Debate or Vote: Which Yields Better Decisions in Multi-Agent Large Language Models?
Authors: Hyeong Kyu Choi
Jerry Zhu
Sharon Li
Venue: NeurIPS 2025 spotlight ()
Keywords: multi-agent debate
majority voting
LLM
Agentic AI
Primary area: deep_learning

Abstract: Multi-Agent Debate (MAD) has emerged as a promising paradigm for improving the performance of large language models through collaborative reasoning. Despite recent advances, the key factors driving MAD’s effectiveness remain unclear. In this work, we disentangle MAD into two key components–Majority Voting and inter-agent Debate–and assess their respective contributions. Through extensive experiments across seven NLP benchmarks, we find that Majority Voting alone accounts for most of the performance gains typically attributed to MAD. To explain this, we propose a theoretical framework that models debate as a stochastic process. We prove that it induces a martingale over agents’ belief trajectories, implying that debate alone does not improve expected correctness. Guided by these insights, we demonstrate that targeted interventions, by biasing the belief update toward correction, can meaningfully enhance debate effectiveness. Overall, our findings suggest that while MAD has potential, simple ensembling methods remain strong and more reliable alternatives in many practical settings. Code is released in https://github.com/deeplearning-wisc/debate-or-vote.

Debate or Vote: Which Yields Better Decisions in Multi-Agent Large Language Models?
Hyeong Kyu Choi
Xiaojin Zhu Sharon Li*
Department of Computer Sciences, University of Easconsin-Madison (froilanchoi, jerryzhu, sharonli]@cs wisc.
Abstract
Multi-Agent Dehate (MAD) has emerged promising paradigm for improving the performance of large language models through collaborative reasoning. Despite recent advances, the key factors driving MAD's effectiveness remain unclear. In this work, we disentangle MAD into two key components-Majoriy Voting and inter-agent Debate and assess their respective contributions. Through extensive experiments across seven NLP benchmarks, we find that Majority Voting alone accounts for most of the performance gains typically attributed MAD. To explain this, we propose a theoretical framework that models debate as stochastic process. We prove that it induces a martingale over agents' belief trajectories, implying that debate alone does not improve expected correctness. Guided by these insights, we demonstrate that targeted interventions, by biasing the belief update toward correction, can meaningfully enhance debate effectiveness. Overall, findings suggest that while MAD has potential, simple ensembling methods remain strong and more reliable alternatives in many practical settings. Code is released in sttps://ge thub. com www.///e///ee/eeeceeeeeeeeeeccee wwwllleennnnniiiiee. wwsc/debate-eeeeeeeee.eeeeeeeeed -o-tte...-vv.ee
Introduction
"Out of intense complexities, intense simplicities emerge."
W. CHURCHILL
Throughout history, humans have relied on deliberation to resolve ambiguity, challenge assumptions, and seek better answers. From courtrooms and panels to scientific collaborations, group reasoning plays central role in decision-making. This process-where individuals reflect, revise, and converge through interaction-has long been seen hallmark of intelligent behavior. Inspired by this, recent work has explored whether large language models (LLMs) might similarly benefit from structured interaction. Multi-Ager Agent Debate (MAD) has emerged are prompted to discuss a shared question, each updating their answer based on the responses their peers [1-6]. The hope is that, like human deliberation, such interaction will improve reasoning and lead to better outcomes.
popular framework: multiple LLM agents
At its core, MAD integrates two key ingredients: the use of multiple agents ("Multi-Agent") and their interaction through iterative discussions ("Debate").. Recent work has introduced increasingly sophisticated variants variants-ranging from diverse communication protocols [3, 7, 8]. designing efficient and effective system architectures [1, 2, 9, 10], and assigning varied roles or personas to agents [1 13]. Despite these advances, the underlying mechanisms behind MAD's effectiveness remain unclear. natural step toward understanding MAD's 's performance is disentangle the contribution of each component- are the gains primarily due to meaningful communication between agents, or simply the result of aggregating multiple outputs? Answering this question is important because
Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Question
1.0
Single Agent
oO
6D
Best MAD
0.8
Majority Voting
Accuracy
Question
0.6
X
0.4
0.2
Final Answer
Final Anewer
Arithmetics
GSMBK
Pro Medicine
CSQA
HellaSwag
Formal Logic
HH-RLHH
Majority Voting
Multi-Agent Dehute
Figure I: Majority Voting vs, MAD overview.
Figure Majority Voting is the main contributor to MAD.
informs whether the growing complexity of MAD design is justified by tangible benefits. If most of the performance gain stems from ensembling--ia -ieiinnnn....................... aggregating diverse outputs from independent agents-tthr simpler methods like majority voting may suffice, avoiding additional computational and architectural overhead (see Figure for visual comparison).
To better understand the relative contributions of ensembling vs. interaction, we conduct an extensive empirical study quantifying each component's 's effect. Specifically, we measure the contribution of the "Multi-Agent" component using the performance achieved through Majority Voting, i.e., the aggregated output of agents before any debate rounds occur. We then compare this baseline to the final performance after multiple rounds of "Debate", allowing us to isolate the additional benefit introduced by inter-agent communications. Surprisingly, we find that Majority Voting accounts for most of the performance gains in MAD. In fact, in most cases, majority voting without any debate performs on par with MAD, as seen in Figure 2. To ensure the broad applicability of our findings, our evaluation spans seven diverse benchmarks across multiple tasks and models.
Beyond empirical observations, we introduce theoretical framework in Section that rigorously explains how agents' uncertainty and belief updates shape collective decision-making in both voting and debate. At its core. the framework models each agent as a stochastic process governed by a Corchle-Coomound--We Muutinmmmali (DCM) distribution, capturing internal uncertainty through a Dirichlet belief prior and output randomness via Multinomial sampling- This closely mirrors the behavior of real-world I.IMs, which produce different outputs for the same question due to uncertainty and stochastic generation process (e.g., via temperature or nucleus sampling). Within this framework, we characterize MAD a Bayesian posterior belief update process and prove that it induces a martingale over agents' belief in the correct answer- meaning the expected belief remains unchanged over debate rounds. This implies that debate itself does not systematically improve or degrade beliefs on average; rather, belief evolution driven by stochastic peer influence. In other words, we prove formally that majority vote does essentially all the work, which explains empirical findings.
Our theoretical framework further sheds light on the new design principles to improve MAD (Section 5). In particular, highlights the importance of controlling the martingale by biasing belief updates toward correct signals during debate. We operationalize this insight through several interventions, where correct responses exert more influence than misleading ones, leading to improvements over standard MAD. We summarize our contributions and significance follows:
1. We comprehensively demonstrate that Majority Voting is effective Multi-Agent Debate, when evaluated across seven representative benchmark datasets. We further expand our investigation to more general MAD settings, including configurations with larger and more capable agents, heterogeneous agent populations, and open-ended natural language tasks.
2. We develop new theoretical framework that reveals majority voting's 's success probability, and rigorously characterizes multi-agent debate martingale process. This framework lays a principled foundation for future work to better understand MAD systems.
3. Our theoretical analysis informs that debate alone does not improve beyond majority voting. By designing strategies that help preserve correct responses across debate rounds, achieve

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notable improvements in multi-agent debate performance. This sheds light on future research effectively improve MAD systems.
2 Preliminaries
Multi-Agent Debate collaboratvve framework in which multiple language model agents engage in structured inleraction-typicallyy in the form of iterative exchanges or discussions- solve task such as question answering or text generation [1-6]. In typical MAD protocol, each agent independently generates initial response and then engages series of debate rounds. At round I, an agent receives the original question along with responses from its peers at round - 1, prompting the model update its response accordingly. This iterative process is designed leverage diverse reasoning paths and peer wisdom, potentially enhancing the overall decision quality. After all rounds of debate, the final answer is typically derived through an aggregation mechanism, such majority voting. Specific prompts are provided in Appendix B.1.
Debate Voting: Formalization. Let X' denote the input space (e.g., natural language questions), and y the output space (e.g.. free-form or multiple-choice answers). We consider a set of N language model agents, denoted by (a1,-.., , ,aN where each agent defines stochastic function fi X y that produces an initial response Vi.0 ~ fi (x) for input x E X.
In the Majority Voting setting, the initial responses { are directly aggregated using a voting function V : yN + Y/ to obtain the final prediction, typically returning the most frequent answer.
yo 0y.......... UN.0).
In contrast, Multi-Agent Debate introduces T rounds of iterative communication. We formalize the communication structure of debate undirected graph G, where each node corresponds to agent and edges indicate which agents observe one another. At round t 2 1, each agent a, observes responses from set of neighboring agents at the previous round and updates its answer accordingly. We define the response set of neighbors available to agent a; at round as:
R (U 1 E ((i))
where (i) S (1, (1....,N) the fully connected setting, (Nii) = {1..... N }). The response update given by:
NN is the index set of neighbors observable to agent a; including itself (e.g.,
yi. .: R. R.!O)
where D denotes single round of debate. The iterative debate process over T' rounds can be expressed as a function composition:
Vrr (D D . D) R.) D(T) (x).
The final aggregated output after 7' rounds is:
yr U(U1,T,322.T ,UNTT)
We adopt the simultaneous-talk protocol [3], where all agents update in parallel based on the previous round's responses. Following the common setup in prior works, we focus on homogencous agent settings, i.e., all agents share the same underlying model architecture or behavior. This allows us to isolate the effect inter-agent communication and contrast MAD directly with simple majority voting. Our goal is to contrast the performance of MAD against simple majority voting and assess whether iterative inter-agent communication provides measurable improvements beyond ensembling alone. We will extend to the heterogeneous setting in Section 6.
3 Is Debate Really Necessary? A Closer Look at Debate vs. Voting
Multi-agent debate is often regarded as promising mechanism for enhancing LLM performance via collaborative deliberation. But how much of its effectiveness truly comes from the debate itself-and how much is simply due to aggregating multiple answers? To address this question, we dissect ensembling and inter-agent communication-ant and present
MAD into two empirical evidence revealing that simple majority voting accounts for most of the observed gains. We begin by explaining the experimental setup in the following section.

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Table I: Majority Voting Multi-Agent Debate. Benchmark performances are measured in Accuracy.
Qwen2.5- 3-7B-Instrucl
Methods
Arithmettccs
GSM8K
MMLU
MMLU HellaSwag CommonSensee HH-RLHF Average
(Pro.Med.)
(Form.Log.)
QA
Single-Agent
aggle- baseline
0.8140
0.8713
0.7868
0.4905 4.08 0.7880
0.8153
0.4773 1:01| 0.7205
Multi-Agent
Decentralized (?
0.7600
0.8867
0.8051
0.5556
0.8033
0.8567
0.4967
0.7377
Decentralized MAD (?' - 3)
0.6700
0,8533
0.8051
0,5000
0,8000
0.8500
0,5000
0.7112
Decentralized MAD (I'
0.6700
0.8333
0.8051
0,4762
0,8000
0.8433
0.5067
0,7050
Sparse MAD (T 2)
0.8400
0,9033
0,8051
0,4762
0,7967
0.8367
0.4733
0.7330
Sparse MAD
3)
0,8100
0.8833
0.8162
0.4365
0.7967
0.8367
0,4733
0.7218
Sparse MAD (T 5)
0.7900
0,8700
0,8088
0,4365
0,7900
0.8333
0.4833
0,7160
Centralized MAD (?' = 2)
0.4300
0,7300
0.8162
0.4762
0,8100
0.8567
0,4667
0.6551
Centralized MAD (7'
0,4800
0.7367
0.8162
0,4603
0,8100
0.8500
0,4733
0 0.6609
Centralized MAD (T' 5)
0.5500
0.7200
0.8125
0.4444
0.8133
0.8467
0.4833
0.6672
Majority Voting
0.9900
0.9400
0.7941
0.5397
0.8033
0.8300
0.4867
0.7691
Llama3, --sttrrttttt
Methods
Arithmetics
GSM8K
MMLU (Pro.Med.) (Form.Log.)
MMLU
HellaSwag CommonSense HH-RLHF Average
QA
Single-Agent
Single-agent baseline
0.7320 a m 0,7393 1 A1 0.7441 = A1 0,3794 1.00 0.6267 1. 18
0.6767 m
0.4440 +. oz 0.6203
Multi-Agent
Decentralized MAD (?' = 2)
0.8200
0.7933
0.7868
0.5238
0.6767
0.7267
0,5233
0.6929
Decentralized MAD (T-
0,8300
0.7933
0.7684
0,5000
0.6300
0.7033
0.5267
0.6788
Decentralized MAD (I' 5)
0,8500
0.7800
0.7463
0.5000
0.6300
0.7000
0,5267
0.6761
Sparse MAD
2)
0,8133
0.8015
0,4683
0.6767
0.7567
0.5267
0.6990
Sparse MAD (T' 3)
0,8500
0.7967
0.7831
0,4206
0,6233
0.7467
0,5333
0.6791
Sparse MAD (T' 5)
0,8500
0,7667
0.7868
0.4365
0.6233
0.7233
0,5400
0.6752
Centralized MAD (T' 2)
0.7300
0.6400
0.6949
0,3810
0,6000
0.7400
0,4800
0.6094
Centralized MAD (T' = 3)
0.7700
0.6200
0.6507
0.3730
0.6133
0.7200
0,4900
0.6053
Centralized MAD (T
0.7300
0.5933
0.5846
0.3413
0,5800
0.6967
0,4433
0.5670
Majority Voting
0.8900
0.8733
0.8199
0.5159
0.6933
0.7800
0.4967
0.7242
3.1 Experimental Setup
Baselines. The key distinction among multi-agent debate methods typically lies in the design of the debate function D. particularly in how agents communicate and the roles they assume. To comprchensively evaluate these variations, we consider the following representative approaches: (1) Decentralized MAD [2], where each agent observes all other agents' responses from the previous round. (2) Sparse MAD [10], variant of Decentralized MAD with sparse communication topology to enhance efficiency. (3) Centralized MAD [14], where a central agent aggregates peer responses and generates the updated response at each round. (4) Majority Voting, which selects the final answer by aggregating initial responses from multiple agents without any debate. This can be viewed as a special case with T = 0. For all the multi-agent approaches, we adopt N = 5 in our main comparison and will ablate on the effect of N. For single-agent baselines, we average across independent runs.
Benchmarks. Following previous MAD literature, we focus on solving six natural language question answering tasks by conducting extensive evaluations across benchmark datasets: (1) Arithmetics, (2) Mathematical Reasoning (Grade School Math 8k [15]), (3) Factual Question
Answering (MMLU Professional Medicine and Formal Logics [16, 17J) (4) Natural Langauge
Inference (HellaSwag [18]), (5) Commonsense Reasoning (CommonsenseQA [19]), and (6) Alignment
Labeling (HH-RLHF [20]), where we adopt the "AI labeler alignment" practice [21]. similar to [10].
For fairness in comparison, all baselines evaluated on the same data subsets. More details are provided in Appendix A.2.
3.2 Key Observations
Majority voting is surprisingly strong. In Table 1, we compare the performance of single-agent, MAD, and majority voting approaches across seven benchmark datasets using the Qwen2.5-77 Instruct [22] and literature, we compare 2- and 3-round, along with a prolonged 5-round debate setting among five agents. Interestingly, while MAD consistently outperforms the single-agent bascline, it does not
[23] models. Consistent with the typical choice of existing

