Breaking the Martingale Curse: Multi-Agent Debate via Asymmetric Cognitive Potential Energy
Authors: Yuhan Liu, Juntian Zhang, Yichen Wu, Martin Takac, Salem Lahlou, Xiuying Chen, Nils Lukas
Organizations: Mohamed bin Zayed University of Artificial Intelligence · Gaoling School of Artificial Intelligence, Renmin University of China · Harvard University
Multi-Agent Debate (MAD) has emerged as a promising paradigm for enhancing large language model reasoning. However, recent work reveals a limitation:standard MAD cannot improve belief correctness beyond majority voting; we refer to this as the Martingale Curse. This curse arises because correlated errors cause agents to converge toward erroneous consensus, where debate merely reinforces collective mistakes rather than filtering noise. We propose AceMAD, a framework that breaks the Martingale Curse by harnessing asymmetric cognitive potential energy to transform MAD from a random walk into a directed convergence process with positive drift. Through a peer-prediction mechanism, agents predict their peers' belief distributions, revealing asymmetric cognitive potential: truth-holders not only know the correct answer but also anticipate the crowd's misconceptions, while the hallucinating majority remains blind to their collective error. This asymmetry creates a potential energy gap that we quantify via strictly proper scoring rules. We prove this cognitive potential manifests as information-theoretic superiority and, under nonlinear aggregation, converts into submartingale drift toward truth, directly breaking the Martingale Curse. Experiments on challenging subsets across six benchmarks show AceMAD recovers sparse truth signals even when initial majorities are incorrect, substantially outperforming baseline methods.
Figures & tables
Figure 1: Breaking Martingale Curse via Peer Prediction. (Left) Standard MAD fails as the majority converges on a common misconception (“D”), drowning out the truth. (Right) AceMAD recovers the truth (“C”) by incentivizing agents to predict peer belief. The truth-holder is identified by correctly anticipating the crowd’s behavior, allowing the system to filter out correlated noise.
Methods
TruthfulQA
ARC-C
BBH
LogiQA
MedQA
MMLU-Pro
Average
Majority Voting
11.2
23.1
22.1
12.8
9.4
5.5
14.0
Decentralized MAD
15.7 ( ↑ 4.5)
38.9 ( ↑ 15.8)
37.6 ( ↑ 15.5)
13.1 ( ↑ 0.3)
21.0 ( ↑ 11.6)
6.6 ( ↑ 1.1)
22.1 ( ↑ 8.1)
Centralized MAD
14.3 ( ↑ 3.1)
38.9 ( ↑ 15.8)
40.3 ( ↑ 18.2)
11.3 ( ↓ 1.5)
21.3 ( ↑ 11.9)
6.3 ( ↑ 0.8)
22.1 ( ↑ 8.1)
Sparse MAD
17.5 ( ↑ 6.3)
38.0 ( ↑ 14.9)
38.3 ( ↑ 16.2)
11.6 ( ↓ 1.2)
20.3 ( ↑ 10.9)
7.3 ( ↑ 1.8)
22.2 ( ↑ 8.2)
Table 1: Performance Comparison on Challenging Subsets ( N=5 ). We compare Majority Voting with three MAD variants . ↑ and ↓ indicate relative gains and losses. Overall, MAD variants consistently outperform the Majority Voting baseline across the benchmarks.
Figure 2: Impact of Agent Scaling. Performance on challenging subsets improves with group size ( N=1→5 ), indicating that multi-agent debate effectively filters correlated noise.
Figure 3: The AceMAD: Converting Asymmetric Cognitive Potential into Submartingale Drift. The process iterates through four phases: (1) Argumentation: Agents exchange arguments ( mi(t) ) in a shared context; (2) Signal Extraction: Agents privately commit to their self-belief ( pi ) and peer-prediction ( qi ),revealing asymmetric second-order cognition; (3) Verification: Brier Score Si quantifies the cognitive potential gap; (4) Non-linear Amplification: Exponential weight updates convert potential into directional drift, progressively amplifying truth-holders’ influence until they dominate the aggregate, breaking the Martingale Curse.
Methods
ARC-C
LogiQA
MMLU-Pro
TruthfulQA
MedQA
BBH
Average
Closed-Source Model: GPT-4o-mini
Single Agent
20.37
16.25
6.42
31.88
11.61
33.79
20.05
Majority Voting
23.15
21.88
5.50
17.94
23.23
22.07
18.96
Decentralized MAD
41.67
22.19
6.88
33.63
23.23
44.14
28.62
Centralized MAD
41.67
22.19
8.72
39.51
24.19
39.66
29.32
Sparse MAD
43.52
22.19
8.72
32.74
23.23
47.24
29.61
Table 2: Main results on challenging subsets ( N=5 ). We evaluate AceMAD ( T∈{2,3,5} ) against different baselines across six benchmarks using both GPT-4o-mini and open-source models. Performance gains over the best baseline are shown in parentheses.
