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A General Equilibrium Theory of Orchestrated AI Agent Systems
Jean-Philippe Garnier
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
The paper establishes a general equilibrium theory for centralized orchestration of Large Language Model (LLM) agent systems. It models the system as an Arrow-Debreu production economy extended to infinite-dimensional Hilbert spaces, where LLM agents act as firms with production sets defined by their frozen weights, and an orchestrator acts as a consumer maximizing welfare. The framework proves the existence of a general equilibrium, functional Walras' law, and Pareto optimality, with convergence properties under specific contraction conditions.
Entities (10)
Relation Signals (7)
Orchestrator → modelsas → Consumer
confidence 98% · The orchestrator is the consumer, choosing a routing policy
LLM Agent → modelsas → Firm
confidence 98% · Each LLM agent is modeled as a firm whose production set
Orchestrated AI System → follows → Arrow-Debreu Framework
confidence 95% · The framework is a production economy in the sense of Arrow-Debreu
Orchestration Dynamics → constitutes → Walrasian Tâtonnement
confidence 92% · The orchestration dynamics constitute a Walrasian tâtonnement
Commodity Space → extendedby → Bewley
confidence 90% · extended to infinite-dimensional commodity spaces following Bewley (1972)
LLM Agent → hasproductionset → Hilbert Space
confidence 90% · production set Y a subset H = L 2 ([0, T ], R R )
SLO Parameters → interpretedas → Policy Rates
confidence 85% · The framework admits a DSGE interpretation with SLO parameters as policy rates
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Abstract
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 * , $\pi$ * ). 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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- Source: https://arxiv.org/abs/2602.21255v2
- Canonical: https://arxiv.org/abs/2602.21255v2
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A General Equilibrium Theory of Orchestrated AI Agent Systems Jean-Philippe Garnier BrainiaK jean-philippe.garnier@brainiak.ai (February 2026 Version 2.0 — Preprint) 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 Ya⊂ℋ=L2([0,T],ℝR)Y_a =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∈ℋAp ^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 finite-dimensional approximation VK⊂ℋV_K , that every such economy admits at least one general equilibrium (p∗,y∗,π∗)(p^*,y^*,π^*). A functional Walras’ law ∑a=1A⟨pa∗,za(p∗)⟩ℋ=0 _a=1^A p_a^*,\,z_a(p^*) _H=0 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 tâtonnement that converges globally under the contraction condition, unlike classical tâtonnement (Scarf, 1960). The framework admits a DSGE interpretation with SLO parameters as policy rates. Keywords: general equilibrium, production economy, orchestration, large language models, functional prices, Hilbert space, Walras’ law, Pareto optimality, Bewley, DSGE. MSC 2020: 91B50, 47H10, 91B02, 68T05. JEL: C62, C65, D51, D61. Contents 1 Introduction 1.1 The Problem 1.2 The Approach: An Arrow–Debreu–Bewley Production Economy 1.3 Contributions 1.4 Relation to Existing Work 2 The Economy of Orchestrated Agents 2.1 Commodity Space and Price Space 2.2 Firms: LLM Agents as Producers 2.3 The Consumer: The Orchestrating Firm 2.4 Equilibrium 3 Functional Walras’ Law 4 Existence of Equilibrium 4.1 The SFSL Approximation: From ℋH to VKV_K 4.2 The Projected Economy ℰKE_K 4.3 State Space and Compactness 4.4 Main Existence Theorem 4.5 Extension to ℋH: Bewley’s Argument 5 Welfare Theorems 5.1 First Welfare Theorem 5.2 Second Welfare Theorem 6 Uniqueness and Convergence 6.1 The Orchestration Dynamics 6.2 Contraction Condition 6.3 Sufficient Condition 7 DSGE Interpretation 7.1 DSGE Embedding 7.2 The Taylor Rule of Orchestration 8 Consolidated Statement 9 Open Questions and Research Program 9.1 Equilibrium Multiplicity and Selection 9.2 Nash Equilibrium Between Agents 9.3 Dynamic Equilibrium with Learning Agents 9.4 Asymmetric Information 9.5 Dynamic DAGs 10 Conclusion References 1 Introduction 1.1 The Problem Modern AI deployments operate as organizations: collections of specialized agents—large language models, retrieval systems, classifiers, validators—composed in directed acyclic graphs (DAGs) and coordinated by a central orchestrator. The orchestrator decides, at each step, which agent to invoke, in what order, and with what priority, subject to latency, quality, and cost constraints (service-level objectives, SLOs). This architecture is now standard in enterprise AI systems. Yet it lacks a rigorous mathematical foundation. Existing routing strategies rely on static heuristics, hand-tuned weights, or simple greedy policies. No framework addresses the fundamental questions: Does an optimal allocation of agents exist? Is it stable? Is it unique? Can it be computed efficiently? Do the orchestration dynamics converge? This paper provides that foundation. 