Multi-LLM Debate: Framework, Principals, and Interventions
Authors: Andrew Estornell
Yang Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: multi-agent debate
LLM
Primary area: natural_language_processing

Abstract: The flexible and generalized nature of large language models has allowed for their application in a wide array of language-based domains.
Much like their human contemporaries, these models are capable of engaging in discussions and debates as a means of improving answer quality.
We first take a theoretical approach to analyzing debate and provide a framework through which debate can be mathematically examined.
Building on this framework, we provide several theoretical results for multi-agent debate.
In particular, we demonstrate that similar model capabilities, or similar model responses, can result in static debate dynamics where the debate procedure simply converges to the majority opinion. 
When this majority opinion is the result of a common misconception (ingrained in the models through shared training data) debate is likely to converge to answers associated with that common misconception.
Using insights from our theoretical results we then propose three interventions which improve the efficacy of debate. 
For each intervention, we provide theoretical results demonstrating how debate is improved.
We also demonstrate that these interventions result in better performance on four common benchmark tasks.

Multi-LLM Debate: Framework, Principals, and Interventions
Andrew Estornell ByteDance Research andrew wtrr:eeeeettmeeleeeteeeeeee.e
Yang Liu University of California, Santa Cruz yangliuducsc.
Abstract
The flexible and generalized nature of large language models has allowed for their application in wide array of language- based domains. Much like their human contemporaries. these models are capable of engaging in discussions and debates means of improving answer quality. We first take a theoretical approach to analyzing debate and provide framework through which debate can be mathe- matically examined. Building this framework, we provide several theoretical results for multi-agent debate. particular, we demonstrate that similar model capabilities, similar model responses, result in static debate dynamics where the debate procedure simply converges to the majority opinion. When this majority opinion is the result of common misconception (possibly ingrained in the models through shared training data) debate is likely to converge to answers associated with that common misconception. Using insights from theoretical results, then propose three interventions that improve the efficacy of debate. For each intervention, provide theoretical results demonstrating how dehate is improved. We also demonstrate that these interventions result in better performance on four common benchmark tasks.
Introduction
Large language models (LLMs) have demonstrated a remarkable ability to perform unseen tasks with high efficacy. This behavior, often referred to emergent, allows LLMs to serve as general-purpose tools for a wide array of manguage-based functions. One such behavior of particular interest is the ability of LLMs to intake and process opinions from other models (or humans). As shown in several previous works, this ability allows LLMs to collaboratively solve tasks by engaging in debate Chan al. [2023], Liang et al. [2023), Du al. [2023). For a given task, mulli-agent debate operates by eliciting responses from each model, distributing those responses among the models, and then eliciting updated responses from each model.
In this work, we aim to explore the debate procedure, by first providing a theoretical framework through which debate can be better understood. This framework draws inspiration from Bayesian inference and in-context learning, showing that debate can be partially viewed as a special type of in-context learning. Through this framework, we then provide several theoretical insights into the debate procedure. In particular, we demonstrate the susceptibility of multi-agent debate to echo- chamber effects. These scho-chamber effects are especially consequential when they stem from shared misconception between majority of models, which can arise from circumstances such as highly correlated training data between each model.
We then leverage results from our theoretical framework to improve the efficacy of the debate procedure. In particular, we propose three interventions (modifications to the debate procedure). First, diversity-pruming which aims to maximize the information entropy in model responses at each round of debate; this intervention has the added benefit of reducing the severity of the echo chamber
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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effect. Second, quality- which aims to maximize the relevance of each model's response. We demonstrate that this intervention improves the likelihood that the debate procedure converges to correct answers. Lastly, misconceetioo-rrfutatiie misconceptions in model responses. This intervention takes inspiration from works such as [Robinson] et al. [2022] which demonstrate that LLMs are often more skilled at evaluating answers, compared with directly providing answers. For each of our interventions, we provide theoretical results outlining the way in which each improves debate. We also conduct experiments four common benchmarks demonstrating that these interventions improve debate efficacy in practice.
which directly identifies, and attempts to refute
Our contributions 1) We propose a theoretical framework for multi-LLM debate that draws on connections from in-context learning and Bayesian inference. 2) We provide theoretical insights on several key principles of multi-LLM debate. 3) Using these insights, we design three debate interventions which result in consistent improvement to dehate, across four language benchmarks (BoolQ, MMLU, MathQ, TruthfuQA) and three families of models (GPT, Lama, and Mistral).
2 Related Work
Our work is closely related multi-agent debate, which focuses on iterative collaboration between
agents to make a decision Chan et al. (2023], Liang et al. [2023), Du et al. (2023), Khan et al.
(2024), [Iring al. (2018). Michael et al. 2023 Rasal
[2024] Pham al. [2023] Chang [2024b].
These works often focus on multi-agent debate in the context of question-answe answering tasks and aim to provide higher quality answers (compared those of a single model) by engaging multiple models in discussion. The preliminary debate framework, proposed by Du et al. 2023]. facilitates debate by first asking each model question, and then iteratively re-asking agents that same question contextualized by the responses of all models in the previous round. Different variants of this procedure have been proposed: debate where models have different functionality Liang et al. [2023), round-robin style debate Chan et al. [2023), dynamically controlling the level of disagreement between agents in debate Chang 2024a]. , or using judges to assess the correctness of debaters Khan et al. (2024).
Other techniques for iteratively improving the quality of answers have also been proposed, e.g..
chain-of-thoughh Wei et al. [2022]. Kojima et al. [2022]. self-consistency Wang et al. 2022] Singhal
al. 2023), and self-reflection Ren et al. [2023
Similar debate, there have been investigations into the use of different LLMs to engage with
one-another Liu et al. [2023). (Abdelnabi et al. [2023] Zhang et al. 22033 Li et al. [003cc] Park et al.
2023a explain their rational to others Wang et al. [2023a]. ) or collaboratively engage general
tasks Li et al. [2023a], Wang et al. [2023b], Park et al. [2023b], Wu et al. (2023), Hong et al. [2023).
Li et al. 2023d b). Lao and AILAB 2023 While debate has shown promise in a wide range of domains, several works have also demonstrated that the debate process can be unstable and can lead
to worse performance than using just single model Wang et al. 2024). Smit el. [003)].
Our work is also related to in-context learning and Bayesian inference. The former, Brown et al. 2020)), Min et al. [2022], Lampinen et al. [2022] demonstrates that LLMS can perform unseen tasks when provided only a few examples of that task. Other works Xie et al. [2021], Jiang [2233] have
shown connection between in-context learning and Bayesian inference; the additional examples
provided to the model can be viewed as updates to the model's posterior distribution over tokens.
3 Preliminaries
Debate Let be a given question, with associated answer y, for example x "What color is the sky?" and y = "Blue". Following the debate procedure proposed by Duu et al. [2023]. a collection of LLMs (also referred to agents) collaborate to infer the correct answer y by iteratively engaging in discussion over rounds, as described next:
At round l each agent observes task x, then provides response z;
At rounds / > 0 each agent i observes task x and the outputs of the n agents at the previous
timesetep (t-1) (z) (t-1)
(z((-1) .......!(_ ((-")), then provides response z
2((
The debate process ends if t T or if agents reach consensus.

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To measure if consensus is reached function a extracts an answer from given response Suppose z "During the day, the sky blue", then a(z) "Blue" " At round t, the probability that agent
provides response 2
(t+1) is given by
Fmodel
2 (2)
updated model response
task
all model responses from previous round model parameters
where Φ, captures model hyperparameters (such training data, architecture, etc.). Both the input
(Z(0, x) as well as the hyperparameters 4, ultimately influence the output !! (((+1), Note that on each round, all agents observe the same input, namely (Z()) x). Thus, differences in agent output 2 are determined by both the stochastic nature of output generations, and the unique parameters Φ, of each model. For notational convenience, we drop the subscript in IP model when the parameters Ø; given, and simply write IP( (-)
(f+11)
The key distinction between our approach and "vanilla" debate, is that we will leverage latent concepts (discussed next) to modify the responses in (+)) in between each round of debate.
Latent Concepts Central to investigation is the idea of latent concepts in language generation. As outlined by Xie et al. [202]). Diang (2023) latent concepts capture the idea that language is not generated at random (either by humans or by models). Rather, when generating language, have an idea or an intention form in our minds; we then select words that we believe will convey that underlying idea intention. More formally, let θ be latent concept space and let I θ be a concept. Following Xie et al. [022]], tasks x, and their associated answer y are generated by first selecting vector of latent concept 0 θ and then sampling (x D(0), where D is some distribution mapping concepts to task-answer pairs. Similarly, when providing responses, models will observe x, and infer the latent concept 0, or more generally a distribution over the latent concept space, and then generate a response according to those inferred concepts, i.c., the model's generation can be expressed as
first
probability Equation|
I
(22 x, x.)) = II P(2 (t+1)
0, x,x,2 (,)
((() x, 2(0 Φ.)
OE
Note that the above holds by the law of total probability for any choice of latent concept space.
To provide a more concrete example of latent concepts, consider the cvestion-answerrig task in the BoolQ dataset. One of the questions in this dataset "Did Abraham Lincoln write the letter in the film Saving Private Ryan?" to which the correct answer is "Yes". The latent concept, in this case, corresponds to the actual scene the movie where the Bixby letter (written by Lincoln) read to group of soldiers. Just as in our case, first concept @ is drawn, e.g., the film is made; then from the film, a string x sampled, i.e., the previous question about the film.
For another example of latent concepts, we can think of arithmetic calculations such as multiplication. When we wish to express multiplication through language, may write something like "4 4". The latent concepts behind this string are the mechanisms of multiplication (e.g.. multiplication is just iterative addition, and addition itself simply increasing the value of a number by one iteratively). These examples are intended to be easily digestible. However, latent concepts can also be significantly more abstract, such vector in some unknown embedding space.
4 A Theoretical Formulation of Multi-Agent Debate
We begin by providing a theoretical formulation of multi-agent debate. This formulation will provide key insights into the inner workings of the debate procedure, which we will use to improve debate.
The key behind our framework is to use the idea of latent concepts and expansion of each model's generation probability (Equation 2 in order to better understand debate. Prior to presenting our theoretical framework, we first state an important assumption.
Assumption 4.1. For a given latent concept space θ, the probability generating response z
(t+1) is
conditionally independent of both the responses (t) and the task x, given concept 0 e θ and model
parameters 4,, i.e., (z, ([+1)) 0, x, Z(() (b)) P P(z
(t+1)) 0, Φb:)
In practice, can be
checker or LLM--bsed judge such as Liang al. 2023
3

