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Breaking the Martingale Curse: Multi-Agent Debate via Asymmetric Cognitive Potential Energy

Yuhan Liu, Juntian Zhang, Yichen Wu, Martin Takac, Salem Lahlou, Xiuying Chen, Nils Lukas

TL;DR

AceMAD is proposed, 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, and recovers sparse truth signals even when initial majorities are incorrect, substantially outperforming baseline methods.

Abstract

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.

Breaking the Martingale Curse: Multi-Agent Debate via Asymmetric Cognitive Potential Energy

TL;DR

AceMAD is proposed, 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, and recovers sparse truth signals even when initial majorities are incorrect, substantially outperforming baseline methods.

Abstract

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.
Paper Structure (44 sections, 9 theorems, 54 equations, 5 figures, 6 tables, 1 algorithm)

This paper contains 44 sections, 9 theorems, 54 equations, 5 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Standard Multi-Agent Debate
  4. The Debate Protocol.
  5. The Challenging Interval
  6. Debate as a Martingale Process
  7. When Does Debate Matter?
  8. Experimental Setup
  9. Benchmarks.
  10. Baselines.
  11. Key Empirical Observations
  12. Debate is Essential in the Challenging Interval
  13. Theoretical Framework
  14. Asymmetric Cognitive Potential Energy in MAD
  15. Interpretation.
  16. ...and 29 more sections

Key Result

Proposition 2.2

In a closed debate system without external oracle signals, if the belief update rule is linear (as in Eq.(rule)) and symmetric, the sequence of the average belief in the ground truth, denoted as $\mu_t = \bar{p}^{(t)}(y^*)$, forms a martingale:

Figures (5)

  • 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.
  • Figure 2: Impact of Agent Scaling. Performance on challenging subsets improves with group size ($N=1 \to 5$), 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 ($m_i^{(t)}$) in a shared context; (2) Signal Extraction: Agents privately commit to their self-belief ($p_i$) and peer-prediction ($q_i$),revealing asymmetric second-order cognition; (3) Verification: Brier Score $S_i$ 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.
  • 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 ($N \le 10$), 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.

Theorems & Definitions (18)

  • Definition 2.1: Challenging Interval
  • Proposition 2.2: Martingale Property of Standard MAD
  • Definition 4.1: Debate Observable Space
  • Theorem 4.2: Information-Theoretic Manifestation of Cognitive Potential
  • Definition 4.3: Peer Prediction Scoring
  • Theorem 4.4: Cognitive Potential Energy Separation
  • Definition 4.5: Nonlinear Potential-to-Influence Conversion
  • Theorem 4.6: Submartingale Convergence: Breaking the Curse
  • Theorem E.1: Blackwell dominance of AceMAD
  • proof
  • ...and 8 more