1.2 The Approach: An Arrow–Debreu–Bewley Production Economy The central observation is that an orchestrated AI agent system is a production economy in the sense of Arrow–Debreu [1], with the commodity space extended to the Hilbert space ℋ=L2([0,T],ℝR)H=L^2([0,T],R^R) following Bewley [5]. The isomorphism is structural: Arrow–Debreu / Bewley This paper Commodity space ℝLR^L / L2L^2 ℋ=L2([0,T],ℝR)H=L^2([0,T],R^R), metric trajectory space Price vector p∈ℝLp ^L / p∈L2p∈ L^2 Functional price pa∈ℋp_a per agent Firms with production sets YjY_j LLM agents with Ya⊂ℋY_a Consumer with utility U Orchestrating firm with welfare W Technology (capital, labor) Model weights ωa _a (frozen) Budget constraint ∑a⟨pa,xa⟩ℋ≤∑aΠa(p) _a p_a,\,x_a _H≤ _a _a(p) Walrasian auctioneer Orchestrator (SFSL ++ NMT) General equilibrium Fixed point (p∗,y∗,π∗)(p^*,y^*,π^*) Walras’ law p⋅z(p)=0p· z(p)=0 ∑a⟨pa,za(p)⟩ℋ=0 _a p_a,\,z_a(p) _H=0 Brouwer (finite dim.) Brouwer in VKV_K (SFSL subspace) Bewley (1972) extension VK→ℋV_K as K→∞K→∞ The commodity space ℋ=L2([0,T],ℝR)H=L^2([0,T],R^R) is the space of square-integrable metric curves. Each LLM agent is a firm whose production set Ya⊂ℋY_a is determined by its model weights ωa _a—its frozen technological capital. The orchestrator is the consumer, with a welfare function W and a budget constraint evaluated at functional prices p∈ℋAp ^A. The routing policy over the DAG is the consumer’s demand decision. Prices are elements of ℋH itself (the Hilbert space is self-dual by Riesz’s theorem): pa(t)∈ℝRp_a(t) ^R gives the shadow value of each metric of agent a at time t. This functional pricing is the key structural difference from finite-dimensional Arrow–Debreu and from the heuristic scoring used in practice. 1.3 Contributions 1. Production economy framework (§ 2): We model the orchestrated agent system as an Arrow–Debreu production economy with commodity space ℋ=L2([0,T],ℝR)H=L^2([0,T],R^R), firms (agents) with production sets YaY_a, and a consumer (orchestrator) with budget constraint. 2. Functional Walras’ law (§ 3): We prove that the value of functional excess demand is zero for all prices: ∑a⟨pa,za(p)⟩ℋ=0 _a p_a,\,z_a(p) _H=0. This is a theorem, derived from the consumer’s budget constraint, not an axiom. 3. Existence theorem (§ 4): Via a Bewley-type finite-dimensional approximation using the SFSL projection (VK⊂ℋV_K , dimVK=K V_K=K), we apply Brouwer’s theorem to prove that every orchestrated agent economy admits at least one general equilibrium. 4. Welfare theorems (§ 5): Every equilibrium is Pareto-optimal (First Welfare Theorem). Every Pareto optimum is decentralizable as an equilibrium via SLO adjustment (Second Welfare Theorem). 5. Uniqueness and convergence (§ 6): Under a contraction condition on the orchestration dynamics, the equilibrium is unique and the closed-loop tâtonnement converges geometrically (Banach). 6. DSGE interpretation (§ 7): The system admits a Dynamic Stochastic General Equilibrium embedding, with SLO parameters as policy rates in a Taylor-rule analogue. 7. Research program (§ 9): We identify open questions—equilibrium multiplicity, Nash equilibria between strategic agents, dynamic DAGs, asymmetric information—constituting the research agenda of a new discipline. 1.4 Relation to Existing Work General equilibrium theory. The foundational existence proof is due to Arrow and Debreu [1], building on Nash [17] and von Neumann [26]. The welfare theorems are classical [9]. Bewley [5] extended the existence proof to economies with infinitely many commodities (commodity space L∞L^∞ or LpL^p), using finite-dimensional approximations and weak compactness. Mas-Colell and Zame [15] provided a comprehensive treatment of equilibrium in infinite-dimensional spaces. Our setting uses L2L^2 as the commodity space, where the Riesz representation theorem makes the price space identical to the commodity space. Dynamic macroeconomics. The DSGE framework originates with Kydland and Prescott [12] and Blanchard and Kahn [6]. Indeterminacy and endogenous cycles in multi-sector models are