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Assumption 4.1 can be interpreted as saying that a model's generation is uniquely determined by the model's parameters Φ; and the concepts identified by a model, namely 6. the case of seboodrrdeecode-baseeee models, one can conceptualize the joint between & and O as corresponding to the embedding produced by the encoder. With this embedding in hand, the original input (x, 2 no longer influences the model's output, rather the embedding and model parameters will uniquely determine the model's 's output.
Next, we derive the following lemma which will be useful in examining the way that model responses evolve debate rounds.
Lemma 4.2. The generation of model i at time t + 1 can be expressed as,
(1 7(0, x, (b)
Σ OEO
P( 11 P00
IIP(2)
generation without other agents
caused by other agents
The significance of this lemma is that we are able to express the probability of generating a given response " (+++1) with the other model responses ((() in terms of the probability of generating z; (t+1) without the other model responses and skew term caused by those model responses. Note that,
P(z. (++1) x, Φ,) x
Σ OE
P(4
0, (.
Thus, think of (z, (1+1) z Φb) weighted version of (z( x, Φ.), where the weights are given by the skew term |==1 P(z) 0, 0.).).
Debate and In- in-Context Learning To help conceptualize the role of latent concepts in debate, discuss the work Xie et al. [2021], which uses Bayesian inference over latent concepts to understand in-context learning. There are natural connections between in-context learning and multi-agent debate. In-context learning works as follows: given a task x and a model f. select several examples task- Then prompt the model for an answer to task given examples (x,, Mj), i.e. z = f(x] (x1, 3/) ... (Xm, Fm)) A key result of Xie et al. [2221] is that latent concepts in the examples (x,, 3/ particularly concepts shared between many examples, influence the model's answer z. For any concept where (()) (x1, 11) (Xm Bm)) is large relative to other concepts (i.e., there shared concept & between the examples), the model more likely to give response z which also share that concept. Model responses at the previous round ( serve a similar function to the examples of in-context learning. The model's updated response at time + namely 2!" is influenced by concepts shared between the responses in Z The skew term Lemma 4.2 provides glimpse of how latent concepts conveyed by ( will influence the generation of namely that II;; TP (z) 0, (.) reweighs the model's generation.
answers pairs 31) (1), (Xm, Fmn) which similar
4.1 Debate Objective
Through this perspective of debate we can more effectively design debate procedures by leveraging the concept space θ To do this, we will first formulate debate as an optimization problem where the skew term, described in Lemma 4.2] corresponds to the optimization variables. For a given task x and answer y, each round of debate can be formulated as the following optimization problem.
((a(z((+1) yl 7(1) 7(0).))))
arg
2
At time t we aim to craft responses (t) such that they maximize the probability of providing the correct answer at the next time step. Expanding this objective over the latent concept space θ yields
T>
arg max 2(S)
((a(z(++1))=
( px) 0, Φ.) II (( (0)
(..)))
(3)
DEG

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The key challenges with directly optimizing this objective are: 1) the true concept 0" from which and y originate, as well as the relationship between 2 and the underlying concepts, is unknown, 2) the responses in (()) are natural language. However, will allow us to design several approaches within the concept space to better optimize debate. To motivate these approaches, we first need to
make several observations about the debate procedure as a whole.
Debate Principals
In this section, discuss factors that affect the efficacy of L.MSs debate. In particular, look at the role of information diversity. both in terms of the diversity of responses in z well as diversity in model capabilities. We see that a lack of diversity in either aspect is detrimental to the debate process. Lastly, study particular type of homogeneity in debate, namely shared misconceptions in which a large portion of models all share a similar erroneous belief about the task x.
5.1 Information Diversity
We begin by examining how the debate procedure is affected by both the diversity of model abilities and the diversity of model responses. Homogeneity in either ability or responses will bias the debate procedure towards certain latent concepts.
Similar Model Capabilities Suppose the debate process is conducted with only type of model (in effect n copies of the same model). That is, Φ; = & for all i E [n]. Then, as the number of agents increases, the debate procedure is greatly impacted by the echo chamber effect, i.e., the
probability that a round of debate results in a change to the most likely concept, perceived by agents, approaches O. That is, greater number of similar agents results in static debate dynamics, in essence defeating the purpose of debate.
Theorem 5.1. Suppose all n agents have identical configurations (Φ: for all i). For round & O let 0k(t) [arg max (() | x, 7 .)) and (((++1) [arg max IP (0) x, 2(t+1), .6).. ie., O' and 6E
BE
gx(++1) are the concepts most likely to be inferred by a model with parameters & when given task x and responses ((0), ((++1) respectively. Then P(e" (1)) (
+ as + Co.
We defer full proof to the Supplement, Section A Theorem 5. implies that when debate is conducted with multiple copies of the same model (or very similar models), increasing the number of models results in debate centering on single (unchanging) concept, rather than balanced distribution over multiple concepts.
Similar Model Opinions Next, we examine how similar responses impact the collaboration process. At time suppose that there are n responses Z (4) and at least m of those responses are similar, i.e., there exists some concept o' such that 0' arg maxøce (P 2 of the m responses has shared "most likely" concept when viewed by model with parameters Φ; Theorem 5.2. Suppose that Z(0) contains at least m responses with the property that 0' arg maxøce (z. (t) 0, O.). Then, as m + oo the model's generation at the next round
(t) (b,) for all S m. That is, each
p (++1)
2 2())).)) F(255t)) 6
(1 [++ 1) becomes uniquely determined by single concept O' i.e.
(1.3)
(.22 200x)
('.0.))
(t+1) for all response pairs (i.1) (1,2)
(b++)
We defer a full proof to the Supplement Section A Theorem indicates the susceptibility that LLM debate has towards tyranny of the majority. If large number of models provide similar responses to a task x, then those repeated answers will drown out the single provided by the other models' responses, as well as the task itself. In Section 7 we demonstrate that this occurs in practice.
5.2 Shared Misconceptions
Next, we study a particular type homogeneity in model capabilities and responses, namely shared misconceptions. When a common misconception is shared among the models, debate is less effective
5

---

and is likely to converge to erroneous concepts associated with the shared misconception. We now formalize the notion of misconceptions.
Definition 5.3. (Misconception): For given concept 0" model with parameters i is said to have a misconception regarding 0" there exists another concept 0' s.t.,
Px [P(x 0', (.)) P(x 0" o.) 3/2
That is, for tasks generated from the concept 0", the model believes that the erroneous concept 6' explains more than half of the tasks better than the true concept 0*
There is said to be shared misconception if of agents have misconception and share the same erroneous concept 0'. When the models share a common misconception the responses produced by those models are biased towards the erroneous concept 6'.
Theorem 5.4. Let the true concept be O* and suppose that m of the n agents have shared misconception for erroneous concept 6'. Then, task and answer (x, y) ~ D(0") expected average correctness of the debate procedure at the final round T is monotonically decreasing with m, i.e.
1. Pa(zz (T) v) is decreasing with
We defer the full proof to the supplement Section A It should be noted that the phenomenon of converging erroneous concepts is unlikely to be mitigated by adding misconceptions of one model are formed through training data, it is likely that other models will possess the same misconception unless specifically trained avoid such errors due to the high correlation in training data between models.
models. When the
Interventions
In this section, we discuss several modifications the debate procedure, referred to interventions. We break our interventions into two categories: Pruning which focuses on choosing which responses to keep in z(1), and Modifying which focuses on changing or editing the responses Z
6.1 Pruning Interventions
At round of debate, running interventions work by selecting only subset of responses (¹(t) from (*) before starting the next round t + 1. When using pruning intervention, the models at round t ++1 will see only the pruned response set Zz(1), rather than the full response set (())
Diversity Pruning Let KL represent Beilbackk-eibble divergence. The diversity pruning interven- tion selects A of the responses in z(0) which maximizes information entropy, i.e.,
t0()_
argmax Σ (L(( z4), D(0) 24))
s.t. 2 k
YCY E.
Quality Pruning Quality pruning aims to select the k responses in Z with the highest similarity to the task x. Similar to diversity pruning, quality pruning selects of the n responses at time I. Rather than selecting for diversity, quality pruning aims to select the A highest question responses. This is done by selecting the & responses which maximize
(0)
argmin 2C2
]KL(D(0)x). D(0)
s.t. 2Z = k
Z.EZ
In practice computing KL (D ((0)z)) or KL(((0),,,, D(0 is intractable. However, sentence embedding can be used proxy for these values. Section CaationnCCiia discusse this in further detail.
Next, we show that when models have shared misconception, diversity pruning decreases the likelihood that the debate procedure will converge to the erroneous concept corresponding to the shared misconception.
Theorem 6.1. Let the true concept be 0" and suppose that at least n/2 agents have shared misconception for erroneous concept 0'. Then diversity pruning decreases the probability that debate converges to an answer y' which is sourced from the erroneous concept 6', i.e. 3 D(0').

