PaperlayerView exact revision on arXiv

Source-linked AI summary

Bias Control and Injection for Multi-Agent Aggregation

Sungjae Lee, Soheil Abbasloo, Jabin Koo, Jungseul Ok

arXiv:2610.21290v1cs.AIcs.CLcs.MA

TL;DR

When agents disagree, aggregation must determine whose answer to trust despite non-semantic biases that can obscure reasoning quality. The paper controls presentation bias for initial decisions and injects bias against selected answers to assess reliability. Across mathematical benchmarks and heterogeneous model combinations, the framework improves accuracy and token efficiency over non-oracle methods, and also improves aggregation in debate and Mixture-of-Agents.

  • Problem

    Agreement-based aggregation can favor shared errors over correct minority answers, while LLM decisions can be distorted by frequency, style, position, and other non-semantic biases.

    Source: §1 · pp. 1–2

  • Method

    The test-time framework balances and style-normalizes one reasoning trace per distinct answer, then places each agent’s selected answer in the minority to assess decision reliability.

    Source: §4 · p. 5

  • Results

    Across four model combinations, the framework achieves the highest average accuracy, beating the strongest non-oracle baseline by 3.2–6.2 percentage points; it also improves accuracy in debate and Mixture-of-Agents.

    Source: §1 · p. 2, §5.2 · p. 6

  • Takeaways & Limitations

    The framework improves multi-agent answer aggregation without validation data or prior knowledge of relative model strength, and can be applied without modifying debate or Mixture-of-Agents processes.

    Source: §1 · p. 2, §7 · p. 9

    Evaluation focuses on mathematical reasoning with four open-weight 8B–14B models and fixed upstream reasoning histories; broader domains, larger models, and interventions during interaction or synthesis remain future directions.

    Source: §8 · p. 9

Abstract

from arXiv · show

Multi-agent reasoning with large language models (LLMs) raises a central question for collective decision-making: when agents disagree, whose decision should be trusted? Resolving such disagreements is difficult because agents' decisions are susceptible to non-semantic biases such as majority pressure and response-style preferences, which can obscure the quality of the underlying reasoning. Our analysis shows how bias can be both mitigated and exploited: bias control improves initial answer selection, while bias injection elicits decision instability that serves as a reliability signal. Building on these findings, we introduce a test-time aggregation framework that requires neither validation data nor prior knowledge of relative model strength. The framework implements bias control by presenting one style-normalized reasoning trace per distinct answer, and bias injection by placing each aggregating agent's initially selected answer in the minority. Overall, our framework achieves higher average accuracy and better token efficiency than competing non-oracle aggregation methods across three mathematical reasoning benchmarks and four heterogeneous LLM combinations. It also improves final-answer accuracy when applied to multi-agent debate and Mixture-of-Agents, without modifying their interaction or synthesis processes.

1 INTRODUCTION

Multi-agent reasoning can produce conflicting solutions, and final aggregation may miss a correct answer among them. The paper proposes controlling presentation bias to improve initial selections and injecting bias against those selections to assess reliability.

1 INTRODUCTION

  • Correct answers appear in 83.6% of sampled response pools on average, but majority voting selects them in only 72.2%.The paper notes that majority voting can overlook correct answers with limited support.Source: §1 · p. 1, Figure 1 · p. 1
  • The framework first compares one style-normalized trace per distinct answer under balanced frequency, then places each selected answer in the relative minority to test its reliability.It requires neither validation data nor prior knowledge of relative model strength.Source: Figure 2 · p. 2, §1 · p. 2
  • The analysis finds that style normalization improves initial answer selection, while bias injection changes incorrect decisions more often than correct ones.The resulting instability is a more informative reliability signal than reversing candidate order or repeating the query.Source: §1 · p. 2
  • Across three mathematical reasoning benchmarks and four heterogeneous LLM combinations, the framework achieves higher average accuracy and better token efficiency than competing non-oracle methods.It also improves final-answer accuracy in debate and Mixture-of-Agents without modifying their interaction or synthesis processes.Source: §1 · p. 2

2 RELATED WORK

Prior work combines agent outputs through interaction, consensus, majority voting, consistency, synthesis, and verification. This section situates the paper at the intersection of final-answer aggregation and non-semantic biases in multi-agent judgments.

Multi-Agent Reasoning

  • Multi-agent approaches generate, exchange, critique, synthesize, or revise solutions to combine reasoning diversity and complementary strengths (Du et al., 2023; Wang et al., 2025; Choi et al., 2026a; Tian et al., 2026; Cui et al., 2026; Zhu et al., 2026).Interactions do not consistently help and may converge on incorrect consensus or amplify early errors (Wang et al., 2024b; Smit et al., 2023; Choi et al., 2026a; Zhu et al., 2026; Zhu et al., 2025; Choi et al., 2025; Okawa, 2026; Cui et al., 2026).Source: §2 · pp. 2–3

Final-Answer Aggregation

  • Final-answer methods include debate consensus, validation-informed model selection, layered synthesis, answer scoring, rationale selection, and verifier judgments (Du et al., 2023; Choi et al., 2026a; Wang et al., 2026a; Wang et al., 2025; Ai et al., 2025; Cui et al., 2026; Wang et al., 2026b; Lifshitz et al., 2025).Majority voting and consistency-based aggregation can fail with correlated errors or low-frequency correct answers (Choi et al., 2026a; Ai et al., 2025; Cui et al., 2026; Wang et al., 2026b).Source: §2 · p. 3

Biases in Multi-Agent Aggregation

  • LLM judgments can be influenced by frequency, position, style, and model identity, with heterogeneous systems also producing different trace styles (Ma et al., 2025; Zhu et al., 2025; Choi et al., 2025; Wang et al., 2024a; Shi et al., 2025; Feuer et al., 2025; Choi et al., 2026b; Chen et al., 2025).Prior work has mostly analyzed or mitigated bias, while some work exploits it to examine robustness or derive uncertainty signals (Ma et al., 2025; Shi et al., 2025; Li et al., 2024; 2025; Ye et al., 2025; Yang et al., 2026b;a; Koo et al., 2026; Yang et al., 2026c; Badshah et al., 2026).Source: §2 · p. 3

3 MOTIVATION: INTERVENING ON BIASES

The section tests whether presentation-bias interventions can improve initial answer selection and whether adverse bias against a selected answer can signal its reliability. It analyzes independently sampled mathematical-reasoning pools and finds benefits from both style normalization and adverse-frequency injection.

3 MOTIVATION: INTERVENING ON BIASES

  • The analysis uses 220 problems from OlymMATH, AIME, and HMMT, with three completed traces from each of four models and four three-model combinations.It focuses on instances not resolved by unanimous model-level agreement; participating models also judge the competing answers.Source: §3 · p. 3

3.1 CONTROLLING BIASES IN CANDIDATE PRESENTATION

  • For initial decisions, the method retains one trace per distinct answer and rewrites each into common plain prose, controlling answer frequency and source-dependent presentation patterns.The rewriting prompt instructs the rewriter to preserve trace length, reasoning content, numerical values, equations, and final answer.Source: §3.1 · pp. 3–4
  • Style normalization improves pooled answer-selection accuracy and maintains or improves accuracy in 9 of 12 agent–combination settings.The comparison uses paired decisions on the same problems, candidate answers, and source traces, with only the normalized condition presenting rewritten traces.Source: §3.1 · p. 4, Table 1 · p. 4

3.2 INJECTING BIAS TO ELICIT DECISION INSTABILITY

  • The instability signal is whether a controlled decision changes under an adverse presentation, not whether the perturbed query produces a better answer.Adverse-frequency presents the initial choice once and its strongest competing answer twice, while keeping the set of answer hypotheses fixed.Source: §3.2 · p. 4
  • Adverse-frequency injection changes 59.8% of incorrect decisions and 19.1% of correct decisions, a 40.7-point gap that exceeds repeated inference and position swap.Figure 3 presents flip rates conditioned on initial correctness; the section reports a substantial selectivity gap even when the added trace is an exact copy.Source: Figure 3 · p. 4, §3.2 · p. 4
  • When agents disagree, the framework tests their controlled decisions with adverse-frequency bias and combines observed decision holds with cross-agent support without changing upstream reasoning.Figure 4 depicts the reasoning-history responses grouped by answer before aggregation.Source: Figure 4 · p. 5

