{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T03:48:41Z","timestamp":1777002521857,"version":"3.51.4"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"vor","delay-in-days":13,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PID-2020-112581GB-C21"],"award-info":[{"award-number":["PID-2020-112581GB-C21"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003030","name":"Ag\u00e8ncia de Gesti\u00f3 d\u2019Ajuts Universitaris i de Recerca","doi-asserted-by":"publisher","award":["2017 SGR 78"],"award-info":[{"award-number":["2017 SGR 78"]}],"id":[{"id":"10.13039\/501100003030","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Comb Optim"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This paper considers resolute decision rules in which each voter may vote \u201cyes\", \u201cabstain\" or vote \u201cno\", and the outcome is \u201cyes\" or \u201cno\". The model we consider is more general than that of simple games since the input admits abstention or indecision, but it is more specialized since it assumes the properties of monotonicity and anonymity. Many subclasses of these resolute decision rules have been studied in the literature from an axiomatic point of view. The purpose of this work is to enumerate these subclasses as a function of the number of voters.<\/jats:p>","DOI":"10.1007\/s10878-026-01412-9","type":"journal-article","created":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T09:51:12Z","timestamp":1773481872000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the enumeration of resolute majority rules"],"prefix":"10.1007","volume":"51","author":[{"given":"Josep","family":"Freixas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6183-061X","authenticated-orcid":false,"given":"Dani","family":"Samaniego","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,3,14]]},"reference":[{"key":"1412_CR1","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/S0165-1765(02)00014-9","volume":"75","author":"G Asan","year":"2002","unstructured":"Asan G, Sanver R (2002) Another characterization of the majority rule. Econ Lett 75:409\u2013413","journal-title":"Econ Lett"},{"key":"1412_CR2","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/s00355-013-0787-2","volume":"43","author":"D Bubboloni","year":"2014","unstructured":"Bubboloni D, Gori M (2014) Anonymous and neutral majority rules. Soc Choice Welfare 43:377\u2013401","journal-title":"Soc Choice Welfare"},{"key":"1412_CR3","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/j.mathsocsci.2015.04.002","volume":"76","author":"D Bubboloni","year":"2015","unstructured":"Bubboloni D, Gori M (2015) Symmetric majority rules. Math Soc Sci 76:73\u201386","journal-title":"Math Soc Sci"},{"key":"1412_CR4","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.mathsocsci.2016.08.007","volume":"84","author":"D Bubboloni","year":"2016","unstructured":"Bubboloni D, Gori M (2016) Resolute refinements of social choice correspondences. Math Soc Sci 84:37\u201349","journal-title":"Math Soc Sci"},{"key":"1412_CR5","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s10203-020-00294-8","volume":"44","author":"D Bubboloni","year":"2021","unstructured":"Bubboloni D, Gori M (2021) Breaking ties in collective decision making. Decisions Econ Finan 44:411\u2013457","journal-title":"Decisions Econ Finan"},{"key":"1412_CR6","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/0165-1765(88)90003-1","volume":"28","author":"DE Campbell","year":"1988","unstructured":"Campbell DE (1988) A characterization of majority rule. Econ Lett 28:307\u2013310","journal-title":"Econ Lett"},{"key":"1412_CR7","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s001990050318","volume":"15","author":"DE Campbell","year":"2000","unstructured":"Campbell DE, Kelly JS (2000) A simple characterization of majority rule. Econ Theor 15:689\u2013700","journal-title":"Econ Theor"},{"key":"1412_CR8","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/0165-4896(96)00815-3","volume":"32","author":"F Carreras","year":"1996","unstructured":"Carreras F, Freixas J (1996) Complete simple games. Math Soc Sci 32:139\u2013155","journal-title":"Math Soc Sci"},{"key":"1412_CR9","doi-asserted-by":"publisher","first-page":"949","DOI":"10.1162\/JEEA.2008.6.5.949","volume":"6","author":"P Dasgupta","year":"2008","unstructured":"Dasgupta P, Maskin ES (2008) On the robustness of majority rule. J Eur Econ Assoc 6:949\u2013973","journal-title":"J Eur Econ Assoc"},{"key":"1412_CR10","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/BF01263275","volume":"26","author":"DS Felsenthal","year":"1997","unstructured":"Felsenthal DS, Machover M (1997) Ternary voting games. Internat J Game Theory 26:335\u2013351","journal-title":"Internat J Game Theory"},{"key":"1412_CR11","doi-asserted-by":"publisher","first-page":"12627","DOI":"10.1007\/s00500-020-05422-5","volume":"25","author":"J Freixas","year":"2021","unstructured":"Freixas