Demystifying Multi-Agent Debate: The Role of Confidence and Diversity

TL;DR

MAD often exhibits a belief martingale, $\mathbb{E}[p_{i,t} \mid \mathcal{F}_{t-1}] = p_{i,t-1}$, which limits gains over majority voting. The authors introduce two interventions grounded in human deliberation: (i) diversity-aware initialization to broaden the initial hypothesis pool and (ii) confidence-modulated debate with calibrated confidence signals that weight updates, yielding a submartingale characterized by $\mathbb{E}[p_{i,t} \mid \mathcal{F}_{t-1}] \ge p_{i,t-1}$. Theoretical results show that diversity raises the prior probability of success without altering dynamics, while confidence-modulated updates produce a positive drift toward the correct answer. Empirically, across six QA benchmarks and two LLMs, these methods consistently outperform vanilla MAD and majority vote, with larger gains on harder tasks, illustrating that principled, training-free adjustments can substantially enhance MAD effectiveness. $

Abstract

Multi-agent debate (MAD) is widely used to improve large language model (LLM) performance through test-time scaling, yet recent work shows that vanilla MAD often underperforms simple majority vote despite higher computational cost. Studies show that, under homogeneous agents and uniform belief updates, debate preserves expected correctness and therefore cannot reliably improve outcomes. Drawing on findings from human deliberation and collective decision-making, we identify two key mechanisms missing from vanilla MAD: (i) diversity of initial viewpoints and (ii) explicit, calibrated confidence communication. We propose two lightweight interventions. First, a diversity-aware initialisation that selects a more diverse pool of candidate answers, increasing the likelihood that a correct hypothesis is present at the start of debate. Second, a confidence-modulated debate protocol in which agents express calibrated confidence and condition their updates on others' confidence. We show theoretically that diversity-aware initialisation improves the prior probability of MAD success without changing the underlying update dynamics, while confidence-modulated updates enable debate to systematically drift to the correct hypothesis. Empirically, across six reasoning-oriented QA benchmarks, our methods consistently outperform vanilla MAD and majority vote. Our results connect human deliberation with LLM-based debate and demonstrate that simple, principled modifications can substantially enhance debate effectiveness.

Figures (3)

• Figure 1: Illustration of two human-inspired ingredients for effective MAD: (a) diverse initial answers increase the chance the correct hypothesis is present; (b) explicit confidence sharing enables confidence-weighted updates that steer beliefs toward the correct answer.
• Figure 2: Sankey plots of correctness transitions from round 1 to round 5 (Qwen): The confidence-modulated debate (b) shows improvement on answer revision (blue), indicating better belief drift to correctness.
• Figure 3: (a) Correlation between initial answer diversity and dataset difficulty (measured by model accuracy) over 62 datasets (including 57 MMLU sub-tasks). (b) Relationship between dataset difficulty and debate performance gain. MAD tends to yield larger gains on more difficult datasets.

Theorems & Definitions (13)

• Proposition 1: Diversity-aware initialization improves prior success
• Theorem 1: Confidence-weighted debate yields a submartingale
• Theorem 2: Effect of diversity-aware initialization
• proof
• Lemma 1: Convex combination form
• proof
• Definition 1: Confidence-weighted expected correctness
• Lemma 2: Decomposition of confidence-weighted correctness
• proof
• Corollary 1