Recent work in the field of multi-agent systems has sought to use techniques and concepts from the field of formal methods to provide rigorous theoretical analysis and guarantees on complex systems where multiple agents strategically interact, leading to the creation of the field of equilibrium analysis, which studies equilibria concepts from the field of game theory through a complexity-theoretic lens. Multi-agent systems, however, are complex mathematical objects, and, therefore, defining them in a precise mathematical manner is non-trivial. As a result, researchers often considered more restrictive models that are easier to model but lack expressive power or simply omit critical complexity-theoretic results in their analysis. This paper addresses this problem by carefully analyzing and contrasting complexity-theoretic results in the explicit model, a mathematically precise formulation of the models commonly used in the literature, and the circuit-based model, a novel model that addresses the problems found in the literature. The utility of the circuit-based model is demonstrated through a comprehensive analysis that considers upper and lower bounds for the realizability and verification problems, the two most important decision problems in equilibrium analysis, for both models. By conducting this analysis, we see that problematic issues that are endemic to the explicit model and the equilibrium analysis literature as a whole are adequately handled by the circuit-based model.

翻译：近年来，多智能体系统领域的研究尝试采用形式化方法领域的技术与概念，为多个智能体进行策略性交互的复杂系统提供严格的理论分析与保证，由此催生了均衡分析这一新兴领域。该领域通过计算复杂性理论的视角研究博弈论中的均衡概念。然而，多智能体系统是复杂的数学对象，因此以精确的数学方式对其进行定义并非易事。这导致研究者往往采用限制性更强、更易建模但表达能力不足的模型，或在其分析中直接忽略关键的计算复杂性理论结果。本文通过系统分析与对比显式模型（文献中常用模型的数学精确定义形式）与基于电路的新型模型（针对现有文献问题提出的模型）中的计算复杂性理论结果，以解决上述问题。我们通过对两个模型中可实现性问题与验证问题（均衡分析中两个最重要的判定问题）的上界与下界进行综合分析，证明了基于电路的模型的有效性。通过此项分析，我们发现基于电路的模型能够妥善处理显式模型及整个均衡分析文献中普遍存在的关键问题。