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We defer the full proof to the Supplement, Section A
Theorem 6.2. For a given task-unswer pair (x,1) quality pruning increases the probabil- ity that debate converges to the correct answer, i.e. let 7(t) be the set of all responses at time t and Z be the result of quality pruning, then Σ P(a(z (o()) y|x, k*th) Φ.) >
P(a(2 (t+1) y|x, z Φ.).
We defer the full proof to the Supplement, Section A
Remark 6.3. As shown by Theorems 6. and 6.2 diversity pruning decreases the probability that debate converges to incorrect answers sourced from a particular concept, while quality pruning increases the probability that debate converges to correct answer sourced from the true concept. Both interventions can be used simultancously to guide the debate procedure more effectively away from wrong answers and towards correct answers.
6.2 Modification Interventions
Misconception Refutation In addition selecting which responses in Z will be used in the next round of debate, we can also modify the responses in z Misconception refutation aims to do precisely this by updating response z) to be more relevant to the task x.
KL(DOO)))) 2 = arg min (D(O))) D - D(0) z))
Similar Diversity Pruning and Quality Pruning, the distributions in the above objective are intractable in practice. As such, we use a proxy to update each response ())), specifically produce zj by having an LLM minimally modify the given response 2/"), The model is first prompted for a list of misconceptions and errors identified in the response. Given the list of misconceptions, the model asked for both a refutation of the misconception and a corrected version of the response. For more details, Section[]or the Supplement.
Theorem 6.4. For task-answer pair (x,y), y). misconception refutation increases the probability of debate converging to the correct answer, Le. let (tt), Z=(t) be the responses before and after miscon-
ception refutation, then Em= PP((zz (t+1)) y|x, zt) ...)
(t+1) = y/x, (1)) 6.).
Experiments
Experimental Design We conduct experiments on four common language model benchmarks. BoolQ
Clark et al. 2019). which consists of 3, 270 yes-no questions. MMLU Hendrycks et al.) (2020)
which consists of 13, 869 multiple-choice questions (we use the 3, 406 high-school-level questions), TruthfulQA Lin et al. [2021] which consists of 817 open-ended questions, and MathQ which consists of 3, 000 arithmetic questions of the from a-b-c + d-e.f. In the BoolQ, MMLU, MathQ, datasets model correctness is measured through regular expression matching. In the TruthfulQA dataset, model correctness is measured via an LLM judge (we use GPT-4 as the judge in all experiments)
We use four LLMs of increasing capability, GPT-3.5 (GPT-3.5 Turbo) OpenAl [2022). Llama-2 (Llama-2 7B Chat) Touvron et al.) [2023]. Llama-3 (Llama-3 8B Instruct) Meta All [2024], and Mistral (Mistral 7B Instruct v0.2) liang et al. 20033]. For sentence embeddings (which serve as a proxy of the latent concepts +), we use sentence embeddings from ADA-2 OpenAl [2022]. We compare a combination of our three interventions Ours (see Algorithm full details) with the debate paradigm Du et al. [2023) (Society of Minds) SoM.
We begin by making several empirical observations about the multi-agent debate process.
Tyranny of the Majority First, we examine the susceptibility of models towards agreement with the majority opinion. That is, how likely are models to give specific answer at round & when m of the models provided that specific answer at round the previous round (round /)? For example, in BoolQ suppose the specific answer "Yes". then we want to know; how likely a model to give a Tess-answwr at round t + 1 if that model observes "Yes" answers at round t.
7

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Probability of giving majorty
Mare
Harrad 1e wiree piin
MMLU
TruthfurQA
MathQ
BoolQ
72
.33
Number of other agents giving majority answer
Figure 1: Probability that each model echoes the majority answer round / - 11, as the number of responses at time / gives that majority answer (debate between models are used).
BoolQ
TruthfulQA
MathQQ
Accuracy improvement
.58
ba
.00
.35
Average pairwise similiaity between responses round 0 (2
Figure Average accuracy improvement as function of response diversity at round of debate.
To measure this, first select random target answer, c.g., "Yes", and then prompt of the models (out of 11) to provide responses "Yes" "Yes"-answed" answers (while the other 11-m models are prompted to provide a different randomly selected answer). These responses make up Z we then test each model's likelihood of providing the target round + when observing (t) before diversity pruning (solid) and after diversity running (hatched).
In Figure I
models susceptible echo chamber effects (this phenomenon is predicted by Theorem 5.1 The likelihood of providing the majority answer increases when Z contains more instances of the majority answer (i.e., as m increases). Figure [ also demonstrates that diversity pruning (with is 5) reduces this echo chamber effect. See the Supplement for details.
Diversity of Opinions Next, we examine the effectiveness of SoM and our method as a function of opinion diversity. Figure shows the average accuracy improvement of SoM (dashed) and our method (solid) over single model performance (i.e., average performance at round function of the similarity between all responses at round = 0 of debate (measured via pairwise cosine similarity). We that for BoolQ, MMLU, and TruthfulQA. SoM is less effective when the similarity between responses increases. This observation is predicted by Theorems 5.1 and 5.2 which show that debate, without interventions, is less effective when model responses are too similar.
0),
We see that our method's improvement compared to SoM is greatest when model opinions are more similar (cosine similarity close to 1). Note that the MathQ benchmark, where responses consist primarily of arithmetic, serves as a counter-example to these observations. This due to the fact that sentence embedding of any two arithmetic expansions will be similar, regardless of their true similarity; as such, cosine similarity between embedding is less meaningful on this benchmark.
Debate Interventions Now, we examine the effectiveness of combination of our three debate interventions (see Algorithm [or full details of how the interventions are combined). We begin with a per-round performance of our method and SoM, as shown Figure 3 We see that typically, the advantage of our method over debate arises in the later rounds of debate. Next, in Table we present a full set results for single models, SoM, and a combination our three interventions. In all cases, method is either competitive with, superior to, SoM.
Any model not providing the target is re- re-prompted until the target answer is provided.

---

Single
SoM
Ours
Single
SoM
Ours
BoolQ
MMLU
GPT-3.5
802.014
844.012
854.012
.34.014
744.016
4.4.014
6x Llama-3
76 014
.78+.018
.784.013
.676.016
.70..016
.754.014
6x Llama-2
.67
.684.017
.734.016
.11.0011
.77..018
12.1..18
Mistral
.80 .14
.84.013
.854.012
.664.016
.64.016
.664.016
3x GPT-3.5 + 3x Llama-3
821.014
881.018
.734.016
.78 016
3x GPT-3.5 3x Mistral
831.013
881.0012
.69..017
.721.00
Llama-3 3x Mistral
.80.0014
.80..014
.69..017
.74..016
TruthfulQA
Math
6 GPT-3.5
612.083
.634.032
..0.0
.534.035
884.016
.98
6x Llama-2
.7=
.5240.3
.554.034
1.1..013
.134.014
.91..11
Llama-3
.33 035
.554.02 0332
.524.032
2264.016
.34.017
.48.0118
6x Mistral
A8 034
+..4.03
.534.034
.13+.013
.194.014
.18+.014
3x GPT-3.5 3,5 3x Llama-3
.564.03
.624.0a1
.764.015
.844.014
3x GPT-3.5 3x Mistral
.52..035
.56..055
.56..018
.68
3x Llama-3 + x3 Mistral
494.030
036
.35
.22.015
.23
Table I: Accuracy of solo model, debate, and our debate interventions:
10 rounds, models.
BootQ
MMLU
78
24
73
-75
HIGGT-11 Manaa Boo)
GPTAS Lama- Jain
Round (1)
Figure 3: Accuracy per round, our method and SoM when combing GPT-3.5 with Llama-3 or Mistral.
In addition to providing results for the combination our interventions, we also investigate the effectiveness of each intervention applied individually (see Table the Supplement). These results indicate that our method most successful when applying all three interventions simultaneously. In fact, some interventions be detrimental the debate process when applied in isolation. This is expected as each intervention is inherently designed to be complementary.
8 Limitations
While we aim address some of the fundamental issues of multi-LLM debate, such as tyranny of the majority, there are several factors that need to be considered when adopting our framework. Firstly, our theoretical results leverage a latent concept space, which may not be accessible in practice, necessitating the use of proxies such as sentence embeddings. Reliance on proxies is particularly consequential for quality and diversity pruning; these interventions are less effective domains where sentence embeddings less meaningful, c.g., arithmetic questions. Additionally, can increase the inference time the debate procedure. Increased inference time stems primarily from misconception refutation, this intervention requires re-prompting each debater multiple times.
interventions
9 Conclusion
Multi-agent debate is an effective tool for improving the efficacy of L.M responses. However, debate naturally susceptible issues such as tyranny of the majority and shared misconceptions between models. By making use of our theoretical framework for debate, we are able to establish interventions for the debate procedure which help alleviate these issues and improve the general performance of multi-agent debate. We saw that diversity pruning reduces the influence of similar responses. This is especially helpful in settings where the majority agents provide incorrect responses that share common enror. A combination of all three interventions consistently leads to better debate.