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reliably surpass the much simpler majority voting strategy. Notably, in most cases, majority voting performs on par with MAD. To further assess the impact model capacity, we additionally evaluate the more capable (wen2.5-32BBInnss
model in Section 6. Although overall performance improves in the MAD setting, the majority voting strategy continues to account for most of the performance gains. These findings suggest that the effectiveness of MAD is largely driven by model ensembling. rather than the iterative debate process itself.
To gain deeper insight into the effect of MAD components, we present an ablation study in Figure 3. It illustrates the effect of varying the number of Qwen2.5 agents participating in each round of debate, from N = 1 to N = 5. Overall, increasing the number of agents generally leads to improved performance. The trend suggests that MAD's effectiveness may stem primarily from the ensemble effect of multiple agents. Our next section formalizes our observations.
0300
GIVEK
Frg Meddc
HelloSwog
Number of Agents
Figure 3: Accuracy improves with agents.
4 Theoretical Analysis
To better understand the dynamics underlying our empirical findings in Section 3, we now turn to a formal analysis of multi-agent debate and majority voting. Our theoretical framework allows to capture how agent uncertainty and belief updates shape the collective decision-makee and voting, grounded in Bayesian principles. Specifically, for given input question, we consider a population of N agents, each generating a response from a finite set A, which may represent multiple- choice question options or a set of plausible completions for open- tasks. We model each agent as an idealized generative model governed by a Dirichlet- Socellet-Cooounddd Compound-Multioniia (DCM) distribution. This closely mirrors how practical LLM systems produce different outputs for the same question due to uncerluinty and sampling variability. In particular, the Dirichlet prior captures the agent's internal belief over possible answers, while the Multinomial models the stochastic generation process (e.g., via temperature or nucleus sampling). This distribution is thus natural choice because it encapsulates both internal uncertainty and output randomness, while also providing a principled Bayesian framework for belief updates across debate rounds-cnabling analytical study of dynamics during the debate process. Below we provide the mathematical details of the DCM model.
in both debate
Definition 1. (Agent Response Generation via DCM) At round t, each agent is associated with belief vector ai,t (a
(k) (K) RK, where each entry a reflects the agent's 's belief in response option k E A. To generate response ye,, the agent follows two-step process:
ait (a.......
....,,.,.
(Belief sampling) 0 Dirichlet((x..))
(Response generation) Vs.t Categorical(O...)
The marginal probability of generating any particular response Vi. E A- after integrating the randomness in i given by P(y.t k a..)) = a EjEA "i,
(k)
Before analyzing the dynamics debate, we first consider the base case where agents respond independently without interaction. The following characterizes the success probability of majority voting under this condition, based solely on the agents' homogeneous initial beliefs, 0x,0 = a = (a ...... .......(K)) (K)). Without loss of generality, let answer index be the correct option. We assume that the correct answer has the largest belief (1), and all the other beliefs are ordered such that (1)) > (2) 2 ... (K).
Theorem 1. (Majority Voting Success Probability) Let 0= (0t) (0())...,0K)) a Σ dj denote the mean of the Dirichlet distribution, Dirichleu(a),, and define the margin A = 0, 0z If N > K/A?, then the probability that majority voting selects the answer is lower bounded as:
P(ynv = =) exp -
A
2
VK
VN
5

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Qwen2.5-77 5--5B-----ntuct
Llama3.
0.8
Mean Accuracy
3.4
Aridhmeics
6SMBK
Formal Logic
HellSwag
CommonsenzeGA
Debate Round
Debate Round
Figure 4: Martingale process of the mcan agent accuracy across debate rounds.
This result highlights the magnifying effect of majority voting: even when the correct answer only marginally more likely than the alternative answers, the lower bound on the success probability asymptotically approaches 1 as N scales. Notably, this holds even when 0 < 2 as long it remains the most probable choice option. In practice, we recognize that MAD systems often operate with small number of agents due to computational constraints. We provide sharper analysis specialized this realistic regime that applies to arbitrary N Theorem 1.A (Appendix C), without constraining its size. This complementary result provides insights into the reliability of majority voting in more practical, resource-constriing settings.
Remark 1.
1
Next, we analyze the multi-agent debate performance by formalizing how each agent's ' belief a evolves through debate. Specifically, each agent observes its neighbors' responses and performs Bayesian posterior update of its belief accordingly.
Definition 2. (Bayesian Belief Update from Neighbor Responses) Let Lyja-1 i e N())} be the set of responses observed by agent i from its neighbors N (() at round t. These responses induce a count vector C,2 (c (1) .....
(K) ...),, NK where Q denotes the number of neighbors who selected response k. Then, the agent updates its Dirichlet parameter as: Oii Ciit-1 + Ci
Under this formulation, each round of multi-agent debate corresponds Bayesian update step under the conjugacy of the Birichlet-Multionne model.
Lemma 1. (Bayesian Conjugacy in Multi- Agent Debate) At round after observing responses from its neighbors (i), the agent aggregates these into a count vector C, as in Definition Then, by Bayesian conjugacy, the posterior distribution over 0,, remains Dirichlet:
0g, (0),-1))eNN() Dirichlet(aa,-11--
Then, for an agent i with neighbors N (i). let P., ((1) denote its belief in the correct answer at debate round f. Under Bayesian conjugacy,
(1)
(1)
(1) G.
i
0
Pt.
Σ
This agent belief's evolution throughout debate forms a martingale process (Theorem 2), with full proof in Appendix D.
Theorem 2. (Martingale Behavior of Multi-Agent Debate)
For any agent i round > if
Σ
P = Pt,-1
JEN(S)
then sequence (20 forms martingale. That is, the expected belief at the next round equals the current belief:
E Elpc. [c-1] Pi,t11.

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Theoretical insights: Majority vote does essentially all the work. This theorem highlights fundamental property of multi-agent debate under Bayesian update of the DCM model: the agent's s belief in the correct answer behaves like a martingale- that is, its expected value remains unchanged across rounds. This result is closely related to the classical Pólya Urn scheme [24]. Intuitively, this means that debate itself does not systematically improve or degrade an agent's belief on average; instead, belief updates are driven entirely by the stochastic influence of peer responses. While some debate trajectories may lead to stronger belief in the correct answer (i.e., correction), others may lead to weaker belief (i.e., subversion). While these local fluctuations affect the posterior counts, the expected belief in the correct answer remains equal to the initial Pr, = ( without any debate, when agents are homogeneous and fully-connected (Appendix D.3.1). This implies that, under our voting
i
theoretical model, debate alone does not necessarily improve the initial accounts for the primary performance gains. Our theory thus aligns well with our empirical findings in Section 3. Furthermore, in Appendix D.4, we extend our analysis to more general setting in which each agent's expected belief evolves over the course of the debate, such that the condition in Theorem 2 may no longer hold. In this context, we introduce the notion of "collective intelligence" and demonstrate that it forms a martingale process even with heterogeneous agents.
Martingale behavior is supported empirically. We empirically examine whether the sequence {PL h220 exhibits martingale behavior. For each benchmark and debate round t, we estimate p. as the mean accuracy of the five agents". As shown in Figure 4, the resulting trajectories are essentially flat, which consistent with the theoretical property that the expected value of a martingale remains unchanged over time. Raw values for mean accuracy are provided in Table 6 (Appendix E).
Generalized interpretation of the Bayesian update step. In Lemma and Theorem 2, we define the update dynamics in terms of the count vector C, Our framework can be generalized to open-ended tasks and capture the heterogeneous influence of each agent's 's response, with minor interpretational adjustments. For open-ended tasks, although responses are not strictly countable in categorical form, they can be represented in distributional or similarity count vector can be viewed more broadly, e.g., soft histogram over clustered response types, an embedding- -baeed semantic agreement measure, or a weighted similarity score between textual outputs. In such settings, "consensus" is better defined semantically rather than symbolically: agents independently produce explanations or rationales that are semantically aligned, this can be considered consensus even when surface forms differ. This notion can be operationalized using thresholds on embedding similarity, Levenshtein distance, or overlap in reasoning chains, among other metrics.
spaces. Here, the
5
How Does Theory Inform Improved Design of MAD?
The martingale property in our theory reflects a neutral expectation over time, underscoring that without additional bias towards correct signals, debate alone does not guarantee convergence to the truth. Any additional benefit arises from local asymmetries in the observed stochastic process {p-) Hence, to improve the effectiveness of MAD, we explore alternative designs that facilitate correction and/or suppress subversion in the debate process.
5.1 Belief Update by Biasing Towards Correct Signal
To investigate how targeted intervention in the belief update process can promote convergence to the correct answer, first consider an oracle-style method that explicitly biases updates toward the correct signal. In this variant, once an agent produces the correct answer in any debate round, it becomes "locked" in that state; that is, its belief vector is no longer updated by subsequent peer responses. Formally, if agent i outputs the correct answer at round f, we use this response in all subsequent rounds E' > t. This update mechanism amplifies the asymmetry in favor of correction: correct signals persist and accumulate over rounds, while incorrect signals can still be revised. Consequently, the system's is dynamics depart from the neutral martingale behavior discussed in Section 4 and instead exhibit a directional drift toward the correct answer.
2 Note that this quantity is different from the accuracy in Table 1, which is the ratio of correct responses after voting.
7

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Table 2: Improved design of MAD.
The model Owwe2.5-7BB-Intteee
Methods
Arithmetics
GSMBK
MMLU (Pro.Med.)
MMLU (Form.Log)
HellaSwag
CommonSense QA
HH-RLHF
Average
Decentalized MAD (7'
2)
AaD-vanilla
0.7600
0.8567
0.5238
0.8033
0.8567
MADD-Cofommist
0.9200
0.9200
0.8015
0,5397
0,8000
0,8600
0.4967
0.7625
Follower
0.9300
0,8088
0.5317
0.8100
MAD-oracle
0.9400
0.9667
0.8897
0.6587
0.8333
0.8933
0,5833
0.8236
Decentrlizze MAD (7
3)
MAD-vanilla
0.6700
0.8533
0,8051
0,5000
0.8000
0,8500
0.5000
0.7112
MAD- Conformist
0.9000
0.9133
0.8015
0.5635
0,8033
0.8567
0.4933
0.7617
MAD-Follower
0.9100
0.9200
0.8015
0,5476
0.8067
0,8567
0.5000
0.7632
MAD oracle
0.9400
0.9667
0.8897
0.6746
0.8333
0.8933
0.5833
0.8259
Desentrliized MAD
MAD-vanilla
0.6700
0.8333
0
0,5000
0,8000
U.8433
0.5067
0.7084
MAD-Confomiss
0,8900
0.9133
0.8088
0.5079
0,8000
0.8500
0.4967
0.7524
MAD-Follower
0.9000
0.9133
0.7978
0.5397
0,8033
0,8533
0.4967
0.7577
MAD-oracle
0,9400
0.9667
0.8897
0.6825
0.8333
0.8933
0.5967
0. 0.8289
Sparse MAD (7
MAD-vanilla
0,8400
0,9033
0.8051
0,4683
0.7967
0.8367
0.4733
0.7319
MAD-Confomiist
0.9100
0.9235
0,8015
0,5238
0,8000
0,8300
0,4833
0.7531
MAD-Follower
0.9200
0.9233
0.7941
0.5079
0,8000
0.8267
0.4833
0.7508
MAD-oracle
0,9200
0.9735
0,9007
0.61
0.8267
0,9000
0.6333
0.8236
Sparse MAD (7 - 3)
MAD-vanilla
0.8100
0,8833
0,8162
0,4206
0.7967
0.8367
0,4733
0.7195
MAD-Confomiss
0.9100
0.9233
0.8155
0.5000
0,8033
0.8367
0,4633
0.7499
MAD-Follower
0.9100
0.9267
0,8015
0,5159
0,8000
0.8267
0.4667
0.7496
MAD-oracle
0.9200
0.9733
0.9007
0.6429
0.8267
0,9000
0.6467
0. 038300
Sparse MAD
MAD vanilla
0.7900
0.8700
0,8088
0,4365
0.7900
0,8333
0.4833
0.7160
MADDConformist
0.9200
0,8162
0,4444
0.7967
0,8333
MAD Follower
0.9200
0.9233
0.8225
0,5000
0,8033
0,8333
0.4767
0.7527
0.9767
0,9007
0.8267
We refer this method as MAD-oracle, and report its performance in Table 2. This variant yields substantial improvements over standard MAD, and always surpasses the Majority Voting baseline by large margin. For instance, in Decentralized MAD with T = 5 rounds, accuracy on MMLU (Form. Log.) increases from 0,5000 to 0.6825. Although this approach is not feasible in practice- since the true answer not available-it reveals an upper bound on the benefit achievable by incorporating bias toward correct signals in the belief update process, In the next subsection, we investigate more realistic alternative that aims suppress subversion without direct access to ground truth.
5.2 Belief Update Guided by the Majority Vote
In practice, one would not have access to the oracle setting where correct answers are preserved by assumption. To approximate this behavior, we introduce simple modifications the MAD update rule that leverage the positive signals from majority voting. Our design rationale is guided by our theoretical saalysis-whiih shows that majority voting provides more reliable estimate of the correct answer than any single agent, as it aggregates marginal advantages across the population. This suggests that using the majority response as proxy for the ground truth can help steer belief updates biasing the system toward correction without needing oracle access.
in the right direction-effecttin To explore this idea, we propose two lightweight interventions that incorporate the majority vote into agents' belief dynamics. Specifically, we evaluate two strategies: (1) MAD-Conformin response matches the majority vote the previous round, retains that response; (2) MAD-Follower:: with 30% probability, the agent adopts the majority response from the previous round, and otherwise samples new one. As shown in Table 2, these strategies consistently outperform the MAD-vanilla baseline. While they do not reach the oracle's upper bound performance, they demonstrate that simple, theory-informed modifications can yield meaningful improvements- -poiitinnoinnngiiiin to a promising direction for future work to close the gap.
if an agent's s
8