4 PROPOSED METHOD

The framework routes unanimous upstream agreement directly to an answer and otherwise uses bias-controlled decisions followed by a problem-specific instability test. A final rule combines first- and second-pass decisions without requiring validation data, model-strength estimates, or a designated aggregator.

4 PROPOSED METHOD

  • The upstream agreement gate returns an agreed answer when every participating model has the same unique modal answer; otherwise, it invokes additional aggregation.Within-model ties do not satisfy the gate.Source: §4 · p. 5
  • Bias control retains one trace per distinct answer, presents each once, and rewrites traces into a shared plain-prose form before aggregators select answers.Unanimous controlled selections are returned; disagreement triggers bias injection.Source: §4 · p. 5
  • Bias injection places each agent’s controlled choice in the minority using a distinct trace for the strongest-supported competing answer, or reverses candidate order if none is available.The answer hypotheses remain unchanged during adverse-frequency injection.Source: §4 · p. 5
  • Final aggregation selects an answer retained by at least one agent and supported by a second-pass majority; if none qualifies, it uses controlled-decision or combined-stage votes.Original response frequencies help construct candidates but do not count as final votes.Source: §4 · p. 5

5 EXPERIMENTS

The experiments evaluate four heterogeneous three-model combinations on mathematical reasoning benchmarks, comparing accuracy and token costs against aggregation baselines. The framework leads non-oracle methods in average accuracy, lies near the observed accuracy–cost frontier, and improves final-answer accuracy in debate and Mixture-of-Agents.

5.1 EXPERIMENTAL SETUP

  • The evaluation uses four three-model combinations of open-weight reasoning models and 220 problems from OlymMATH, AIME 2024–2025, and February HMMT 2024–2025.Final-answer accuracy and mean input-and-output tokens per problem are the primary measures; the primary response budget is k = 3.Source: §5.1 · p. 6

5.2 EXPERIMENTAL RESULTS

  • Most Ours configurations with k ∈ {2, 3, 4} lie on or near the observed accuracy–cost Pareto frontier across the four model combinations.Figure 5 reports macro-average accuracy across OlymMATH, AIME, and HMMT against total input-and-output tokens per problem.Source: §5.2 · p. 6, Figure 5 · p. 7
  • 3.2–6.2 percentage points over the strongest non-oracle baseline, Ours achieves the highest average accuracy across all four model combinations.It also achieves the best or tied-best non-oracle accuracy in 12 of 12 benchmark–combination pairs and outperforms the oracle reference by 1.8–4.3 points.Source: §5.2 · p. 6, Table 2 · p. 7
  • Ours achieves the highest recovery while retaining most majority-voting-correct answers, with rates macro-averaged across benchmarks.Source: Figure 6 · p. 7
  • Applying Ours to debate and Mixture-of-Agents improves final-answer accuracy at moderate additional inference cost while leaving upstream reasoning histories fixed.The method is applied when specified disagreement or answer-change conditions are met.Source: §5.2 · p. 8

6 ANALYSIS

The analyses examine style normalization, bias injection, and their ablations. They find that normalization reduces representation distance beyond formatting removal, while adverse-frequency injection yields more selective and accurate retained decisions than the tested alternatives.

