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The Nakamura numbers for computable simple games

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  • Masahiro Kumabe
  • H. Reiju Mihara

Abstract

The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked.

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Bibliographic Info

Article provided by Springer in its journal Social Choice and Welfare.

Volume (Year): 31 (2008)
Issue (Month): 4 (December)
Pages: 621-640

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Handle: RePEc:spr:sochwe:v:31:y:2008:i:4:p:621-640

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References

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  1. H. Reiju Mihara, 2003. "Nonanonymity and sensitivity of computable simple games," Game Theory and Information 0310006, EconWPA, revised 01 Jun 2004.
  2. H. Reiju Mihara, 1994. "Arrow's Theorem and Turing Computability," Public Economics 9408001, EconWPA, revised 23 Aug 1994.
  3. Andjiga, Nicolas Gabriel & Mbih, Boniface, 2000. "A note on the core of voting games," Journal of Mathematical Economics, Elsevier, vol. 33(3), pages 367-372, April.
  4. Truchon, Michel, 1995. "Voting games and acyclic collective choice rules," Mathematical Social Sciences, Elsevier, vol. 29(2), pages 165-179, April.
  5. Rajat Deb, 2004. "Rights as alternative game forms," Social Choice and Welfare, Springer, vol. 22(1), pages 83-111, 02.
  6. