{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T04:13:34Z","timestamp":1782447214945,"version":"3.54.5"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2008,3,11]],"date-time":"2008-03-11T00:00:00Z","timestamp":1205193600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Soc Choice Welfare"],"published-print":{"date-parts":[[2008,12]]},"DOI":"10.1007\/s00355-008-0300-5","type":"journal-article","created":{"date-parts":[[2008,3,10]],"date-time":"2008-03-10T10:30:03Z","timestamp":1205145003000},"page":"621-640","source":"Crossref","is-referenced-by-count":6,"title":["The Nakamura numbers for computable simple games"],"prefix":"10.1007","volume":"31","author":[{"given":"Masahiro","family":"Kumabe","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. Reiju","family":"Mihara","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2008,3,11]]},"reference":[{"key":"300_CR1","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0304-4068(99)00003-8","volume":"33","author":"NG Andjiga","year":"2000","unstructured":"Andjiga NG and Mbih B (2000). A note on the core of voting games. J Math Econ 33: 367\u2013372","journal-title":"J Math Econ"},{"key":"300_CR2","volume-title":"Social choice and individual values","author":"KJ Arrow","year":"1963","unstructured":"Arrow KJ (1963). Social choice and individual values, 2nd edn. Yale University Press, New Haven","edition":"2"},{"key":"300_CR3","doi-asserted-by":"crossref","DOI":"10.3998\/mpub.14228","volume-title":"Positive political theory I: collective preference","author":"D Austen-Smith","year":"1999","unstructured":"Austen-Smith D and Banks JS (1999). Positive political theory I: collective preference. University of Michigan Press, Ann Arbor"},{"key":"300_CR4","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF00179983","volume":"12","author":"JS Banks","year":"1995","unstructured":"Banks JS (1995). Acyclic social choice from finite sets. Soc Choice Welf 12: 293\u2013310","journal-title":"Soc Choice Welf"},{"key":"300_CR5","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF00303169","volume":"6","author":"J Bartholdi III","year":"1989","unstructured":"Bartholdi III J, Tovey CA and Trick MA (1989a). Voting schemes for which it can be difficult to tell who won the election. Soc Choice Welf 6: 157\u2013165","journal-title":"Soc Choice Welf"},{"key":"300_CR6","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/BF00295861","volume":"6","author":"JJ Bartholdi III","year":"1989","unstructured":"Bartholdi III JJ, Tovey CA and Trick MA (1989b). The computational difficulty of manipulating an election. Soc Choice Welf 6: 227\u2013241","journal-title":"Soc Choice Welf"},{"key":"300_CR7","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s00355-003-0278-y","volume":"22","author":"R Deb","year":"2004","unstructured":"Deb R (2004). Rights as alternative game forms. Soc Choice Welf 22: 83\u2013111","journal-title":"Soc Choice Welf"},{"key":"300_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0304-4068(88)90022-5","volume":"17","author":"JS Kelly","year":"1988","unstructured":"Kelly JS (1988). Social choice and computational complexity. J Math Econ 17: 1\u20138","journal-title":"J Math Econ"},{"key":"300_CR9","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0165-4896(90)90004-Q","volume":"20","author":"V Kolpin","year":"1990","unstructured":"Kolpin V (1990). Equivalent game forms and coalitional power. Math Soc Sci 20: 239\u2013249","journal-title":"Math Soc Sci"},{"key":"300_CR10","unstructured":"Kumabe M, Mihara HR (2007) Computability of simple games: a complete investigation of the sixty-four possibilities. MPRA Paper 4405, Munich University Library"},{"key":"300_CR11","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.jmateco.2007.05.012","volume":"44","author":"M Kumabe","year":"2008","unstructured":"Kumabe M and Mihara HR (2008). Computability of simple games: a characterization and application to the core. J Math Econ 44: 348\u2013366","journal-title":"J Math Econ"},{"key":"300_CR12","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/0165-4896(88)90003-0","volume":"16","author":"AA Lewis","year":"1988","unstructured":"Lewis AA (1988). An infinite version of Arrow\u2019s Theorem in the effective setting. Math Soc Sci 16: 41\u201348","journal-title":"Math Soc Sci"},{"key":"300_CR13","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s001990050157","volume":"10","author":"HR Mihara","year":"1997","unstructured":"Mihara HR (1997). Arrow\u2019s Theorem and Turing computability. Econ Theory 10: 257\u201376","journal-title":"Econ Theory"},{"key":"300_CR14","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0304-4068(98)00061-5","volume":"32","author":"HR Mihara","year":"1999","unstructured":"Mihara HR (1999). Arrow\u2019s theorem, countably many agents and more visible invisible dictators. J Math Econ 32: 267\u2013287","journal-title":"J Math Econ"},{"key":"300_CR15","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.mathsocsci.2004.04.001","volume":"48","author":"HR Mihara","year":"2004","unstructured":"Mihara HR (2004). Nonanonymity and sensitivity of computable simple games. Math Soc Sci 48: 329\u2013341","journal-title":"Math Soc Sci"},{"key":"300_CR16","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF01763051","volume":"8","author":"K Nakamura","year":"1979","unstructured":"Nakamura K (1979). The vetoers in a simple game with ordinal preferences. Int J Game Theory 8: 55\u201361","journal-title":"Int J Game Theory"},{"key":"300_CR17","volume-title":"Classical recursion theory: the theory of functions and sets of natural numbers","author":"P Odifreddi","year":"1992","unstructured":"Odifreddi P (1992). Classical recursion theory: the theory of functions and sets of natural numbers. Elsevier, Amsterdam"},{"key":"300_CR18","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/S1574-0110(02)80012-1","volume-title":"Handbook of social choice and welfare, vol 1, chap 8","author":"B Peleg","year":"2002","unstructured":"Peleg B (2002). Game-theoretic analysis of voting in committees. In: Arrow, KJ, Sen, AK and Suzumura, K (eds) Handbook of social choice and welfare, vol 1, chap 8, pp 395\u2013423. Elsevier, Amsterdam"},{"key":"300_CR19","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/S0304-4068(98)00063-9","volume":"32","author":"MK Richter","year":"1999","unstructured":"Richter MK and Wong K-C (1999). Computable preference and utility. J Math Econ 32: 339\u2013354","journal-title":"J Math Econ"},{"key":"300_CR20","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1002\/bs.3830070104","volume":"7","author":"LS Shapley","year":"1962","unstructured":"Shapley LS (1962). Simple games: an outline of the descriptive theory. Behav Sci 7: 59\u201366","journal-title":"Behav Sci"},{"key":"300_CR21","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-02460-7","volume-title":"Recursively enumerable sets and degrees: a study of computable functions and computably generated sets","author":"RI Soare","year":"1987","unstructured":"Soare RI (1987). Recursively enumerable sets and degrees: a study of computable functions and computably generated sets. Springer, Berlin"},{"key":"300_CR22","first-page":"168","volume":"192","author":"Y Tanaka","year":"2007","unstructured":"Tanaka Y (2007). Type two computability of social choice functions and the Gibbard-Satterthwaite theorem in an infinite society. Appl Math Comput 192: 168\u2013174","journal-title":"Appl Math Comput"},{"key":"300_CR23","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0165-4896(94)00770-9","volume":"29","author":"M Truchon","year":"1995","unstructured":"Truchon M (1995). Voting games and acyclic collective choice rules. Math Soc Sci 29: 165\u2013179","journal-title":"Math Soc Sci"},{"key":"300_CR24","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1006\/jeth.1999.2602","volume":"88","author":"M Hees van","year":"1999","unstructured":"van Hees M (1999). Liberalism, efficiency and stability: some possibility results. J Econ Theory 88: 294\u2013309","journal-title":"J Econ Theory"},{"key":"300_CR25","first-page":"1285","volume-title":"Handbook of game theory, vol 2, chap 36","author":"RJ Weber","year":"1994","unstructured":"Weber RJ (1994). Games in coalitional form. In: Aumann, RJ and Hart, S (eds) Handbook of game theory, vol 2, chap 36, pp 1285\u20131303. Elsevier, Amsterdam"}],"container-title":["Social Choice and Welfare"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00355-008-0300-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00355-008-0300-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00355-008-0300-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T10:28:03Z","timestamp":1559039283000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00355-008-0300-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,11]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["300"],"URL":"https:\/\/doi.org\/10.1007\/s00355-008-0300-5","relation":{},"ISSN":["0176-1714","1432-217X"],"issn-type":[{"value":"0176-1714","type":"print"},{"value":"1432-217X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3,11]]}}}