J (2021) On the enumeration of Boolean functions with distinguished variables. Soft Comput 25:12627\u201312640","journal-title":"Soft Comput"},{"key":"1412_CR12","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1080\/02331934.2022.2154126","volume":"73","author":"J Freixas","year":"2024","unstructured":"Freixas J (2024) On the enumeration of some inequivalent monotone Boolean functions. Optimization 73:1253\u20131266","journal-title":"Optimization"},{"key":"1412_CR13","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.ejor.2012.10.017","volume":"226","author":"J Freixas","year":"2013","unstructured":"Freixas J, Kurz S (2013) The golden number and Fibonacci sequences in the design of voting structures. Eur J Oper Res 226:246\u2013257","journal-title":"Eur J Oper Res"},{"key":"1412_CR14","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/s10479-013-1348-x","volume":"222","author":"J Freixas","year":"2014","unstructured":"Freixas J, Kurz S (2014) Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum. Ann Oper Res 222:317\u2013339","journal-title":"Ann Oper Res"},{"key":"1412_CR15","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s10479-011-0863-x","volume":"193","author":"J Freixas","year":"2012","unstructured":"Freixas J, Molinero X, Roura S (2012) Complete voting systems with two classes of voters: weightedness and counting. Ann Oper Res 193:273\u2013289","journal-title":"Ann Oper Res"},{"key":"1412_CR16","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s10479-021-03999-0","volume":"302","author":"J Freixas","year":"2021","unstructured":"Freixas J, Pons M (2021a) An extension and an alternative characterization of the May\u2019s theorem. Ann Oper Res 302:137\u2013150","journal-title":"Ann Oper Res"},{"key":"1412_CR17","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1007\/s11238-021-09814-3","volume":"91","author":"J Freixas","year":"2021","unstructured":"Freixas J, Pons M (2021b) On anonymous and weighted voting systems. Theor Decis 91:477\u2013491","journal-title":"Theor Decis"},{"key":"1412_CR18","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/j.dam.2021.03.011","volume":"297","author":"J Freixas","year":"2021","unstructured":"Freixas J, Samaniego D (2021) On the enumeration of simple bipartite games. Discret Appl Math 297:129\u2013141","journal-title":"Discret Appl Math"},{"key":"1412_CR19","doi-asserted-by":"publisher","first-page":"2111","DOI":"10.1007\/s10479-023-05621-x","volume":"336","author":"J Freixas","year":"2024","unstructured":"Freixas J, Samaniego D (2024) Some characterizations of resolute majority rules. Ann Oper Res 336:2111\u20132124","journal-title":"Ann Oper Res"},{"key":"1412_CR20","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s00355-003-0212-3","volume":"21","author":"J Freixas","year":"2003","unstructured":"Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welfare 21:399\u2013431","journal-title":"Soc Choice Welfare"},{"key":"1412_CR21","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1016\/j.geb.2008.09.035","volume":"69","author":"J Freixas","year":"2009","unstructured":"Freixas J, Zwicker WS (2009) Anonymous yes-no voting with abstention and multiple levels of approval. Games Econom Behav 69:428\u2013444","journal-title":"Games Econom Behav"},{"key":"1412_CR22","doi-asserted-by":"publisher","first-page":"940","DOI":"10.1111\/j.1540-5907.2006.00225.x","volume":"50","author":"RE Goodin","year":"2006","unstructured":"Goodin RE, List C (2006) A conditional defense of plurality rule: generalizing May\u2019s theorem in a restricted informational environment. Am J Political Sci 50:940\u2013949","journal-title":"Am J Political Sci"},{"key":"1412_CR23","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/j.mathsocsci.2015.06.003","volume":"77","author":"L Hoots","year":"2015","unstructured":"Hoots L, Powers RC (2015) Anonymous and positively responsive aggregation rules. Math Soc Sci 77:9\u201314","journal-title":"Math Soc Sci"},{"key":"1412_CR24","doi-asserted-by":"publisher","first-page":"1489","DOI":"10.1111\/iere.12394","volume":"60","author":"S Horan","year":"2019","unstructured":"Horan S, Osborne MJ, Sanver MR (2019) Positively responsive collective choice rules and majority rule: a generalization of May\u2019s theorem to many alternatives. Int Econ Rev 60:1489\u20131504","journal-title":"Int Econ Rev"},{"key":"1412_CR25","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s11238-016-9563-y","volume":"82","author":"H Jeong","year":"2017","unstructured":"Jeong H, Ju B (2017) Resolute majority rules. Theory And Decision 82:31\u201339","journal-title":"Theory And Decision"},{"key":"1412_CR26","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.dam.2023.03.004","volume":"334","author":"S Kurz","year":"2023","unstructured":"Kurz S, Samaniego D (2023) Enumeration of simple games with two equivalence classes of players. Discret Appl Math 