{"id":"https://openalex.org/W4306411865","doi":"https://doi.org/10.1007/s00291-022-00694-9","title":"Two-step Shapley-solidarity value for cooperative games with coalition structure","display_name":"Two-step Shapley-solidarity value for cooperative games with coalition structure","publication_year":2022,"publication_date":"2022-10-17","ids":{"openalex":"https://openalex.org/W4306411865","doi":"https://doi.org/10.1007/s00291-022-00694-9"},"language":"en","primary_location":{"id":"doi:10.1007/s00291-022-00694-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00291-022-00694-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00291-022-00694-9.pdf","source":{"id":"https://openalex.org/S2502551423","display_name":"OR Spectrum","issn_l":"0171-6468","issn":["0171-6468","1436-6304"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"OR Spectrum","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00291-022-00694-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5042922444","display_name":"Rong Zou","orcid":null},"institutions":[{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]},{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["CN","NL"],"is_corresponding":false,"raw_author_name":"Rong Zou","raw_affiliation_strings":["Department of Applied Mathematics, University of Twente, Enschede, 7500, AE, The Netherlands","School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China","Department of Applied Mathematics, University of Twente, Enschede, The Netherlands"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, University of Twente, Enschede, 7500, AE, The Netherlands","institution_ids":["https://openalex.org/I94624287"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","institution_ids":["https://openalex.org/I17145004"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China","institution_ids":["https://openalex.org/I17145004"]},{"raw_affiliation_string":"Department of Applied Mathematics, University of Twente, Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004218295","display_name":"Wenzhong Li","orcid":"https://orcid.org/0000-0002-9199-3655"},"institutions":[{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenzhong Li","raw_affiliation_strings":["School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","institution_ids":["https://openalex.org/I17145004"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China","institution_ids":["https://openalex.org/I17145004"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052226987","display_name":"Marc Uetz","orcid":"https://orcid.org/0000-0003-4223-2435"},"institutions":[{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Marc Uetz","raw_affiliation_strings":["Department of Applied Mathematics, University of Twente, Enschede, 7500, AE, The Netherlands","Department of Applied Mathematics, University of Twente, Enschede, The Netherlands"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, University of Twente, Enschede, 7500, AE, The Netherlands","institution_ids":["https://openalex.org/I94624287"]},{"raw_affiliation_string":"Department of Applied Mathematics, University of Twente, Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030146233","display_name":"Genjiu Xu","orcid":"https://orcid.org/0000-0002-1472-3950"},"institutions":[{"id":"https://openalex.org/I17145004","display_name":"Northwestern Polytechnical University","ror":"https://ror.org/01y0j0j86","country_code":"CN","type":"education","lineage":["https://openalex.org/I17145004"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Genjiu Xu","raw_affiliation_strings":["School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, 710072, Shaanxi, China","institution_ids":["https://openalex.org/I17145004"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an, China","institution_ids":["https://openalex.org/I17145004"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5052226987"],"corresponding_institution_ids":["https://openalex.org/I94624287"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.5469,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.80686883,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"45","issue":"1","first_page":"1","last_page":"25"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11182","display_name":"Auction Theory and Applications","score":0.9728000164031982,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10841","display_name":"Economic and Environmental Valuation","score":0.9474999904632568,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/shapley-value","display_name":"Shapley value","score":0.9603448510169983},{"id":"https://openalex.org/keywords/solidarity","display_name":"Solidarity","score":0.7490832805633545},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.6647135615348816},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.6441975831985474},{"id":"https://openalex.org/keywords/subgame-perfect-equilibrium","display_name":"Subgame perfect equilibrium","score":0.4536523222923279},{"id":"https://openalex.org/keywords/cooperative-game-theory","display_name":"Cooperative game theory","score":0.4420255720615387},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3528698682785034},{"id":"https://openalex.org/keywords/game-theory","display_name":"Game theory","score":0.32905498147010803},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3243798017501831},{"id":"https://openalex.org/keywords/political-science","display_name":"Political science","score":0.06970787048339844},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.051225364208221436}],"concepts":[{"id":"https://openalex.org/C199022921","wikidata":"https://www.wikidata.org/wiki/Q240046","display_name":"Shapley value","level":3,"score":0.9603448510169983},{"id":"https://openalex.org/C2780641677","wikidata":"https://www.wikidata.org/wiki/Q815726","display_name":"Solidarity","level":3,"score":0.7490832805633545},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.6647135615348816},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.6441975831985474},{"id":"https://openalex.org/C175983524","wikidata":"https://www.wikidata.org/wiki/Q1615834","display_name":"Subgame