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A General Equilibrium Theory of Orchestrated AI Agent Systems

Jean-Philippe Garnier

TL;DR

It is proved, via Brouwer's theorem applied to a finitedimensional approximation V K $\subset$ H, that every system of large language model (LLM) agents operating under centralized orchestration admits at least one general equilibrium.

Abstract

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.

A General Equilibrium Theory of Orchestrated AI Agent Systems

TL;DR

It is proved, via Brouwer's theorem applied to a finitedimensional approximation V K H, that every system of large language model (LLM) agents operating under centralized orchestration admits at least one general equilibrium.

Abstract

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 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 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 H, that every such economy admits at least one general equilibrium (p * , y * , * ). 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.
Paper Structure (35 sections, 11 theorems, 19 equations)

This paper contains 35 sections, 11 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. The Problem
  3. The Approach: An Arrow--Debreu--Bewley Production Economy
  4. Contributions
  5. Relation to Existing Work
  6. The Economy of Orchestrated Agents
  7. Commodity Space and Price Space
  8. Firms: LLM Agents as Producers
  9. The Consumer: The Orchestrating Firm
  10. Equilibrium
  11. Functional Walras' Law
  12. Existence of Equilibrium
  13. The SFSL Approximation: From H to VK
  14. The Projected Economy EK
  15. State Space and Compactness
  16. ...and 20 more sections

Key Result

Theorem 3.1

Under Assumptions ass:production--ass:consumer, for all price vectors $p \in \mathcal{X}$: Equivalently, $\left\langle p,\, z(p) \right\rangle_{\mathcal{X}} = 0$: the value of aggregate functional excess demand is zero for all prices.

Theorems & Definitions (34)

  • Remark 2.1
  • Definition 2.2: Production set
  • Remark 2.3: The "as if" principle
  • Definition 2.4: Orchestrated General Equilibrium
  • Remark 2.5
  • Theorem 3.1: Functional Walras' Law
  • proof
  • Remark 3.2: Interpretation
  • Remark 3.3: Comparison with finite-dimensional Walras
  • Remark 4.1: The theoretical role of SFSL
  • ...and 24 more