Wrong but Useful: Trajectory Value Beyond Answer Correctness in Multi-Agent Messages

Chih-Hsuan Yang; Anjir Ahmed Chowdhury; Cheng-Hau Yang; Weijian Zheng; Fernando Llorente; Xiaolong Ma; Xinyang Li; Eliu A. Huerta

cs.AIAug 14, 2026

Wrong but Useful: Trajectory Value Beyond Answer Correctness in Multi-Agent Messages

Authors: Chih-Hsuan Yang, Anjir Ahmed Chowdhury, Cheng-Hau Yang, Weijian Zheng, Fernando Llorente, Xiaolong Ma, Xinyang Li, Eliu A. Huerta,

Organizations: Argonne National Laboratory, Lemont, IL, USA · University of Houston, Houston, TX, USA · Brookhaven National Laboratory, Upton, NY, USA · University of Chicago, Chicago, IL, USA

Abstract

Multi-agent reasoning systems often use agreement, confidence, or automated scores to decide which messages should shape a final answer. Such filtering assumes that a message likely to be correct is also worth keeping. Yet a wrong answer can contain a useful decomposition, constraint, or scientific principle. We test this distinction with Diverse Hypothesis Deliberation (DHD), a controlled measurement protocol that caches five independently generated messages and replays the same downstream solver, called the integrator, with each message available or hidden. The replay comparison measures a message's trajectory value: whether making the message available helps or harms subsequent reasoning. Across five mathematics and science benchmarks and two openly available model families, gpt-oss-120b and gemma-4-31B-it, wrong-helpful messages appear in every benchmark-model combination. Among wrong-answer messages that change final correctness, more than four in ten changes are helpful in each model. Controlled repeats show that the number of repeatable message effects is unlikely to arise from replay variation alone (p=0.0002). A focused intervention on repeatable wrong-helpful messages finds that the complete message works best, while retaining its reasoning preserves more success than retaining only its answer; the source of the complete-message advantage remains open. Within the same problem, repeated trajectory-value evidence also identifies a better keep-or-remove choice than answer correctness alone. Answer correctness is therefore informative but does not determine trajectory value. DHD measures this missing property and produces reusable labels for learning when agents should listen.

Figures & tables

Figure 2: The Diverse Hypothesis Deliberation (DHD) protocol. A recruiter assigns five problem-specific roles without seeing the ground-truth answer. Each hypothesizer independently produces one structured message from its role, with no access to peer messages or the ground-truth answer. An integrator synthesizes a selected subset of the cached messages into one final answer, without being told which proposed answers are correct. An evaluator then compares the integrator’s answer with the ground-truth answer without observing agent reasoning.

Figure 3: Two replay mechanisms provide complementary observations of trajectory value from the same cached messages. (a) In-pool leave-one-out (LOO) replay hides $h_{i}$ and compares the result with full-pool integration; (b) single-message replay shows only $h_{i}$ and compares the result with independent solving. Each matched replay pair yields a helpful, neutral, or harmful observation.

Figure 4: Direction of in-pool LOO effects among replays that flip final correctness. The colored bars answer: among flips, which direction did they take? Helpful and harmful shares therefore sum to

100\%

within each row, and bar labels give the event counts. The row label separately reports the flip rate among all eligible messages, including neutral replays. Helpful means the full pool is correct and the reduced pool is wrong; harmful means the reverse. More than four in ten flips involving wrong-answer messages are helpful in each model family; across all messages with wrong proposed answers, helpful flips account for

6.3\%

in OSS and

3.2\%

in Gemma.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Candidate availability and integration outcomes across model families. Each panel reports independent-solver accuracy, $K{=}5$ integrator accuracy, and whether at least one message has an evaluator-correct proposed answer. Candidate availability measures answer availability rather than reasoning usability; an integrator can also synthesize an answer absent from the individual messages. The Gemma panel uses its matched complete-LOO sample, whose messages are all answer-bearing.

Figure 7: Off-diagonal in-pool LOO outcomes across benchmarks and model families. Left: helpful share among flips from messages with wrong proposed answers. Right: harmful share among flips from messages with correct proposed answers. Large numbers give these conditional shares; smaller lines give the flip rate among all eligible messages and the number of flips. Neutral replays are excluded only from the conditional share. Cross-benchmark levels are descriptive because tasks and baselines differ.