6.1 UNDERSTANDING STYLE NORMALIZATION

  • Full normalization reduced mean representation distance by 8.2%, versus 1.7% for formatting removal, and reduced source-identification accuracy, particularly for formatting features.The matched traces share a problem and answer; Figure 8 compares cosine distances in the Phi representation space, which is not used for rewriting.Source: Figure 8 · p. 8, §6.1 · p. 8

6.2 UNDERSTANDING BIAS INJECTION

  • Adverse-frequency injection retained 37.5% of controlled decisions at 46.9% conditional accuracy, a 16.6-point gain over the 30.3% controlled baseline.Its retained-set accuracy exceeded the tested position-swap and repeated-inference conditions, though it retained fewer decisions.Source: §6.2 · p. 8, Table 3 · p. 8
  • Exact-copy injection produced a 37.3-point incorrect–correct flip gap, versus 32.5 for repeated inference and 34.3 for position swap; distinct traces yielded 40.7 points.The comparison examines instances with upstream disagreement and holds initial controlled decisions fixed across conditions.Source: §6.2 · p. 9, Table 4 · p. 9

6.3 ABLATION

  • Removing normalization lowered accuracy from 63.4% to 61.3%; adverse-frequency injection reached 41.6%, versus 37.6% without a second pass, 39.2% with repeated inference, and 39.3% with position swap.The injection variants use the same final aggregation rule on instances where controlled decisions disagree.Source: §6.3 · p. 9

7 CONCLUSIONS

The paper concludes that a training- and validation-free framework can both control and exploit biases in multi-agent aggregation. It reports improved accuracy and token efficiency in independent reasoning and improved final-answer aggregation in debate and MoA.

7 CONCLUSIONS

  • Bias control reduces frequency and style influences on initial decisions, while bias injection uses decision instability as a problem-specific reliability signal.Source: §7 · p. 9
  • Across three mathematical reasoning benchmarks and four heterogeneous model combinations, the framework improves accuracy and token efficiency.Source: §7 · p. 9
  • Applying the framework to debate and Mixture-of-Agents improves final-answer aggregation without modifying their upstream processes.Source: §7 · p. 9

8 LIMITATIONS

The evaluation is limited to mathematical reasoning with four 8B–14B open-weight models, and bias interventions are applied only at final aggregation with upstream histories fixed.

8 LIMITATIONS

  • Evaluation covers mathematical reasoning with four open-weight models in the 8B–14B range.Broader domains and larger models remain untested in this evaluation.Source: §8 · p. 9
  • Bias interventions are studied at final aggregation while upstream reasoning histories remain fixed.This scope also applies when the procedure is used with Debate and Mixture-of-Agents.Source: §8 · p. 9
  • Extending interventions to interaction or synthesis is future work, with potential effects on solution diversity and agent agreement.Source: §8 · p. 9

AI USE STATEMENT

The paper discloses AI assistance for editing, formatting, and code, and presents an aggregation algorithm that routes agreement before applying bias control, bias injection, and final aggregation.

AI USE STATEMENT

  • Generative AI assisted with manuscript editing and formatting and experimental and analysis code; authors reviewed and verified all outputs.The authors take responsibility for the final content.Source: AI USE STATEMENT · p. 10

A AGGREGATION ALGORITHM

  • The algorithm first returns an answer when the upstream agreement gate is satisfied; otherwise, it groups traces by answer and normalizes one trace per answer for judges.The normalized candidates are ordered by answer frequency, and unanimous initial judge selections are returned directly.Source: A AGGREGATION ALGORITHM · p. 13
  • When initial judge selections disagree, the algorithm probes each selection by placing it in the minority, then uses probe and initial votes to determine the final answer.It returns an answer with probe support and a strict majority of probe votes; otherwise, it uses the strict-majority initial answer or the most-voted answer.Source: A AGGREGATION ALGORITHM · p. 13

B EXPERIMENTAL SETTINGS

The experimental settings specify response-generation sampling, stopping and resampling rules, and target length ranges for normalized traces.

B EXPERIMENTAL SETTINGS

  • CoT sampling follows official model-card settings where available for temperature, top-p, and top-k.The table is identified as summarizing the main hyperparameters.Source: B EXPERIMENTAL SETTINGS · p. 13, Table 5 · p. 14
  • Generation stops after k completed responses or 2k attempts per model, and truncated responses are discarded and resampled within that limit.Tokens from all attempts count toward reported cost.Source: B EXPERIMENTAL SETTINGS · p. 13
  • The normalizer targets 85–115% of original trace length, while accepted normalized traces may fall within 80–120%.Source: B EXPERIMENTAL SETTINGS · p. 13

C.1 ROBUSTNESS TO THE STYLE NORMALIZER

Across the reported robustness and diagnostic analyses, the framework retains strong aggregation performance, while normalization reduces—but does not remove—generator-specific signals.