perfect equilibrium","level":3,"score":0.4536523222923279},{"id":"https://openalex.org/C2781416736","wikidata":"https://www.wikidata.org/wiki/Q105354050","display_name":"Cooperative game theory","level":3,"score":0.4420255720615387},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3528698682785034},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.32905498147010803},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3243798017501831},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.06970787048339844},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.051225364208221436},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s00291-022-00694-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00291-022-00694-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00291-022-00694-9.pdf","source":{"id":"https://openalex.org/S2502551423","display_name":"OR Spectrum","issn_l":"0171-6468","issn":["0171-6468","1436-6304"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"OR Spectrum","raw_type":"journal-article"},{"id":"pmh:oai:ris.utwente.nl:openaire_cris_publications/148de47a-9817-4b16-a9e0-36470148601e","is_oa":true,"landing_page_url":"https://research.utwente.nl/en/publications/148de47a-9817-4b16-a9e0-36470148601e","pdf_url":"https://ris.utwente.nl/ws/files/287936169/s00291_022_00694_9.pdf","source":{"id":"https://openalex.org/S4406922991","display_name":"University of Twente Research Information","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Zou, R, Li, W, Uetz, M & Xu, G 2023, 'Two-step Shapley-solidarity value for cooperative games with coalition structure', OR Spectrum, vol. 45, pp. 1\u201325. https://doi.org/10.1007/s00291-022-00694-9","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:RePEc:spr:orspec:v:45:y:2023:i:1:d:10.1007_s00291-022-00694-9","is_oa":false,"landing_page_url":"http://link.springer.com/10.1007/s00291-022-00694-9","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:ris.utwente.nl:publications/148de47a-9817-4b16-a9e0-36470148601e","is_oa":false,"landing_page_url":"http://www.scopus.com/inward/record.url?scp=85139933573&partnerID=8YFLogxK","pdf_url":null,"source":{"id":"https://openalex.org/S4406922991","display_name":"University of Twente Research Information","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"doi:10.1007/s00291-022-00694-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00291-022-00694-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00291-022-00694-9.pdf","source":{"id":"https://openalex.org/S2502551423","display_name":"OR Spectrum","issn_l":"0171-6468","issn":["0171-6468","1436-6304"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"OR Spectrum","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","score":0.4699999988079071,"display_name":"Peace, Justice and strong institutions"}],"awards":[{"id":"https://openalex.org/G1448560474","display_name":null,"funder_award_id":"71671140","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2402172580","display_name":null,"funder_award_id":"72071159","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4271117568","display_name":null,"funder_award_id":"202006290073","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322725","display_name":"China Scholarship Council","ror":"https://ror.org/04atp4p48"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4306411865.pdf","grobid_xml":"https://content.openalex.org/works/W4306411865.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W204900244","https://openalex.org/W954238337","https://openalex.org/W1981022005","https://openalex.org/W1983688774","https://openalex.org/W2038413652","https://openalex.org/W2057683529","https://openalex.org/W2068985771","https://openalex.org/W2081372027","https://openalex.org/W2087834145","https://openalex.org/W2093715448","https://openalex.org/W2102099143","https://openalex.org/W2104131557","https://openalex.org/W2138324664","https://openalex.org/W2152669793","https://openalex.org/W2196139862","https://openalex.org/W2584155954","https://openalex.org/W2776200353","https://openalex.org/W2974452980","https://openalex.org/W3012313914","https://openalex.org/W3013249975","https://openalex.org/W3031753297","https://openalex.org/W3123605639","https://openalex.org/W4213026308","https://openalex.org/W4238768247","https://openalex.org/W4300576158"],"related_works":["https://openalex.org/W3204498529","https://openalex.org/W2187416006","https://openalex.org/W1981630704","https://openalex.org/W2134138723","https://openalex.org/W1524149928","https://openalex.org/W1538386239","https://openalex.org/W1894585900","https://openalex.org/W2006581498","https://openalex.org/W2013832345","https://openalex.org/W1998066849"],"abstract_inverted_index":{"Abstract":[0],"This":[1],"paper":[2],"proposes":[3],"an":[4,88],"alternative":[5],"for":[6,11,92],"the":[7,29,43,55,63,70,76,80,101],"two-step":[8,50],"Shapley":[9,60],"value":[10,25,61,78,94],"cooperative":[12],"games":[13],"with":[14],"coalition":[15],"structure":[16],"that":[17,31],"has":[18],"earlier":[19],"been":[20],"proposed":[21],"by":[22],"Kamijo.":[23],"The":[24],"is":[26,66],"based":[27,41],"on":[28,42],"idea":[30],"within":[32,83],"a":[33,49,58],"union":[34],"of":[35,57,79],"players,":[36],"worth":[37],"should":[38],"be":[39],"distributed":[40,67],"solidarity":[44,77],"principle.":[45],"Specifically,":[46],"we":[47],"propose":[48],"Shapley-solidarity":[51],"value,":[52],"in":[53,62],"which":[54],"surplus":[56],"union\u2019s":[59,71],"quotient":[64],"game":[65],"equally":[68],"among":[69],"members,":[72],"and":[73,95],"players":[74],"obtain":[75],"respective":[81],"subgame":[82],"their":[84],"union.":[85],"We":[86],"provide":[87],"intuitive":[89],"procedural":[90],"characterization":[91],"this":[93],"give":[96],"three":[97],"axiomatizations":[98],"to":[99,103],"pinpoint":[100],"differences":[102],"comparable":[104],"values.":[105]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2026-03-27T14:29:43.386196","created_date":"2025-10-10T00:00:00"}