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Table 3: Results on a larger model.
Table 4: Heterogeneous persona agents. Single-
agent is averaged over 5 personas (prompts in B.3).
Qwen2.5-328 -ann2.5532B-nnstreee
Methods
GSM8K
HellaSwag
Qwen2 1an2l-------nuuee
Single-Agen
Methods
MMLJU (Pro.Med))
Single-agent haseline
0.7566
0.8620
Single-Agent
Single-agent baseline
0,8047
0,7890
Decentralized MAD Multi-Agent
0,9400
0.8633
Multi-Agent
Decentralized MAD (T = 3)
0.9367
0.8600
Decentralized MAD (T
2)
0.8033
0.8419
Decentralized MAD (T' 5)
0.9367
Decentralized MAD (T' -3
0.7733
0.8382
Sparse MAD (T =
0.8433
0.8667
Decentalizzed MAD (7
0.7433
0.8382
Sparse MAD (f' -3)
0.9367
0.8633
Sparse
Sparse MAD =
0.8667
0.8346
Sparse MAD
5)
0.9333
0.8667
Sparse MAD (I'
0.8567
0.8382
Centralized MAD (T = 2)
0.8000
0.8667
Contralized MAD (7
0.5933
0,8051
Centralized MAD (7
0.8333
0.8667
Centralized MAD (2' = =3)
0.5901
0.7978
Centralized MAD (7 = 5)
0.8333
0.8667
Centralized MAD (T
0.6000
0.7978
Majority Voting
0.9433
0.8667
Majority Voting
0.9367
0.8235
Extended Experiments to General Settings
In this section, we broaden the scope of our investigation in Section 3 to more general settings, evaluating whether the key observation (i.e., majority vote is as effective debate) holds on larger model size, heterogeneous agents, and open-ended question formats.
Consistent observations in a larger and more capable model. To assess the generality of our findings, we extend our evaluation more capable language models. Specifically, we test our setup on the Dwen2.5-33B-IIsse
model [22], using two representative tasks: GSM8K and HellaSwag, As shown in Table 3, the results confirm our earlier observations that the performance of Majority Voting remains comparable to that of multi-agent methods. This suggests that our claim is not limited smaller models, but also holds in high-capacity LLMs.
Heterogeneous Agents. While our primary focus has been on homogeneous agent settings, an important question remains: Do our findings also extend to heterogeneous agent configurations? To investigate this, we evaluate MAD systems composed of agents with distinct personas, as shown in Table 4. Following the optimal persona sets identified for "college mathematics" and "clinical knowledge" via the "agent selection algorithm" introduced in [9], we construct diverse agent roles for each task. For GSM8K, the team includes ceneerl-puupose "Assistant" alongside specialized roles: "Mathematician". "Lawyer", "Economist", and "Programmer", For the MMLU Professional Medicine subset, include "Doctor", "Psychologist" "Mathematician" and "Programmer", In practice, we implement this by assigning each agent a system prompt that encodes a specific role or persona-e.3 Mathematician, Programmer, Lawyer-usite in [9] (see Appendix B.3 for the prompts). Even in these heterogeneous settings, Majority Voting mostly paralleled MAD variants. However, several MAD results on Pro. Med. show larger gains, suggesting the potential benefit of assigning diverse personas in task-specific MAD systems.
the same prompt templates provided
Evaluation on open-ended text generation tasks. In our previous experiments, we mainly focused closed-ended question answering tasks, which are the primary focus of previous works on MAD. A natural follow-up question is whether our findings will hold in open-ended tasks, such as free-form text generation. To explore this, we evaluate MAD on a text summarization task using subset of CNN/DailyMail dataset [25]. Unlike classification tasks, applying Majority Voting in summarization is not straightforward due to the lack of discrete answer choices. Instead, we report the best- performing agent at each debate round, as shown in Table 5. Interestingly, we observe that ROUGE-1 and ROUGE-L scores remain relatively invariant across rounds, suggesting that the key observations from closed-ended tasks may also extend to open-ended tasks like summarization.
Table 5: Open-ended text generation. Qwen2.5-7B-Insste on Decentralized MAD.
Methods
Rouge-1
Rouge-I.
Best Single-agent
0,2760
0.1871
MAD (T = 1)
0.2686
0.1814
MAD (T 2)
0.2773
0.1867
MAD (T 3)
0.2825
0.1852

---

7 Related Works
Recently, there has been growing interest in multi-agent systems (MAS). Several survey papers have reviewed state-of-the-ard LL.M-based MAS approaches [14, 26-28]. Within MAS, MAD has emerged as a particularly promising approach for enhancing the performance of single-agent benchmarks. In the following, we discuss the strengths and limitations of current MAD systems.
Pros of Multi-Agent Debate. A key strength of MAD lies in its iterative discussion process, which has the potential to enhance both factual accuracy and reasoning quality. Building on this paradigm, several works have proposed MAD-based approaches for variety of tasks [1-6]. To further advance MAD systems, [7] introduced enhancements grounded in debate theory, while [29] developed the Peer Rank and Peer Discussion mechanisms to select appropriate agent pairs for debate. Many studies have focused on designing effective communication architectures and protocols to improve efficiency and effectiveness [3, 8-10, 30, 31J. Other works have emphasized the importance of diversity in MAD systems, leveraging heterogeneous LLM agents [11]. injecting distinet personas into each agent [9, 12, 13]. or enabling text generation with controlled diversity [32, 33]. Additionally, learning- based methods have been explored to optimize MAD dynamics [9, 34, 35].
Cons of Multi-Agent Debate. While MAD systems are widely used as effective tools for solving
various tasks, recent studies have raised concerns about their actual effectiveness. For instance, [36] conducted an in-depth analysis identifying 14 distinct failure modes in MAD systems, and [37] found that MAD does not consistently outperform single-agent approaches. Similarly, [38] showed that
LLM agents are not self-corrective enough for MAD to be successful, and [39] reported that MAD performs no better than advanced single-agent reasoning methods and highlighted its sensitivity to hyperparameters. [40] echoed these concerns, showing that well-prompted single agents can sometimes outperform MAD. More specifically, [7] and [37] observed occurrences of subverted or incorrect answers in MAD debates, while [41] showed that MAD systems often converge to the majority opinion, even when that opinion reflects common misconceptions. On a slightly different gear, [42] compared various decision protocols and showed that multiple rounds of MAD actually decreases performance.
this work, we perform systematic comparison between MAD and simple majority vote, and provide theoretical foundation to understand how the success probability evolves throughout debate, shedding light on future design of improved MAD.
8 Conclusion
In this study, we provided comprehensive analysis of MAD and its core components. To investigate this, we conducted extensive experiments on seven benchmarks. Contrary to prevailing assumptions, observe that most performance gains of MAD stem from majority voting rather than the debate process itself. To support this finding, we introduce a theoretical framework that characterizes debate dynamics as martingale process, which preserves the expected success probability of each agent over time. These insights suggest that ensembling strategies like majority voting remain strong and often more reliable, highlighting the need preserve correct answers during inter-agent debate. Overall, our work sheds light on the key mechanisms underlying MAD and offers concrete directions for improving its design.
Acknowledgement
The authors would like to thank Leitian Tao and Xuanming Zhang for their valuable comments on the manuscript. Hyeong Kyu Choi and Sharon Li are supported in part by the AFOSR Young Investigator Program under award number FA9550-23-1-00 84, National Science Foundation under awards 2IS-2237037 and 155-2331669, Office of Naval Research under grant number N00014-23-1- 2643, Schmidt Sciences Foundation, Open Philanthropy. A Alfred P. Sloan Fellowship, and gifts from Google and Amazon. Xiaojin Zhu was supported in part by NSF grants 2202457, 2331669, 1836978, 2023239, ARO MURI 191 17, and AF CoE FA9550-18-1-016
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Appendix
Table of Contents
A Experimental Details
21
A.1 Hyperparameters and Resources
21
A.2 Dataset Details
21
B Prompt Templates
22
B.1 MAD Templates
22
Task Templates
22
Persona Prompts
23
Special Case of Theorem
24
Proofs
25
D.1 Proof of Theorem
25
D.2 Proof of Lemma
27
D.3 Proof of Theorem 2
28
D.4 Collective Intelligence
31
E Martingale Process Empirical Investigation
32
F Proper Evaluation Matters
32
G Closed-source LLM Evaluation
33
H Broader Impact, Limitations and Future Works
33
A Experimental Details
A.1 Hyperparameters and Resources
Hlperparameters. To enable stochastic sampling from homogeneous agents, we set the sampling temperature 1.0, and use nucleus sampling probability of 0.9, which means that sampling is done dynamic set of likely tokens that together account for 90% of the total probability. Furthermore, we generate a maximum of 512 tokens for all models experimented in the paper.
Resources. All experiments were conducted using either RTX A6000 or RTX A100 GPUs.
A.2
Dataset Details
Here, we provide dataset details and the number of samples utilized our experiments.
Arithmetics comprises 100 arithmetic questions, in the form of "What is the result of a +bcctd- e/f?" The values f are randomly sampled from integers between 0 to 30.
GSM8K [15] comprises high-quality grade school math word problems intended to test mathematical multi-step reasoning. We randomly subsample 300 questions from the original test split.
MMLU (Professional Medicine) [16, 17] a benchmark specialized in evaluating reasoning abilities in medical domains at professional level. Specifically, the dataset requires medical concepts, clinical reasoning, and biomedical knowledge to answer questions. We use the entire test split comprised of
272 questions.
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MMLU (Formal Logic) [16, 17] is designed to evaluate a model's proficiency in formal reasoning. symbolic manipulation, and logical analysis. We use the entire split comprised of 126 question for evaluation.
HellaSwag [18] is natural language inference (NLJ) benchmark dataset, in the context of sentence complction tasks. That is, the benchmark tests whether a model can choose the most plausible continuation of given context from multiple options, which is task that demands just linguistic proficiency but also real-world knowledge and reasoning. We randomly subsample 300 questions from the original test split.
QAmmonsenseQA [19] multiple-choice question answering dataset designed to evaluate a model's 's ability to apply commonsense knowledge in natural language understanding. We randomly subsample 300 questions from the original validation split.
HH-RLHF [20] is collection of human-annotated data designed to train and evaluate language models for alignment with human preferences, focusing on helpfulness and harmlessness. The dataset is annotated with relative preferences, comprising 'chosen' and 'rejected' sample pairs. Similar to the "AI labeler alignment" practice [21], we ask the LLM agent to select the example that is more helpful and less harmful. To avoid selection bias [43-45], we randomly shuffle the order of "chosen" and "rejected" in the input prompt. We use a random subset 300 pairs from the original test split.
CNN/DailyMauie [25] is a dataset for abstractive text summarization. It was originally constructed from news articles published by CNN and the Daily Mail, aimed for evaluating models that generate concise summaries of long-form text. We use random subset of 30 samples from the test split of dataset version 3.0.0.
B
Prompt Templates
Here, we provide all the prompt templates used for our experiments.
B.1 MAD Templates
The following is the prompt template for multi-agent debate. For brevity, assume 3 agents are in the debate.
These are the recent opinions from other agents:
One of the agents' response:
<agent 2's response from the previous round>
One of the agents' response:
<agent 3's response from the previous round>
This was your most recent opinion:
<agent 1ss response from the previous round>
Use these opinions carefully as additional advice to revise your recent opinion to give your final answer to the question:
<question>
Make sure to state your final answer in curly brackets at the very end of your response, just like: "[final answer: 12.34)".
For GSM8K, we give slightly different demonstrative example: "[final answer: 123]", and for the other MCQ datasets, we give: "I "[final answer: (A))". For the CNN/Daily Mail dataset, we instruct the model to 'Make sure to provide your summary after stating "# Summary #".²
B.2 Task Templates
For completeness, we provide the exact input format used for each dataset. These templates correspond to the <question>>> field in the MAD prompt structure. Datasets not listed below follow the original question format provided in their respective sources without modification.
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Arithmetics
What the result a+bbc+
MMLU Professional Medicine Formal Logic
<question>
(A) <option 1>
(B) <option 2>
(C) <option 3>
(D) coption 4>
HellaSwag
Can you choose the option that best follows:
"<context>"?
(A) <option 1>
(B) <option 2>
(C) <option 3>
(D) <option 4>
CommonsenseQA
<question>
(A) <option 1>
(B) <option 2>
(C) <option 3>
(D) coption 4>
(E) <option 5>
HH-RLHF
Which of the following is relatively more helpful and/or harmless data?
(A) <chosen> or <rejected>
(B) <rejected> or <chosen>
CNNs
Summarize the following in three sentences:
<article>
B.3 Persona Prompts
To assign heterogeneous personas to agents, use system prompt that specifies the role each agent should assume. We adopt the persona descriptions from [9], which are provided below:
Asiittant: You are a super-intelligent Al assistant capable of performing tasks more effectively than humans.
Mathematician: You are mathematician. You are good at math games, arithmetic calculation, and long-term planning.
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You are economist. You are good at economics, finance, and business. You have experience on understanding charts while interpreting the macroeconomic environment prevailing across world economies.
Economist:
You are programmer. You are good at computer science, engineering, and physics. You have experience in designing and developing computer software and hardware.
Programmer:
Lawyer: You are lawyer. You are good at law, politics, and history.
Psychologist: You are a psychologist. You are good at psychology. sociology, and philosophy. You give people scientific suggestions that will make them feel better.
Doctor: You are doctor and come up with creative treatments for illnesses or diseases. You are able recommend conventional medicines, herbal remedies and other natural alternatives. You also consider the patient's s age, lifestyle and medical history when providing your recommendations.
C Special Case of Theorem
In this section, we analyze a special case of Theorem I, where the probability of the correct answer, denoted by 0,, exceeds 3. In other words, each agent independently makes correct decision with probability at least 0.5. It naturally follows that 1
2 2 1 1-0 = E 2 max;e(2.....)) 0,, indicating that the margin term A in Theorem requires A 2 0, which does not rely on the number of agents N or the set size of possible answers k.
To formalize this, we define Po = 0, to represent the initial probability that an individual agent selects the correct answer. Also, for simplicity, let X1, , XN be independent Bernoulli random variables, where X; ~ Bernoullippo). The average correctness across agents is then given by the empirical mean X Ei=1 X,, which corresponds to the fraction of agents who vote correctly. Under this setup. we analyze the lower bound of the Majority Voting success probability as follows:
Theorem 1.A (Majority Voting Success Probability). Let X1.X2.... .X, be independent Bernoulli random variables with X, Bernoulli(p)), where po E (0, 1) is the probability that each agent is correct. Let X = En= X; be the fraction of correct agents among N independent agents. If Po > 2/2 the probability that majority vote is successful is lower bounded as follows:
2 exe -2N Po - 2 1
2
8 2
Proof of Theorem I.A. Since X; ~ Bernoull(pp). we have E[X,] = p. and the variables are independent and bounded: 0 < X; < 1. Define S, E Xi. so E[S.] np, and X S. so
E[X] p.
We apply Hoeffding's Inequality to bound the deviation of S, from its mean. Hoeffding's 's Inequality states that for independent random variables X; with a; Sx S bú, and S, = Σm= X₁:
21
P(S, E(S,) 2 () S exp
1== (b; 04)3
2t
P(S, 118n-1156 E[Sp) s -t) 5 exp
- (b: - a;
Here, a; 0, b: so (b. a; 1, and Σ (b.
a,,² n. Let t
[((cn) (
P(S, - np S -en) S exp
exp(-2ne"
n
Rewrite terms of X:
P(S. - -p < -En) = SSn - 2 -e = PP( <
24

