[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84169-en":3,"doc-seo-84169-105":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},84169,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","What Semivalues Cannot See: The Information Content of Anonymous Marginal Values","Semivalues share a common kernel: games that all anonymous marginal value rules cannot distinguish at once, yet are nonzero from four players onward. The paper answers what useful structure this kernel contains by expressing joint semivalue information in Harsanyi-dividend coordinates as each player’s total synergy across coalition sizes, forming closed-circuit arrangements. It proves mixed-difference audits recover exactly degree-≤ d dividend-slice harmonics with closed-form dimensions, characterizes audit defeat at c≥2d+2, and shows anonymity as the binding axiom within the marginal framework. ","arXiv :2607 .070 13v 1 [ cs .GT] 8 Jul 2026  \nWhat Semivalues Cannot See: The Information Content of Anonymous Marginal Values  \nMatthew Fried∗  \nAbstract  \nThe semivalue family shares a common kernel: games invisible to every anonymous marginal value at once, nonzero from four players (Kleinberg and Weiss, 1985; Amer, Derks and Gim´enez, 2003) . Crisman and Orrison (2015) ask what useful structure this kernel carries; this paper gives a concrete answer. In Harsanyi-dividend coordinates the joint information of all semivalues is exactly each player’s total synergy at each coalition size, so the kernel is synergy arranged in closed circuits. We prove: order-≤ d mixed-difference audits recover exactly the degree-≤ d dividend-slice harmonics, with closed-form dimension at every rung; nonzero blind games fail superadditivity, monotonicity, and core existence, yet distinct convex games with identical values under every semivalue exist from four players, with exact perturbation thresholds; the positive weighted Shapley family attains full information 2n − 1, so anonymity is the binding axiom within the marginal framework; and a coalition of size c defeats every audit of order ≤ d precisely when c ≥ 2d+2, within the convex class for small perturbations. An exhaustive census at n = 5 exhibits non-isomorphic voting rules with identical values under every semivalue power index; no weighted game participates in any collision, prompting a swing-rigidity conjecture. Measured against the theory, classical cooperative games sit at 0 .90 to 1 .00 visibility to the family versus 0 .089 for a random game.  \nKeywords: semivalues, Shapley value, Harsanyi dividends, interaction indices, simple games, power indices  \nMSC Classification: 91A12 JEL Classification: C71  \n1 Introduction  \nBegin with a puzzle every practitioner knows. The Shapley value performs beautifully on the classical applications (airport landing fees, bankruptcy division, where it coincides with O’Neill’s random-arrival rule [17], and cost sharing on networks), yet feels inadequate the moment complementarities take center stage: team formation, data markets, feature attribution. The standard explanations are heuristic (computational cost, axiom debates) . This paper gives an exact one. Every anonymous semivalue is a system of totals; a game is a system of arrangements; we compute precisely which arrangements totals can express, measure the classical games against that subspace, and find most of them above 0 .90 visibility versus 0 .089 for a random game. This suggests the methodology succeeded where it was applied because where it was applied was unrepresentative, and no internal signal could ever have revealed this, because the missing part is, by the structure of the methods themselves, the part that produces no signal.  \n∗ SUNY Farmingdale. Email: [friedm1@farmingdale.edu](friedm1@farmingdale.edu)  \nThe axiomatic program of value theory, initiated by Shapley [18] and systematized by Dubey, Neyman, and Weber [11] and Weber [22], classifies solution concepts by the axioms they satisfy. This paper proposes and executes a complementary classification: by the information they extract. Each value is a linear functional of the game; each family of values spans a subspace of the dual of game space; the dimension and the identity of that subspace are computable invariants of the family. We compute them for the classical families and find the results sharp enough to reorganize how one thinks about the axioms themselves.  \nRelated work, and the division of labor. The structural core of this story is classical, and we use it rather than claim it. Kleinberg–Weiss decomposed game space under Sn and parametrizedall symmetric linear values [4–7]; Hern´andez-Lamoneda–Ju´arez–S´anchez-S´anchez [3] gave the modern dissection, from which it is a Schur-lemma consequence that symmetric linear values factor through the trivial-plus-standard constituents of each slice; and Amer–Derks–Gim´enez [1] con","cbCaiozWPeK88Rqu","https://ap.wps.com/l/cbCaiozWPeK88Rqu","pdf",369738,1,15,"English","en",105,"# Introduction\n## Axiomatic program and complementary classification\n## Related work and division of labor\n# Contributions\n## Kernel in Harsanyi coordinates\n## Price list via order-≤ d interaction functionals\n## Economic structure of the kernel\n## Cause via weighted Shapley values","[{\"question\":\"What does the paper mean by the common kernel of semivalues?\",\"answer\":\"It refers to nonzero cooperative games that remain invisible to every anonymous marginal value simultaneously, with this phenomenon becoming nontrivial starting from four players.\"},{\"question\":\"How does the paper connect semivalues to Harsanyi-dividend coordinates?\",\"answer\":\"In those coordinates, the combined information from all semivalues equals each player’s total synergy at every coalition size, so the kernel corresponds to synergy organized in closed circuits.\"},{\"question\":\"When do coalitions defeat all mixed-difference audits of order ≤ d?\",\"answer\":\"A coalition of size c defeats every audit of order ≤ d precisely when c ≥ 2d+2 (within the considered convex class under small perturbations).\"}]",1784193611,38,{"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},"what-semivalues-cannot-see-the-information-content-of-anonymous-marginal-values","",{"@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/what-semivalues-cannot-see-the-information-content-of-anonymous-marginal-values/84169/",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","2026-07-28","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 does the paper mean by the common kernel of semivalues?","Question",{"text":75,"@type":76},"It refers to nonzero cooperative games that remain invisible to every anonymous marginal value simultaneously, with this phenomenon becoming nontrivial starting from four players.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper connect semivalues to Harsanyi-dividend coordinates?",{"text":80,"@type":76},"In those coordinates, the combined information from all semivalues equals each player’s total synergy at every coalition size, so the kernel corresponds to synergy organized in closed circuits.",{"name":82,"@type":73,"acceptedAnswer":83},"When do coalitions defeat all mixed-difference audits of order ≤ d?",{"text":84,"@type":76},"A coalition of size c defeats every audit of order ≤ d precisely when c ≥ 2d+2 (within the considered convex class under small 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