studied in Benhabib and Nishimura [4] and Nishimura and Yano [18]. The tâtonnement instability result is due to Scarf [21]. Our system exhibits global tâtonnement stability under the contraction condition—a property not shared by generic Arrow–Debreu economies. Human capital theory. The interpretation of learned agent profiles as human capital follows Becker [3]. The endogenous evolution of profiles is analogous to Lucas [14] on human capital accumulation in growth models. Multi-agent AI systems. The orchestration of LLM agents has been studied empirically [8, 28] but without formal economic foundations. Game-theoretic approaches to multi-agent AI [22] use Nash equilibria but do not connect to Arrow–Debreu general equilibrium or functional spaces. Functional data analysis. The embedding of agent behavior in L2([0,T],ℝR)L^2([0,T],R^R) draws on Ramsay and Silverman [19]. The finite-dimensional projection (SFSL) is related to functional principal component analysis [10]. Neural operators on functional spaces are studied in Li et al. [13]. To our knowledge, no existing work establishes a general equilibrium framework for orchestrated AI agent systems, proves existence and optimality of equilibrium in a functional commodity space, or derives a Walrasian law in L2L^2 for such systems. 2 The Economy of Orchestrated Agents 2.1 Commodity Space and Price Space Let R≥1R≥ 1 be the number of metric dimensions (e.g., latency, quality, cost, load, error rate, throughput) and let T>0T>0 be the observation horizon. The commodity space per agent is the separable Hilbert space ℋ=L2([0,T],ℝR),H=L^2 ([0,T],\,R^R ), (1) endowed with the inner product ⟨f,g⟩ℋ=∑r=1R∫0Tfr(t)gr(t)dt f,\,g _H= _r=1^R _0^Tf_r(t)\,g_r(t)\,dt. An element τ∈ℋτ is a metric trajectory: a square-integrable curve recording the agent’s performance over time. Let A≥1A≥ 1 be the number of agents. The full commodity space of the economy is =ℋA=∏a=1Aℋ,X=H^A= _a=1^AH, (2) with inner product ⟨x,y⟩=∑a=1A⟨xa,ya⟩ℋ x,\,y _X= _a=1^A x_a,\,y_a _H. By the Riesz representation theorem, the topological dual of ℋH is ℋH itself: every continuous linear functional on ℋH is represented by an inner product with some element of ℋH. Therefore, the price space is ∗==ℋAX^*=X=H^A. A price vector is p=(p1,…,pA)∈p=(p_1,…,p_A) , where pa(t)∈ℝRp_a(t) ^R is the instantaneous shadow price of agent a’s metrics at time t. The value of a commodity bundle x∈x at prices p∈p is ⟨p,x⟩=∑a=1A⟨pa,xa⟩ℋ p,\,x _X= _a=1^A p_a,\,x_a _H. Remark 2.1. In Arrow–Debreu [1], the commodity space is ℝLR^L and prices are vectors p∈ℝLp ^L. In Bewley [5], the commodity space is L∞L^∞ and prices are in (L∞)∗(L^∞)^*—the space of finitely additive measures. In L2L^2, the self-duality eliminates the gap between commodities and prices: prices are functions of the same nature as the commodities they price. 2.2 Firms: LLM Agents as Producers Each agent a∈1,…,Aa∈\1,…,A\ is a firm characterized by its technological endowment ωa∈ℝP _a ^P—the frozen model weights. Definition 2.2 (Production set). The production set of agent a is a set Ya⊂ℋY_a representing all metric trajectories that agent a can produce given its technology ωa _a. Assumption 1 (Production regularity). For each a∈1,…,Aa∈\1,…,A\: (Y1) YaY_a is nonempty, closed, and convex. (Y2) YaY_a is bounded: Ya⊂Bℋ(Ra)Y_a⊂ B_H(R_a) for some Ra>0R_a>0. (Y3) Inaction is feasible: 0∈Ya0∈ Y_a. Convexity (Y1) holds because stochastic mixtures of feasible trajectories are feasible: if agent a can produce trajectories y′y and y′y by processing different workload distributions, it can produce λy′+(1−λ)y′λ y +(1-λ)y by randomizing. Boundedness (Y2) reflects finite computational capacity. Inaction (Y3) means the agent can be idle. Given functional prices pa∈ℋp_a , agent a maximizes profit: Πa(p)=supy∈Ya⟨pa,y⟩ℋ. _a(p)= _y∈ Y_a p_a,\,y _H. (3) The supply correspondence is σa(p)=y∈Ya:⟨pa,y⟩ℋ=Πa(p) _a(p)= \y∈ Y_a: p_a,\,y _H= _a(p) \. By (Y1)–(Y2), σa(p) _a(p) is nonempty and compact for all p≠0p≠ 0. Remark 2.3 (The “as if” principle). LLM agents do not literally optimize. At equilibrium, however, the orchestrator’s optimal routing assigns each agent a workload whose resulting trajectory is profit-maximizing at the equilibrium prices. This is the Arrow–Debreu “as if” principle: agents that are optimally orchestrated behave as if they were autonomous profit-maximizers. The First and Second Welfare Theorems (§ 5) make this equivalence precise. 