[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85525-en":3,"doc-seo-85525-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85525,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","A General Equilibrium Theory of Orchestrated AI Agent Systems","Establishes a general equilibrium theory for large language model (LLM) agent systems under centralized orchestration, formulated as an Arrow–Debreu production economy extended to an infinite-dimensional commodity space (Hilbert space of metric trajectories). Each LLM agent is modeled as a firm with a production set in the Hilbert space, while the orchestrator acts as the consumer selecting routing policies over an agent DAG to maximize welfare under functional-price budgets. Proves equilibrium existence, a functional Walras law, welfare theorems, decentralizability, and uniqueness with geometric convergence under contraction.","arXiv :2602 .2 1255v2 [ cs .GT] 13 Jul 2026  \nA General Equilibrium Theory of Orchestrated AI  \nAgent Systems  \nVersion 2.1 (Corrected Price Simplex and Tâtonnement Normalization)  \nJean-Philippe Garnier  \nBrainiaK  \n[jeanphi. garnier@brainiak. tech](jeanphi. garnier@brainiak. tech)  \nMarch 2026  \nAbstract  \nWe 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, extended to an infinite-dimensional commodity space following Bewley. Each LLM agent is modeled as a firm whose production set is a subset of the Hilbert space H = L2 ([0, T ], RR ) of metric trajectories. The orchestrator is the consumer, choosing a routing policy over the agent DAG to maximize system welfare under a budget constraint evaluated at functional prices.  \nThis corrected version makes one point explicit: in the finite-dimensional projected economy, the admissible price set is the positive simplex  \n∆AKp−1 = (p ∈ RK : pa,k = 1 ) ,  \nand the price update in the tâtonnement is the Euclidean projection onto that simplex after positive truncation. It is not an L2 renormalization onto a unit sphere. This clarification preserves the compact-convex geometry required by Brouwer’s theorem and aligns the numerical implementation with the economic interpretation of prices as nonnegative shadow values.  \nWe prove existence of equilibrium, a functional Walras law, Pareto optimality, decentralizability of Pareto optima, and uniqueness with geometric convergence under a contraction condition. The orchestration dynamics define a Walrasian tâtonnement on a compact convex state space. The framework also admits a DSGE interpretation in which SLO parameters act as policy rates.  \nKeywords: general equilibrium, production economy, orchestration, large language models, functional prices, Hilbert space, Walras law, Pareto optimality, Bewley, DSGE.  \nMSC 2020: 91B50, 47H10, 91B02, 68T05 .  \nJEL: C62, C65, D51, D61 .  \nContents  \n1 Introduction 2  \n1.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2  \n1.2 The approach: an Arrow–Debreu–Bewley production economy . . . . . . . . . . 3  \n1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.4 Relation to existing work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n2 The economy of orchestrated agents 4  \n2.1 Commodity space and price space .......................... 4  \n2.2 Firms: LLM agents as producers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2.3 The consumer: the orchestrating firm . . . . . . . . . . . . . . . . . . . . . . . . 5  \n2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n3 Functional Walras law 5  \n4 Existence of equilibrium 6  \n4. 1 The SFSL approximation from H to VK . . . . . . . . . . . . . . . . . . . . . . . 6  \n4.2 The projected economy EK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6  \n4.3 State space and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6  \n4.4 The corrected equilibrium map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n4.5 Extension to the full Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n5 Welfare theorems 8  \n5. 1 First welfare theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n5.2 Second welfare theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n6 Uniqueness and convergence 9  \n6.1 The corrected orchestration dynamics . . . . . . . . . . . . . . . . . . . . . . . . 9  \n6.2 Contraction condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n6.3 A sufficient condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n7 DSGE interpretation 10  \n7.1 DSGE embedding .................................... 10  \n7.2 The","cbCaitXmKWLB7rpm","https://ap.wps.com/l/cbCaitXmKWLB7rpm","pdf",352730,3,1,13,"English","en",105,"# Introduction\n## The problem\n## The approach: an Arrow–Debreu–Bewley production economy\n# The economy of orchestrated agents\n## Commodity space and price space\n## Firms: LLM agents as producers\n## The consumer: the orchestrating firm\n## Equilibrium\n# Functional Walras law\n# Existence of equilibrium\n# Welfare theorems\n# Uniqueness and convergence\n# DSGE interpretation\n## DSGE embedding\n## The Taylor rule of orchestration\n# Consolidated statement\n# Open questions and research program\n# Conclusion","[{\"question\":\"How is an orchestrated LLM agent system modeled in the paper?\",\"answer\":\"It is treated as a production economy: each LLM agent is a firm with a production set inside a Hilbert space of metric trajectories, and the orchestrator is the consumer choosing a routing policy to maximize welfare under budget constraints.\"},{\"question\":\"What is the corrected price update rule in the finite-dimensional approximation?\",\"answer\":\"Admissible prices lie in the positive simplex, and tâtonnement updates are performed by Euclidean projection onto that simplex after positive truncation, rather than performing an L2 normalization onto a unit sphere.\"},{\"question\":\"Which main equilibrium properties are proven?\",\"answer\":\"The paper proves existence of equilibrium, a functional version of Walras’ law, Pareto optimality, decentralizability of Pareto optima, and uniqueness with geometric convergence under a contraction condition. It also frames the orchestration dynamics as a Walrasian tâtonnement on a compact convex state space.\"}]",1784204177,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-general-equilibrium-theory-of-orchestrated-ai-agent-systems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-general-equilibrium-theory-of-orchestrated-ai-agent-systems/85525/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How is an orchestrated LLM agent system modeled in the paper?","Question",{"text":75,"@type":76},"It is treated as a production economy: each LLM agent is a firm with a production set inside a Hilbert space of metric trajectories, and the orchestrator is the consumer choosing a routing policy to maximize welfare under budget constraints.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the corrected price update rule in the finite-dimensional approximation?",{"text":80,"@type":76},"Admissible prices lie in the positive simplex, and tâtonnement updates are performed by Euclidean projection onto that simplex after positive truncation, rather than performing an L2 normalization onto a unit sphere.",{"name":82,"@type":73,"acceptedAnswer":83},"Which main equilibrium properties are proven?",{"text":84,"@type":76},"The paper proves existence of equilibrium, a functional version of Walras’ law, Pareto optimality, decentralizability of Pareto optima, and uniqueness with geometric convergence under a contraction condition. It also frames the orchestration dynamics as a Walrasian tâtonnement on a compact convex state space.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]