---

P(X sp S -mne)
We want the majority vote to be successful, i.c., X > First, consider the complementary event:
1 P X = Ss (p(p-p12)) )
and set c p 1, 2'
p-t= 2
7 =
2
Since p > 2. > Substitute e:
P X 3
1
exp
-2n
1
Thus, the probability of successful majority vote is:
2) 2
2 - exp P x = - x < 2
-2n
Proof completed.
Under this alternative formulation, the term p serves conservative approximation of the margin A, since all competing answer choices are necessarily assigned probabilities less than J. Then, note that the dominant term in the resulting lower bound remains consistent with that of the orignal Theorem 1, scaling O(N A?).
D Proofs
D.1 Proof of Theorem
Theorem 1. (Majority Voting Success Probability) Ler 0
O(K)) a - Oj denote the mean of the Dirichlet distribution, Dirichlet(a)), and define the margin A = 0 02 If N 2 then the probability that majority voting selects the answer 1 is lower bounded as:
A
P(ymv =)
2 exp
N
VN
Proof of Theorem 1. We proceed in several steps to establish the result.
Step 1: Define the Empirical Distribution
The empirical distribution of votes given by p - c/N = f2,. .p)), where ùú = .cj/N
the fraction of agents selecting answer j. The true distribution is = 0. so Esp,] = 0y. The mojority-voted answer corresponds to the index with the largest Ny:
Vnv arg max P/- (E(
Our goal is to show that P1 > b. for all # with high probability under the given conditions.
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Step 2: Establish Key Lemma
To connect the empirical distribution to majority voting, we introduce the following lemma.
Lemma 2. If [lp 1-p--pl p < A, then Ymv = 1.
Proof of Lemma 2. We prove this by contrapositive. Suppose ymn # 1. Then, there exists some j * 1
such that Pj 2 P1- Compute the I norm contribution involving classes and j:
PP P1l Pg P3 2 ((p: pi) (p: -5)] 1p
Pj (pi β2)l-
Since p1 01. Pj f and A 01 -22 and given 01 2 0g for all j 2 we have p1 Pj 2
02 A (since 02 02 for j 2). Given púj 2 pl. compute:
Pí P3 S 0 and p1 Pj A4
so:
P1 - Pj (pi 2.) A 0 A.
Thus:
(p1 - 51) (p, - 1,) A
implying:
lp1 P1 + P3j p. 2 A. Since ||p - -Pl1 = Ex lp; - p.l 2 |1 - 21 + |p, - pyl. it follows that:
|p- p N A.
Hence, if Vav + 1, then | pll1 2 A. Conversely, if | pll1 A, then Bay 1. Proof of Lemma 2 completed.
This lemma implies:
Pyy = 1) 2 P( p|l1 < <)
Step Bound the L1 Deviation
Since 0= a E/= aj is the mean of the categorical distribution induced by the Dirichlet model, and agents draw answers independently from 0, the counts с ~> Mult inltiommial(,, 0). Note that the Dirichlet-multinomial assumption in the original proof simplifies multinomial distribution here because 0 is fixed as the mean.
We need to bound P([ 1Plp--1111 2 A). We use concentration inequality for multinomial distributions. known result (e.g., Proposition 19 in [46]) provides:
(v*+))
(
1
(c+))
p
pll 2 VK
S exp(-Ne3),
for some > This bound accounts for the K K-dimensional nature of the distribution. Set the threshold equal to A:
1
VK
te
A.
VN
Solve for
1
A
+e=
N
VK
K
VN
The condition N > K/A² ensures 0:
implies
V NN > VK or N > K,
VK
which matches the theorem's condition. Since A > 0, this condition guarantees the bound is meaningful.
26

---

Substitute é into the inequality:
A -N
(5
--
P(Ip pll 2 A) S exp
vN)
Thus:
(-N
4
P ( pl1 < < 2 exp
VK
VN
Step 4: Conclusion
From Lemma 2:
P(yme 1) P(II p|l < A).
Combining with the probability bound, derive:
A VK
Py 1) exp
exp
N
This matches the statement of Theorem 1. Proof completed.
D.2 Proof of Lemma
Lemma (Bayesian Conjugacy in Multi-Agent Debate) At round t, after observing responses from its neighbors (i), the agent aggregates these into count vector Ci. in Definition 2. Then, by Bayesian conjugacy, the posterior distribution over 0., remains Dirichlet:
Out (1)t-1))eNN() Dirichlet(Qs,r-11+ + Cc.t)
Proof of Lemma 1. We aim show that the posterior distribution of Oi,,, the agent i's belief over the answer space at round given the observed responses (Uy.-1))eN(o) from its neighbors, is Dirichlet distribution with updated parameters Ci,,-- Ci,c.
First, consider the prior belief of agent at the start of round which is based on its belief from the previous round t - 1. The prior distribution over the parameter Og,, is:
0,t Dirichlet(a (a
(K) Oit-)
where 0;,111 (a (1)
(K) 1 are positive real numbers representing the prior parameters inherited from the previous round, and the density is proportional to:
K
p(O..))
πI m=
mm)
1-1
with Σ g((m) 1.
At round i, agent i observes the responses (U),-1))eeN() from its neighbors, where Uja-1
(1...., K} represents the answer chosen by neighbor at round f- 1. According to Definition 2, these responses are aggregated into count vector C,,,
(c_)) .c...... ....,... ......))). where:
(k)
(m) i,t
Σ JEN(()
1 -0----m]] m],
att and E (m) = (i))), the number of neighbors. Assuming each neighbor's response Uj is
drawn independently from a categorical distribution parameterized by 0,,, the likelihood of observing the count vector C₁ given or,, follows multinomial distribution:
Ci 0g, Multinomial((((())) 0..).
27

---

with the likelihood proportional to:
K
(m) Le
p(c,a 0s,e) x
II m=1
g(m))
By Bayesian conjugacy, the posterior distribution of Oi, given the observed counts is proportional to the product of the prior and the likelihood:
p(O2. Ci,e) x p(O),) p(ce, (0,,).
Substituting the expressions:
K
K
K II m=1
II
(m)) glm)
n km=1
(m)
p(O,, c2,)) x
m=1
This form is characteristic of Dirichlet distribution. Thus, the posterior is:
(K) (K)) 0 (Bit- Dirichlet(a (1) + Ser.... ,,,,,,,,,1111 Gi
(1)
or equivalently:
0i. (Vit-1) Dirichlet(Qs,-11
Since Ca, is derived from the neighbor responses (ıj.-- beN(() as specified, and the update follows from the conjugacy property the Dirichlet and multinomial distributions, the lemma holds. Proof completed.
D.3 Proof of Theorem
Theorem 2. (Martingale Behavior of Multi-Agent Dehate)
For any agent i round t if
1
Σ
W(
PY--1 Pi,t11
JEN(S)
then sequence (pra)e2o a martingale. That is, the expected belief at the next round equals the current belief:
E
Elpce a+-1l = Pi
Proof of Theorem 2. For agent i with neighbors Mii), let Pt, = x(1) denote its belief in the correct answer at debate round t. Under Bayesian conjugacy,
(1) Oi1
Pt. := 000
0 + Gt
ΣK
K
()
ΣK
(0 ()
o
Here, we prove that the sequence {1r, is martingale by showing that the conditional expectation of the belief in the correct answer at round t given the filtration up to round t - 1 equals the current belief. sequence {Xe]c20 is martingale if [E[X, Fi-1] Xt-1. where F (ax, 01, a4-1) represents the history of belief parameters.
By definition, write agent's belief in the correct answer at timestep and as:
(1)
a
Pit-1
(2)
Σラ=1 ii1-1
(1)
(1)
0, 1 Ci
Pi, =
(3)
Σ K
j=1 a i + [V(i)
28

---

where (1) Ctt is the number of neighboring agents that has selected the correct answer at the previous debate round. Then, the expectation of pt conditioned on the filtration Fi-1 is:
0,1-1 + 2,8
Elp Fe-1] E
Fi-1
(4)
1 1 (W(1))
(1)
0,t-1 + Blc (1) Fi-1l
(1)
(5)
K
1 + (i)l
9 + BBCEENiA 1| 11 F--1]
(6)
Σ/= 9 + W((i)
1 + E EjENTO Ps.1-1
(7)
ΣØ= 1,111 + Wi(i))
12-1 + +W((i)l Pr,-1
(by the condition in Theorem 2).
(8)
K Σラ= - [V(i)
From equation (2), we have
K (1) Pi,t1 Σ 0
(9)
1.6-1
j=l
and by plugging equation (9) into equation (8), arrive
E Elp
Fi-1] Pi,-1
(10)
Proof completed.
D.3.1 When does (1) in Theorem 2 Hold?
In order for multi-agent debate be a martingale process, the condition in Theorem equation (1), which states the relation between the belief of an agent with its peers' beliefs, should hold. Here, we expand our discussion exploring debate topologies that satisfy (1), therefore the martingale property.
Proposition 1. (Sufficient Conditions for (1)) If the debate topology consists of agents that either (i) are homogeneous and fully- fullyccomeettd,, or (ii) are fully-isolated, or (iii) form isolated cliques whose agents in each clique are homogeneous, then the debale process is martingale.
proof of Proposition We prove one by one that each case satisfies (1).
(i) Homogeneous agents with fully-connected communication topology
This setting best represents
experimental setting: Decentralized MAD [2]. In this setup, since the agents are homogeneous, their initial beliefs are identical: 01,0 = 02200 = ... Q, Also, since
assuming a fully-connected communication topology, the count vector C₁ that counts the neighboring agents' answers (including itself), is identical across agents. Therefore, 0it 0,,--1 + Ct Oyyy holds for all i, By dividing the equation with |a.,ell1, get = pr VrP. Then, (1) is satisfied.
(ii) Fully-isolated structure with M(i)) (i) for all
This an extreme case where none of the agent is connected to peer. That is, an agent will only refer to its own previous answer to update its response, which is effectively an iterative self- refinement approach. In this case, each agent's belief parameter evolves independently through Oi 04,--1 + 13 1 = 1}. where without loss of generality, answer 1 is the correct answer. Hence, it can be easily shown that each agent follows martingale process, satisfying (1). Note that this holds regardless of the homogeneity and heterogeneity of the agents.
(iii) Isolated cliques, where agents in clique homogeneous
This scenario is a mixture of the above two cases. Within each clique, the agents' belief vectors evolve with respect to case above. Then, the entire topology of multiple cliques will follow case 2,
29

---

where each clique's collective belief evolves independently of other cliques. Hence, as a whole, (1) is satisfied.
In all three cases (i)-(iii), (1) satisfied. Then, by Theorem 2, a multi-agent debate system with dehate topologies equivalent to (i)-(iii) will have the martingale property.
Proposition provides several sufficient conditions under which equation (1) holds. There may. however, exist other debate graph topologies that also satisfy this condition. complete characterization of such cases is left for future work.
Remark 2.
D.3.2 When does (1) in Theorem 2 Not hold?
In addition to the cases where the condition in Theorem 2 hold, is also important examine the debate topologies under which (1) does not hold. Here, we discuss two cases: line graph and ring graph.
Line graph
A debate topology that consists of three agents connected as a line may not satisfy (1), as shown below:
(1)
Let the agents be homogeneous and their initial beliefs in the correct answer be 00i00 2,200 $3.0 generated, could happen that c = (1, 0, 0). This means only agent produced the correct answer. In terms of Dirichlet parameter updates, have:
(1) 1, and assume ||a|,o|| = ||a22,olı = ||a.,lll 2. Since the count vector is randomly
1.0
1
P1,1
(11)
|1
+
2
2
(1)
0,2
2
P2.1
(12)
+ 3
5
(1)
3,0
1
Ps.1
(13)
||00,0|
+ 4
In this case, P2,1 * (PI,1 + P2,1 + PPa.). Hence, this serves as a counterexample to condition (1) for future time steps.
Ring graph
debate topology that consists of four agents forming a ring graph may not satisfy (1), as shown below:
Let the agents be homogeneous and their initial beliefs in the correct answer be a 02200
(1)
(1)
ag,0 04,0 1, and assume ||c1,,||1
(1) (1)
||020011
||a.,0 ||04,o||1
2. If only agent got
30

