] / 4.59 [2.45, 6.96]
Qwen3-4B / MMLU-Redux	8.70 [3.50, 13.72] / 6.36 [2.34, 10.42]	5.98 [3.78, 8.32] / 5.50 [3.46, 7.69]
OLMo3-7B / GPQA	1.92 [.46, 3.35] / 1.39 [.17, 2.58]	1.70 [1.16, 2.31] / 2.01 [1.35, 2.77]
OLMo3-7B / MMLU-Redux	2.84 [.87, 4.85] / 2.24 [.78, 3.78]	1.93 [1.26, 2.63] / 2.10 [1.41, 2.83]

Appendix

Table 9: Truth sensitivity of answer-stripped trace rationales. Entries are mean margin changes with 95% confidence intervals, shown as Standard / Verify.

Backbone	Dataset	Prompt	Neutral	Correct support	Answer withheld	Shuffled rationale
Qwen3-4B	GPQA	Standard	.32	.50	.32	.28
Qwen3-4B	GPQA	Verify	.32	.46	.34	.14
Qwen3-4B	MMLU-Redux	Standard	.28	.58	.26	.28
Qwen3-4B	MMLU-Redux	Verify	.38	.42	.32	.26
OLMo3-7B	GPQA	Standard	.30	.80	.30	.30
OLMo3-7B	GPQA	Verify	.30	.86	.30	.30

Appendix

Table 10: Free-generation accuracy after three peer messages. Each entry evaluates the same 50 questions. Correct support is bolded.

Round	Verify Vote	Standard Vote	$\Delta$ Vote	Verify Hit	Standard Hit	$\Delta$ Hit
R0	76.04	76.04	+0.00	83.95	83.95	+0.00
R1	77.95	77.81	-0.14	85.38	84.89	-0.49
R2	78.19	78.12	-0.08	82.20	83.38	+1.18
R3	78.37	78.16	-0.21	80.44	82.98	+2.54
R4	78.10	78.43	+0.32	79.56	82.04	+2.48
R5	78.21	78.13	-0.08	79.26	81.62	+2.37

Appendix

Table 11: Round-level prompt-style dynamics averaged over the 15 matched dataset-method cells. R0 is identical because both controls use the same seed answers before debate.

\Delta

is Standard minus Verify. All values are percentages.

Dataset	Method	N	Verify Final	Standard Final	$\Delta$	Standard R5 Vote	Standard R5 Hit
AMC12	Vanilla Debate	739	92.96	92.42	-0.54	92.29	93.78
	AC-Greedy Debate	739	91.61	91.34	-0.27	91.75	94.18
	SC-Greedy Debate	739	91.88	92.02	+0.14	91.88	93.50
MATH500	Vanilla Debate	300	91.33	92.00	+0.67	91.67	93.00
	AC-Greedy Debate	300	91.67	92.67	+1.00	93.33	94.00
	SC-Greedy Debate	300	92.00	91.67	-0.33	91.67	93.33

Appendix

Table 12: Prompt-style control for the five-round new-seed debate runs. Verify Final is the completed verify_then_answer debate prompt; Standard Final repeats the same datasets, seed banks, verifier settings, and vote_then_judge_tiebreak finalizer with the normal standard debate prompt.

\Delta

is Standard Final minus Verify Final. All accuracy values are percentages.

Scope	Qwen3-4B	OLMo3-7B
Four shared datasets, pooled	$r=.770,\ \rho=.787$	$r=.548,\ \rho=.648$
Five MMLU subjects, pooled	$r=.568\ [.125,.787]$	$r=.635\ [.219,.805]$

Appendix

Table 14: Association between initial headroom

H=P_{0}-V_{0}

and final lift

G=F-V_{0}

on the two backbones. The MMLU row reports a 95% paired question-bootstrap interval for Pearson

r

Backbone	Method	R0 vote	Headroom	Recovery rate	Recovered	New correct	Damage	Gain	Final
Qwen3-4B	SC-Greedy	72.33	11.67	45.71	5.33	0.67	2.00	$+4.00$	76.33
Qwen3-4B	AC-Greedy	75.33	8.33	48.00	4.00	0.33	3.00	$+1.33$	76.67
OLMo3-7B	SC-Greedy	55.33	14.67	36.36	5.33	0.67	3.00	$+3.00$	58.33
OLMo3-7B	AC-Greedy	54.00	15.33	26.09	4.00	0.33	2.00	$+2.33$	56.33

Appendix

Table 15: AMC12-to-MMLU-Redux transfer decomposition. Recovery rate is conditional on R0 headroom; the contribution columns are unconditional percentage points.

Selector	Pool $N=50\to 100\to 200\to 500$	Construction $m=25\to 50\to 100\to 150$
AC-Greedy	$84.06\to 84.08\to 84.29\to 84.12$	$83.22\to 83.59\to 84.12\to 84.38$
SC-Greedy	$83.95\to 84.03\to 84.01\to 83.81$	$83.30\to 83.36\to 83.81\to 83.85$

Appendix

Table 17: Sensitivity to candidate-pool size

N

and construction-set size

m

. Each arrow follows the increasing values shown in the column header.