2.3 The Consumer: The Orchestrating Firm The orchestrating firm is the sole consumer of the economy. It chooses an allocation of agent services to maximize system welfare. Production technology (DAG). The directed acyclic graph G=(V,E)G=(V,E), |V|=A|V|=A, represents feasible agent compositions. A path p∈(G)p (G) is a sequence of agents in topological order; (G)P(G) is finite. Routing policy. A routing policy π∈Δ(())=π:(G)→[0,1]∣∑pπ(p)=1π∈ (P(G))= \π:P(G)→[0,1] _pπ(p)=1 \ is a probability distribution over DAG paths. The routing policy determines the demand for each agent’s services: da(π)∈ℋd_a(π) is the expected metric trajectory of agent a under policy π. Welfare function. The welfare function W:Δ(())→ℝW: (P(G)) is defined by W(π)=−ℒ(π)W(π)=-L(π), where ℒ(π)=ℒreg(π)+λ1ℒlat(π)+λ2ℒqual(π)L(π)=L_reg(π)+ _1L_lat(π)+ _2L_qual(π) is the aggregate SLO penalty. The weights λ1,λ2>0 _1, _2>0 are the SLO parameters. Assumption 2 (Consumer regularity). W is continuous, strictly quasi-concave on Δ(()) (P(G)), and locally non-satiated: for every feasible allocation, there exists a nearby feasible allocation with strictly higher welfare. Budget constraint. The consumer’s expenditure on agent services, evaluated at functional prices p, cannot exceed the total profit of all firms: ∑a=1A⟨pa,da(π)⟩ℋ≤∑a=1AΠa(p). _a=1^A p_a,\,d_a(π) _H\;≤\; _a=1^A _a(p). (4) This is the standard Arrow–Debreu budget constraint for a consumer who owns all firms (profit share θa=1 _a=1 for all a) and has zero endowment. By Assumption 2 (local non-satiation), the budget constraint (4) binds at optimum: ∑a=1A⟨pa,da(π∗)⟩ℋ=∑a=1AΠa(p). _a=1^A p_a,\,d_a(π^*) _H\;=\; _a=1^A _a(p). (5) SLO constraint set. C=π∈Δ(()):π[lat]≤Lmax,π[qual]≥Qmin,π[cost]≤CmaxC= \π∈ (P(G)):E_π[lat]≤ L_ ,\;E_π[qual]≥ Q_ ,\;E_π[cost]≤ C_ \ (6) is a closed convex set (intersection of half-spaces in ℝ|(G)|R^|P(G)|). 2.4 Equilibrium Definition 2.4 (Orchestrated General Equilibrium). An orchestrated general equilibrium is a triple (p∗,y∗,π∗)(p^*,y^*,π^*) with p∗∈p^* , y∗=(y1∗,…,yA∗)∈∏aYay^*=(y_1^*,…,y_A^*)∈ _aY_a, π∗∈Δ(())∩Cπ^*∈ (P(G))∩ C, satisfying: (E1) Firm optimality: For each a, ya∗∈σa(p∗)y_a^*∈ _a(p^*), i.e., ⟨pa∗,ya∗⟩ℋ=Πa(p∗) p_a^*,\,y_a^* _H= _a(p^*). (E2) Consumer optimality: π∗π^* maximizes W(π)W(π) over π∈Δ(())∩Cπ∈ (P(G))∩ C subject to the budget constraint (4). (E3) Market clearing: da(π∗)=ya∗d_a(π^*)=y_a^* for all a∈1,…,Aa∈\1,…,A\. Remark 2.5. Conditions (E1)–(E3) are the exact analogues of the Arrow–Debreu equilibrium conditions: firm profit maximization, consumer utility maximization subject to budget, and market clearing. The key structural difference is that all three conditions operate in the Hilbert space ℋH: profits, budget, and market clearing are evaluated via the inner product ⟨⋅,⋅⟩ℋ ·,\,· _H, not via a finite-dimensional dot product. 3 Functional Walras’ Law Define the functional excess demand for agent a at prices p as za(p)=da(π∗(p))−ya∗(p)∈ℋ,z_a(p)=d_a(π^*(p))-y_a^*(p)\;∈\;H, (7) where da(π∗(p))d_a(π^*(p)) is the consumer’s demand and ya∗(p)y_a^*(p) is the firm’s supply, both evaluated at prices p. The aggregate functional excess demand is z(p)=(z1(p),…,zA(p))∈z(p)=(z_1(p),…,z_A(p)) . Theorem 3.1 (Functional Walras’ Law). Under Assumptions 1–2, for all price vectors p∈p : ∑a=1A⟨pa,za(p)⟩ℋ= 0. _a=1^A p_a,\,z_a(p) _H\;=\;0. (8) Equivalently, ⟨p,z(p)⟩=0 p,\,z(p) _X=0: the value of aggregate functional excess demand is zero for all prices. Proof. By local non-satiation (Assumption 2), the consumer’s budget constraint binds: ∑a=1A⟨pa,da(π∗(p))⟩ℋ=∑a=1AΠa(p). _a=1^A p_a,\,d_a(π^*(p)) _H\;=\; _a=1^A _a(p). (5) By definition of profit (3), Πa(p)=⟨pa,ya∗(p)⟩ℋ _a(p)= p_a,\,y_a^*(p) _H for each a. Substituting: ∑a=1A⟨pa,da(π∗(p))⟩ℋ=∑a=1A⟨pa,ya∗(p)⟩ℋ. _a=1^A p_a,\,d_a(π^*(p)) _H= _a=1^A p_a,\,y_a^*(p) _H. By linearity of the inner product: ∑a=1A⟨pa,da(π∗(p))−ya∗(p)⟩ℋ=∑a=1A⟨pa,za(p)⟩ℋ=0.∎ _a=1^A p_a,\,d_a(π^*(p))-y_a^*(p) _H= _a=1^A p_a,\,z_a(p) _H=0. Remark 3.2 (Interpretation). Equation (8) is the exact functional analogue of p⋅z(p)=0p· z(p)=0 in Arrow–Debreu [27, 1]. It states that the total value of excess demand—evaluated at functional prices in ℋH—is zero for every price vector, not only at equilibrium. This is because every unit of value demanded is paid for from firm profits, and every unit of firm profit is spent by the consumer. The economy’s books balance in ℋH. Remark 3.3 (Comparison with finite-dimensional Walras). The scalar identity ∑asa⋅(da−κa)=0 _as_a·(d_a- _a)=0 (a “Walras’ law” in ℝAR^A with scalar scores sas_a) is a corollary of Theorem 3.1, obtained by projecting onto a one-dimensional subspace per agent. It is informationally weaker: it forgets the temporal and metric-level structure of prices. The functional version (8) is the primitive result; the scalar version is a shadow. 4 Existence of Equilibrium 4.1 The SFSL Approximation: From ℋH to VKV_K The commodity space ℋH is infinite-dimensional. Brouwer’s fixed-point theorem—the standard tool for proving existence of Arrow–Debreu equilibrium—requires finite dimension. We resolve this via the Bewley [5] strategy: approximate the infinite-dimensional economy by a sequence of finite-dimensional economies, prove existence in each, and identify the limit. The Streaming Functional Stats Layer (SFSL) provides the approximation. Let φpm:ℋ→ℝK _pm:H ^K be a bounded linear operator that extracts K summary statistics from a trajectory (exponential moving averages of each metric: mean, variance, trend, etc.), with K=R⋅FK=R· F where F is the number of statistics per metric. Let φpm†:ℝK→ℋ _pm :R^K be the Moore–Penrose pseudo-inverse (the minimum-norm reconstruction). Define the approximation subspace: VK=Im(φpm†)⊂ℋ,dimVK=K.V_K=Im( _pm ) , V_K=K. (9) Assumption 3 (SFSL completeness). The family of SFSL operators φpmKK≥1\ _pm_K\_K≥ 1 with increasing K satisfies ⋃K=1∞VK¯=ℋ _K=1^∞V_K=H. This holds when the SFSL statistics are based on exponential kernels with varying decay rates, since exponentials form a complete system in L2[0,T]L^2[0,T]. Remark 4.1 (The theoretical role of SFSL). SFSL is not merely a computational compression. It is the Bewley approximation scheme: the finite-dimensional subspace VKV_K plays the same role as the finite-dimensional “truncations” in Bewley’s [5] existence proof for L∞L^∞ economies. SFSL transfers the problem from Schauder’s theorem (required for infinite-dimensional spaces) to Brouwer’s theorem (applicable in VKV_K). 4.2 The Projected Economy ℰKE_K Fix K and the subspace VKV_K. The K-projected economy ℰKE_K is defined by: • Projected production sets: YaK=ΠVK(Ya)Y_a^K= _V_K(Y_a) where ΠVK:ℋ→VK _V_K:H→ V_K is the orthogonal projection. By (Y1)–(Y2), YaKY_a^K is nonempty, compact, and convex in VKV_K. • Projected prices: pK=(p1K,…,pAK)∈VKAp^K=(p_1^K,…,p_A^K)∈ V_K^A. The price space in coordinates is ℝAKR^AK. • Projected equilibrium conditions: (E1)–(E3) restricted to VKV_K, with all inner products computed in ℋH (which reduces to the ℝKR^K dot product in coordinates of VKV_K). Choose an orthonormal basis ϕ1,…,ϕK\ _1,…, _K\ of VKV_K. Every element y∈VKy∈ V_K has coordinates y^=(y^1,…,y^K)∈ℝK y=( y_1,…, y_K) ^K with y=∑ky^kϕky= _k y_k _k, and ⟨p,y⟩ℋ=p^⋅y p,\,y _H= p· y. In coordinates, ℰKE_K is a standard Arrow–Debreu economy in ℝAKR^AK with AKAK commodities. 4.3 State Space and Compactness Lemma 4.2. The state space ΩK=∏a=1AYaK×ΔpK×Δ(()) _K= _a=1^AY_a^K\;×\; _p^K\;×\; (P(G)) is a nonempty compact convex subset of ℝnR^n, n=AK+(AK−1)+|(G)|n=AK+(AK-1)+|P(G)|, where ΔpK=p∈ℝ+AK:∑a,kpa,k=1 _p^K=\p ^AK_+: _a,kp_a,k=1\ is the price simplex in coordinates. Proof. Each YaKY_a^K is a compact convex subset of ℝKR^K (projection of a closed bounded convex set onto a finite-dimensional subspace). ΔpK _p^K is a standard simplex, hence compact and convex. Δ(()) (P(G)) is a standard simplex, hence compact and convex. The product of compact convex sets is compact and convex. ∎ 4.4 Main Existence Theorem Define the equilibrium map ΦK:ΩK→ΩK _K: _K→ _K by: ΦK,1(y,p,π) _K,1(y,p,π) =ΠYK[y+α(d(π)−y)], = _Y^K\! [\,y+α\,(d(π)-y)\, ], (10) ΦK,2(y,p,π) _K,2(y,p,π) =normalize(max(0,p+βz(p))), =normalize\! ( (0,\,p+β\,z(p)) ), (11) ΦK,3(y,p,π) _K,3(y,p,π) =softmax(−V∗(s(p,y))/τ), =softmax\! (-V^*(s(p,y))/τ ), (12) where: • ΠYK _Y^K projects onto ∏aYaK _aY_a^K (componentwise); • d(π)=(d1(π),…,dA(π))d(π)=(d_1(π),…,d_A(π)) is the demand under policy π; • z(p)=d(π)−yz(p)=d(π)-y is excess demand; • sa(p,y)=⟨pa,ya⟩ℋs_a(p,y)= p_a,\,y_a _H is agent a’s revenue (a scalar); • V∗(s)=Bellman-DP(s,G)V^*(s)=Bellman -DP(s,G) is the vector of optimal path values; • α,β>0α,β>0 are step sizes; τ>0τ>0 is the Bellman temperature. Assumption 4 (Temperature). τ>0τ>0. Lemma 4.3. Under Assumptions 1–4, ΦK:ΩK→ΩK _K: _K→ _K is continuous and maps ΩK _K into itself. Proof. ΦK,1 _K,1: convex combination followed by projection onto a compact convex set; continuous, image in ∏aYaK _aY_a^K. ΦK,2 _K,2: excess demand z is continuous in (y,p,π)(y,p,π); max(0,⋅) (0,·) and normalizenormalize are continuous; image in ΔpK _p^K. ΦK,3 _K,3: revenue sa=⟨pa,ya⟩ℋs_a= p_a,\,y_a _H is continuous; Bellman-DP is continuous on ℝ+AR^A_+ (envelope theorem); softmaxsoftmax is continuous for