Methods
ARC-C
LogiQA
MMLU-Pro
TruthfulQA
MedQA
BBH
Average
Decentralized MAD
41.67
22.19
8.72
33.63
23.23
44.14
28.93
AceMAD (Different LLMs)
56.48 ( ↑ 14.81)
20.94 ( ↓ 1.25)
9.63 ( ↑ 0.91)
37.22 ( ↑ 3.59)
38.71 ( ↑ 15.48)
74.14 ( ↑ 30.00)
39.52 ( ↑ 10.59)
AceMAD (Different Persona)
60.19 ( ↑ 18.52)
23.13 ( ↑ 0.94)
12.08 ( ↑ 3.36)
46.19 ( ↑ 12.56)
23.23 ( ↑ 0.00)
76.21 ( ↑ 32.07)
40.17 ( ↑ 11.24)
AceMAD (Different Thinking System)
61.11 ( ↑ 19.44)
22.19 ( ↑ 0.00)
12.08 ( ↑ 3.36)
43.95 ( ↑ 10.32)
23.23 ( ↑ 0.00)
37.59 ( ↓ 6.55)
33.36 ( ↑ 4.43)
Table 3: Impact of heterogeneity configurations on AceMAD performance. Results demonstrate the effectiveness of inducing diversity via cross model mixing, persona-driven role-playing, and cognitive system variation to satisfy.
Figure 4: Ablation study of AceMAD . We compare the full protocol against two variants: (1) Self-Belief Only , which ignores second-order peer forecasts, and (2) Standard MAD , which uses uniform weighting. The results highlight that peer prediction is essential for filtering out incorrect majority beliefs.
Figure 5: Impact of agent scaling on accuracy. The performance trajectory exhibits two distinct regimes: (1) Scaling Growth ( N≤10 ), where increasing the group size enhances the probability of surfacing sparse truth signals, and (2) Scaling Decay ( N>10 ), where an oversized majority reinforces correlated noise, leading to a diminished signal-to-noise ratio and collective fallacies.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Methods
ARC-C
LogiQA
MMLU-Pro
TruthfulQA
MedQA
BBH
Average
Majority Voting
75.00
22.19
31.80
52.47
43.23
52.47
46.19
Decentralized MAD
78.70
27.50
35.78
60.99
56.45
60.99
53.40
Centralized MAD
75.93
46.25
36.54
84.75
56.77
62.78
60.50
Sparse MAD
75.00
47.19
35.78
62.78
57.10
84.75
60.43
AceMAD (T=3)
80.56 ( ↑ 1.86)
54.06 ( ↑ 6.87)
38.50 ( ↑ 1.96)
78.03 ( ↓ 6.72)
60.00 ( ↑ 2.90)
90.00 ( ↑ 5.25)
66.86 ( ↑ 6.36)
Appendix
Table 4: Generalization across models: Performance comparison using the DeepSeek-V3.1 model ( N=5 ). AceMAD consistently outperforms all baseline variants, demonstrating its robustness across different LLM architectures.
Methods
TruthfulQA
MedQA
BBH
Average
Single Agent
27.80
28.06
17.93
24.60
Majority Voting
30.49
27.42
17.59
25.17
Decentralized MAD
33.18
27.74
16.90
25.94
Centralized MAD
37.53
27.42
16.55
27.17
Sparse MAD
33.63
26.45
16.55
25.54
AceMAD (T=3)
36.77
29.68
22.76
29.74
Appendix
Table 5: Performance comparison on the Llama-3.1-8B-Instruct model ( N=5 ). This table evaluates the effectiveness of AceMAD using an open-source model backbone across all benchmarks, demonstrating that the peer-prediction mechanism remains robust across different model architectures.
Methods
ARC-C
LogiQA
Average
MAD
81.06
49.15
65.11
AceMAD
85.07 ( ↑ 4.01)
53.41 ( ↑ 4.26)
69.24 ( ↑ 4.13)
Appendix
Table 6: Performance comparison on the full benchmarks ( N=5 ). Unlike the focus on the Challenging Interval , this table reports results across the entire test sets of ARC-C and LogiQA. AceMAD demonstrates consistent gains even in these mixed-difficulty regimes, with improvements over the baseline shown in parentheses.