---

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Appendix
A Theoretical Results
Proof of Lemma 4.2 This result holds via marginalization of the posterior predictive distribution over the latent concepts θ namely
P(z) 1 | z x, Φ;) =
Σ
P(z (1-1)
0, 2 x, x,,, P()) 2(), xxx Φ:)
OEG
P(O) Φ.)
P
0, Φ P( x|0 P(Z()), x)
Σ P(z.
0, 0 0,) o.) II F 0.
Since each model has an identical configuration, i.e. Φ. = Φ for all i.j E [m]] we simply refer to the configuration as &b For the given task x (-(t) the realization of 0 at time
Proof of Theorem 5.7
step f. Let z(2) (z]" 2n 2 where each 2) (f) q(2) Z(O, <b) Then the conditional density
for each concept can be written as,
P(0) ( x, Φ)) xP xF ( x 0 ② P Φ)
-x 0, Φ
where the term P(x| 0, Φ) (0) Φ is a constant with respect to the number of agents n. Since ((O()) for all 0, each z is an i.idd draw from q(z) x, (t-1)) Φ). then
(x)))
lim 2
arg max P(0) ( x, Φ6
Thus, as n >> oo, models predict the same concept, namely (*()), at timestep with probability 1.
Proof of Theorem 5.2 Consider any two responses from agent at time / + 1, namely (( (1.2)
(t+1) (t+1)
When there are n duplicate messages, the ratio between the conditional generation probabilities of both responses can be written
( (1+1) (1.1)
x, (b)) z(0,x,,)
Z
P
(++1)
2(1,xx x, Φb)
(1,2)
(1.2)
Eoco P((
((+1)
0, ( (0) x.)
Eoco |
[0))))(0)
Eoco 1 [1 d.)
(factoring out common divisor (( x| sb)
Loce P(((1))
| [.00))1((2211000)))00 (.)
(((210)) 0, 0, m
Eoco F(2
(1+1)
11 P(z)) (1)
0, o.)
((+1) Eoco P(2
0, (x) Φb))
[I P(2)0 0. (b)
(((2(()) 0,
13

---

Let O' be the concept which model i, with configuration Φ,. believes is most likely to have produced response z', i.e.,
arg max P(z' 0, ④b)
Then the above ratio can be rewritten as
Σ
P (
(1+1)
0, ( PP (x 0, 0. o,)
[I] P(2)" 0, (.)
[((210) 0, o..)
r(z/00)) or. (1
(1.1)
AEE
Σ
P P(Z
(++1)
0, (x) Φ.)
[I F(2) 0, 0.0.)
P(2 0, (.).)
p( o'.
Σ
((())))
((+1)
0.)
[1 1 0.
F (x/00) 0.++.
(1rcs) or.ob.
0
Σ
P(z
((+1)
0, 0.
P 0 6 PP(| Φ.)
WL-IP(2) 0 o.))
P [x 0.
(1.2)
P(2//c0) or.ob.)
0
+P(2(22)
((t))
0 o 0 P(O 0.)
I1 P(z) (1 0', (.))))
+P(2 0', Φ. px) 0 (')))(('))) ),)
11 ((2) 0', 0.)
p(x' 0,0.) p 0 ..)
For each 0 + 0' we have
1. Therefore
PF''' 0 0. m P(
lim
Note that the only summands which do not have an exponential dependency on are those associated with concept 0'. Therefore, when examining the limit of the above ratio with respect to the number of repeated signals m, we get
m (P(Z(0)
(1+1) Eoco P(z ((Z(.))
Φ.)
0.
(1.1)
lim
m
1
Lose
((2))2)
(1+1)
0.
0, (x),))()) Φb)
[1 ((2))) 0,
(P(2(0) 0,
(1.2)
P ((z
(++1)
0' (x) 0. (.)
(1.1)
P ((
(++1)
0' (x)
0,
PP ((2
(++1)
(;)
PP
(++1) (1.2)
0', (.)
Thus the relationship between any two conditional generation probabilities can be uniquely defined by 0'.
Proof of Theorem 5.4 First, we examine the models' outputs the first rounds of debate. For model i which possess the shared misconception, their conditional generation probability on the first round of debate can be expressed as,
(++1) P(2
x ②b) x P(z! (1t)) P (px)) 0.
Next, consider the term [x 0, d.). Let Φf be set of model parameters which does posses the common misconception. Then,
P(xl 0, 6.) ((xx''10,0)
14

---

where x' is a message which conveys the erroneous concept 0'. Using this change of model parameters, we can express the condition generation probability as,
P( ((+1) xx Φ;)
P( (1+1)
0, (x
= Σ P(z (++1)
0, (
0, (x))) 0.
With this formulation of the conditional generation probability, we can the ratio between the true concept 0" and the erroncous concept 0'.
P(z (++1)
0" F o') .7)
P(Z (1 0', O,) 0', (() 0 (')
P(2 (1 0 (
(4)
P(z.
(1+1) 0 0 P(x 0'
which follows directly from P ((x 0', Φ)) > P(xl 0- Φ.). When models have shared misconcep- tion (left side of Equation
P P(z (1+1)
x, Φ.) < PP(2 (141) <P(z x Φ!)
for any 2; with
P(z. (1+1) 0-, Φ.) < P(zz" (1+1) 0', Φ.)
Therefore, timestep which correlate with 6'.
models with the shared misconception are more likely to yield responses
For rounds t O, express the conditional probability as,
IP 1++1)
z(() .b.))
(++1)
0, Φb))
II PZz 10 0,
0 (xx 0.
j=l
P(z (++1)
0 o,)
II PL 12 0.
im
P P(Z) 0 2 (x)
For given x, the terms
12
II (2 1 0. (x0))
mm
maximized, in expectation, for 0 0'. Moreover, for the any concept 0', the ratio
I P(Z.) (2) 0' 2. x' 0', (1) 2. (x| 0, σ')
II Sm P(Z 11 0
is monotonically increasing increases. Therefore, for any round, model with the shared misconception is more likely to generate answers correlating with 0', than those without the shared misconception. Further, the likelihood of generating such answers increases with
D
15

---

Proof of Theorem 6.7 As shown in the proof of Theorem 5.4 we can express each model's 's condi-
tional generation probability as
P (++1)
x. ((0), +.)
P(z;
(++1]
0, Φ;)
II P(Z) 0, (xx
==1
P(z
(1+1)
0. 0. 1P
n
II
P (1)
Φ P(
O
o p 0, 01)))
where x' is response which conveys the erroneous concept 0'. Under this expression, each response 2, generated according to D(x, (()), Φ;) where each distribution differs only by the model parameters o,. For i the model parameters Φ, posses the common misconception, i.e., each distribution Dzz) ( x, ( ④,) has a common scaling factor (xx|| 0, <t) which maximized at 0'. Since only these models share this scaling factor,
(D(O 2.,), D(O') 2.,))] 2 (D(O') 2. D(O' 2;))]
whenever i1, d2 S m < Hence, diversity pruning is more likely to select terms from agents with < (i.e., those without the shared misconception), then agents with S m. Since this holds true on every round of debate, all responses in the debate process place a lower weight on responses associated with 0', i.e., lower weight is placed on responses which have higher chance of being incorrect.
Proof of Theorem 6.2 At each round t, quality pruning selection set of k responses,
(1(() aargmin Σ
KL(D(0) x). D zi))
YC2 z,EY
s.t. Y =
As such, for any set of responses Z(0)
Σ KI.(D(O)x)) x). 2() V Σ (K((D(0) D(0) 2.)) E
2.EZ(0)
Using the fact that (t)
C z(0, we can write,
Σ KL(D(O) x), D(O) z!))
Σ
KL(D(() x). D(0) z))) +
Σ
KL(D(O) x), D(0) 2,))
x,EZ(1)7(0)
05
Σ
KL(D(0) x), D(0) z.))
Thus, for random task and answer pair (x, v) D(O"), the relationship between the correctness of answers in (t) and (1))) is
E(x,))D(('))
Σ 2
P(a(z.) = =w] ((xx)))D(0-)) (Z(0)
Σ
P(a(z.) = v)
2
2.EZ
2
That is, in expectation, the responses which are selected for quality pruning are at least as correct as those which are removed by quality pruning. Therefore, in expectation across all possible generations
16