Selector	Construction signal	Requires	R0 hit@4	R0 vote	R5 final
AC-Greedy	Correct-answer coverage	Labels	84.67	74.67	75.67
SC-Greedy	Answer-cluster coverage	Clusters	82.00	68.67	75.33
Hidden-Prefix	Representation-cluster coverage	Neither	$84.30\pm 0.69$	$75.07\pm 1.03$	$77.00\pm 1.25$
TopAcc	Individual accuracy	Labels	$83.60\pm 0.62$	$73.93\pm 1.07$	$75.40\pm 0.80$

Appendix

Table 18: Fixed-society Qwen3-4B MMLU-Redux results. Hidden-Prefix and TopAcc report mean

\pm

SD across 10 paired generation seeds; AC-Greedy and SC-Greedy are the submitted fixed runs and are included only as reference points.

Figure 4: Label-free five-round dynamics. This appendix counterpart to Fig. 3 shows methods that do not use labeled training diagnostics for society selection. Solid lines show round-level majority-vote accuracy before finalization; dashed hollow legend entries marked “Oracle” show proposal-hit coverage.

Method	Round	Vote	Hit	Final	$H$	$G$	$\rho$	$R$	$I$	$L$
Vanilla Debate	R0	75.65	83.03	77.92	7.39	2.27	0.31	53.49	9.94	5.00
	R1	77.85	85.05	77.92	7.20	0.07	0.01	43.10	1.23	4.25
	R2	77.84	81.86	77.92	4.02	0.08	0.02	24.33	0.14	1.58
	R3	78.05	80.07	77.92	2.03	-0.13	-0.06	43.39	0.60	1.37
	R4	78.13	79.04	77.92	0.92	-0.21	-0.23	38.89	0.13	1.08
	R5	77.99	78.98	77.92	0.99	-0.08	-0.08	10.00	0.13	0.33

Appendix

Table 19: Round-level recovery diagnostics averaged over the five benchmarks. For each round, Vote and Hit are the round-specific majority-vote and proposal-hit accuracies, while Final is the end-of-run vote_then_judge_tiebreak accuracy. We define

H=P-V

G=D-V

\rho=G/H

R=\Pr(D=1\mid P=1,V=0)

I=\Pr(D=1\mid P=0,V=0)

, and

L=\Pr(D=0\mid V=1)

, where

D

is the final decision. All values except

\rho

are percentages; averages are unweighted over benchmarks.

Dataset	Method	N	Init Vote	R5 Vote	Final	Init Hit	R5 Hit	Lift	Judge Ties
MATH500	Vanilla Debate	300	91.33	91.33	91.33	93.33	91.67	+0.00	1
MATH500	AC-Greedy Debate	300	91.33	91.33	91.67	94.33	91.67	+0.33	2
MATH500	SC-Greedy Debate	300	92.00	92.00	92.00	93.67	92.67	+0.00	0
OlympiadBench	Vanilla Debate	474	80.38	81.65	81.22	83.54	82.28	+0.84	7
OlympiadBench	AC-Greedy Debate	474	79.11	81.43	81.43	82.91	81.86	+2.32	7
OlympiadBench	SC-Greedy Debate	474	80.59	80.59	80.80	83.33	81.22	+0.21	5

Appendix

Table 20: Five-round debate results for the new seed teams. Final is accuracy after vote_then_judge_tiebreak ; Lift is Final minus Init Vote. Bold marks the best final accuracy within each dataset. Average rows report the unweighted mean across the five benchmarks.

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LLM Evaluation Multi-Agent Debate LLM Decision-Making Inference-Time Compute Allocation Majority Voting

Open paper AskPDF arXiv

When should a language model answer directly, sample and vote, or engage in multi-agent debate? Recent work shows voting often explains much of the gain attributed to debate, while selective-debate systems activate deliberation only on uncertain examples. We ask: under a matched ceiling on generated tokens (960 per example), how much per-example routing headroom exists, and how much is recoverable from cheap pre-deliberation signals? We evaluate greedy decoding, three-sample voting, and a two-agent critique-revise debate on MuSiQue and GSM8K using Llama 3.1 8B Instruct and Ministral 3 8B Instruct. On MuSiQue, an oracle selecting the correct protocol per example gains +14.0 and +13.7 pp over the best fixed one. The best fixed protocol is model- and dataset-dependent: each (model, dataset) cell has a different winner. This headroom is hard to recover from cheap ex-ante signals. A vote-entropy threshold is the only controller that directionally beats the best fixed protocol on both models (+1.3 and +1.7 pp), though individual paired-bootstrap CIs include zero. A joint analysis (meta-analysis +1.6 pp, p=0.125; Bayesian P(both>0)=0.59) is directionally consistent but not significant. Learned controllers (LR, GBT) do not outperform the threshold. The key finding is structural: vote entropy predicts where debate is safe, not where debate is needed. High entropy sharply reduces debate backfire, but 66% of debate-helpful examples (31/47) occur when voting is unanimous but wrong. A single-prompt self-critique probe on Llama flips the answer in 127/127 unanimous cases, yielding zero mutual information with the debate-helpful label; we cannot rule out a prompt-compliance artifact, but either interpretation disqualifies the probe as a router. Recovering the remaining headroom requires behavioral probes that avoid format-compliance confounds at the 8B scale.

Abstract

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