[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85702-en":3,"doc-seo-85702-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85702,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","How Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak Multi-Agent Swarm","A swarm of cheap, unreliable agents can reach a correct consensus when a small number of costly “oracle” correctors are placed well. The work models consensus on a graph where each oracle pins one node toward truth with concave strength tied to its cost, and evaluates quality via coherence. Results show the coherence remains submodular, enabling cost-benefit greedy to achieve within 1−1/e of the budget-optimal placement and yielding a budget–correctness frontier with closed forms in special cases.","arXiv :2607 .09765v 1 [ cs .AI ] 7 Jul 2026  \nHow Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak  \nMulti-Agent Swarm  \nIgor Itkin  \nIndependent Researcher, Tel Aviv, Israel  \n[ig. itkin@gmail. com](ig. itkin@gmail. com) ORCID 0009-0004-9513-8463  \nJuly 2026  \nAbstract  \nA cheap swarm of unreliable agents can be steered to a correct consensus by a few strong, expensive “oracle” correctors. We ask how much one must spend, and where to place the oracles. We model the swarm as a consensus on a graph in which each oracle pins one node toward the truth at a cost-coupled, concave strength, and measure quality by the coherence H (R) = trM(R)−1 . Our first result is that H stays submodular (each added oracle helps less than the last) even when the oracles differ in strength, so a cost-benefit greedy comes within 1 − 1/e of the best placement at any budget. Inverting the budget gives the budget–correctness frontier B ⋆ (ε), the least spend that guarantees an ε-correct consensus: closed-form on the complete graph, and a minimal oracle count k⋆ when oracles cost the same. Whether a budget then buys a few strong oracles or many medium ones is decided by one scalar, the curvature of the cost–quality law:  \ndiminishing returns favour spreading, sharply increasing returns favour concentration. Measured on the Qwen3 ladder (0 .6–32B), the law is concave for factual and math verification (replicatedon Gemma-4) but convex for emergent code tracing, so the verdict is genuinely task-dependent.  \nCode and data: [https://github.com/YehudaItkin/budgeted-oracle-placement](https://github.com/YehudaItkin/budgeted-oracle-placement).  \n1 Introduction  \nDeployed multi-agent large language model (LLM) systems increasingly pair two kinds of agent: a large pool of cheap, individually unreliable workers, and a few strong, expensive models used as verifiers or correctors. The strong models cost more but also correct harder. This raises a budgeting question that the consensus literature has not asked: given a fixed budget, how many strong correctors should one buy, and where should they be placed, to guarantee that the swarm reaches a correct answer?  \nPractice already treats agent assembly as budgeted selection: Yuan et al. cast the choice of tools and sub-agents as a knapsack over heterogeneous-cost, heterogeneous-quality components and solve it online [4] . That work optimizes an empirically measured success rate with a generic online-knapsack competitive ratio, over a static set with no interaction structure. We supply the dynamical foundation that line of work omits. The agents form a consensus loop on a graph, not a static pool, and the objective is the provable tracking error of that loop, not a measured success rate. The decision is where on the graph each oracle sits, because a node’s worth depends on its  \nposition. Finally, the oracle’s cost is coupled to its pinning strength through a law w (c), so buying more correction costs more. This coupling is what turns placement into a budgeting problem. It yields a (1 − 1/e) submodular guarantee and a budget–correctness frontier B⋆(ε), closed-form on the complete graph and greedy-computable elsewhere. Its equal-cost limit is the minimal oracle count k⋆ .  \nThe model and the placement machinery extend our companion analysis of delayed verification [1], whose homogeneous, cardinality-constrained corrector placement is the special case ci ≡ 1, wi ≡ w recovered in Corollary 1 .  \nThis paper makes five contributions.  \n• A cost-coupled corrector model (Section 3): a grounded-Laplacian swarm in which an oracle of cost c pins toward truth with concave strength w (c); correctness is the coherence bound H (R) ≤ ε, and the object of interest is the minimal budget B⋆(ε) .  \n• Submodularity under cost-coupled pins (Theorem 1): the coherence objective [10] is a classically submodular leader-selection criterion [9]; we show the error reduction ρ (R) = H (∅) − H(R) stays monotone submodular u","cbCaiqbYjjmZmeDi","https://ap.wps.com/l/cbCaiqbYjjmZmeDi","pdf",1203552,1,30,"English","en",105,"# Introduction\n## Contributions\n# Related work","[{\"question\":\"What problem does the paper address about multi-agent swarms?\",\"answer\":\"It studies how to budget for a few strong, expensive correctors so that a weak swarm of many unreliable agents still reaches a correct consensus.\"},{\"question\":\"How is the swarm modeled and how is “correctness” measured?\",\"answer\":\"The swarm forms a consensus loop on a graph where oracle correctors pin nodes toward the truth with cost-coupled concave strength. Correctness is evaluated using a coherence objective that bounds the tracking error.\"},{\"question\":\"What optimization guarantee does the paper provide for choosing oracle placements under a budget?\",\"answer\":\"Because the coherence objective stays submodular even with heterogeneous oracle strengths, a cost-benefit greedy strategy achieves a placement within 1−1/e of the best budget-feasible solution.\"}]",1784205696,76,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"how-much-does-correctness-cost-budgeted-placement-of-strong-correctors-in-a-weak-multi-agent-swarm","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/how-much-does-correctness-cost-budgeted-placement-of-strong-correctors-in-a-weak-multi-agent-swarm/85702/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"What problem does the paper address about multi-agent swarms?","Question",{"text":75,"@type":76},"It studies how to budget for a few strong, expensive correctors so that a weak swarm of many unreliable agents still reaches a correct consensus.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the swarm modeled and how is “correctness” measured?",{"text":80,"@type":76},"The swarm forms a consensus loop on a graph where oracle correctors pin nodes toward the truth with cost-coupled concave strength. Correctness is evaluated using a coherence objective that bounds the tracking error.",{"name":82,"@type":73,"acceptedAnswer":83},"What optimization guarantee does the paper provide for choosing oracle placements under a budget?",{"text":84,"@type":76},"Because the coherence objective stays submodular even with heterogeneous oracle strengths, a cost-benefit greedy strategy achieves a placement within 1−1/e of the best budget-feasible solution.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":21,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]