---

the answer correct at = O,
(I)
01,0
2
P1,
(14)
|0100||
+ 3
5
(1)
2,
2
P2,1
(15)
002,0
3
5
(1)
3,0
1
P3,1
| 11
(16)
(1)
04,0
P1,1
|04,oll1
+
5
(17)
In this case, P3.1 + (P2.1 + Ps.1 + P1.1). Hence, this serves as a counterexample to (1). Notably, this ring graph corresponds to the Sparse MAD baseline [10]. Although this topology does not theoretically guarantee the martingale property, we empirically demonstrate in Appendix E that holds approximately. We conjecture that the dynamies for short debate rounds does not deviate greatly from a martingale.
D.4 Collective Intelligence
If the agents are heterogeneous and fully-connected, the debate is necessarily martingale since the condition (I) for individual agent beliefs may not hold. However, we find that the "collective intelligence" may follow martingale. Let's define the collective belief of the correct answer
Σ. Qt
(18)
Σ. Oit
(K)
Then, the expected collective belief is:
Σ, a + Blc (1) Fi-1l)
Elq: Fi]
(19)
Σ a/ + E, Wi(i)
(K)
Σ, 1 + +E|E 11 = (F-
(20)
ΣnA - Σ, IN(1)
This collective belief will be martingale if and only if
Σ, EjeN(e) R[1 11 1F.
(21)
Σ.
is satisfied in (20). This condition is equivalent to
E Ejenc Pia-1
E.a
(22)
ENN(1
En, a
Then, sufficient condition for this to hold is, for all
Ee a Constant, [WO)= =
(k)
meaning that the magnitude of the belief vector should be constant across agents, and that all agents should be connected each other. When these conditions are met, (qt (ro follows martingale process.
31

---

E Martingale Process Empirical Investigation
Here, we provide the raw mean accuracy values used for Figure 4 is provided in Table 6.
directly evaluate the martingale process by comparing the Pi,- 1. as measured by the mean accuracy values across agents at step - I, with the expected value the next step Elpr. F(-1)) which is measured by running 5 independent runs on the subsequent debate round. In the following two tables, we report the results with Sparse MAD [10], consisting of Qwen2.5-7B agents and Llama3. agents, three rounds of debate evaluated on the Formal Logic benchmark. observe that Pi Elpr, Ft-1] both cases.
As further investigation,
Table Raw mean accuracy values for Figure 4. Top row is from Tiwe2.55-BB-Inseie Instruct, and the bottom row from Llama3. 1 I-st-lnttruct. The MAD structure is the fully connected Decentralized MAD [2].
Rounds
Arithmetics
GSMSK
Pro. Med.
Formal Logic
HellaSwag
CommonsensoQA
HI1-RLHI
0,5400
0.7740
0,8022
0.4937
0.7900
0.8233
0,4613
0,4880
0.7267
0.8015
0,4492
0,7880
0,8280
0,4600
3
01.4880
0.7240
0.8029
0.4254
0.7880
0.8333
0.4547
0.4860
0.7287
0.8015
0.4159
0.7880
0.8340
0,4560
0.7293
0.8015
0.4048
0.7873
0.8340
0,45660
Rounds
Arithmeties
GSM8K
Pro. Med.
Formal Logic
HellaSwag
Commonsense(A
HH-RLHF
0.6340
0.6553
0.7199
0.3667
0.6487
0.7167
0.5060
2
0.6980
0.6533
0.6978
0,3492
0.6287
0.7307
0,5173
0.6920
0.6520
0.6728
0.3397
0.5953
0.7220
0.5060
0,7300
0.6413
0.6662
0.3254
0,5820
0.7067
0.4987
0.7280
0.6380
0.6581
0.3175
0,5647
0.6953
0.4973
Table 7: Swwn2.55-B-Innsrrue on Formal Logic
Table 8: Llama3.1 1-88---nstttrctt on Formal Logic
Round
PLt-1
Elp,, [F,--1]
Round
Pi.-1
Elp.. Fr,c-1
0.363
(0.024)
0.451
0.446 (0.016)
2
0.400
0.410 (0.011)
2
0.443
0,452 (0.003)
0.371
0.379 (0.011)
3
0,479
0.490 (0.016)
F Proper Evaluation Matters
Another key takeaway from our study is that careful evaluation is critical for accurately assessing the utility of MAD. We find that the method used extract final answers from free-form model responses significantly affect measured performance- sometimes even reversing conclusions. While prior works have reported consistent gains from MAD over majority voting [2. 3, 10], we find these results may be partially driven by error-prone answer extraction, where rule-based parsing can fail even when the model's Iss response is correct. For example, a model's output may be correct, but incorrectly marked incorrect purely due failures in parsing, rather than actual reasoning mistakes.
To improve the reliability of answer extraction, we explicitly instructed each agent to append its final answer using standardized format-for example, "(final answer: ")". This strategy substantially reduces parsing failures and yields more reliable evaluations (see Appendix B.1 for prompt details). Once final answers extracted from each agent using the same protocol, we select the majority answer as the final response.
Table 9: Effect of different answer extractors. Wwen2.5-7B-Insteer Decentralized MAD.
GSMBK
Methods
Ours
Prior [2]
Single-agent
0.8713 0.8867
0.6620
MAD (7 = 2)
0.7533
Majority Voting
0.6700
Table 9 shows that the extraction strategy significantly impacts performance. Our evaluation protocol improves single-agent accuracy and, on GSM8K, even outperforms MAD when the latter is evaluated using the prior strategy from [2]. These results show that our strategy reveals model's true capability more faithfully, and caution against attributing improvements to debate when they may stem from superficial formatting gains. Without rigorous and consistent evaluation, we may incorrectly estimate MAD's benefits and obscure whether inter-agent communication truly enhances decision quality.
32

---

G Closed-source LLM Evaluation
We extend our experiments to a closed-source LLM setting. Specifically, we conducted additional evaluations using three GPT-4 agents across four benchmarks. In Table 10, the overall trends remain consistent with those observed in our open-source model experiments, supporting the generality of our findings.
Table 10: Further experiments on GPT-4
Arithmetics
CSQA
HellaSwag
HH-RLHF
Majority Voting
0.9967
0.8721
0.9078
0.5612
Decentralized MAD (T
0.9867
0.8788
0.9078
0,5580
Decentralized MAD (T' 2)
0.9867
0.8784
0,9044
0.5577
Decentralized MAD (I' 0.9833
0.8780
0.9044
0.5459
H Broader Impact, Limitations and Future Works
Broader Impact. Our findings highlight an important perspective on Multi-Agent Debate, showing that much of its effectiveness can be achieved through simpler, more accessible methods like Majority Voting. This opens the door to building more efficient and scalable collaborative AI systems without sacrificing performance. Moreover, by identifying the MAD as a martingale process, we offer actionable insights that can help make future MAD systems more robust and trustworthy. We believe our work contributes a new perspective on debate- -aaade Al frameworks that both principled and practical. Ultimately, this supports the broader goal of making Al systems more reliable, collaborative, and aligned with human reasoning.
Limitations and Future Works. As discussed in Section 2, our study primarily focuses on the Simultaneous Talk protocol [3], where agents generate and share their responses concurrently in each debate round. While this setting is widely adopted in prior work, it does not capture the full spectrum of possible communication strategies within multi-agent systems, Alternative protocols, such as One-by-One, where agents respond sequentially, or where a summarizer agent oversees and summarizes the state the debate, introduce different dynamics that may influence the rates of subversion and correction. Investigating these alternative protocols in depth remains an important direction for future work. Furthermore, our theoretical framework relies the assumption of agent homogeneity and may directly generalize to heterogeneous settings. A promising direction for future work is to extend the martingale analysis to account for heterogeneous agents with differing prior beliefs reasoning capabilities.
33


