C ortex D ebate: Debating Sparsely and Equally for Multi-Agent Debate (2025)

Abstract: Nowadays, single Large Language Model (LLM) struggles with critical issues such as hallucination and inadequate reasoning abilities. To mitigate these issues, Multi-Agent Debate (MAD) has emerged as an effective strategy, where LLM agents engage in in-depth debates with others on tasks. However, existing MAD methods face two major issues: (a) too lengthy input contexts, which causes LLM agents to get lost in plenty of input information and experiences performance drop; and (b) the overconfidence dilemma, where self-assured LLM agents dominate the debate, leading to low debating effectiveness. To address these limitations, we propose a novel MAD method called ”CortexDebate”. Inspired by the human brain’s tendency to establish a sparse and dynamically optimized network among cortical areas governed by white matter, CortexDebate constructs a sparse debating graph among LLM agents, where each LLM agent only debates with the ones that are helpful to it. To optimize the graph, we propose a module named McKinsey-based Debate Matter (MDM), which acts as an artificial analog to white matter. By integrating the McKinsey Trust Formula, a well-established measure of trustworthiness from sociology, MDM enables credible evaluations that guide graph optimization. The effectiveness of our CortexDebate has been well demonstrated by extensive experimental results across eight datasets from four task types.

## introduction recently, inspired by human cooperation, many multi-agent interaction methods @xcite @xcite have been proposed to further improve the reasoning results of llms. these methods aim to address critical issues faced by single llm agent, such as hallucination and poor reasoning ability. among these methods, multi-agent debate (mad) @xcite has emerged as one of the most promising strategies, as it can effectively improve the performance of llm agents through the debating process among them. although previous mad methods have achieved promising results, they still suffer from two major shortcomings. as shown in figure 1 , firstly, in these methods, each llm agent is often required to debate with all other llm agents, which causes its input context to expand significantly as the number of agents and debating rounds increase. consequently, since single llm agent usually struggles to handle lengthy input contexts @xcite , it may get lost in the vast amount of input information, leading to a significant performance drop. secondly, prior mad methods determine the debating influence of each llm agent simply according to its own confidence, which may lead to the overconfident llm agents gradually dominating the entire debating process. as a result, the potential useful information provided by other "weak" llm agents may be ignored. such unequal debate is harmful to debating effectiveness, as also confirmed by @xcite . therefore, inspired by the human cognition theory (thiebaut de @xcite , this paper proposes a new mad approach named cor-texdebate, which mimics the working mode of the human brain cortex. as revealed by (thiebaut de @xcite , given a problem, the human brain tends to establish a dynamic and sparse network among different cortical areas, and this network is gradually optimized by a specialized module named white matter. during the optimization process, the white matter focuses more on the influence between paired areas rather than the performance of a single cortical area. by treating llm agents as cortical areas in human brain, our proposed cortexdebate establishes a sparse and directed debating graph, where the nodes represent participated llm agents and the edges carry information transmission between two since three llm agents reach a consensus, the final answer is 23. llm agents. each directed graph edge is assigned a weight that reflects how much the performance of the tail llm agent is expected to be improved by debating with the head llm agent. therefore, each tail llm agent will not debate with the head llm agents which do not help improve its performance. it means that the edges with small weights in the debating graph will be removed, resulting in a sparse graph. as a result, the length of context input to such tail llm agent will also be reduced. to optimize the edge weights of the debating graph, akin to the white matter dynamically governing the optimization of sparse graph among different cortical areas in human brain, our cor-texdebate introduces a module named mckinseybased debate matter (mdm) that serves as the artificial white matter. to alleviate the overconfidence dilemma present in prior works, mdm considers both the performance of head llm agent and the performance improvement expectation of tail llm agent in deciding each edge weight. specifically, mdm innovatively introduces mckinsey trust formula @xcite to calculate edge weights, which has been widely used in sociology to evaluate the level of trustworthiness of a person through four aspects, including credibility, reliability, intimacy, and self-orientation. among them, the first two evaluate individual abilities, while the last two evaluate the collaboration effectiveness with others. therefore, this formula may suppress overconfident llm