C.1 ROBUSTNESS TO THE STYLE NORMALIZER

  • Replacing Qwen3-8B with GLM-Z1-9B or Nemotron-Nano-9B changes average accuracy by −2.4 to +1.3 percentage points across four model combinations.These results suggest comparable performance with alternative style normalizers.Source: §C.1 · p. 14, Table 6 · p. 15

C.2 TIE-BREAKING IN SLM-MUX

  • Ours exceeds the test-set Best-SC oracle SLM-MUX variant by 3.8–6.4 percentage points across all four model combinations.The oracle variant has higher average accuracy than Pooled MV in every combination, and both SLM-MUX variants use identical responses and inference cost.Source: §C.2 · pp. 14–15, Table 7 · p. 15

C.3 AGGREGATION UNDER UPSTREAM DISAGREEMENT

  • 5.8–9.5 percentage-point gains over the strongest non-oracle baseline accompany the highest average accuracy in all four model combinations under upstream disagreement.The comparison uses budget-matched baselines; Single-Best-SC is an oracle reference.Source: §C.3 · p. 15, Table 8 · p. 16

D.1 CROSS-GENERATOR CONVERGENCE ACROSS REPRESENTATION SPACES

  • 8.2% and 8.5% mean-distance reductions in Phi and GLM follow full normalization, versus 1.7% and 2.2% after formatting removal.Figure 10 compares matched traces across Phi, GLM, and Qwen representation spaces; full normalization produces larger reductions, particularly in Phi and GLM.Source: Figure 10 · p. 16, §D.1 · p. 16

D.2 SOURCE-GENERATOR ATTRIBUTION

  • 94.1% to 72.2% formatting-feature attribution accuracy falls most after normalization, while substantial source information remains.Rhetorical-marker attribution falls from 88.8% to 79.7%; these feature-group results measure source distinguishability, not contributions to representation-level convergence.Source: Table 9 · p. 17, §D.2 · p. 17

D.3 ADDITIONAL ANALYSIS OF FINAL AGGREGATION

Answers both retained under adverse-frequency injection and supported by a probe majority have the highest conditional accuracy. The examples and prompts show how trace normalization and adverse-frequency presentations are implemented.

D.3 ADDITIONAL ANALYSIS OF FINAL AGGREGATION

  • 49.1% conditional accuracy for answers with both a hold and majority probe support exceeds 18.1% with majority support alone and 23.5% with a hold alone.Table 10 reports conditional accuracy by hold count H(a) and second-pass support count S(a).Source: Table 10 · p. 17, §D.3 · p. 17

D.4 EXAMPLES OF STYLE NORMALIZATION

  • The normalization examples preserve reasoning steps, calculations, and final answers while making the traces’ wording and presentation more consistent, including for an erroneous competing trace.The examples include a correct answer of 6 and a competing answer of 1.Source: §D.4 · p. 18, Table 11 · p. 18, Table 12 · p. 21
  • In the tiling example, the trace concludes that the tiling is unique even when rotations and reflections count as distinct.The passage states that flipping the entire tiling leaves the grid unchanged, so there is no alternative.Source: §D.4 · p. 22

E PROMPTS

  • The style-normalization prompt changes wording and formatting while preserving reasoning content and final answers, with a length constraint limiting deviation from the original trace.Source: Figure 11 · p. 23
  • After a controlled decision selects an answer other than 197, the injection prompt adds an existing trace so 197 becomes the higher-frequency competitor without changing the problem, hypotheses, or decision instruction.The second-pass decision uses the same instruction; only candidate presentation changes.Source: E PROMPTS · pp. 23–24, Figure 12 · p. 24

Related papers

Loading 2610.21290v1…