Reviews:
[{"review_id": "qePMJbZfxm", "reviewer": "Reviewer_HZhH", "summary": "In this paper, the authors breakdown MAD into two components, majority voting and inter-agent debate. Through extensive experiments across 7 NLP benchmarks, they show that majority voting is responsible for most, if not all ,of the improvement seen in MAD techniques vs single agent numbers. \n\nTo explain this, they develop a theoretical framework to model the debates as a stochastic, peer-influenced process. They prove that an unguided debate process induces a martingale over the agents' belief trajectories, i.e. expected correctness does not improve withe debate alone. Empirical results in the paper, support this as well. \n\nBased on this, the paper proposes MAD-Conformist and MAD-Follower which use majority votes to guide the belief update process to bias the belief update process towards correctness. Both methods show a strong improvement over standard MAD techniques across a variety of tasks.", "questions": "In a majority of the results, it seems that correctness scores decrease with more debate rounds. Do the authors have an insight into why that is the case? Is it due to the degradation of initial strong majorities or a few agents changing their response causing weaker majorities to fade? Is it an overall breakdown of agreement, i.e. as the debate progresses, the models disagree with each other more rather than agree?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n - This paper does a great job of breaking down why MAD does and doesn't work as expected. It provides strong empirical proof for majority voting being the largest contributor to any MAD improvements and develops a sound mathematical explanation for why that is the case. Modeling the agents as DCMs and the debate process as a martingale provides helps understand how MAD works. \n - The paper's claims are substantiated by a comprehensive set of experiments. It multiple models of different sizes and model families, diverse benchmarks (from reasoning to QA), and various MAD configurations (decentralized, sparse, etc.), which strongly supports the generalizability of the findings.\n - The paper includes simple but impactful improvements to MAD, incorporating majority voting to bias the agents' belief update process towards correctness. \n - This paper shows that sometimes simpler is better, and serves a reminder of the crucial role a good baseline plays in contextualizing improvements. \n\n\n### Weaknesses\n - The theoretical framework and most empirical results are based on homogenous agents. While this is useful for simplicity, it somewhat defeats the purpose of a debate. Since all agents share the same beliefs, there is less variety in the responses generated by them. [Estornell et. al.](https://proceedings.neurips.cc/paper_files/paper/2024/file/32e07a110c6c6acf1afbf2bf82b614ad-Paper-Conference.pdf) show that a lack of diversity in responses or model capabilities is detrimental to the debate process.\n- The lack of diversity is somewhat addressed through the heterogenous agents experiment in section 6 but the models are all still similarly capable. It would also be helpful to include a t-SNE visualization of the response embeddings to confirm that the diverse personas result in truly diverse responses. \n - The experimental setup is based on a relatively simple MAD design (simultaneous-talk, simple debate). The paper doesn't evaluate against more sophisticated methods involving structured debates, where the results might not generalize. \n - Limited novelty for MAD-Conformist and MAD-Follower: [Smit et. al.](https://arxiv.org/pdf/2311.17371) propose an agreement modulation mechanism to precisely control what proportion of responses agree with each other and show that that high inter-agent agreement leads to improved correctness.  The paper would be more impactful if it contextualized its proposed methods against similar related work.", "was_revised": true, "final_justification": "I appreciate the explanation on how the framework allows for diversity in the debate and the lexical diversity analysis of the different personas.", "rebuttal": "We thank the reviewer for the constructive feedback and for complimenting the strong and extensive empirical results.\n\nBelow, we address the key concerns.\n\n---\n\n`A1` *Clarification on agent setting*\n\nWe understand the reviewer’s concern and fully agree that diversity is valuable in multi-agent debate settings. We would like to clarify the key considerations of out setup:\n\n- Homogeneous agents allow us to **isolate the core mechanisms**, rigorously dissecting the contribution of inter-agent *debate* vs. simple *majority voting*, without introducing confounding factors such as varied model capabilities or role-specific behaviors. This clean setup enables both a rigorous theoretical analysis and a principled understanding of where performance gains originate.\n- The homogeneous agent setting is the most commonly studied setup in prior MAD work (e.g., [1,2,3,4,5]), **making our theoretical framework broadly relevant and well-grounded in the established research landscape**. **For a theoretical foundation paper, it's important to first fully understand the essential ground**, which can later inform and extend to more advanced/specialized cases.\n\nThat said, we want to emphasize that **our framework does not preclude diversity**:\n\n1. **Stochastic response diversity does arise in our setting**: As Estornell et al. rightly emphasize, response diversity is important for productive debate. Crucially, _such diversity can still emerge in homogeneous agent setups_. Even when agents share the same base prompt and belief prior, stochastic sampling (e.g., temperature tuning or top-p sampling) induces variability in responses—allowing agents to explore different reasoning paths and generate disagreement.\n2. **Our theory does allow capturing different beliefs**: As defined in Definition 1 of our paper, each agent samples its belief $\\boldsymbol{\\theta}\\_{i,t}$ from a Dirichlet distribution. Even if all agents share the same prior concentration parameter $\\boldsymbol{\\alpha}$, the sampled beliefs $\\boldsymbol{\\theta}\\_{i,0}$ differ across agents due to the inherent randomness in the Dirichlet process. This means that agents can hold distinct initial beliefs and produce diverse responses, enabling non-trivial interactions during debate.\n3. Lastly, **our theoretical model and empirical results remain applicable and informative, even as richer forms of agent heterogeneity are introduced**. As you recognized, we *do* study heterogeneous agents in Section 6 (Table 4), where we assign different personas (e.g., \"Mathematician\", \"Lawyer\") following established practices. Even in these diverse settings, we observe that simple Majority Voting remains highly competitive, further supporting our claim that ensembling is a primary driver of MAD performance. For these reasons, we believe our work lays a strong foundation for future research in this direction. \n\n\n[1] C. Chan et al., \"ChatEval: Towards Better LLM-based Evaluators through Multi-Agent Debate\" (ICLR 2024)\n\n[2] Y. Du et al., \"Improving Factuality and Reasoning in Language Models through Multiagent Debate\" (ICML 2024)\n\n[3] Y. Li et al., \"Improving Multi-Agent Debate with Sparse Communication Topology\" (EMNLP-findings 2024)\n\n[4] Q. Wang et al., \"Rethinking the Bounds of LLM Reasoning: Are Multi-Agent Discussions the Key?\" (ACL 2024)\n\n[5] G. Zhang et al., \"Cut the Crap: An Economical Communication Pipeline for LLM-based Multi-Agent Systems\" (ICLR 2025)\n\n\n---\n\n`A2` *Do diverse personas result in diverse responses?*\n\nWe appreciate the reviewer’s interesting suggestion to plot the t-SNE projections. We looked further into this, but since persona is not an explicit attribute that is easily captured in output embeddings, we found that t-SNE projections do not show meaningful differences between single- and multi-persona setups.\n\nAs an alternative, we retrieved the frequency table of the mean Levenshtein distance, which is computed as:\n\n$$\nD = \\frac{1}{N^2}\\sum_{i=1}^N\\sum_{j=1}^N \\text{Levenshtein}(y_i, y_j),\n$$\n\nwhere $y_i$ is the response from agent $i$ (see table below). 30 samples from GSM-8K were evaluated on the Qwen2.5 agents. The table reveals a clear shift under the multi-persona configuration, **indicating greater lexical diversity in the generated responses**.\n\n| **bins** | **Single-persona** | **Multi-persona** |\n|:---:|:---:|:---:|\n| $350 \\leq D <400$ | 13 | 1 |\n| $400\\leq D < 450$ | 11 | 16 |\n| $450 \\leq D < 500$ | 6 | 10 |\n| $500 \\leq D < 550$ | 0 | 2 |\n| $550 \\leq D < 600$ | 0 | 1 |\n\n---\n\n`A3` *Structured debate methods*\n\n\nWe appreciate the reviewer’s point. Structured debate formats (role-based, judge-mediated systems, etc.) are indeed a powerful line of work. Our focus, however, is to understand the fundamental dynamics of inter-agent communication, which is why we adopt a minimal, decentralized protocol to isolate core mechanisms without added structure.\n\nThat said, **our theoretical framework is not restricted to simplistic settings**. **The DCM-based model applies broadly to any debate mechanism** where agents update beliefs based on peer responses. Notably, the belief update rule in Theorem 2 can be better generalized as:\n$$\np_t :=  \\frac{\\alpha_{i,t-1}^{(1)} + w^{(1)} \\cdot c_{i,t}^{(1)}}{\\sum_{j=1}^K ( \\alpha_{i,t-1}^{(j)} + w^{(j)} \\cdot c_{i,t}^{(j)})},\n$$\nwhich includes a flexible weight term $w$ that can capture structured signals—such as those from prompt design, fine-tuning, or human-guided adjudication. Structured systems like judge-based debate can be seen as increasing $w^{(1)}$ for more credible or correct responses, introducing a directional bias that helps overcome the stagnation highlighted in our model. \n\nWe also want to highlight that **other reviewers have explicitly recognized the impact and foundational value of our framework**:\n\n> “_I believe that the formulation introduced in the paper could have a **moderate-to-high impact** on the MAD community – a field which has suffered from imprecise and ad-hoc approaches in the past_.” — *Reviewer 6aic*\n\n\nIn summary, **our framework offers a general foundation for understanding belief dynamics, and can serve as a diagnostic tool even in structured settings**. By revealing when and why debate fails to improve accuracy, it provides diagnostic insight that can guide the design of more effective coordination strategies. We see integration with structured protocols as a valuable direction for future work.\n\n---\n\n`A4` *Novelty of MAD-Conformist and MAD‑Follower—needs better context like agreement modulation (Smit et al.).*\n\nWe thank the reviewer’s suggestion to better contextualize our interventions within related work. We acknowledge that Smit et al.’s work on agreement modulation is closely related and highlights an important connection to our proposed interventions. Our interventions—MAD-Conformist and MAD-Follower—are theoretically motivated by our martingale analysis (Theorem 2), aiming to address the core stagnation issue in MAD dynamics through lightweight, generalizable mechanisms.\n\nTo further clarify positioning, we will explicitly compare our designs with related strategies such as Agreement Modulation (Smit et al.), All-Agents Draft (AAD) and confidence-weighted voting (Kaesberg et al.) in the revised paper. We believe this framing will highlight the novelty and complementary nature of our work, both theoretically and practically.\n\nThank you again for this constructive suggestion, which we believe strengthens the paper.\n\n---\n\n`A5` *Why does accuracy often decrease with more debate rounds?*\n\nAccording to our theoretical framework (Theorem 2), belief updates under a DCM process are expectation-neutral; i.e., in the absence of external directional signals, agents are equally likely to drift toward or away from the correct answer over successive debate rounds. However, **when generalizing the belief update rule (see `A3`), it is possible that the dynamics become skewed**: if the update weight $w$ is larger for incorrect responses—e.g., due to persuasive but flawed reasoning, the expected correctness can decline over time.\n\nTo capture this phenomenon, we analyze the subversion and correction rates of Qwen2.5 agents across debate rounds (see table below). Here, subversion rate denotes the proportion of responses that were correct in the previous round but became incorrect in the next, while correction rate tracks the opposite. Overall, the subversion rate seems to overwhelm correction rates, suggesting that real-world LLM agents may be slightly more frequently persuaded away from the correct answer—an empirical indicator that the weights $w$ assigned to incorrect responses are higher in practice.\n\n| **Dataset** | **Round** | **Correction Rate (%)** | **Subversion Rate (%)** |\n|:---:|:---:|:---:|:---:|\n| Arithmetics | 2 | 0.40 | 2.00 |\n| | 3 | 0.00 | 1.80 |\n| | 5 | 0.20 | 3.00 |\n| GSM8K | 2 | 0.33 | 1.33 |\n| | 3 | 0.40 | 1.73 |\n| | 5 | 0.40 | 2.13 |\n| Pro. Medicine | 2 | 0.96 | 0.96 |\n| | 3 | 0.96 | 1.10 |\n| | 5 | 0.96 | 1.03 |\n| Formal Logic | 2 | 1.43 | 1.90 |\n| | 3 | 1.90 | 2.38 |\n| | 5 | 1.75 | 2.06 |\n| HellaSwag | 2 | 0.33 | 0.40 |\n| | 3 | 0.33 | 0.73 |\n| | 5 | 0.40 | 1.20 |\n| CSQA | 2 | 0.73 | 1.00 |\n| | 3 | 0.53 | 1.40 |\n| | 5 | 0.60 | 1.13 |\n| HH-RLHF | 2 | 1.33 | 2.13 |\n| | 3 | 1.40 | 2.53 |\n| | 5 | 1.27 | 2.53 |"}, {"review_id": "UP2hwMRnnk", "reviewer": "Reviewer_hHcS", "summary": "This paper explores the effectiveness of Multi-Agent Debate in improving the performance of large language models. It contrasts MAD with simple Majority Voting through extensive experiments across seven NLP benchmarks. The key findings are that Majority Voting accounts for most of the performance gains typically attributed to MAD, and that debate alone does not systematically improve expected correctness. The paper also presents a theoretical framework modeling debate as a stochastic process and proves that it induces a martingale over agents’ belief trajectories. Based on these insights, the authors propose interventions to enhance debate effectiveness by biasing belief updates toward correct signals.", "questions": "1. Why is the Dirichlet-Compound-Multinomial distribution chosen for modeling LLM response generation? Are there related papers or statistical data supporting this choice?\n\n2. What is the base model used for Figure 3?\n\n3. Can an average result column be added to Table 1 and Table 2?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. This paper puts forward an interesting and significant argument that Majority Voting is responsible for most of the performance gains usually associated with MAD, and it substantiates this argument through both experimental and theoretical analysis.\n\n2. The paper compares Majority Voting and various MAD techniques across over seven NLP datasets, and the experimental results largely align with the theoretical proof.\n\n3. The conclusions drawn in this paper hold certain implications for enhancing the design of MAD architectures.\n\nWeaknesses:\n\n1. The paper mainly conducts comparative experiments on the Qwen2.5-7B-Instruct and Llama3.1-8B-Instruct base models. The experimental results on Llama3.1-8B-Instruct show a certain degree of instability, and experiments on larger-scale bases are only performed on some benchmarks, making it impossible to fully understand the model's patterns. Additionally, the authors do not test on other closed-source LLMs.\n\n2. The conclusions of the paper mainly target closed-ended question answering tasks, and the applicability of Majority Voting techniques to open-ended text generation tasks is limited. Whether there is a Majority Voting architecture suitable for text generation tasks remains a topic of discussion.", "was_revised": true, "final_justification": "The authors have provided responses to the questions I raised earlier regarding the Dirichlet-Compound-Multinomial distribution and the verification of model stability in their rebuttal. After rechecking, I confirm that my overall score of this paper remains consistent with the assessment conclusions presented in the initial review.", "rebuttal": "We thank the reviewer for the constructive feedback and for **acknowledging the value of our experiments and theoretical analyses**.\n\nBelow, we address the key concerns.\n\n---\n\n`A1` *Instability of LLaMA3.1-8B-Instruct results*\n\nThank you for the question! We would like to emphasize that this behavior is entirely consistent with our theoretical framework, which models MAD as a martingale process—meaning that the expected correctness of agents does not improve with additional debate rounds unless directional signals are injected.\n\nAs shown in Table 1 in the manuscript, LLaMA occasionally shows gains from MAD, but those gains are modest and non-monotonic, aligning with our martingale-based explanation (Theorem 2). This demonstrates that instability is not a flaw of the experiment, but a key empirical support for our theoretical claim: MAD, in its vanilla form, does not consistently lead to correctness improvements. We will clarify this interpretation more explicitly in the revised manuscript.\n\n---\n\n`A2` *Further experiments on larger-scale models*\n\nThank you for this suggestion. In response, we have conducted additional experiments using the larger-scale Qwen2.5-32B-Instruct model across four representative benchmarks. As shown in the table below, the effect of Decentralized MAD varies. For Arithmetics, GSM8K, and HellaSwag, we observe slight performance degradation or oscillation across rounds, while for Professional Medicine (MMLU), MAD improves over majority voting. These results further support the generality of our martingale-based analysis: even with highly capable agents.\n\n\n|  | Arithmetics | GSM8K | Pro. Medicine | HellaSwag |\n|:---:|:---:|:---:|:---:|:---:|\n| Majority Voting | 1.000 | 0.9433 | 0.9007 | 0.8767 |\n| Decentralized MAD ($T = 2$) | 0.9733 | 0.9367 | 0.9154 | 0.8767 |\n| Decentralized MAD ($T = 3$) | 0.9667 | 0.9200 | 0.9154 | 0.8733 |\n| Decentralized MAD ($T = 5$) | 0.9700 | 0.9300 | 0.9191 | 0.8667 |\n\n---\n\n`A3` *Further experiments on closed-source models*\n\nIn response to your request for experiments on closed-source models, we conducted additional evaluations using three GPT-4 agents across four benchmarks. The overall trends remain consistent with those observed in our open-source model experiments, supporting the generality of our findings.\n\n|   | Arithmetics | CSQA | HellaSwag | HH-RLHF |\n|:---:|:---:|:---:|:---:|:---:|\n| Majority Voting | 0.9967 | 0.8721 |0.9078 | 0.5612 |\n| Decentralized MAD ($T = 1$) | 0.9867 | 0.8788 | 0.9078 | 0.5580 |\n| Decentralized MAD ($T = 2$) | 0.9867 | 0.8784 | 0.9044 | 0.5577 |\n| Decentralized MAD ($T = 3$) | 0.9833 | 0.8780 | 0.9044 | 0.5459 |\n\n---\n\n\n`A4` *Application to open-ended tasks*\n\nGreat point! We agree that multi-agent debate becomes more nuanced in open-ended or long-form NLP tasks. That said, we believe the core intuition of our belief update framework remains applicable, with a few key adaptations. In particular, we discuss below: (i) extending the update rule and (ii) defining consensus in open-ended tasks:\n\n**(i) Extending the update rule**\n\nAlthough open-ended responses are not easily countable in a strict categorical sense, they can still be modeled in distributional or similarity-based forms. Specifically, the core mechanism of our Bayesian update rule,\n$$\np_t :=  \\frac{\\alpha_{i,t-1}^{(1)} + c_{i,t}^{(1)}}{\\sum_{j=1}^K ( \\alpha_{i,t-1}^{(j)} + c_{i,t}^{(j)})},\n$$\ndoes not inherently require hard categorical responses. In open-ended settings, the count vector $c_{i,t}$ can be interpreted more general—for example, as\n\n- a soft histogram over clustered response types,\n- an embedding-based semantic agreement metric,\n- or a weighted similarity score between textual outputs.\n\nThus, **the Bayesian belief update process can be extended by letting $c_{i,t}$ reflect aggregated semantic signals** rather than strict response counts.\n\n**(ii) Defining consensus in open-ended tasks**\n\nSimilar to the extended update rule for open-ended tasks, **consensus is better defined semantically, not symbolically**. For example, if agents independently generate similar explanations or rationales, we may consider this as semantic consensus, even if the surface forms differ. This can be operationalized via thresholds on embedding similarity, Levenshtein distance, or overlap in reasoning chains etc.\n\nOur current experiments use discrete agreement on multiple-choice question tasks, but we believe the above adaptations can extend naturally to open-ended contexts. We view this direction as an exciting next step for generalizing the MAD framework beyond classification.\n\n---\n\n`A5` *Rationale for choosing the Dirichlet-Compound-Multinomial model for LLM response generation*\n\nThis is an insightful question. We adopt the Dirichlet-Compound-Multinomial (DCM) formulation as the backbone of our belief-update model, grounded in both theoretical rigor and empirical observations:\n\n- **Principled Bayesian foundation**: The Dirichlet distribution is the conjugate prior of the Multinomial distribution, making it a straightforward and well-established choice for modeling the nature of LLM text generation dynamics. In our setup, an agent’s belief over $K$ possible responses is updated as it receives signals from other agents, and the DCM provides a closed-form way to model this update: an agent starts with a Dirichlet prior $\\alpha$, observes peer responses, and updates beliefs based on the posterior. This mirrors classical Bayesian updating and lends a principled probabilistic structure to belief evolution in a multi-agent setting.\n\n- **Empirical alignment with LLM dynamics**: As shown in Figure 4 of our manuscript, the evolution of $p_t$ over rounds of decentralized debate aligns well with the expectation-neutral behavior predicted by DCM. That is, belief trajectories exhibit symmetric drift under majority influence, without systemic bias toward any particular answer choice. This symmetry and neutrality are core predictions of DCM and match what we observe across multiple rounds of real LLM-agent interactions.\n\n---\n\n`A6` *What is the base model used for Figure 3?*\n\nThe base model for Figure 3 is Qwen2.5-7B-Instruct. We will make sure to include this in our revised version. Thank you for pointing this out.\n\n---\n\n\n`A7` *Can an average result column be added to Tables 1 and 2?*\n\nOf course! We will add the average columns in our revised version. We appreciate your suggestion."}, {"review_id": "5lramtODOu", "reviewer": "Reviewer_6aic", "summary": "The paper disentangles multi-agent debate (MAD) into two key components – Majority Voting and inter-agent Debate to assess the contribution arising from each. It demonstrates that the vast majority of the performance comes from Majority Votes, rather than from any agent debate. The paper goes on to introduce a mathematical framework and two theorems that help explain their empirical results, and highlight why in these homogeneous-agent Q&A tasks, inter-agent debate will not improve the expected correctness, i.e. Debate induces a martingale over agents’ belief trajectories, which in expectation remains unchanged across rounds.", "questions": "How do the empirical and theoretical findings presented in the paper align with paragraph “Pros of Multi-Agent Debate”? \n\nFor example, the conclusions of the theoretical analysis is that MAD induces “a martingale process, which preserves the expected success probability of each agent over time.” However, also stated on page 9: MAD “has the potential to enhance both factual accuracy and reasoning quality”. Please can you clarify how both these passages can hold?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "On Quality: \n\nThe paper presents strong motivation, and is well written. Clear and concise background is provided on the Majority Voting and Multi-Agent Debate, citing relevant works. It was a pleasure to read. \n\nThe paper does well to couple an empirical study (with 7 datasets and multiple strong LLM agents), with a theoretical formulation that explains the empirical findings. This is a particularly strong part of the paper, as to the best of my knowledge, many papers in this field do not formally define MAD with as much care as taken here. For example, careful attention is given to define MAD strategies as undirected graphs, where edges indicate which agents can observe one another. \n\nThe empirical evaluation is also conducted with care, and although the code is not released, it feels highly reproducible given the level of detail provided.