[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-179936-105":59,"doc-detail-179936-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","m-n-agreement-tractable-agreement","⟨M, N,ε,δ⟩-Agreement - Tractable Agreement","","This document introduces a novel agreement framework called ⟨M, N,ε,δ⟩-agreement, designed for scenarios involving multiple tasks and agents. The framework offers a comprehensive solution by satisfying M agreements for N agents, with each agreement requiring ⟨εj, δj⟩-agreement for task j. This approach is shown to be the most general, encompassing various existing agreement protocols as special cases, including Aumann (1976), Aaronson ⟨ε,δ⟩ (2005), Almost CP (Hellman and Samet 2012; Hellman 2013), CIRL (Hadfield-Menell et al. 2016), Iterated Amplification (Christiano et al. 2018), Debate (Irving et el. 2018; Cohen et al. 2023, 2025), and Tractable Agreement (Collina et al. 2025). The document details Algorithm 2, CONSTRUCTCOMMONPRIOR, and Algorithm 1, which outlines the ⟨M, N,ε,δ⟩-Agreement process. The latter involves refining partitions, constructing common priors, and running an ⟨ε,δ⟩-agreement protocol. The proposed framework is distinguished by its ability to address all considered agreement metrics, including historical consistency, bounded rationality, asymmetry, and noise, marking a significant advancement in the field of multi-agent agreement systems.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/m-n-agreement-tractable-agreement/179936/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/m-n-agreement-tractable-agreement/179936.png","ImageObject",300,407,{"name":92,"@type":93},"Oliver","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-02",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the main contribution of this document?","Question",{"text":113,"@type":114},"The document introduces and formalizes a novel ⟨M, N,ε,δ⟩-agreement framework, which is the most general formulation for multi-agent agreement systems.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What existing agreement protocols are encompassed by the new framework?",{"text":118,"@type":114},"The ⟨M, N,ε,δ⟩-agreement framework encompasses Aumann (1976), Aaronson ⟨ε,δ⟩ (2005), Almost CP, CIRL, Iterated Amplification, Debate, and Tractable Agreement.",{"name":120,"@type":111,"acceptedAnswer":121},"What are the key features of the proposed ⟨M, N,ε,δ⟩-agreement framework?",{"text":122,"@type":114},"The framework addresses all considered agreement metrics, including historical consistency, bounded rationality, asymmetry, and noise, while allowing for M agreements across N agents.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},179936,1788338803,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},8796095461610,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","|  |  |  |  |\n| --- | --- | --- | --- |\n\n| Framework | No-CPA | Approx | Multi-M | Multi-N | Hist. | Bnd. | Asym. | Noise | Upper | Lower |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| Aumann (1976) | × | × | × | × | ✓ | × | × | × | × | × |\n| Aaronson ⟨ε,δ⟩ (2005) | × | ✓ | × | ✓ | ✓ | ✓ | × | ✓ | ✓ | ✓ |\n| Almost CP (Hellman and Samet 2012; Hellman 2013) | ✓ | × | × | ✓ | ✓ | × | × | × | × | × |\n| CIRL (Hadfield-Menell et al. 2016) | × | ✓ | × | × | × | ✓ | × | ✓ | ✓ | × |\n| Iterated Amplification (Christiano et al. 2018) | ✓ | ✓ | × | × | ✓ | ✓ | × | ✓ | ✓ | × |\n| Debate (Irving et el. 2018; Cohen et al. 2023, 2025) | ✓ | × | × | × | ✓ | ✓ | × | ✓ | ✓ | × |\n| Tractable Agreement (Collina et al. 2025) | ✓ | ✓ | × | ✓ | ✓ | ✓ | × | × | ✓ | × |\n| ⟨M, N,ε,δ⟩-agreement (Ours) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |\n\n\n| Algorithm 1: ⟨M, N,ε,δ⟩-Agreement |\n| --- |\n| Require: N agents with initial partitions {Πi,j0 } for each task j ∈ [M]; protocol P; CONSTRUCTCOMMONPRIOR defined in Algorithm 2; ⟨ε,δ⟩-agreement protocol A\u003Cbr>Ensure: Agents reach ⟨εj ,δj ⟩-agreement for all M tasks\u003Cbr>1: for j ← 1 to M do\u003Cbr>2: t ← 0\u003Cbr>3: repeat\u003Cbr>4: t ← t + 1\u003Cbr>5: for all agent i ∈ [N] do\u003Cbr>6: send mi,tj via P\u003Cbr>7: Πi,tj ← REFINEPARTITION 􀀀 Πi,tj−1, m·j,t 􀀁\u003Cbr>8: end for\u003Cbr>9: CPj ← CONSTRUCTCOMMONPRIOR􀀀{Πi,tj} , 10: {τi,tj}􀀁\u003Cbr>11: until CPj  INFEASIBLE\u003Cbr>12: Condition all agents on CPj\u003Cbr>13: RUNCPAGREEMENT (A, P , CPj , fj ,εj ,δj)\u003Cbr>14: end for |","cbCaijtYMYA2HmiX","https://ap.wps.com/l/cbCaijtYMYA2HmiX","pdf",780122,21,"English","# Algorithm 1: ⟨M, N,ε,δ⟩-Agreement","[{\"question\":\"What is the main contribution of this document?\",\"answer\":\"The document introduces and formalizes a novel ⟨M, N,ε,δ⟩-agreement framework, which is the most general formulation for multi-agent agreement systems.\"},{\"question\":\"What existing agreement protocols are encompassed by the new framework?\",\"answer\":\"The ⟨M, N,ε,δ⟩-agreement framework encompasses Aumann (1976), Aaronson ⟨ε,δ⟩ (2005), Almost CP, CIRL, Iterated Amplification, Debate, and Tractable Agreement.\"},{\"question\":\"What are the key features of the proposed ⟨M, N,ε,δ⟩-agreement framework?\",\"answer\":\"The framework addresses all considered agreement metrics, including historical consistency, bounded rationality, asymmetry, and noise, while allowing for M agreements across N agents.\"}]","⟨M, N,ε,δ⟩-Agreement - Tractable Agreement | PDF",53]