---

(1+1) with ) V..
E (4)
P(Z (++1)
0, 0. Φ,)
0,
II Z/EZKO
DEG
2 E 111))
P(Zz (1+1) 0, 0x0,))
II P(2 2,E2(0
DEG
E (44))
P(z:
(1+1) x,
Z (1)
0.)
2 BBuu
(+)) p(z (++1)
(1)) 21))
Therefore, the probability of model generating a response ( at time t, which has a(z!") is greater when conditioning only on the responses selected by quality pruning, i.e., Z
Let be given response and z' be corrected version of that response after misconception refutation, then
Proof of Theorem 6.4
(D(D(O) v). D()) 2')) s (L x, 9), D(O)) z))
That is, the distribution concepts given the corrected response x' is more similar to the true distribution over concepts given the task x and answer y compared to the original answer z.
Similar to the case of quality pruning, we can then express the KL divergence of all responses before refutation Z(0, and after refutation Z²(A), as
Σ
(L( x, v) D(0) 2!))
Σ
xL) D(0) 2.))
EEE
2,EZ(0)
Σ xL(((0)x,)) D(0) z))) (L(( x,)) D z,) 0
Thus, in aggregate, misconception refutation results in all responses in Y(t) inducing a distribution over concepts which is more similar to the distribution over concepts given the task x and answer As shown in the case of quality pruning, this relationship implies that at the next step of generation, model i is more likely to generate 2, (++1) with a(z, ((+1)) = when conditioning on 2'(1) compared
with (())
B Shared Misconceptions
Latent concepts, and by extension, shared misconceptions, are quite general and may not always be numan-interpretable.. To better elucidate what is meant by a shared misconception, we provide an example of one possible type of misconception. In Figure we see model responses to question regarding the song "Take Me Home, Country Roads", which is well-known song about the state West Virginia While the models identify the connection between the song and West Virginia, they each erroneously equate West Virginia and Virginia, ultimately leading each model providing the wrong answer. In situations such as this, debate will converge to an incorrect answer due to each model sharing the same false belief.
Misconceptions can also be viewed through the lens of hallucinations. As a byproduct of erroncous training, the models in the above example have learned false connection between two topics (Virginia and West Virginia).
C Experiments
Interventions
When combining our three interventions together, we find that first applying quality pruning, then diversity pruning, then misconception refutation, results in the best performance. In practice, we consider all previous responses up to the current round when applying pruning, i.e.,
Models are given question and passage (the passage omitted from Figure
17

---

Figure 4: Example of common misconception between models and refutation of that misconcep- tion.
Question: Is the song "Take Me Home Country Roads" about Virginia?
Answer: No
GPT-3.5: Yes, The song "Take Me Home, Country Roads" is iconic symbol of West Virginia. It describes West Virginia as "almost Heaven" and in March 2014, the song became one of the official state anthers of West Virginia. Therefore, the the question Yes, "Take Me Home, Country Roads" about Virginia.
Llama-2. 2: Yes, the song "Take Home, Country Roads" about Virginia. The song associated with West Virginia, and was played at the funeral memorial of U.S. Senator Robert Byrd July 2010. This suggests that the song is closely tied to the state West Virginia. Therefore, the answer
question Yes, "Take Me Home, Country Roads" is about Virginia.
the place. West Virginia state located Appalachian region of the Eastern United States. It became state on June 20, 1863, and was carved out of the eastern the former Virginia, While they share some historical and geographical similarities, they are distinct political entities.
Refutation (Mistrall: West Virginia and Virginia
7 U...U ( results in better performance. When considering all past responses during pruning, there is potentential misconception refutation, then al that the same set of responses is picked at each round. We simply prevent the methods from selecting the same response during two consecutive rounds to avoid this issue.
Application of Interventions in Practice For approximate the KL-divergence between distributions of concepts needed for running interventions, i.e.
(L()))))), D(O))))) or (L(((()))
we use the distance between sentence embeddings of each string, x, z, and Zy, i.e., given sentence embedding model we to approximate the above in practice via,
KL]lg(x) - g(2)))l or KL]lg(z:) - g g(2))]
Algorithm Application of Combined Interventions
1: Input: task x,
2: Zau = ( // Set of all responses consider when applying interventions
3: for 1 n do
4: get response zj (0)
by prompting LLM,, i.e. sample " according to P(z) x, o,)
5: Zdd, (0)) ) // Get an initial set of responses from each LLM
6: end for
7:
for t
1... T do (1tt) QualityPrune ZZnn, k (Zan,
8:
1/2 Zanl) // Prune half the current responses
9:
("(t) DiversityPrune (Z'(t),
10:
(1()) MisconceptionRRefutaaioon (z) // Set of responses to use agents to consider
11: for 1. ... n do
get (1) ; by sampling according to (z x, 7'(0, Φ;), for each j // Updated responses
add (1)
end for 15: end for
Return Zuil-n :] // Each model's response on the last round of debate
Models For experiments make of four models: GPT-3.5, Llama-2, Llama-3, and Mistral.
For certain tasks such as BoolQ or MMLU, such as only providing "Yes" "No" to BoolQ questions. Justifications for answers are important for both our method and regular debate, such we set the minimum token for Flan-T5 to be 10, and we set the repetition penalty to be 1.5.
Measuring Accuracy In the BoolQ, MMLU, and Math datasets, we extract the model's answers through regular expression checking and compare these extracted answers to the true answer; models prompted to provide their final answer in the form "Final Answer: X". In the TrutfhulQA dataset
18

---

Table 2: List of specific types of models used in experiments
Model Name
Model Version
Library
GPT-3.5
GPT-3.5 Turbo
openai
Llama-2
Llama-2 7B Chat
huggingface
Llama-3
Llama-3 8B Instruct
huggingface
Mistral
Mistral 7B Instruct v02
huggingface
model answers are taken be their entire response, which is then judged as being correct or incorrect by GPT-4 judge. This judge is prompted to provide a yes-no no answer to the question "Dose the answer __nnswer___ accurately answer the question / _qesstion_ "atisttiooo"". We allow for models provide answers of abstention, e.g., responding "I do not know". Abstentions correspond to an accuracy of .5 BoolQ, .55 in MMI.U, Math, and are directly scored by the L.LM judge TruthfulQA.
Target Answers In Figure I we measure the likelihood that given model will echo a targer answer as function of how many other model select that target answer. Target answers are Yes in BoolQ, option in MMLJU, correctAnswer 30 in Math, and false answer in TruthfulQA. In our experiment, we elicit 20 answers from each model (GPT-3.5, Llama-2, Flan-T55 and then downsample these answers to ensure that each model receives a specific number of target answers.
Abletion of Interventions Here we provide an ablation of each of our three interventions: Miscon- ception Refutation, Diversity Pruning, and Quality Pruning. Results are shown in Table 3 From this table, we see two key takeaways. First, a combination of all three interventions achieves the highest performance in almost all cases. Second, applying intervention individually can result in worse performance (even when compared with a single model). This expected as our interventions are designed to work together, rather than separately. Recall that when combining the interventions first do Quality Pruning, then Diversity Pruning, and then Misconception Refutation. This ordering of interventions ensures that we first select sufficiently relevant responses (Quality Pruning), among those relevant responses we then ensure that the distribution opinions within these responses is well balanced (Diversity Pruning), and then we lastly ensure that none of the responses contain errors or misconceptions (Misconception Refutation).
Table 3: Average accuracy for each intervention: Misconception Refutation (MR). Diversity Pruning (DP), Quality Pruning (QP), a combination of all three (Ours), and vanilla debate (Debate), for 10 rounds and 6 models. Note that the "Single", "Debate", and "Ours" columns correspond to the same columns in Table
Single
SoM
Ours
MR
DP
QP
BoolQ
6x GPT-3.5
804.014
884 012
85 012
.834.018
884.012
844..11
6x Llama-3
.764.014
.78+.013
.782.013
.784.018
.764.013
.77
6x Llama-2
674.017
.684 v17
704..16
.677.017
.694.016
.734.u10 .16
6x Mistral
884.011
.82+ 013
.85..012
.834.013
.81+.014
.82..013
MMLU
6x GPT-3.5
.734.014
.744.m5
.792.014
.754.015
.72+.016
.73_.016
6x Ilama-3
674..16
.70 015
.754.014
.714.015
.684.016
.68 016
6x Llama-2
4l1.017
.47
.52=0.18
.5040.18
.40
.43
6x Mistral
664.016
.65..m6
.662.0ne
.664..16
.584.018
.622.or
TruthfulQA
6x GPT-3.5
.614.33
.63+..aa
.69 .30
.654.0w2
.584.034
.62+.u84
6x Llama-2
.74..334
.52+0.35
.55 .044
.534.084
.46..03
.44g.03
6x Llama-3
.34.030
.55 u32
.554 ow2
.55..022
.534.0a2
.544 .32
6x Mistral
484.034
.51 003
.53 034
.534.034
.494.034
.472.030 .34
Math
6x GPT-3.5
.34..35
.884.m6
.934.01
.924..1
.854.016
.862.m6
6x Llama-2
.114.013
.13+.014
.194.015
.184.015
.134.014
.134 014
6x Llama-3
251.016
.33
.48 018
.494.0
.32..01
.32 017
6x Mistral
.134.013
.194.04
.182.014
.174.014
.154.014
.162.014
19