\n\nThe Extended Experiments section preempted multiple of my questions – although a more thorough experiment into Open-Ended tasks would be welcomed (see below questions). \nHow would one consider extending the Bayesian update rule to account for the additional context an agent could provide to another, going beyond a simple count vector $c_{i,t}$ which would be required for open-ended tasks? \n\nOn Impact: \n\nI believe that the formulation introduced in the paper could have a moderate-to-high impact on the MAD community – a field which has suffered from imprecise and ad-hoc approaches in the past. \n\nHowever, I believe the paper can be more forthright with the assumptions made in the Theoretical Analysis section – which lead to the main conclusions of the paper. \nFor example, Line 172: “We assume that the correct answer has the largest belief $\\alpha^{(1)}$”. \nThis feels like an overly strong assumption, and one that removes much of the \nsurprise of the findings.\nDoes this imply that the probability of the correct response is lower-bounded by Theorem 1 only in cases when the homogenous agent correctly assigns the largest $\\alpha^{(1)}$ belief to that answer in round 0? If this is the case, are we surprised that majority voting provides the largest contribution to the aggregated response - given that the homogenous agent already “knows” the answer?  \n \nSurely the more interesting setting is the one where the underlying agent (or subset of agents) does not assign the largest belief to the correct response in the initial round – and how their belief can be updated over multiple rounds of debate? The theorem does not cover this more challenging case, and therefore one should be careful with the claims made, Line: 60: “we prove formally that majority vote does essentially all the work, which explains our empirical findings.” \n\nSimilarly, when we consider the agent’s Dirichlet parameter update, (Line 190) we are only considering the counts associated with the $K$ possible responses – this seems reductive and overly restrictive. For example, one would want an agent to be persuaded by the additional context and explanation another agent provides, biasing its belief away from its largest $\\alpha$ value towards the correct response. Do you consider this setting?  \n\nFor these reasons, it’s not clear to me that the theoretical results presented will hold when more complex and open-ended tasks are performed. How would a consensus even be defined in long form NPL tasks? \n\nOn Clarity: \n\nHeterogeneous Agents: The manuscript would benefit from more context provided in this section. For example, what are “optimal persona sets” and how is this implemented in practice? \nIt would also be interesting to expand the theoretical analysis to this setting, but appreciate that might be future work.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive and thoughtful feedback! We are deeply encouraged by and grateful for your words:\n\n> \"_I believe that the formulation introduced in the paper could have a moderate-to-high impact on the MAD community – a field which has suffered from imprecise and ad-hoc approaches in the past_.\" \n\nwhich is exactly what prompted us to write this paper :)\n\nBelow, we address your key concerns.\n\n\n---\n\n`A1` *Assumption that the correct answer has the largest initial belief $\\alpha^{(1)}$*\n\nYou are raising an excellent question here. We would like to emphasize that **this assumption reflects only a weak prior preference**, not a strong or certain belief. For example, suppose a single agent assigns a belief of 0.01 to the correct answer and 0.009 to the next-best option. While the agent’s accuracy is only 1%, Theorem 1 shows that with enough agents, majority voting can boost correctness to nearly 100%, which is the core benefit of majority voting. Importantly, this works even if the correct option is only slightly favored, no matter how small that margin is.\n\nMoreover, **we provide an alternative version of the theorem in Appendix D, which does not assume that the correct option has the largest initial belief**. Theorem 1.A establishes a lower bound on the probability that the correct answer is selected by majority vote, without assuming any belief constraints. This highlights that the benefit of voting is not solely due to prior confidence, but also emerges from the statistical structure of ensembling. \n\n\n\nLastly, we want to emphasize that **this assumption is not required for the general formulation of our belief dynamics or for understanding the behavior of multi-agent debate more broadly**. Theorem 2 does not rely on this assumption. The martingale property holds regardless of whether the correct option initially has the highest belief or not. It tells us that in expectation, beliefs do not drift toward the correct answer over time unless external structure or directional bias is introduced. This explains why naive multi-agent debate may not consistently outperform majority voting—debate alone, without guidance, cannot correct a misaligned initial prior.\n\nIndeed, the most important and challenging setting is where the agents do not initially favor the correct answer. **Our theoretical framework helps us understand what structural mechanisms are needed to shift beliefs in such settings**. It also highlights the limitations of relying solely on peer interactions and motivates our design of guided interventions. \n\n---\n\n`A2` *Extending the Bayesian update rule beyond count vectors and defining \"consensus\" for open-ended tasks*\n\nThis is an insightful and important question! We agree that multi-agent debate becomes more nuanced in open-ended or long-form NLP tasks. That said, we believe the core intuition of our belief update framework remains applicable, with a few key adaptations. Below, we address both (i) extending the update rule and (ii) defining consensus in open-ended tasks:\n\n**(i) Extending the update rule**\n\nAlthough open-ended responses are not easily countable in a strict categorical sense, they can still be modeled in distributional or similarity-based forms. Specifically, the core mechanism of our Bayesian update rule,\n$$\np_t :=  \\frac{\\alpha_{i,t-1}^{(1)} + c_{i,t}^{(1)}}{\\sum_{j=1}^K ( \\alpha_{i,t-1}^{(j)} + c_{i,t}^{(j)})},\n$$\ndoes not inherently require hard categorical responses. In open-ended settings, the count vector $c_{i,t}$ can be interpreted more generally—for example, as\n\n- a soft histogram over clustered response types,\n- an embedding-based semantic agreement metric,\n- or a weighted similarity score between textual outputs.\n\nThus, **the Bayesian belief update process can be extended by letting $c_{i,t}$ reflect aggregated semantic signals** rather than strict response counts.\n\n**(ii) Defining consensus in open-ended tasks**\n\nSimilar to the extended update rule for open-ended tasks, **consensus is better defined semantically, not symbolically**. For example, if agents independently generate similar explanations or rationales, we may consider this as semantic consensus, even if the surface forms differ. This can be operationalized via thresholds on embedding similarity, Levenshtein distance, or overlap in reasoning chains etc.\n\nOur current experiments use discrete agreement on multiple-choice question tasks, but we believe the above adaptations can extend naturally to open-ended contexts. We view this direction as an exciting next step for generalizing the MAD framework beyond classification.\n\n---\n\n`A3` *Limiting counts to $K$ possible responses is restrictive ... one would want an agent to be persuaded by the additional context from other agents, biasing its belief towards the correct response. Do you consider this setting?*\n\nAbsolutely! We agree that persuasion via explanation is a desirable property of debate. However, we respectfully clarify that the limitation to $K$ discrete responses is not inherently at odds with such persuasive dynamics.\n\nThe use of $K$ responses in our framework is simply a modeling abstraction to make the belief update process tractable and interpretable. This $K$-way formulation can be readily generalized to larger or even infinite response spaces (e.g., open-ended generation tasks discussed in Q1), without invalidating the core martingale result. What matters is how agents map peer responses—whether discrete or continuous—into belief adjustments.\n\n---\n\n`A4` *What are “optimal persona sets” and how is this implemented in practice?* \n\nAs noted in Lines 272–276, the \"optimal persona sets\" refer to a collection of agent roles selected via the \"agent selection algorithm\" introduced in [1]. This method identifies a combination of personas that are empirically effective when collaborating in multi-agent tasks.\n\nIn practice, we implement this by assigning each agent a system prompt that encodes a specific role or persona—e.g., Mathematician, Programmer, Lawyer—using the same prompt templates provided in [1]. We agree that our manuscript would benefit from more explicit implementation details, and we will include a full description of the relevant details in the revised version.\n\n[1] Z. Liu, et al. \"Dynamic llm-agent network: An llm-agent collaboration framework with agent team optimization\" (COLM 2024).\n\n---\n\n`A5` *How do the empirical and theoretical findings presented in the paper align with paragraph “Pros of Multi-Agent Debate”?*\n\nThank you for this observation. First, we would like to clarify that the paragraph titled \"Pros of Multi-Agent Debate\" appears in the Related Works section and is intended as a summary of claims made in prior literature—not the central claim of our own work.\n\nIn contrast, our paper critically examines those assumptions and challenges the notion that multi-agent debate in its current form consistently improves factual accuracy or reasoning quality. Our theoretical and empirical results show that, without carefully designed mechanisms or external interventions, MAD can stagnate or even degrade performance over time due to its martingale dynamics.\n\nWe appreciate the opportunity to clarify this and will revise the manuscript to clearly distinguish the claims of prior work from our contributions."}, {"review_id": "kZIFgJDqOv", "reviewer": "Reviewer_qiU5", "summary": "This paper investigates the core components of Multi-Agent Debate (MAD), arguing that its performance gains stem primarily from the aggregation of multiple agent outputs (i.e., Majority Voting) rather than the iterative debate process. While prior work has already raised empirical questions about the universal effectiveness of MAD compared to other ensembling methods [2] and has highlighted the power of voting protocols over extended debate rounds [3], this paper's main contribution is the introduction of a formal theoretical framework to explain why this is the case. The authors model the debate as a martingale process over agent beliefs, demonstrating that debate alone does not systematically improve expected correctness. \n\nThe modeling assumes that each agent follows a Dirichlet-Compound-Multinomial (DCM) process. The belief update rule is based on counting responses from neighboring agents. However, using simple count vectors for belief updates can make the complex interactions in LLM-based debates, and this modeling choice receives no further empirical validation. Only two observations inform the introduction of the assumption: the stochasticity of LLM outputs and the interaction between agents. The only supporting evidence is the stability of average accuracy over rounds, which is a minimal sign of the martingale behavior claimed.\n\nThe authors propose two MAD variants that bias belief toward correct responses. They observe locking correct answers, which breaks the martingale property and improves performance. Since this process needs access to ground truth, they propose heuristic methods using majority-voted responses. However, these methods often perform similarly to or worse than majority voting, limiting their practical value.\n\nOverall, the paper presents a clean theoretical framework, but the modeling assumptions are strong and under-justified, and the proposed improvements offer limited advantage in practice.\n\n[1] Tongxuan Liu, et al. \"GroupDebate: Enhancing the Efficiency of Multi-Agent Debate Using Group Discussion.\" arXiv preprint arXiv:2409.14051 (2024)\\\n[2] Andries Smit, et al. \"Should we be going MAD? A Look at Multi-Agent Debate Strategies for LLMs.\" Proceedings of the 41st International Conference on Machine Learning (2024)\\\n[3] Lars Benedikt Kaesberg, et al. \"Voting or Consensus? Decision-Making in Multi-Agent Debate.\" arXiv preprint arXiv:2502.19130 (2025)", "questions": "**1**. The modeling assumes that belief updates can be represented by simple response counts, using the Dirichlet-Compound-Multinomial process. Can you provide empirical evidence or ablation results showing that this update rule reflects how real LLM responses evolve during debate?\n\n**2**. The only support for the claimed martingale behavior is the stable mean accuracy over rounds. Did you check other signals such as $p_t$ directly or further support the theoretical claim?\n\n**3**. The proposed methods that use majority-voted results often perform similarly to or worse than plain majority voting. How do you interpret this outcome, and under what conditions would your method provide clear advantages?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### **Strengths**\n\n**Novel Theoretical Framework**: The primary strength of this paper is its novel theoretical framework. By modeling the multi-agent debate as a DCM process and martingale argument it provides a principled and rigorous explanation for why majority voting explains the success of MAD. \n\n**Clarity and Organization**: The paper is exceptionally well-written and organized. The theoretical concepts, including the Dirichlet-Compound-Multinomial (DCM) model and the martingale proof, are presented with clarity, making the core arguments accessible.\n\n### **Weaknesses**\n\n**Limited Empirical Novelty**: The paper's central empirical conclusion, that majority voting is often as good as or better than complex debate while important, is not entirely novel. Recent studies have already shown that MAD is highly sensitive to hyperparameters and does not reliably outperform other prompting strategies [2]. Furthermore, the effectiveness of voting protocols and the potential negative effects of increasing debate rounds have been systematically demonstrated [3]. Thus, the paper's experiments seem as a confirmation of these existing findings rather than a new discovery.\n\n**Strong and Underjustified Modeling Assumptions**: The theoretical framework, while elegant, rests on the strong assumption that the complex, semantic-rich interactions of LLM agents can be simplified to a Dirichlet-Compound-Multinomial (DCM) process updated by simple response counts. However, even standard prompt tuning stragies like reformulating question or refining response can adjust DCM $\\alpha$ and marginal probability. The paper provides limited empirical validation that this model accurately captures the dynamics of real-world LLM debates, beyond showing that the average accuracy remains stable over rounds.\n\n**Limited Practical Advantage of Proposed Methods**: The proposed interventions (e.g., MAD-Conformist), designed to break the martingale property, do not consistently or significantly outperform the simple majority voting baseline. This limits the practical impact of the theoretical insights, as the simpler, more efficient method remains the more compelling choice in most scenarios, a point also relevant in the context of improving MAD's efficiency [1].", "was_revised": true, "final_justification": "The authors have successfully addressed my primary technical concern by providing convincing empirical evidence for their theoretical framework; while practical limitations remain, I now recognize the significant value of their foundational contribution and have raised my score accordingly", "rebuttal": "We thank the reviewer for the constructive feedback and for acknowledging the novelty of our theoretical framework, and also complimenting the clarity of our manuscript. \n\nBelow, we address your key concerns.\n\n---\n\n`A1` *Clarification on empirical finding*\n\n\nThank you for this constructive feedback, and we appreciate the opportunity to better clarify the positioning of our work in relation to prior works.\n\nWe agree that recent empirical studies—including Smit et al. [1] and Kaesberg et al. [2]—have made valuable contributions by revealing important limitations of multi-agent debate (MAD). These works raise fundamental questions about the robustness and efficacy of MAD, and we believe our paper offers a complementary and timely response to precisely address those concerns.\n\nSpecifically, the goal of our empirical analysis is not to claim novelty in observing MAD's limitations, but rather to **motivate the need for a theoretical framework that can explain and predict when and why such limitations arise**. This is the central contribution and novelty of our work, as you also recognized. We develop a formal model of belief dynamics under debate and derive analytical results (e.g., Theorem 1 and 2) that shed light on how agent responses evolve and aggregate over time. This framework offers the **first principled explanation for this observation** -- something not accounted for in prior works. The value and novelty of our work are also recognized by peer reviewers:\n\n> \"_I believe that the formulation introduced in the paper could have a moderate-to-high impact on the MAD community – a field which has suffered from imprecise and ad-hoc approaches in the past_.\" ---- Reviewer 6aic\n\n\nLastly, we recognize that we should have cited and discussed these earlier works more directly in our motivation. We will revise the manuscript to better acknowledge their contributions, clarify how our work builds upon them, and make more explicit that our goal is to provide theoretical foundations for previously observed empirical patterns—thereby moving toward a deeper and more principled understanding of multi-agent systems.\n\n[1] Andries Smit, et al. \"Should we be going MAD? A Look at Multi-Agent Debate Strategies for LLMs.\" Proceedings of the 41st International Conference on Machine Learning (2024)\n\n[2] Lars Benedikt Kaesberg, et al. \"Voting or Consensus? Decision-Making in Multi-Agent Debate.\" arXiv preprint arXiv:2502.19130 (2025)\n\n\n---\n\n`A2` *Clarification on modeling assumptions*\n\nThank you for this thoughtful critique. We agree that modeling the rich and nuanced dynamics of LLM interactions is inherently challenging, and any abstraction—including ours—comes with limitations. Our goal is not to fully replicate the full semantic depth of LLM debates, but rather to **identify and formalize core statistical properties** that govern belief evolution and aggregation in multi-agent settings at a reasonable level of abstraction, and at the same time providing tractable and interpretable analysis.\n\nThe Dirichlet-Compound-Multinomial (DCM) framework is an ideal choice, as we **justified in our manuscript (L150-L162)**:\n\n> \"..._DCM closely mirrors how practical LLM systems produce different outputs due to uncertainty and sampling variability. In particular, the Dirichlet prior captures the agent’s internal belief over possible answers, while the Multinomial models the stochastic generation process (e.g., via temperature or nucleus sampling). This distribution is thus a natural choice because it encapsulates both internal uncertainty and output randomness, while also providing a principled Bayesian framework for belief updates across debate rounds—enabling analytical study of dynamics during the debate process...._\"\n\n\nThis proper abstraction enables us to derive formal insights about the role of belief dynamics, which **empirically align with key behavioral trends observed in real-world LLM-based MAD systems (more evidence discussed in `A3` below, where we directly verified $p_t$)**.\n\nWe acknowledge that prompt tuning strategies----such as rephrasing or refinement----can influence how responses are generated and interpreted. And the sampling variation induced by prompting is precisely what DCM can capture mathematically. To account for such variation, **our framework is flexible enough to accommodate this by allowing a variable weight $w$ in the belief equation from Theorem 2**:\n$$\np_t :=  \\frac{\\alpha_{i,t-1}^{(1)} + w^{(1)} \\cdot c_{i,t}^{(1)}}{\\sum_{j=1}^K ( \\alpha_{i,t-1}^{(j)} + w^{(j)} \\cdot c_{i,t}^{(j)})}.\n$$\nThis weight can be adjusted via prompt design, fine-tuning, or human-guided signals, providing flexibility to model more realistic agent behaviors.\n\nWe believe our heterogeneous-agent setup in Table 4 partially reflects this idea—for instance, a Programmer agent might be more influenced by the reasoning of a Mathematician than that of a Lawyer. Yet even in this more nuanced setting, performance trends remain largely stagnant, reinforcing the core idea of our theory.\n\nOverall, we view our theoretical framework as **a sufficiently general foundation that can incorporate richer dynamics and guide future improvements to debate protocols**.\n\n---\n\n`A3` *Empirical evidence for DCM. Did you check signals such as $p_t$ directly?*\n\nYes, we did! In fact, **Figure 4 presents the mean $p_t$ values across debate rounds** (with raw values provided in Appendix E). Importantly, $p_t$ reflects the single-agent belief in the correct answer at debate round $t$, and the mean $p_t$ is the average among agents ($y$-axis in the figure). This measure differs from the overall task accuracy, which the reviewer may have misinterpreted. This trajectory captures the agent's evolving belief and directly aligns with the DCM update model used in our theoretical analysis. As shown in Figure 4, **the mean $p_t$ values remain largely stationary over rounds, providing direct empirical support for the martingale behavior----consistent with the theoretical result in Theorem 2**.\n\n---\n\n\n`A4` *Practical impact of theoretical insights*\n\nWe appreciate the reviewer’s concern and would like to clarify on the practical impact of our theoretical analysis. One of the key insights of our analysis is that debate has the potential to substantially outperform simple voting when appropriately guided. This is clearly demonstrated through the MAD-Oracle. **These gains indicate that there is considerable headroom in the design space of MAD, and that our theory helps uncover this significant potential**.\n\nThe purpose of our proposed interventions—MAD-Conformist and MAD-Follower—is not to fully bridge this gap, but to empirically validate concrete predictions derived from our theory. These methods are **intended to show promise**, demonstrating that even minimal, theory-driven modifications can improve upon vanilla MAD. **Our aim is not to claim state-of-the-art performance or argue against the utility of alternative approaches (including [1])**, but rather to establish a foundation for designing more effective strategies grounded in a deeper understanding of belief dynamics.\n\nWe see these methods as **initial, illustrative steps**—proofs of concept that MAD performance can be steered by principled design. Far from limiting the practical value of our theory, these findings highlight its role in enabling targeted improvements. As we objectively and transparently note in the manuscript (L256–258):\n\n> \"While they do not reach the oracle’s upper bound performance, they demonstrate that simple, theory-informed modifications can yield meaningful improvements—pointing to a promising direction for future work to close the gap.\"\n\n[1] Andries Smit, et al. \"Should we be going MAD? A Look at Multi-Agent Debate Strategies for LLMs.\" Proceedings of the 41st International Conference on Machine Learning (2024)\n\n ---\n \n`A5` *When will the interventions beat majority voting?*\n\nThis is a very insightful question. The bottleneck in our current interventions is that they lack reliable estimation of which answer is correct, and thus cannot consistently introduce the directional bias needed to steer belief evolution productively. As a result, their gains are often modest. This stands in contrast to MAD-Oracle, which demonstrates the full potential of such a strategy when the correct signal is known/correctly estimated—achieving substantial improvements over majority voting. \n\nWe expect that future work incorporating better estimators of correctness (e.g., via confidence scoring, calibrated self-reflection, or external signal integration) can realize significantly larger gains. In short, our framework suggests a clear direction for improving MAD: design agents that can more effectively estimate and amplify correctness early in the debate. We will incorporate such a discussion in our revised manuscript. Thanks again for your comment!"}]