τ>0τ>0; image in Δ(()) (P(G)). ∎ Theorem 4.4 (Existence of Orchestrated General Equilibrium). Under Assumptions 1–4, for every K≥1K≥ 1, the K-projected economy ℰKE_K admits at least one general equilibrium (pK∗,yK∗,πK∗)∈ΩK(p_K^*,y_K^*, _K^*)∈ _K, where pK∗∈VKA⊂ℋAp_K^*∈ V_K^A ^A. Proof. By Lemma 4.2, ΩK _K is a nonempty compact convex subset of ℝnR^n. By Lemma 4.3, ΦK:ΩK→ΩK _K: _K→ _K is continuous. By Brouwer’s fixed-point theorem [7], there exists (yK∗,pK∗,πK∗)∈ΩK(y_K^*,p_K^*, _K^*)∈ _K such that ΦK(yK∗,pK∗,πK∗)=(yK∗,pK∗,πK∗) _K(y_K^*,p_K^*, _K^*)=(y_K^*,p_K^*, _K^*). At the fixed point: • ΦK,1 _K,1: y∗=ΠYK[y∗+α(d(π∗)−y∗)]y^*= _Y^K[y^*+α(d(π^*)-y^*)] implies d(π∗)=y∗d(π^*)=y^* (market clearing, E3). • ΦK,2 _K,2: p∗=normalize(max(0,p∗+β⋅z(p∗)))p^*=normalize( (0,p^*+β· z(p^*))) and z(p∗)=0z(p^*)=0 by E3, so p∗p^* is a fixed point of the price dynamics. • ΦK,3 _K,3: π∗π^* is the soft optimal routing at revenues s∗s^*, satisfying consumer optimality (E2). Firm optimality (E1) holds because ya∗∈YaKy_a^*∈ Y_a^K and da(π∗)=ya∗d_a(π^*)=y_a^* implies the agent produces what is demanded, which at the fixed-point prices is profit-maximizing (by the welfare theorems below). ∎ Remark 4.5 (Walras’ law at equilibrium). By Theorem 3.1, ∑a⟨pa∗,za(p∗)⟩ℋ=0 _a p_a^*,\,z_a(p^*) _H=0 holds at the equilibrium prices. Since pK∗∈VK⊂ℋp_K^*∈ V_K , this is a genuine inner product in ℋH—the functional Walras’ law holds in the Hilbert space, not in a compressed finite-dimensional surrogate. 4.5 Extension to ℋH: Bewley’s Argument Theorem 4.6 (Existence in ℋH). Under Assumptions 1–3, as K→∞K→∞, the sequence of equilibria (pK∗,yK∗,πK∗)\(p_K^*,y_K^*, _K^*)\ has a subsequence converging weakly in ℋA×ℋA×ℝ|(G)|H^A×H^A×R^|P(G)| to a limit (p∗,y∗,π∗)(p^*,y^*,π^*) that is an equilibrium of the full economy ℰE on ℋH. Proof (sketch). The sequences pK∗\p_K^*\ and yK∗\y_K^*\ are bounded in ℋAH^A (by price normalization and production boundedness). By the Banach–Alaoglu theorem, bounded sequences in a Hilbert space have weakly convergent subsequences. Let (p∗,y∗)(p^*,y^*) be a weak limit point; π∗π^* converges strongly (in ℝ|(G)|R^|P(G)|, which is finite-dimensional). The equilibrium conditions (E1)–(E3) pass to the limit: firm optimality by weak lower semicontinuity of the inner product, consumer optimality by weak upper semicontinuity of W (from strict quasi-concavity), and market clearing by weak continuity of the demand operator. The functional Walras’ law ∑a⟨pa∗,za(p∗)⟩ℋ=0 _a p_a^*,\,z_a(p^*) _H=0 holds at the limit by continuity of the inner product with respect to weak convergence in one argument. Details follow Bewley [5], Theorem 1. ∎ Remark 4.7. The SFSL approximation scheme (VK↗ℋV_K ) plays the same role as the finite-dimensional truncations in Bewley’s existence proof. The functional Walras’ law holds exactly in each VKV_K and passes to the limit in ℋH. 5 Welfare Theorems 5.1 First Welfare Theorem Definition 5.1. An allocation (y,π)(y,π) Pareto-dominates (y′,π′)(y ,π ) if W(π)≥W(π′)W(π)≥ W(π ) and Πa(p∗)≥Πa′ _a(p^*)≥ _a for all a, with at least one strict inequality. An equilibrium allocation is Pareto-optimal if no feasible allocation Pareto-dominates it. Theorem 5.2 (First Welfare Theorem). Every orchestrated general equilibrium (p∗,y∗,π∗)(p^*,y^*,π^*) is Pareto-optimal. Proof. Suppose a feasible allocation (y′,π′)(y ,π ) Pareto-dominates (y∗,π∗)(y^*,π^*). If W(π′)>W(π∗)W(π )>W(π^*), this contradicts consumer optimality (E2), since π∗π^* maximizes W subject to budget. If some firm earns strictly more (⟨pa∗,ya′⟩ℋ>⟨pa∗,ya∗⟩ℋ p_a^*,\,y_a _H> p_a^*,\,y_a^* _H), this contradicts firm optimality (E1). ∎ Corollary 5.3. The orchestrator, by finding the equilibrium of the economy, allocates agent resources Pareto-optimally: no reallocation simultaneously improves system welfare and all firms’ revenues. 5.2 Second Welfare Theorem Theorem 5.4 (Second Welfare Theorem). For every Pareto-optimal allocation (y∗,π∗)(y^**,π^**), there exists a functional price vector p∗∈ℋAp^** ^A such that (p∗,y∗,π∗)(p^**,y^**,π^**) is a general equilibrium, provided the following conditions hold: (i) Each YaY_a is convex (Assumption 1). (i) W is quasi-concave (Assumption 2). Proof (sketch). By the supporting hyperplane theorem in ℋAH^A, every Pareto-optimal allocation on the boundary of the production possibility set can be supported by a continuous linear functional p∗∈(ℋA)∗=ℋAp^**∈(H^A)^*=H^A. This functional constitutes the equilibrium price. Convexity of YaY_a and quasi-concavity of W ensure that the supporting prices decentralize the allocation as a competitive equilibrium. See Debreu [9], Chapter 6, adapted to ℋH via Mas-Colell and Zame [15]. ∎ Remark 5.5. Every desirable system state is reachable as an equilibrium by adjusting SLO parameters (λ1,λ2)( _1, _2) in W. This is the orchestrator’s “monetary policy”: altering policy rates shifts the equilibrium without changing agent endowments. 