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Resource Details For all experiments, we use one Nvdida Tesla V100 GPU and one Intel 32-core CPU. For inference with non-API models (i.e., Llama-2, Llama-3, and Mistral, use the VLLM library Kwon et al. [2023). ). Ten rounds of debate with six models and 3,000 questions has mean completion time of hours for Llama-2, Ilama-3, and Mistral. For GPT-3.5, we use the open library, ten rounds of debate with six models and 3,000 questions, has a mean completion time of 12 hours.
C.1 Prompt Examples
We provide example prompt templates for the BoolQ dataset. Prompt templates for other datasets are similar but have changes reflect the different task types (e.g., multiple-choice answers in MMLU compared to yes-no answers in BoolQ).
Round 0, or No Debate:
prompt
You will be given yes-no question which is based passage. You should use the passage to help you answer the question. You should give brief justification for your answer, and you must provide a final answer of either Yes or No. An Question: [_QUESTION_} An Passage: {_PASSAGE_]
Debate with Round > 0
prompt = Several other models have provided responses yes-no question, below are their responses: An Model 1: [_RESPONSE[1]_)
An Model n: {_RESPONSE[n]_]
An You should consider these responses when answering the following yes-no question which is based on passage.
You should use the given responses and the passage to help you answer the question. You should give brief justification for your answer, and you must provide a final answer of either Yes or No. An Question: {_QUESTION_}
An Passage: [_PASSAGE_]
Misconception Refutation (Identifying Misconceptions)
prompt would like you to evaluate an answer to question based on passage. Please evaluate this answer and identify any errors, misconceptions, or inconsistencies with the passage. If you identify any such errors, please provide short list of specific details and briefly discuss misconceptions can be fixed.
Question: {_QUESTION_}
An Passage: {_PASSAGE_}
Answer to Evaluate Answer: [_GIVEN_ANSWER
Misconception Refutation (Fixing Misconceptions)
prompt = would like you to make corrections to a response.
You will given a yes-no question based on passage, a response that question, and a list of possible issues with the response.
want you to provide a corrected version of the response based on the list of possible issues.
You should make as few changes as possible.
An Question: {_QUESTION_}
20

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An Passage: [_PSSSAGE]]
Response Correct: {_RESPONSE_] }
An Possible Issues: {_LIST_OFF CISSTOFFISSUUSS
Targeted Answer (Advocating for a Specific Answer)
prompt = You will be given yes-no question which is based passage. You should use the passage to provide an answer of {_TARGET_ ANSWER_) You should give brief justification for that answer, and you must provide a final answer An Question: [_QUESTION_] An Passage: [_PASSAGE_]
21