Author rebuttal:

Meta-review: * **Summary**:    
  * Ablation study on Multi-Agent Debate components over 7 NLP benchmark.  
  * Show that majority voting is the main driver of improvement over single-agent.  
  * Develop theoretical framework to model debates as stochastic processes.  
  * Prove that process is a martingale over beliefs: correctness does not improve with debate alone. Back this proof with empirical experiments.  
  * Propose two new MAD strategies using majority votes to guide the belief update process towards correctness: both show strong improvement over variety of tasks.  
* **Strengths:**   
  * Great break down of why MAD does and doesn’t work. (HZhH, hHcS)  
  * Strong empirical proof (HZhH, hHcS)  
  * Sound mathematical explanation (HZhH, hHcS)  
  * Novel theoretical formalization it introduces is much-needed in the MAD community (6aic, qiU5)  
  * Theory backed by comprehensive set of experiments, varying model sizes, families, benchmarks tasks, MAD configurations, supporting generalizability (HZhH, hHcS, 6aic)  
  * Provides simple but impactful improvements on MAD (HZhH, hHcS)  
  * Pleasure to read (6aic), Exceptionally well-written and organized (6aic)  
  * Empirical conducted with care and feels highly reproducible thanks to level of detail provided (6aic)  
  * Extended experiments preempted multiple questions of the reviewer (6aic)  
  *   
* **Weaknesses:**  
  * Relies on homogenous agents, which might be over-simplistic. (HZhH, 6aic) → Convincingly covered in rebuttal.  
  * Simple MAD design: would be curious to evaluate on more sophisticated MAD designs involving structured debate (HZhH) → convincingly covered in rebuttal.  
  * Instability of result in experiment with LLaMA (hHcS) → Convincingly explained as a feature, not a bug, in the rebuttal  
  * Could better compare its new MAD strategies against similar work (HZhH) → Added in rebuttal.  
  * Would appreciate more LLMs and more variability at high-scale LLMs, and closed-source LLMs (hHcS) → Both added in rebuttal.  
  * Question about modeling choices / assumptions (HzhH, hHcS, qiU5, wxsp) → Discussed thoroughly in rebuttal.  
  * Limited novelty of discovery that majority voting is as good as or better than debate (wxsp) → Acknowledged in rebuttal, as novelty is in theoretical framework that offers first explanation of this observation.  
  * New modified MAD does not clearly outperform majority voting (wxsp) → acknowledged in rebuttal, shows “proof of concept”  
* **Summary of rebuttal:**  
  * Discussion of Simplicity of homogenous agents (HzhH) used to isolate core mechanism, as a foundational theoretical paper.   
  * Theory still holds true with diversity factored in.  
  * Will add comparison to similar work suggested by (HzhH)  
  * Thorough rebuttal with generalization of formulas and extra evaluation.  
  * Raised (HzhH) score.  
  * Explained LLaMa experiment’s instability as backed by their theory (hHCs)  
  * Added high-scale experiments (hHCs)  
  * Added closed-source experiments (hHCs)  
  * Convincing discussion of the chosen distribution model (hHCs, 6aic)  
  * Discussion of modeling assumptions in the theorem, with alternative provided without that assumptions and clarification thereof in the manuscript (6aic, wxsp)  
  * Extension of update rule (6aic)  
  * Surprisingly, reviewer (6aic) chose to keep his 4 score in spite of what I (as AC) see as a thorough answer by the authors.  
  * Added references (wxsp)  
  * Multiple rounds of discussion with (wxsp), adding new data from experiments, with clarifications as to how experiments validate some precise points of the theoretical framework. → Authors add new experiments showing it, convinces (wxsp) who updates their score.  
* **Missing:**  
  * Would be interesting to discuss generalization beyond close-ended questions, on open-ended text generation (hHcS) → Elaborated in rebuttal.  
* **Most important reason for accept/reject**: On a red-hot topic, a great combination of empirical analysis of a problem, introduction of a much-needed novel formalization, paired with thorough empirical validation, and an exemplary interaction with the reviewers adding abundant new experiments and clarifications that convinced all of them. Thought-provoking, timely, and beautiful\!
* **Why stands out:** Our field desperately needs proper, thorough theoretical analysis such as this one, which directly connect to practice by backing it with empirical evidence such as this one.
Decision: Accept (spotlight)