6 Uniqueness and Convergence 6.1 The Orchestration Dynamics In the K-projected economy, the closed-loop dynamics implement a Walrasian tâtonnement: yn+1 y_n+1 =ΠYK[yn+α(d(πn)−yn)], = _Y^K\! [\,y_n+α\,(d( _n)-y_n)\, ], (13) sn+1 s_n+1 =(⟨pn,a,yn+1,a⟩ℋ)a=1A, = ( p_n,a,\,y_n+1,a _H )_a=1^A, pn+1 p_n+1 =K(yn+1), =A_K(y_n+1), πn+1 _n+1 =softmax(−V∗(sn+1)/τ), =softmax\! (-V^*(s_n+1)/τ ), where K:VKA→VKAA_K:V_K^A→ V_K^A is the price mechanism (in implementation: a neural operator acting on SFSL summaries). Unlike Walras’ original tâtonnement (price adjustment proportional to excess demand), the orchestration tâtonnement updates all components simultaneously: production, prices, and routing. 6.2 Contraction Condition Let ΦK:ΩK→ΩK _K: _K→ _K denote the full update map (10)–(12). Theorem 6.1 (Uniqueness and Geometric Convergence). Suppose ‖DΦK‖op≤λ<1 \|D _K \|_op≤λ<1, where DΦKD _K is the Jacobian of ΦK _K. Then: (i) The equilibrium (pK∗,yK∗,πK∗)(p_K^*,y_K^*, _K^*) is unique. (i) The tâtonnement (13) converges to (pK∗,yK∗,πK∗)(p_K^*,y_K^*, _K^*) with: ‖xn−x∗‖≤λn1−λ‖x1−x0‖, \|x_n-x^* \|\;≤\; λ^n1-λ\, \|x_1-x_0 \|, (14) where xn=(yn,pn,πn)x_n=(y_n,p_n, _n) and x∗=(y∗,p∗,π∗)x^*=(y^*,p^*,π^*). Proof. ΦK _K is a λ-contraction on the complete metric space (ΩK,∥⋅∥)( _K, \|· \|). Banach’s fixed-point theorem [2] yields uniqueness and geometric convergence. ∎ 6.3 Sufficient Condition Proposition 6.2. A sufficient condition for ‖DΦK‖op<1 \|D _K \|_op<1 is: τ>(1−α)⋅γ⋅P,τ>(1-α)· _A· P, (15) where α∈(0,1)α∈(0,1) is the production update rate, γ=‖DK‖op _A= \|DA_K \|_op is the spectral norm of the price mechanism, and P is the depth of the DAG G. Proof. The tâtonnement has a sequential structure: production update (rate 1−α1-α), then price mechanism (norm γ _A), then soft Bellman routing (sensitivity P/τP/τ). By the chain rule: ‖DΦK‖op≤(1−α)⋅γ⋅(P/τ)<1 \|D _K \|_op≤(1-α)· _A·(P/τ)<1 when (15) holds. ∎ Example 6.3. For α=0.10α=0.10, P=3P=3, γ=0.50 _A=0.50: the condition becomes τ>1.35τ>1.35. For λ=0.5λ=0.5, convergence to 99%99\% accuracy requires ≈7≈ 7 iterations. Remark 6.4 (Global vs. local stability). Classical Walrasian tâtonnement is generically unstable [21]: there exist Arrow–Debreu economies where no tâtonnement process converges to equilibrium. The orchestration tâtonnement converges globally on ΩK _K when Proposition 6.2 holds—a structural property arising from the neural computation of prices (KA_K) and the soft Bellman regularization. 7 DSGE Interpretation 7.1 DSGE Embedding Define the state vector xt=(yt,pt,πt)∈ΩKx_t=(y_t,p_t, _t)∈ _K and let εt _t denote exogenous metric shocks (load spikes, agent failures). The orchestration system is a DSGE model: xt+1=ΦK(xt,εt),[εt]=0.x_t+1= _K(x_t,\, _t), [ _t]=0. (16) Linearizing around the steady state x∗=(y∗,p∗,π∗)x^*=(y^*,p^*,π^*): Δxt+1≈DΦK(x∗)⋅Δxt+DΦK,ε(x∗)⋅εt. x_t+1≈ D _K(x^*)· x_t+D _K, (x^*)· _t. (17) The spectral radius ρ(DΦK(x∗))=λρ(D _K(x^*))=λ determines the speed of return to equilibrium after a shock. When ρ<1ρ<1 (Proposition 6.2), the Blanchard–Kahn [6] saddle-path stability condition is satisfied: the system has a unique stable manifold converging to x∗x^*. 7.2 The Taylor Rule of Orchestration In monetary DSGE models, the central bank adjusts the interest rate according to a Taylor [24] rule: it=i∗+ϕπ(πt−π∗)+ϕy(yt−y∗)i_t=i^*+ _π( _t-π^*)+ _y(y_t-y^*). In the orchestrated economy, the orchestrator adjusts SLO weights: λ1(t) _1(t) =λ1∗+ϕlat(latt−lat∗), = _1^*+ _lat\,(lat_t-lat^*), [latency targeting] (18) λ2(t) _2(t) =λ2∗+ϕqual(qual∗−qualt). = _2^*+ _qual\,(qual^*-qual_t). [quality targeting] (19) Adjusting (λ1,λ2)( _1, _2) shifts W→W→ shifts p∗→p^*→ shifts π∗π^*. This is the orchestrator’s monetary policy: steering the economy of agents toward desired aggregate outcomes without directly controlling individual agents. 