用经济学框架统一建模大模型智能体的协同调度，证明其存在稳定均衡。
A General Equilibrium Theory of Orchestrated AI Agent Systems
- 将大模型视为生产者，调度器作为消费者，在无限维空间中构建生产经济模型。
- 证明系统在预算约束下必有均衡解，且功能价格使整体效率最优。
- 适用于需要高效协同的大模型系统设计，如智能客服、自动化流程管理。
我们建立了在集中式调度下运行的大语言模型（LLM）智能体系统的广义均衡理论。该框架是Arrow-Debreu（1954）意义上的生产经济，扩展至无限维商品空间，遵循Bewley（1972）的设定。每个LLM智能体被建模为一个企业，其生产集$Y_a \subset H = L^2([0, T ], \mathbb{R}^R)$表示由冻结模型权重决定的可行度量轨迹。调度器作为消费者，选择在智能体有向无环图上的路由策略，以在功能价格$p \in H^A$下的预算约束下最大化系统福利。这些价格——商品空间希尔伯特对偶中的元素——为每个时刻每个智能体的每项指标赋予影子价值。通过在有限维近似$V_K \subset H$上应用布劳威尔不动点定理，我们证明了此类经济至少存在一个广义均衡$(p^*, y^*, \pi^*)$。功能性的瓦尔拉斯定律作为定理成立：对所有价格，功能性超额需求的价值为零，这是由消费者预算约束所导致，而非构造结果。我们进一步建立了帕累托最优性（第一福利定理）、帕累托最优的可分解性（第二福利定理），并在压缩条件下证明了唯一性与几何收敛性（巴拿赫）。调度动态构成一个瓦尔拉斯拍卖过程，在压缩条件下全局收敛，不同于经典拍卖（Scarf, 1960）。该框架可解释为具有SLO参数作为政策利率的动态随机一般均衡模型。
原文摘要 · Abstract (English)
We establish a general equilibrium theory for systems of large language model (LLM) agents operating under centralized orchestration. The framework is a production economy in the sense of Arrow-Debreu (1954), extended to infinite-dimensional commodity spaces following Bewley (1972). Each LLM agent is modeled as a firm whose production set Y a $\subset$ H = L 2 ([0, T ], R R ) represents the feasible metric trajectories determined by its frozen model weights. The orchestrator is the consumer, choosing a routing policy over the agent DAG to maximize system welfare subject to a budget constraint evaluated at functional prices p $\in$ H A . These prices-elements of the Hilbert dual of the commodity space-assign a shadow value to each metric of each agent at each instant. We prove, via Brouwer's theorem applied to a finitedimensional approximation V K $\subset$ H, that every such economy admits at least one general equilibrium (p * , y * , $π$ * ). A functional Walras' law holds as a theorem: the value of functional excess demand is zero for all prices, as a consequence of the consumer's budget constraint-not by construction. We further establish Pareto optimality (First Welfare Theorem), decentralizability of Pareto optima (Second Welfare Theorem), and uniqueness with geometric convergence under a contraction condition (Banach). The orchestration dynamics constitute a Walrasian t{â}tonnement that converges globally under the contraction condition, unlike classical t{â}tonnement (Scarf, 1960). The framework admits a DSGE interpretation with SLO parameters as policy rates.
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