Figure 8: Per-message leave-one-out rates by

K

and benchmark. Each bar is the share of all eligible LOO messages at that

K

. Helpful (W+, C+) means removal breaks a correct full-pool result; harmful (W-, C-) means removal fixes a wrong full-pool result. Splitting by proposal correctness prevents the ambiguous reading that “harmful” means only wrong-harmful or only correct-harmful.

Benchmark	Model family	$N$	Direct	Iterative	PER	Broadcast
Omni-MATH-2	Gemma	4,181	69.4	85.9	90.6	93.0
Omni-MATH-2	OSS	4,181	56.8	78.8	85.2	89.2
JEEBench	Gemma	515	70.5	81.4	95.9	98.3
JEEBench	OSS	515	41.6	55.3	91.5	95.0
SciBench	Gemma	574	70.6	79.4	91.3	87.8
SciBench	OSS	574	62.4	72.3	87.5	89.7

Appendix

Table 11: System-level final accuracy (%) for four protocols across five benchmarks and two model families. The 40 protocol cells are complete and have no duplicate problem identifiers.

N

is the number of problems in each matched protocol cell. The comparison establishes the broader collaboration context; the protocols have different computational costs and are not equal-budget DHD baselines.

Model	Benchmark	Disagreement [95% CI]	W $+$	W $-$	C $+$	C $-$	Indeterminate
OSS	Omni-MATH-2	9.1% [7.5, 11.0]	7	7	23	13	950
OSS	JEEBench	4.1% [3.0, 5.5]	0	7	2	4	987
OSS	SciBench	4.0% [3.0, 5.4]	1	10	7	3	979
OSS	LAB-Bench	17.5% [15.3, 20.0]	14	9	35	16	926
OSS	MaScQA	1.6% [1.0, 2.6]	1	4	3	4	988
OSS	Pooled	7.3% [6.6, 8.0]	23	37	70	40	4,830

Appendix

Table 17: Controlled repeated replay at

K{=}5

, with 200 problems per benchmark and model. Disagreement is the share of 1,000 identical-input full-pool pairs with different correctness labels; brackets give Wilson 95% confidence intervals (CIs). Each benchmark also contains 1,000 problem–message estimates. W

+

-

and C

+

-

count messages with wrong/correct proposed answers whose repeated-call interval lies wholly above/below zero; all other intervals are indeterminate. These intervals are exploratory and unadjusted for multiple comparisons. One Gemma SciBench message has four rather than five removal blocks because one evaluation was missing; no value is imputed.

Benchmark	Wrong-answer messages	Correct-answer messages	Correct $-$ wrong
Omni-MATH-2	$0.00\;[-2.04,2.07]$	$2.62\;[0.66,4.61]$	$2.62\;[0.14,5.16]$
JEEBench	$-6.35\;[-11.56,-2.00]$	$0.38\;[-0.47,1.21]$	$6.73\;[2.47,11.73]$
SciBench	$-1.69\;[-4.06,0.49]$	$0.53\;[-0.44,1.56]$	$2.23\;[0.06,4.56]$
LAB-Bench	$-1.40\;[-3.65,0.83]$	$4.19\;[1.17,7.28]$	$5.59\;[2.18,8.96]$
MaScQA	$-2.74\;[-8.53,2.07]$	$0.29\;[-0.70,1.19]$	$3.02\;[-1.64,8.56]$
Pooled	$-1.66\;[-2.91,-0.45]$	$1.34\;[0.65,2.02]$	$3.00\;[1.72,4.28]$

Appendix

Table 18: Mean repeated trajectory-value estimate for OSS by proposal correctness (percentage points). Each message’s estimate averages five matched blocks; 95% intervals hierarchically resample problems and matched replay blocks while keeping each problem’s five messages together. The pooled bootstrap is stratified by benchmark. The final column shows the correct-minus-wrong difference. These are repeated-call estimates, not single-realization rates with the disagreement statistic subtracted.

Model	Separated [null]	W $+$ [null]	W $-$ [null]	C $+$ [null]	C $-$ [null]	BH W $+$ /W $-$ /C $+$ /C $-$	Strict W $+$ /C $-$
OSS	170 [83,129]	23 [10,30]	37 [9,26]	70 [20,43]	40 [24,53]	0/0/0/0	7/13
Gemma	169 [45,83]	73 [13,38]	57 [15,42]	29 [0,7]	10 [2,16]	17/20/10/0	60/5

Appendix

Table 19: Cross-model calibration of repeated

K{=}5

effects. Brackets give 95% count ranges from 5,000 within-block permutations. Testing 5,000 messages creates many opportunities for chance discoveries, so the Benjamini–Hochberg (BH) column controls the expected proportion of false discoveries among selected candidates. The more conservative Benjamini–Yekutieli (BY) correction allows arbitrary dependence among messages and retains none. Strict counts require evaluable outputs in every arm and block.