334:26\u201335","journal-title":"Discret Appl Math"},{"key":"1412_CR27","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s00182-012-0327-9","volume":"42","author":"S Kurz","year":"2013","unstructured":"Kurz S, Tautenhahn N (2013) On Dedekind\u2019s problem for complete simple games. Internat J Game Theory 42:411\u2013437","journal-title":"Internat J Game Theory"},{"key":"1412_CR28","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/j.mathsocsci.2005.12.001","volume":"51","author":"B Llamazares","year":"2006","unstructured":"Llamazares B (2006) The forgotten decision rules: majority rules based on difference of votes. Math Soc Sci 51:311\u2013326","journal-title":"Math Soc Sci"},{"key":"1412_CR29","doi-asserted-by":"crossref","unstructured":"Maskin ES (1995) Majority rule, social welfare functions, and game forms. In: Basu K, Pattanaik PK, Suzumura L (eds) Choice, Welfare, and Development, pp 100\u2013109. Clarendon Press 1995","DOI":"10.1093\/oso\/9780198287896.003.0007"},{"key":"1412_CR30","doi-asserted-by":"publisher","first-page":"680","DOI":"10.2307\/1907651","volume":"20","author":"K May","year":"1952","unstructured":"May K (1952) A set of independent, necessary and sufficient conditions for simple majority decision. Econometrica 20:680\u2013684","journal-title":"Econometrica"},{"key":"1412_CR31","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s10479-008-0330-5","volume":"163","author":"FR McMorris","year":"2008","unstructured":"McMorris FR, Powers RC (2008) The majority decision function for trees with 3 leaves. Ann Oper Res 163:169\u2013175","journal-title":"Ann Oper Res"},{"key":"1412_CR32","doi-asserted-by":"publisher","first-page":"5358","DOI":"10.1109\/TIT.2024.3379594","volume":"70","author":"B Pawelski","year":"2024","unstructured":"Pawelski B (2024) On the number of inequivalent monotone Boolean functions of 9 variables. IEEE Trans Inf Theory 70:5358\u20135364","journal-title":"IEEE Trans Inf Theory"},{"key":"1412_CR33","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1016\/j.econlet.2007.12.005","volume":"100","author":"J Perry","year":"2008","unstructured":"Perry J, Powers RC (2008) Aggregation rules that satisfy anonymity and neutrality. Econ Lett 100:108\u2013110","journal-title":"Econ Lett"},{"key":"1412_CR34","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.mathsocsci.2010.03.005","volume":"60","author":"J Perry","year":"2010","unstructured":"Perry J, Powers RC (2010) Anonymity, monotonicity, and quota pair systems. Math Soc Sci 60:57\u201360","journal-title":"Math Soc Sci"},{"key":"1412_CR35","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/0022-0531(80)90002-2","volume":"23","author":"A Rubinstein","year":"1980","unstructured":"Rubinstein A (1980) Stability of decision systems under majority rule. J Econ Theory 23:150\u2013159","journal-title":"J Econ Theory"},{"key":"1412_CR36","doi-asserted-by":"publisher","first-page":"491","DOI":"10.2307\/1909947","volume":"34","author":"AK Sen","year":"1966","unstructured":"Sen AK (1966) A possibility theorem on majority decisions. Econometrica 34:491\u2013499","journal-title":"Econometrica"},{"key":"1412_CR37","unstructured":"Sloane NJA (1964) The Online Encyclopedia of Integer Sequences. http:\/\/oeis.org"},{"key":"1412_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-77645-3","volume-title":"Mathematics and Politics","author":"AD Taylor","year":"2008","unstructured":"Taylor AD, Pacelli A (2008) Mathematics and Politics, 2nd edn. Springer Verlag, New York","edition":"2"},{"key":"1412_CR39","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/S0165-1765(03)00145-9","volume":"81","author":"GJ Woeginger","year":"2003","unstructured":"Woeginger GJ (2003) A new characterization of the majority rule. Econ Lett 81:89\u201394","journal-title":"Econ Lett"},{"key":"1412_CR40","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1080\/0025570X.1995.11996354","volume":"68","author":"SC Young","year":"1995","unstructured":"Young SC, Taylor AD, Zwicker WS (1995) Counting quota systems: a combinatorial question from social choice theory. Math Mag 68:331\u2013342","journal-title":"Math Mag"}],"container-title":["Journal of Combinatorial Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-026-01412-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10878-026-01412-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-026-01412-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T18:59:34Z","timestamp":1774724374000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10878-026-01412-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["1412"],"URL":"https:\/\/doi.org\/10.1007\/s10878-026-01412-9","relation":{},"ISSN":["1382-6905","1573-2886"],"issn-type":[{"value":"1382-6905","type":"print"},{"value":"1573-2886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3]]},"assertion":[{"value":"1 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"31"}}