Belief-Driven Multi-Agent Collaboration via Approximate Perfect Bayesian Equilibrium for Social Simulation

Weiwei Fang; Lin Li; Kaize Shi; Yu Yang; Jianwei Zhang

cs.MAMar 26, 2026

Belief-Driven Multi-Agent Collaboration via Approximate Perfect Bayesian Equilibrium for Social Simulation

Authors: Weiwei Fang, Lin Li, Kaize Shi, Yu Yang, Jianwei Zhang

Organizations: Wuhan University of Technology, Wuhan, China · University of Southern Queensland, Toowoomba, Australia · The Education University of Hong Kong, Hong Kong, China · Iwate University, Morioka, Japan

Abstract

High-fidelity social simulation is pivotal for addressing complex Web societal challenges, yet it demands agents capable of authentically replicating the dynamic spectrum of human interaction. Current LLM-based multi-agent frameworks, however, predominantly adhere to static interaction topologies, failing to capture the fluid oscillation between cooperative knowledge synthesis and competitive critical reasoning seen in real-world scenarios. This rigidity often leads to unrealistic ``groupthink'' or unproductive deadlocks, undermining the credibility of simulations for decision support. To bridge this gap, we propose BEACOF, a belief-driven adaptive collaboration framework inspired by Perfect Bayesian Equilibrium (PBE). By modeling social interaction as a dynamic game of incomplete information, BEACOF rigorously addresses the circular dependency between collaboration type selection and capability estimation. Agents iteratively refine probabilistic beliefs about peer capabilities and autonomously modulate their collaboration strategy, thereby ensuring sequentially rational decisions under uncertainty. Validated across adversarial (judicial), open-ended (social) and mixed (medical) scenarios, BEACOF prevents coordination failures and fosters robust convergence toward high-quality solutions, demonstrating superior potential for reliable social simulation. Source codes and datasets are publicly released at: https://github.com/WUT-IDEA/BEACOF.

Figures & tables

Figure 1. Comparison of three collaboration types in a judicial deliberation task. Scenario: A defendant claims self-defense after striking an attacker. The figure presents three parallel traces for the same dialogue start under different interaction regimes. Fixed cooperation: Agent B blindly agrees, overlooking the timeline where the attacker was disarmed, leading to miscarriage (overlooked excessive force). Fixed competition: Agent B adversarially rejects the premise without constructive correction, producing deadlock and no consensus. Our belief-driven dynamic switching: Agent B first switches to critical competition to highlight the excessive-force issue; Agent A then switches to cooperation to accept the critique and refine the verdict to ``imperfect self-defense.'' The figure illustrates how dynamic switching between collaboration types identifies flaws and enables a nuanced, correct judgment.

Figure 2. Overview of the Belief-driven Adaptive Collaboration Framework (BEACOF). The framework models collaboration as a dynamic game of incomplete information, applicable across diverse scenarios with specific belief dimensions (top panel). The central workflow executes as follows at round

t

: (1) Meta-Agent Coordination: The centralized Meta-Agent utilizes scenario history to generate contextual payoffs

U_{t}

and predict probability distributions over agent collaboration types. (2) Agent Strategic Action: A Participant Agent

i

, conditioned on its private profile and the Meta-Agent’s outputs, computes an approximate best response strategy

c_{i}^{*}

and generates an interaction message

m_{i}^{*}

. (3) Evaluation & Belief Update: The Meta-Agent evaluates

m_{i}^{*}

to produce a capability estimate tuple

(e_{i}^{t},\omega_{i}^{t})

. The dashed callout box on the right details the critical Gaussian belief update mechanism: other peers (e.g., Agent

j

, bottom left) refine their prior belief estimates regarding agent

i

, denoted as

\mathbf{b}_{j}^{t-1}(i)

, by integrating this new evidence

e_{i}^{t}

weighted by confidence scores and a forgetting factor

\lambda

. This cyclic process drives the evolution of beliefs and strategic adaptation. As illustrated in Figure~\ref{fig:framework}, the proposed BEACOF framework operates as a closed-loop dynamic system. The upper panel demonstrates the framework's adaptability, where belief dimensions are instantiated specifically for different scenarios (e.g., legal metrics for court debates vs. emotional metrics for daily chats). The central execution flow at round

t

involves three key phases: \begin{itemize} \itemCoordination (Center Left): The Meta-Agent first analyzes the interaction history to synthesize contextual payoffs

U_t

and predict probability distributions over collaboration types. \itemStrategic Action (Center Right): Conditioned on these global signals and its own private profile (top-right callout), Agent

i

computes an approximate best response strategy

c_i^*

to generate a message

m_i^*

. \itemBelief Evolution (Bottom Right): The cycle closes with the evaluation and update mechanism. As detailed in the dashed callout box, peer agents (e.g., Agent

j

) utilize the Meta-Agent's evaluation tuple

(e_i^t, \omega_i^t)

to refine their Gaussian belief estimates

\mathbf{b}_j^{t-1}(i)

via a confidence-weighted update rule, thereby driving continuous strategic adaptation. \end{itemize}