Diagnostic sample ( $n$ )	Full	Repeat	Remove	Neutral replacement	Hide answer	Hide reasoning
Wrong-helpful (10)	82	81	26	46	64	44
Correct-helpful (3)	100	100	7	67	87	100
Wrong-harmful (3)	0	0	100	67	27	47
Correct-harmful (2)	100	100	100	100	100	70
Indeterminate (4)	50	53	50	50	55	50

Appendix

Table 23: Integrator success (%) in the component-masking diagnostic. Each cell averages five repetitions; the full and repeat columns each additionally average their two identical-input outcomes. Sample membership was fixed from controlled repeated replay before any masking calls, so these rates describe the diagnostic sample rather than population prevalence. The near-identical full and repeat columns provide a local same-input check. For repeatable wrong-helpful cases, hiding the wrong answer preserves more success than hiding the reasoning.

Model	Full $\rightarrow$ cross-fit	Gain [95% CI]
OSS	$71.85\rightarrow 73.53$	$+1.68$ [0.88, 2.50]
Gemma	$83.11\rightarrow 85.73$	$+2.61$ [1.76, 3.54]

Appendix

Table 24: Micro-pooled accuracy (%) on a held-out replay block of the same problem for the cross-fitted one-removal opportunity analysis. Gain intervals are problem-clustered 95% confidence intervals (CIs) from the bootstrap.

Benchmark	OSS	Gemma
Omni-MATH-2	+2.15	+5.05
JEEBench	+2.75	+2.00
SciBench	+1.00	+2.66
LAB-Bench	+1.95	+1.40
MaScQA	+0.55	+1.95

Appendix

Table 25: Cross-fitted accuracy gain over the full five-message pool by benchmark (percentage points). Gemma SciBench uses 199 complete problems; every other cell uses 200.

Model / arm	Observed	All-action	Removal identity
OSS / A	+1.70	[ $-0.78$ , 0.82]	[ $-0.50$ , 0.64]
OSS / B	+1.18	[ $-0.82$ , 0.76]	[ $-0.74$ , 0.32]
Gemma / A	+2.60	[ $-0.60$ , 0.62]	[0.50, 1.26]
Gemma / B	+2.40	[ $-0.58$ , 0.64]	[0.42, 1.18]

Appendix

Table 26: Sensitivity of the cross-fitted gain to the full-pool replicate and two block-preserving shuffled-control tests. Gains and null intervals are in percentage points. All-action permutes all six actions; removal-identity keeps the full action fixed and permutes the five removal identities.

Figure 9: Pairwise agreement among three large language model (LLM) evaluators under the same answer-equivalence evaluation procedure (

N=16{,}724

saved submissions). Percentages are computed on instances with valid judgments from both evaluators; Cohen’s

\kappa

is shown inside each bar.

Multi-Agent Reasoning

Multi-Agent Reasoning studies how multiple AI agents share evidence, model one another, and coordinate decisions on complex tasks.

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Multi-agent LLM systems rely on message passing among specialized agents to accomplish complex tasks. However, an upstream agent may provide useful information or an incorrect answer that causes a downstream agent to override a correct answer supported by its own evidence. Prior work has not clearly separated the benefits of communication from the damage caused by incorrect messages. We study this problem with controlled experiments across five benchmarks and five receivers, keeping the downstream task and evidence fixed while comparing answers under three conditions: no message, the upstream agent's original message, or a message with the opposite conclusion. Our experiments reveal three key findings. First, messages often help when the downstream agent would otherwise answer incorrectly. Second, messages can also hurt: when the downstream agent would answer correctly without a message, an incorrect upstream message changes the answer in up to 32% of cases. Third, in 94% of audited harmful cases, the downstream agent copies the upstream's specific wrong answer--a pattern we term answer substitution. Removing unreliable messages recovers part of the lost accuracy, suggesting that communication should be selective based on upstream reliability and the evidence already available to the downstream agent.

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