Cooperative multi-agent systems require robust mechanisms for credit assignment under uncertainty. Here we introduce a variational framework, termed the Game-Theoretic Free Energy Principle (GT-FEP), that models coalition formation through a Gibbs distribution over interacting agents. Within this framework, we derive a precision-dependent formulation of cooperative credit assignment and show that an agent's Shapley value exhibits a non-monotonic relationship with sensory precision beta, reflecting a trade-off between noisy inference and overconfident local estimation. Motivated by this observation, we propose Adaptive Precision Control (APC), an online adaptation algorithm that dynamically adjusts observation precision using local estimates of cooperative contribution. We evaluate APC on real-world Swiss roundabout trajectory datasets and on a multi-agent control task derived from the same trajectories. Across both settings, APC adapts to changing noise conditions online and achieves performance comparable to the best fixed precision without prior tuning. Our results connect variational inference, cooperative game theory, and adaptive multi-agent coordination, and suggest that precision adaptation can improve robust cooperation under uncertainty.

翻译：协同多智能体系统需要针对不确定性下的信用分配建立鲁棒机制。我们在此提出一个变分框架，称为博弈论自由能原理（Game-Theoretic Free Energy Principle, GT-FEP），该框架通过相互作用智能体上的吉布斯分布对联盟形成过程进行建模。在该框架内，我们推导出依赖于精度的协同信用分配公式，并证明智能体的沙普利值（Shapley value）与感知精度β呈现非单调关系，这反映了噪声推断与过度自信局部估计之间的权衡。受此观察启发，我们提出自适应精度控制（Adaptive Precision Control, APC）——一种在线自适应算法，该算法利用合作贡献的局部估计动态调整观测精度。我们在真实世界的瑞士环岛轨迹数据集以及基于相同轨迹的多智能体控制任务上评估了APC。在这两种场景下，APC均可在线适应变化的噪声条件，并在无需预先调参的情况下达到与最优固定精度相当的性能。我们的研究结果将变分推断、合作博弈论与自适应多智能体协调联系起来，表明精度自适应能够提升不确定性下的鲁棒协同能力。