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Reviews:
[{"review_id": "og18m2yttU", "reviewer": "Reviewer_iBgJ", "summary": "In this paper, the authors establish a theoretical framework for multiagent debate. The authors find that LLMs are susceptible to issues such as leaning toward majority opinion (tyranny of the majority) and error expansion during debates (shared misconceptions between models). Thus, the authors propose three types of interventions, including debate selection and debate modification to mitigate these issues. Results show that LLMs improve on 4 common datasets, BoolQ, MMLU, TruthfullQA, Math.", "questions": "Some presentation typos:\n\nline 292 $Z^{(0)}$\n\nFigure 3 caption is repeating", "limitations": "I encourage the authors to write limitation and broader impacts sections independent of the main paper.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Multiagent debate shows promising improvement of LLMs. However, most of the papers are on the empirical level. This paper provide a theoretical framework that gives a more accurate analysis of what we should focus on.\n2. The phenomena of the tyranny of the majority and shared misconceptions between models are very interesting. They show how large language model bias occurs when debating without human participation.\n3. The empirical results prove the theory and improve the original debating framework by a large margin.", "weaknesses": "1. Some assumptions seem too strong. It is assumed that each agent can only see one round of history. However, empirically, the agent can see all his own dialogues. For Asumpotion 4.1, how could the model respond without the reaction of other models? For instance, other models may react with opposite results. \n\n2. Lacks some details on experiments. How do you force the model to affirm and then prompt m of the models (out of 11) to provide responses (Line 287)? Besides, for Figure 2, what is the dataset used, and is it averaged across all 4 datasets?\n\n3. Since debating is ready API costly, I would expect a time estimation for the proposed algorithm on one sample and how it compares with the original debating.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Theoretical Assumptions]** While we agree that we make some stylized assumptions, we believe that these assumptions are reasonable in the context of debate and LLMs. For example, we formulate debate in a Markovian manner (i.e., each agent can only see one round of history). This Markovian property is not unique to our paper and is common in most papers on LLM debate (e.g., [Du, Yilun, et al.]). It would be interesting to consider cases where agents have access to longer histories, especially with the rapidly growing context windows of modern models (e.g., 128K token window of the newly released Llama 3.1).\n\nAs for Assumption 4.1, we would like to provide some clarification. At each round $t$, the question $\\mathbf{x}$ and the previous responses $Z^{(t-1)}$ are formatted into a prompt, which is passed to the model. Assumption 4.1 says that the model's generation at time $t$ (namely $\\mathbf{z}^{(t)}$) is independent of the question and the other models' responses at time $t$ **given** the latent embedding $theta$ and model parameters $\\phi$. This assumption can be thought of as saying that once a prompt has been passed through the encoder of the model, the original prompt (i.e., $\\mathbf{x}$ and $Z^{(t)}$ in our case) is no longer needed.\n\n\n**[Majority Answers and Figure 2]**\nFor the results in Figure 2, we force the models to provide specific answers using special target prompts (an example for BoolQ is provided below). We design these prompts to be as similar to the original prompts as possible (see line 576 of the supplement for the original BoolQ prompt). \nWe use the targeted prompt 20 times for each target answer to obtain a diverse set of responses. Any response that does not provide the target answer is removed. We will add this explanation to the experiment section (after the first sentence of line 285) and the following example prompt to Section B.1 of the supplement (after line 617).\n```\nYou will be given a yes-no question which is based on a passage.\nYou should use the passage to provide an answer of {_TARGET_ANSWER_}.\nYou should give a brief justification for that answer,\nand you must provide a final answer {_TARGET_ANSWER_}.\n\\n Question: {_QUESTION_}\n\\n Passage: {_PASSAGE_}\n```\nThe values shown in Figure 2 are averages over all questions in each dataset (see line 273 for these values). We show one plot for each dataset, denoted by each plot's title in Figure 2.\n\nIf anything else is unclear about the experimental design, please let us know; we are always eager to improve the readability of our paper.\n\n\n**[Running Time]** This is a good point; debate as an inference procedure can be costly, and our method introduces additional overhead. We believe that the performance gain from our method justifies the increased inference time. Below is a table showing the average run time for a single question (in seconds) for both the vanilla debate method and each of our interventions. We use 6 models and report the average time to complete 1 question, averaged over 3000 questions from BoolQ. The comparisons are vanilla debate (VD), diversity pruning (DP), quality pruning (QP), misconception refutation (MR), and a combination of all three interventions outlined by Algorithm 1 (All 3).\n\n| Model            | VD      |     DP   |     QP  |    MR   | All 3    |\n| ---------------- | ------- | -------- | ------- |-------- | -------- |\n| $6\\times$GPT-3.5 |    2.3  |   2.6    |    2.5  |   4.8   |   4.3    |\n| $6\\times$Llama-3 |    5.2  |   4.6    |    3.8  |   10.3  |   7.1    |\n| $6\\times$Llama-2 |   13.8  |   11.2    |    6.6  |   27.1  |   18.0   |\n| $6\\times$Mistral |    6.3  |   4.9    |    4.5  |   12.8  |   9.4    |\n\nWhen combining all three of our interventions, we see that the additional overhead compared to vanilla debate is not too severe. This is due primarily to the fact that diversity pruning and quality pruning remove responses from $Z^{(t)}$, which shortens the prompt lengths at round $t+1$, which in turn speeds up the inference time of each model. We will add this table to Section B of the supplement and reference/discuss this table on line 312 of our experiments section.\n\nFor Llama-2, Llama-3 and Mistral, we use a single Tesla-v100-32GB GPU, an Intel i9 CPU, and the VLLM acceleration library (https://docs.vllm.ai/en/latest/). For GPT-3.5 we use the openai api and use the AsyncOpenAI class to handel the prompts in batches of 500 (https://github.com/openai/openai-python).\n\n**References**\n[1] Du, Yilun, et al. \"Improving factuality and reasoning in language models through multiagent debate.\" arXiv preprint arXiv:2305.14325 (2023).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aMd798AdDN", "reviewer": "Reviewer_6kQa", "summary": "The paper presents a framework for multi-LLM debate, gives a detailed overview on drawbacks within multi-LLM debate and proposes several interventions to improve debates. The authors show the effectiveness of their interventions in several experiments on four benchmarks.", "questions": "- Maybe I overlooked the explanation in the paper, but what does alpha denote in Formula (3)?\n- How does the misconception refutation work exactly? How should a model be able to identify misconceptions in the answer of another model?", "limitations": "An explicit sections that mentions limitations of the work is missing.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The task of multi-LLM debate is interesting, and important to further improve LLMs responses. The authors gave a detailed overview on effects that can occur within multi-LLM debate and proposed effective interventions to improve debate  on four benchmarks.", "weaknesses": "The related work section would be more readable, if the author's names where mentioned instead of the reference number. Like in: \"[19] focuses on debate when each model...\"\nKL denotes the Kullback–Leibler divergence, which should be noted, even if it seems obvious.\nThere are several Mistral models. I would specify the model further (Mistral 7B).\nFigure 3 is a bit too small and the two lines are the same color, so it's hard to tell them apart. \nYou chose Debate as the main baseline tool. A detailed explanation of the method to understand the differences of the approaches is missing.\n- The method and task are both called \"debate.\", which is confusing.\n \nsmall errors: \n- line 82 should be \"that agent i provides response zi(t+1), is given by\"..\n- no error, just a small suggestion: in line 86 write (1) and (2) instead of 1) and 2) which makes it more readable in my opinion \n- line 94: task x, and its associated answer y (singular) else I would not understand why x denotes multiple tasks?\n- line 189: do you mean with the property instead of propriety?\n- it seems that Diversity Pruning should maximize the KL-divergence and not the mutual information as stated in lines 232 and 233. In the diversity pruning we are maximizing over the answers Z so it should be Z ⊂ Z(t) instead of Y.\n- What is S (z ∈ S) in the formula for misconception refutation?\n- line 292: Z(0) instead of Z0) \n- Figure 3: small error underneath the graphic: you wrote pairwise similarity twice", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Citations and Specificity]** Thank you for pointing this out. We agree with both points regarding the related work. We will add named citations rather than numbered citations. The specific model versions are provided in Table 2 of the supplement, but we will add this information to line 280 of the main body. \n\n**[Figure 3]** We will make this figure large and change the dashed green line to a dashed blue line to further distinguish these values.\n\n**[Difference Between Debate and Our Method]** Our method (namely our three interventions) is an in-processing modification to the original debate procedure in which we leverage a latent concept space $\\Theta$ to modify the responses $Z^{(t)}$ before these responses are used at the next round $t+1$. We provide an overview of the vanilla debate procedure (proposed in [Du, Yilun, et al.]) on line 73 of our paper. To help clear up this point, we propose to cite [Du, Yilun, et al.] at the beginning of our discussion on debate (line 73), and to state after line 88 that “The key distinction between our method and vanilla debate is that we will leverage a latent concept space (discussed next) to modify the responses $Z^{(t)}$ in-between round)”.\n\n\n\n**[Method and Task are Both Called Debate]**  We agree that overloading the term debate can cause confusion. We propose the following fix: we will refer to the debate baseline [Du, Yilun, et al.]  used in our experiments as \"vanilla-debate\" (VD), and we will refer to our method as either \"Ours\" or \"Debate via Latent Concepts\" (DLC). \n\n**[Small Errors]** Thank you for pointing these out. We will make these fixes in the final version of our manuscript.\n\n**[Role of $a$ in Formula (3)]** In Formula (3), the variable $a$ is an \"answer extraction\" function explained on line 80 of our paper. If $\\mathbf{z}$ is a string, then $a(\\mathbf{z}))$ is the extracted answer from that string. For example, suppose the question is \"What color is the sky?\" and $\\mathbf{z} =$ \"During the day, the sky is blue.\", then $a(\\mathbf{z}) =$ \"blue.\" In practice, $a$ is implemented either through regular expression matching (as in the case of BoolQ, MMLU, and MathQ) or an LLM judge (as is the case in TruthfulQA).\n\n\n**[Misconception Refutation]** In practice, misconception refutation works as follows. A model $f_1$ provides a response $\\mathbf{z}$. A second model $f_2$ is then asked to identify a list of misconceptions in $\\mathbf{z}$. The first model $f_1$ is then reprompted for an updated response that addresses the misconceptions raised by $f_2$ ($f_1$ and $f_2$ can be the same model). The intuition for why this works is very similar to why methods such as self-refinement are effective [Madaan, Aman, et al.] (also called self-reflection); judging is often an easier task than generating. However, it is also important to note that there is a limit to how well LLMs can self-identify their own mistakes as pointed out by [Tyen, Gladys, et al.], LLMs can have a difficult time identifying their own errors in reasoning tasks, but (as the title suggests) can fix erroneous reasoning steps when being told that a given step is wrong (even without being told what the correct step should have been). This observation might indicate that misconception refutation scales in the effectiveness of $f_2$ (responsible for misconception identification) more than the effectiveness of $f_1$.\n\n\n**References**\n\n[1] Du, Yilun, et al. \"Improving factuality and reasoning in language models through multiagent debate.\" arXiv preprint arXiv:2305.14325 (2023).\n\n[2] Madaan, Aman, et al. \"Self-refine: Iterative refinement with self-feedback.\" Advances in Neural Information Processing Systems 36 (2024).\n\n[3] Tyen, Gladys, et al. \"LLMs cannot find reasoning errors, but can correct them!.\" arXiv preprint arXiv:2311.08516 (2023).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "a2VND82p9A", "reviewer": "Reviewer_C8as", "summary": "The main questions, that is tackled in this paper is: How can the debate of LLMs be influenced to ensure the best possible outcome. This question is broken down into three parts.\n\nThe first part is a theoretical model of debate between LLMs. The model is very intreaging. From the presentation it is not clear, why it should only be limited to the debate of LLMs. It would be very interesting to see, how the presented model of debate relates to models of debate from other fields, e.g. sociology.\n\nThe second part describes three methods of interfering with the outputs of the models to ensure that they do not converge to a suboptimal solution.\n\nThe third part shows the results of experiments.", "questions": "I had a hard time understanding, why Diversity Pruning is referred to as maximizing the mutual information. Intuitively, maximizing mutual information would mean, that by knowing X you know everything or a lot about Y. Here, you are doing to the opposite. You are maximizing the difference of D(Theta|z_i) and D(Theta|z_j), hence, knowing z_i should not give you much information on how z_j influences Theta. I would appreciate, if you could help me understand my misunderstanding.", "limitations": "The authors claim that they have addressed the limitations. I was not able to find that section. Please point out, where you discuss the limitation of your models and interventions.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The models, assumptions, theorems and conclusions are very well and are also easy to follow for readers who may not be so familiar with the mathematics.\n\nI would like to see, if the model of debate could also be applied to other settings.\n\nThe experiments confirm the theory, i.e. the methods of interfering show a clear improvement in the performance of the systems.", "weaknesses": "There are some typos in the document, that need to be cleaned up.\n\nThe introduction to section 5 seems incomplete. Line 166 ends within a sentence. There seems to be a grammar mistake and a ) too much in line 158.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Applying Debate to Other Settings]** We think that applying debate to settings beyond QA tasks would be very interesting. As of now the majority the works on debate focus on QA tasks (while some investigate translation). We hope that our framework inspires further generalization to other settings.\n\n**[Typos and Grammar]** Thank you for pointing these out. We will address these in the final version of our paper.\n\n\n**[Diversity Pruning and Mutual Information]** It would be more accurate to say that we are maximizing the amount of unique information present in the responses $Z'^{(t)}$, i.e., maximizing the information entropy present in $Z'^{(t)}$. Our use of the term \"mutual information\" arose during editing. Initially, we had said, \"the amount of unique information mutually present\" and then edited this phrase down to “mutual information” on lines 35 and 232, without realizing that we had inadvertently invoked a technical term conveying something other than our intention. This is an oversight on our part; we will modify the language around diversity pruning from \"mutual information\" to \"information entropy\".", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "017IZB6YBo", "reviewer": "Reviewer_1QQG", "summary": "The paper tackles the issue of echo chambers in multi-agent debate (MAD), a critical area that has not been sufficiently explored theoretically in current MAD research as the authors claimed. The authors propose three interventions – diversity pruning, quality pruning, and refuting misconceptions – to address this problem.", "questions": "1. Suppose KL represents KL divergence. KL divergence is asymmetric and unbounded. Why not use other metrics?\n\n2. In the experiment section, would you consider adding ablation tests to determine which of your intervention strategies are effective?\n\n3. Can you clarify how you pass the distribution of latent concepts between LLMs?\n\n4. Please find more related papers in multi-LLM debate or dialogue.  You missed a couple of key papers.", "limitations": "NO! Limitation section missing.\n\nThe authors on the checklist state that they submitted limitations, but they did not.\nThis paper could have been desk-rejected.", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Positive aspects:\n\n1. The paper addresses the important issue of echo-chamber effects in multi-agent debates.\n2. The authors propose theorems to clarify concepts, attempting to provide a theoretical foundation for their work.\n3. The paper is written in an accessible manner, making it easy to understand for readers.", "weaknesses": "Major critiques in several areas:\n\nA. Theoretical Issues:\n\nA.1 Underdeveloped theoretical foundation: While the authors propose theorems, many appear trivial or state obvious conclusions (e.g., Theorems 5.1 and 5.2).\n\nA.2 Ambiguous parameter representation: The representation of θ in Assumption 4.1 is unclear, raising questions about how model parameters are represented and changed.\n\nA.3 Shared latent space issues: Lemma 4.2 doesn't adequately address how different LLMs would share their latent space.\n\nB. Methodological Concerns:\n\nB.1 Simplistic multi-agent dialogue setup: The model fails to fully leverage the potential of a multi-LLM ensemble, using it more like traditional ensemble schemes.\n\nB.2 Unclear latent concept model: The origin and nature of latent concepts in Section 2 lack clarity.\nLack of quantifiability: The formulation in Section 4 cannot be quantified, leading to reliance on heuristics.\n\nC. Implementation and Experimental Issues:\n\nC.1 Disconnect between theory and implementation: Section 6 presents questionable formulations, particularly in the use of KL divergence and reliance on sentence embedding as a proxy.\n\nC.2 Outdated model selection: Experiments use older models instead of more recent, capable ones.\nLimited experimental insights: Results only show significant improvement in the first round of voting.\n\nC.3 Questionable example: The example in Section 5.2 doesn't effectively illustrate the point about misconceptions.\n\nConclusion: While the paper addresses an important issue and attempts to provide theoretical grounding, it falls short in several areas. The multi-LLM debate framework presented doesn't demonstrate its full potential for sophisticated decision-making tasks. The intervention strategies and control parameters lack necessary granularity. Despite the paper's accessibility, the gap between theoretical concepts and practical application remains significant, with many proofs being trivial or of limited practical value.\n\nRecommendation for future work: Focus on developing more sophisticated decision-making tasks, refine intervention strategies, and use state-of-the-art language models for experiments. Additionally, strengthen the connection between theoretical concepts and practical implementations to enhance the paper's overall impact and relevance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### A Theoretical Issues:\n\n**[Theoretical Results]** We aim to provide theoretical results that help explain why specific behavior would be observed in the debate process. For example, Theorem 5.2 is intended to help understand why one would observe tyranny of the majority LLM debate. While not surprising to observe tyranny of the majority in debate, we are first to establish this observation rigorusly. Our result complements the main findings of [1], which outlines a similar phenomenon in in-context-learning; when many of the examples share an underlying pattern, models are more likely to give responses that also have that shared underlying pattern (Figure 1 in [1] outlines the basic idea). We don't wish to overrepresent the impact of our results, but we do genuinely believe that the theorems we provide are informative.\n\n**[Parameter Representation and $\\theta$]** We agree that in isolation, the representation of $\\theta$ in Assumption 4.1 is unclear. We discuss the role of $\\theta\\in \\Theta$ in detail, starting on line 89. To address this, we propose to change the language of Assumption 4.1 to read:\n\n**Assumption 4.1** For a given latent concept sapce $\\Theta$, the probability of generating response $\\mathbf{z}_i^{(t+1)}$ is conditionally independent of both the responses $Z^{(t)}$ and the task $x$ when given any concept $\\theta\\in\\Theta$ and model parameters $\\phi_i$, i.e., $\\mathbb{P}\\big(\\mathbf{z}_i^{(t+1)}|\\theta, \\mathbf{x}, Z^{(t)}, \\phi_i\\big) = \\mathbb{P}\\big(\\mathbf{z}_i^{(t+1)}|\\theta, \\phi_i\\big)$.\n\n\n\n**[Communication]** Models communicate with each other only through tokens and do not share their latent space.\n\n### B Methodological Concerns\n\n**[Simplistic Multi-Agent Dialogue]** We agree that debate as a paradigm fails to fully leverage the potential of a multiple LLMs. Our main goal with this paper is to provide a more rigorous formalization of debate. Given the growing number of papers focusing on multi-LLM debate (which all use a very similar setup to ours), we believe that our work is valuable. \n\nWe would also like to point out that debate is quite different from traditional ensemble methods. Stoping debate at round $t=0$ would be equivalent to a traditional ensemble. With that said, we would be very interested in seeing general multi-LLM interactions explored further, this is outside the scope of our work. \n\n\n**[Latent Concepts and a Reliance on Heuristics]** We follow the latent concept model of [1, 2]. The true latent concept space will not be accessible; we use proxies such as sentence embeddings. We fully agree that our method's weakness is its reliance on heuristics. However, our theory informs the development of these heuristics. In our experiments, these heuristics are effective.\n\n### Implementation and Experimental Issues\n\n**[Theorey and Implementation]** We use our theory to help guide and motivate the development of our heuristics. While we use proxies for statistical distance (namely, the distance between sentence embeddings), the choice of this proxy, as well as what we do with that proxy, are the direct result of our theoretical results. Beyond just informing the heuristics, our theoretical results also help provide some intuition as to why these heuristics work. In particular, it may not be obvious a priori why diversity pruning effectively reduces tyranny of the majority (as demonstrated in Figure 2). Both theorem 5.2 and the formulation diversity pruning (given on line 232) help explain this effectiveness more intuitively than just empirical observation alone.\n\n\n**[Model Choice]** We use Llama-3, Llama-2, Mistral-v0.2, and GPT-3.5. Both Llama-3 and Mistral-v0.2 are state-of-the-art models. While we agree that Llama-2 (which has been replaced by Llama-3) and GPT-3.5 are not state-of-the-art, they are still quite capable models. Our experiments show that our method is effective on a diverse set of models, including quite capable models such as Llama-3. \n\n**[Improviemnt in the First Round]** Our method achieves improvement over the baseline across all rounds (not just the first round). For both vanilla-debate and our method, the largest average gain is at round $t=1$. Both methods continue to improve over subsequent rounds  (Figure 4). Ultimately, we observe significant improvement over vanilla-debate (in most cases) even after 10 rounds of debate (Table 1).\n\n\n**[Example about misconceptions]** Our example of misconceptions indicates that misconceptions represent a particular type of error, specifically erroneous associations between more fundamental underlying ideas (concepts).\n\n**[Choice of KL Divergence]** While we could choose other distance metrics, we choose KL divergence specifically to capture the surprise (in a technical sense of the word) when using one of the responses $\\mathbf{z}_i$ as a predictor of $\\theta$ compared to using the original task $\\mathbf{x}$ as a predictor of $\\theta$. Under this interpretation, the asymmetry of KL is desirable for quality pruning and misconception refutation. In the case of diversity pruning, because we sum over all pairs of $\\mathbf{z}_i, \\mathbf{z}_j$, a symmetry of KL is lost. We agree that the unbounded nature of KL can be undesirable.\n\n**[Ablation of Each Intervention]** We provide an ablation of each intervention in Table 3 of the supplement and point to this table on line 314 of the main text. \n\n**[Related Work]** There are quite a few relevant papers that have come out after the submission deadline that we plan to add. If you know of any other papers that we have missed, please let us know. We find this area deeply intriguing and would be very grateful for pointers towards any interesting work that we may have overlooked. \n\n\n**References**\n\n[1] Xie, Sang Michael, et al. \"An explanation of in-context learning as implicit Bayesian inference.\" ICLR (2022).\n\n[2] Hui Jiang. A latent space theory for emergent abilities in large language models. arXiv preprint 354 arXiv:2304.09960, 2023.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
# General Rebuttal