agents, and also balance individual competence with teamwork ability of llm agents in mad. the effectiveness of our cortexdebate has been well confirmed by the experiments on diverse tasks, including math, world knowledge question answering, reasoning, and long-context understanding. for instance, when compared with the state-of-theart methods, in math task, cortexdebate increases result accuracy (ra) by up to 9.00% on gsm-ic dataset and 10.00% on math dataset, respectively. in reasoning task, cortexdebate increases ra by up to 9.00% on gpqa dataset and 12.33% on arc-c dataset, respectively. besides, apart from achieving high performance, our cortexdebate significantly reduces the length of context input to each llm agent, with a maximum reduction of 70.79%. the main contributions of this paper are summarized as follows: ## related work in a mad system, each llm agent presents its viewpoint and scrutinizes the viewpoints of other llm agents across multiple rounds of debate @xcite . in summary, the existing mad methods can be categorized as two types, namely sequential debate and parallel debate. sequential debate. in these methods @xcite @xcite @xcite , llm agents generate their viewpoints in turn. each llm agent can only obtain the viewpoints of its preceding llm agents. for example, @xcite require two llms to refute each other in turn. in addition to debaters, @xcite add extra roles, such as judge and critic. the judge speaks before debaters to explain the task, and the critic speaks last to summarize debates. however, in a sequential debate system, each llm agent must wait for previous llm agents to finish reasoning before it starts. this makes debating time increase linearly with the number of llm agents, leading to low efficiency which limits the scalability. parallel debate. in these methods @xcite @xcite @xcite @xcite , all llm agents simultaneously generate their viewpoints based on the viewpoints of other llm agents in the last debating round. for example, chan et al. (2023) require llm agents to critique all answers generated in the last debating round and update its answer in each debating round simultaneously. in addition to the answers generated in the last round, @xcite also provide each llm agent with task-related information retrieved from the web. besides, some methods @xcite try to adjust the debating influence of each llm agent to improve the debating effectiveness. for example, chen et al. (2023) require each llm agent to generate the confidence score for its own answer, and then inputs the score to other llm agents along with the answer. since sequential debate systems face the lowefficiency issue mentioned above, our proposed cortexdebate follows the parallel debate framework. compared with existing parallel debating methods which require each llm agent to debate with all others in each round, our cortexdebate dynamically decides the necessary debating agents by establishing a sparse debating graph among all involved llm agents, so that the input context to each agent can be shortened. this is also in contrary to @xcite in which the debating opponents are fixed. furthermore, different from prior methods which determine the debating impact of each llm agent simply based on its own confidence, we introduce the mckinsey trust formula so that both the confidence of each llm agent and the usefulness to its debating component can be evaluated. ## preliminaries in this section, we provide the problem definition for our cortexdebate, and introduce the mckinsey trust formula which plays an important role in our proposed cortexdebate. ## methodology in this section, we introduce the overall framework of our cortexdebate. as shown in figure 2 and algorithm 1, cortexdebate operates in three phases, including initial answer generation, multi-round debate, and final answer generation. unlike existing mad methods that establish fully-connected and fixed graphs among llm agents, our cor-texdebate establishes a sparse and dynamic graph, where each llm agent selectively debates with those that can contribute to its improvement. be-sides, cortexdebate evaluates the performance of llm agents and their usefulness to their debating components, enabling credible graph optimization. ## experiments this section introduces the experimental setup, experimental results, and analysis of our experiments. ## conclusion in this paper, we propose a new mad method termed "cortexdebate" to improve the reasoning abilities of multi-agent interaction systems. specifically, our cortexdebate establishes a sparse debating graph among participating llm agents, which reduces input information burdens of llm agents. besides, by integrating the mckinsey trust formula, our proposed mdm module conducts credible evaluations to gradually optimize the debating graph, making the debating process equal, indepth, and effective. due to the above designs, our method alleviates two major issues faced by existing mad systems (i.e., too lengthy input contexts and overconfidence-caused unequal debates), and shows superior performance to various stateof-the-art mad methods on various typical tasks. in the future, we plan to continue exploring the potential of cortexdebate in large-scale debates and complex tasks (i.e., domain expert systems).