[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86446-en":3,"doc-seo-86446-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86446,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Nonlinear Axiomatic Attribution for Cooperative Games","Shapley value is widely used for player attribution in cooperative games because it satisfies linearity, consistency, equal treatment, and efficiency. Inclusion AUC is commonly employed to judge ranking quality and identify positively contributing players, but Shapley value can be unreliable for this goal. The limitation stems from linearity: as a linear operator it may have a large null space containing non-negligible perturbations that remain indistinguishable. This work designs nonlinear axiomatic attribution methods inspired by the least core, yielding contributions as unique minimizer solutions that approximate utility functions, and reports improved inclusion-AUC performance over Shapley variants relaxing only efficiency.","Nonlinear Axiomatic Attribution for Cooperative Games  \nWeida Li 1 Zhuanghua Liu 1 Yaoliang Yu2,3 Bryan Kian Hsiang Low 1  \n1 Department of Computer Science, National University of Singapore, Republic of Singapore  \n2 School of Computer Science, University of Waterloo, Canada  \n3Vector Institute, Canada  \narXiv :2607 .09869v 1 [ cs .LG] 10 Jul 2026  \nAbstract  \nThe Shapley value is a widely used concept in attribution problems, as it uniquely satisfies the axioms of linearity, consistency, equal treatment, and efficiency. Often, the inclusion AUC metric is used to evaluate the quality of player rankings, in order to identify positively participating players. However, it can be established that the Shapley value is not always reliable for this purpose. The core issue lies in its linearity: the Shapley value acts as a linear operator with an excessively large null space, which is likely to contain non-negligible perturbations that remain indistinguishable to the operator.  \nTo address this limitation, we explore the design of nonlinear axiomatic attribution methods. Inspired by the least core, which is a popular nonlinear substitute for the Shapley value, we introduce a class of nonlinear attribution methods that retain the remaining necessary axioms. Each method yields a contribution vector that is the unique optimal solution to a minimization problem, which aims to approximate utility functions as faithfully as possible.  \nIn terms of the inclusion AUC metric, our experiments demonstrate the potential effectiveness of these methods compared to Shapley value variants that relax only the efficiency axiom. Our code is available at [https://github.com/watml/](https://github.com/watml/)[ ](https://github.com/watml/)nonlinear-axiom.  \n1 INTRODUCTION  \nOriginally, the concept of the Shapley value was introduced by Shapley [1953] to define a fair allocation of contributions among n players who participate in a cooperative game, represented by a utility function U:2[n] → R, where [n] := {1, 2 ,..., n}. It is considered fair because it uniquely satisfies the axioms of linearity, consistency, equal treatment,  \nand efficiency. The effectiveness of the Shapley value and its variants obtained by relaxing the efficiency axiom has been demonstrated by Jia et al. [2019b], Kwon and Zou [2022a], Li and Yu [2023], Wang and Jia [2023] in data attribution, where data are treated as players and U (S) is defined as the performance of a model trained on the subset S of available training data. One such variant, namely the Banzhaf value [Banzhaf III, 1965], was used in context attribution by Cohen-Wang et al. [2024], where U (S) represents the probability of a statement generated by a large language model when only the parts of the context text prescribed by S are used. Its popularity has also been prominently observed in feature attribution, where U (S) is the predicted value of the model when features outside the subset S are considered missing [e.g., Lundberg and Lee, 2017, Kwon and Zou, 2022b] .  \nHowever, Kumar et al. [2020], Yan and Procaccia [2021] have raised concerns about the imposed linearity axiom, as its necessity remains unclear. Meanwhile, Bilodeau et al.[2024], Wang et al. [2024] have theoretically established that the use of the Shapley value can be unreliable by exploiting linearity. Accordingly, a nonlinear alternative, the least core [Shapley, 1971], has attracted attention [Yan and Procaccia, 2021, Gemp et al., 2024] . In general, the least core may contain infinitely many contribution allocations, so we instead consider the egalitarian least core [Arin et al., 2008, Yan and Procaccia, 2021] .1 Still, the egalitarian least core is constrained by the efficiency axiom, which has been empirically shown to be unnecessary, and even unfavorable, by Kwon and Zou [2022a,b]. By relaxing only the efficiency axiom, one arrives at the family of semi-values [Dubey et al., 1981], which includes Beta Shapley values [Kwon and Zou, 2022a] and weig","cbCaiqdd0sIjKD1J","https://ap.wps.com/l/cbCaiqdd0sIjKD1J","pdf",993021,5,1,23,"English","en",105,"# Abstract\n## Background: Shapley and attribution axioms\n## Motivation: limitations of linearity for inclusion AUC\n## Proposed approach: nonlinear axiomatic attribution via least core inspiration\n## Theoretical contributions and pathological cases","[{\"question\":\"Why can the Shapley value be unreliable for maximizing inclusion AUC in cooperative games?\",\"answer\":\"Linearity makes the attribution operator vulnerable: its large null space can contain meaningful perturbations that do not change the operator output, undermining reliable ranking for inclusion AUC.\"},{\"question\":\"What is the main idea behind the proposed nonlinear axiomatic attribution methods?\",\"answer\":\"The methods relax linearity while retaining the remaining desirable axioms, inspired by the least core. Each method produces a contribution vector as the unique optimal solution of a minimization problem that best approximates utility functions.\"},{\"question\":\"How do the experiments compare nonlinear methods with Shapley-based variants?\",\"answer\":\"Experiments using inclusion AUC indicate potential effectiveness of the nonlinear methods compared with Shapley value variants that relax only the efficiency axiom.\"}]",1784211786,58,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"nonlinear-axiomatic-attribution-for-cooperative-games","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/nonlinear-axiomatic-attribution-for-cooperative-games/86446/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why can the Shapley value be unreliable for maximizing inclusion AUC in cooperative games?","Question",{"text":76,"@type":77},"Linearity makes the attribution operator vulnerable: its large null space can contain meaningful perturbations that do not change the operator output, undermining reliable ranking for inclusion AUC.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main idea behind the proposed nonlinear axiomatic attribution methods?",{"text":81,"@type":77},"The methods relax linearity while retaining the remaining desirable axioms, inspired by the least core. Each method produces a contribution vector as the unique optimal solution of a minimization problem that best approximates utility functions.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the experiments compare nonlinear methods with Shapley-based variants?",{"text":85,"@type":77},"Experiments using inclusion AUC indicate potential effectiveness of the nonlinear methods compared with Shapley value variants that relax only the efficiency axiom.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]