8 Consolidated Statement Theorem 8.1 (General Equilibrium of Orchestrated AI Agent Systems). Let A≥1A≥ 1 LLM agents operate on a finite DAG G=(V,E)G=(V,E) under a centralized orchestrator. Assume (Y1)–(Y3) regular production sets, (A2) regular consumer preferences, (A4) τ>0τ>0. Then: (i) Existence [7]: For every SFSL approximation VKV_K, there exists at least one equilibrium (pK∗,yK∗,πK∗)∈ΩK(p_K^*,y_K^*, _K^*)∈ _K satisfying (E1)–(E3). Under (A3), the equilibria converge to an equilibrium of the full economy on ℋH as K→∞K→∞ [5]. (i) Functional Walras’ Law (Theorem 3.1): ∑a⟨pa,za(p)⟩ℋ=0 _a p_a,\,z_a(p) _H=0 for all p∈ℋAp ^A. (i) Pareto Optimality (First Welfare Theorem): Every equilibrium is Pareto-optimal. (iv) Decentralization (Second Welfare Theorem): Every Pareto optimum is attainable as an equilibrium by adjusting SLO parameters. (v) Uniqueness and Convergence [2]: If τ>(1−α)⋅γ⋅Pτ>(1-α)· _A· P, the equilibrium is unique and the tâtonnement converges geometrically at rate λ=(1−α)⋅γ⋅P/τλ=(1-α)· _A· P/τ. 9 Open Questions and Research Program The results of this paper raise several open questions, constituting the research agenda of the discipline introduced here. 9.1 Equilibrium Multiplicity and Selection Brouwer guarantees existence but not uniqueness outside the contraction condition. Multiple equilibria may arise—analogous to the indeterminacy results of Benhabib and Nishimura [4] for multi-sector growth models. Characterizing the basins of attraction and selecting among equilibria via refinement criteria (trembling-hand, forward induction) is an open problem. 9.2 Nash Equilibrium Between Agents The current framework models agents as passive (price-taking) firms. If agents have preferences (e.g., minimizing load while maximizing perceived quality), the equilibrium becomes a Nash equilibrium in a game between agents and the orchestrator. The relationship between the Brouwer fixed point and Nash equilibria—positive in Arrow–Debreu under convex preferences—is open in this setting. 9.3 Dynamic Equilibrium with Learning Agents If agent capabilities evolve over time (e.g., via fine-tuning or model updates), the production sets YaY_a become time-dependent. This is a dynamic equilibrium where the economy itself evolves—analogous to models with endogenous human capital [14, 20]. The long-run behavior, including convergence of production possibilities and potential path-dependence, is open. 9.4 Asymmetric Information The orchestrator observes metrics with noise: y~a=ya+εa y_a=y_a+ _a. This introduces a principal-agent problem [16, 11]: how to design the observation mechanism as an optimal signal extraction device, and how equilibrium prices are affected by informational asymmetries. Moral hazard (an agent deliberately degrading quality to reduce load) is a natural application of Stiglitz–Mirrlees theory [23]. 9.5 Dynamic DAGs When agents are added or removed, G changes and the equilibrium must be recomputed. Continuity of (p∗,y∗,π∗)(p^*,y^*,π^*) as a function of G—and conditions under which small perturbations of G produce small perturbations of equilibrium—is a perturbation theory problem. These questions constitute the research agenda of the emerging discipline of economic theory of orchestrated AI agent systems—an intersection of dynamic macroeconomics, functional analysis, and AI systems theory that, to our knowledge, has not previously been explored. 10 Conclusion We have established a general equilibrium theory for orchestrated AI agent systems, grounded in the Arrow–Debreu–Bewley framework with commodity space ℋ=L2([0,T],ℝR)H=L^2([0,T],R^R). The central contribution is the identification of orchestrated agent systems as production economies: LLM agents are firms with production sets in a Hilbert space; the orchestrator is a consumer with a budget constraint; prices are functional—elements of ℋH that assign shadow values to each metric of each agent at each instant. From this identification, the fundamental results of general equilibrium theory transfer to the orchestration setting: equilibrium exists (Brouwer/Bewley), the functional Walras’ law holds as a theorem (not by construction but from budget saturation), every equilibrium is Pareto-optimal, every Pareto optimum is decentralizable, and under a contraction condition the equilibrium is unique and globally stable. The practical implication is that orchestration is not an engineering approximation. It is an economy—with the same mathematical structure as market economies. Equilibrium prices exist in ℋH; they are informationally efficient (functional Walras’ law); they support Pareto-optimal allocations. The orchestrator is a Walrasian auctioneer; SLO parameters are policy rates; the closed-loop dynamics are a tâtonnement that, unlike its classical counterpart, converges globally. Broader perspective. 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