		Court Debate						Persona Chat			MedQA
		Legal Articles			Judgement Results
Model	Method	precision	recall	F1-score	Charge Acc	Sentence Acc	Fine Acc	Diversity	Consistency	Contradiction	MedQA Acc
	CAMEL	$\underline{43.03}{\pm 2.24}$	$\underline{8.93}{\pm 0.42}$	$\underline{14.27}{\pm 0.72}$	$\boldsymbol{83.67}{\pm 1.53}$	$43.67{\pm 0.58}$	$\boldsymbol{46.33}{\pm 1.53}$	$\underline{34.45}{\pm 0.11}$	$85.45{\pm 2.89}$	$14.55{\pm 2.89}$	$\underline{52.17}{\pm 0.76}$
	MAD	$31.08{\pm 0.58}$	$8.49{\pm 1.08}$	$13.43{\pm 1.51}$	$\underline{79.87}{\pm 1.95}$	$\underline{45.33}{\pm 1.53}$	$35.33{\pm 0.58}$	$32.97{\pm 0.73}$	$\underline{87.94}{\pm 2.65}$	$\underline{12.06}{\pm 2.65}$	$34.50{\pm 0.50}$
	ReConcile	-			-			-			$44.75{\pm 0.35}$
Llama3.1-8B-Instruct	Ours	$\boldsymbol{43.87}{\pm 0.60}$	$\boldsymbol{12.28}{\pm 0.23}$	$\boldsymbol{16.87}{\pm 0.83}$	$\boldsymbol{83.67}{\pm 1.61}$	$\boldsymbol{48.27}{\pm 6.54}$	$\underline{43.50}{\pm 5.41}$	$\boldsymbol{36.85}{\pm 1.85}$	$\boldsymbol{88.08}{\pm 0.32}$	$\boldsymbol{11.92}{\pm 0.32}$	$\boldsymbol{52.23}{\pm 1.86}$

Table 1. Main results across three evaluation scenarios. We report mean

\pm

standard deviation over five runs. Best results are in bold , second-best are underlined . For Contradiction, lower values indicate better performance. All values are in percentage.

Model	Court Debate	Persona Chat	MedQA
Llama3.1-8B-Instruct	0.772	0.404	0.190
Gemma3-12B	0.535	0.742	0.012
Qwen3-30B-A3B	0.446	0.251	0.031

Table 2. Average regret across scenarios and backbone LLMs.

Figure 3. Ablation study (Llama3.1-8B-Instruct): removing belief updates or fixing collaboration type degrades performance across scenarios. This figure presents the results of ablation studies on our adaptive multi-agent collaboration framework using the Llama3.1-8B-Instruct backbone. The left panel shows the impact of removing belief updates, while the right panel illustrates the effect of fixing collaboration types. In both cases, performance metrics across the Court Debate, MedQA, and Persona Chat scenarios are negatively affected, demonstrating the importance of both belief modeling and dynamic type switching in achieving rational collaboration outcomes.

Figure 4. The case study of dynamic collaboration type switching in resolving complex medical reasoning tasks. The framework BEACOF adaptively shifts the interaction type, guiding agents from an initial incorrect consensus to the ground truth. Initially, agents converge on an incorrect diagnosis (Option C) due to reasoning inertia. By dynamically switching the collaboration type in Round 2, the framework stimulates agent B to critically re-evaluate the prompt's constraints rather than simply agreeing. This strategic shift facilitates the discovery of the subtle second-order inference (the drug's off-label use, Option E), demonstrating how adaptive collaboration prevents groupthink and refines decision-making.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Scenario Type	Source	Samples	Key Modality	Avg. Input Length
Court Debate	Adversarial	AgentsCourt ( He et al., 2024 ) & CJO	100	Unstructured Legal Text	$\sim$ 550 tokens
MedQA	Mixed (Coop-Comp)	MedQA-USMLE ( Jin et al., 2020 )	200	Clinical Vignettes (5-Option)	$\sim$ 180 tokens
Persona Chat	Open-Ended	PersonaChat ( Zhang et al., 2018a )	100	Dialogue History	$\sim$ 320 tokens

Appendix

Table 3. Statistical summary of the datasets used in the evaluation scenarios. The “Avg. Input Length” denotes the average token count of the initial context provided to the agents.

Case Cause	Count	Case Cause	Count
Dangerous Driving	28	Traffic Casualty	12
Theft	15	Smuggling	8
Fraud	12	Illegal Business/Gambling	7
Intentional Injury	10	Others (e.g., Assault)	8
Total	100

Appendix

Table 4. Distribution of Case Causes in the Court Debate Dataset. The dataset covers a wide spectrum of criminal offenses to test adversarial robustness.

Answer Key	Count	Percentage (%)
A	43	21.5
B	40	20.0
C	40	20.0
D	49	24.5
E	28	14.0
Total	200	100.0

Appendix

Table 5. Distribution of Correct Answer Keys in the MedQA Dataset. The balanced distribution prevents agents from exploiting positional bias.

Dialogue Length (Turns)	Count	Percentage (%)
11 – 20	22	22.0
21 – 30	49	49.0
31 – 40	18	18.0
$>40$	11	11.0
Total	100	100.0

Appendix

Table 6. Distribution of Dialogue Lengths (Turns) in the Persona Chat Dataset. Long-context interactions challenge the agents’ ability to maintain consistency.

Multi-Agent Collaboration

Multi-agent collaboration studies how autonomous agents coordinate, share information, and combine capabilities to solve tasks together.

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