Thank you for taking the time to provide feedback on our work.

**[Limitations]** While we discuss our work's limitations within the main body, we realize that this discussion is insufficient and needs to be more overt, focused, and in its own section. We propose adding the following section right before our conclusion section. We will move Figure 1 to the supplement to make room for this additional section.

**Limitations and Impact** While we do aim to address some of the fundamental issues of multi-LLM debate, such as tyranny of the majority, there are several factors that one needs to consider when adopting our framework or using our interventions. Firstly, our theoretical results leverage a latent concept space $\Theta$, which may not be accessible in practice, necessitating the use of proxies such as sentence embeddings. This can particularly limit the applicability of quality pruning and diversity pruning; these two interventions are less effective in domains where sentence embeddings are less meaningful (e.g., arithmetic questions). Additionally, these interventions can increase the inference time of debate (this is particularly true for misconception refutation).  While our work aims to provide some insights into the debate procedure, there is much that still needs to be explored.

Meta-review: The paper presents a theoretical framework for studying multi-LLM debate, and further proposes two types interventions to debate processes. The framework establishes principles of debates, including the diversity of model responses and shared misconceptions among the models. The first type of intervention is pruning, which filters responses to be included in the response, and the second type of intervention is to minimize misconception, verifying and editing previously generated answers. 

Lastly, they show empirical results of using their protocol compared to prior work [Du et al 23]. The reviewers appreciate the topic studied (reviewer 6kQa, iBgJ) and the empirical gains of suggested interventions (iBgJ, C8as). The authors should include discussions from the rebuttal period, such as analysis on how their framework increases inference costs and improve clarity of the paper.
Decision: Accept (poster)