{"id":"https://openalex.org/W1970022167","doi":"https://doi.org/10.1007/s00182-009-0179-0","title":"On ordinal equivalence of the Shapley and Banzhaf values for cooperative games","display_name":"On ordinal equivalence of the Shapley and Banzhaf values for cooperative games","publication_year":2009,"publication_date":"2009-09-15","ids":{"openalex":"https://openalex.org/W1970022167","doi":"https://doi.org/10.1007/s00182-009-0179-0","mag":"1970022167"},"language":"en","primary_location":{"id":"doi:10.1007/s00182-009-0179-0","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00182-009-0179-0","pdf_url":null,"source":{"id":"https://openalex.org/S132210109","display_name":"International Journal of Game Theory","issn_l":"0020-7276","issn":["0020-7276","1432-1270"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Game Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082609136","display_name":"Josep Freixas","orcid":"https://orcid.org/0000-0002-9033-9432"},"institutions":[{"id":"https://openalex.org/I4210111014","display_name":"Fundaci\u00f3 CTM Centre Tecnol\u00f2gic","ror":"https://ror.org/026cnzg44","country_code":"ES","type":"company","lineage":["https://openalex.org/I4210111014"]},{"id":"https://openalex.org/I9617848","display_name":"Universitat Polit\u00e8cnica de Catalunya","ror":"https://ror.org/03mb6wj31","country_code":"ES","type":"education","lineage":["https://openalex.org/I9617848"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Josep Freixas","raw_affiliation_strings":["Department of Applied Mathematics III, Engineering School of Manresa, Technical University of Catalonia, EPSEM, Av. Bases de Manresa, 61\u201373, 08242, Manresa, Spain","Technical University of Catalonia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics III, Engineering School of Manresa, Technical University of Catalonia, EPSEM, Av. Bases de Manresa, 61\u201373, 08242, Manresa, Spain","institution_ids":["https://openalex.org/I4210111014","https://openalex.org/I9617848"]},{"raw_affiliation_string":"Technical University of Catalonia","institution_ids":["https://openalex.org/I9617848"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082609136"],"corresponding_institution_ids":["https://openalex.org/I4210111014","https://openalex.org/I9617848"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":null,"fwci":7.6764,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.96723279,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"39","issue":"4","first_page":"513","last_page":"527"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9825000166893005,"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/T11031","display_name":"Game Theory and Applications","score":0.9790999889373779,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.8083053827285767},{"id":"https://openalex.org/keywords/shapley-value","display_name":"Shapley value","score":0.7734336853027344},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.7439389824867249},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.644501805305481},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.5514184832572937},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.4888426661491394},{"id":"https://openalex.org/keywords/bondareva\u2013shapley-theorem","display_name":"Bondareva\u2013Shapley theorem","score":0.45466652512550354},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.44995808601379395},{"id":"https://openalex.org/keywords/game-theory","display_name":"Game theory","score":0.38982099294662476},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3799677789211273},{"id":"https://openalex.org/keywords/combinatorial-game-theory","display_name":"Combinatorial game theory","score":0.3301214575767517},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.26883864402770996},{"id":"https://openalex.org/keywords/repeated-game","display_name":"Repeated game","score":0.21103864908218384},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18350273370742798},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.05101469159126282},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0462307333946228}],"concepts":[{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.8083053827285767},{"id":"https://openalex.org/C199022921","wikidata":"https://www.wikidata.org/wiki/Q240046","display_name":"Shapley value","level":3,"score":0.7734336853027344},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.7439389824867249},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.644501805305481},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5514184832572937},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4888426661491394},{"id":"https://openalex.org/C51074407","wikidata":"https://www.wikidata.org/wiki/Q4941364","display_name":"Bondareva\u2013Shapley theorem","level":5,"score":0.45466652512550354},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.44995808601379395},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.38982099294662476},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3799677789211273},{"id":"https://openalex.org/C102234262","wikidata":"https://www.wikidata.org/wiki/Q1320931","display_name":"Combinatorial game theory","level":4,"score":0.3301214575767517},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26883864402770996},{"id":"https://openalex.org/C202556891","wikidata":"https://www.wikidata.org/wiki/Q1584646","display_name":"Repeated game","level":3,"score":0.21103864908218384},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18350273370742798},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.05101469159126282},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0462307333946228},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s00182-009-0179-0","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00182-009-0179-0","pdf_url":null,"source":{"id":"https://openalex.org/S132210109","display_name":"International Journal of Game Theory","issn_l":"0020-7276","issn":["0020-7276","1432-1270"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Game Theory","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:spr:jogath:v:39:y:2010:i:4:p:513-527","is_oa":false,"landing_page_url":"http://hdl.handle.net/10.1007/s00182-009-0179-0","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:upcommons.upc.edu:2117/9171","is_oa":false,"landing_page_url":"https://hdl.handle.net/2117/9171","pdf_url":null,"source":{"id":"https://openalex.org/S4377196262","display_name":"UPCommons institutional repository (Universitat Polit\u00e8cnica de Catalunya)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I9617848","host_organization_name":"Universitat Polit\u00e8cnica de Catalunya","host_organization_lineage":["https://openalex.org/I9617848"],"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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1526848034","https://openalex.org/W1541826016","https://openalex.org/W1562353621","https://openalex.org/W1593575857","https://openalex.org/W1975833254","https://openalex.org/W1991070015","https://openalex.org/W1999332050","https://openalex.org/W2040217797","https://openalex.org/W2054708620","https://openalex.org/W2062270858","https://openalex.org/W2075810647","https://openalex.org/W2079982674","https://openalex.org/W2090368339","https://openalex.org/W2091481536","https://openalex.org/W2104472147","https://openalex.org/W2134063650","https://openalex.org/W2153675089","https://openalex.org/W2168931195","https://openalex.org/W4213130996","https://openalex.org/W4285719527","https://openalex.org/W4300850178"],"related_works":["https://openalex.org/W1998702299","https://openalex.org/W2117439358","https://openalex.org/W1996049950","https://openalex.org/W1964562120","https://openalex.org/W1501424052","https://openalex.org/W2006581498","https://openalex.org/W4281561852","https://openalex.org/W66826376","https://openalex.org/W1894585900","https://openalex.org/W3194882106"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":4}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2016-06-24T00:00:00"}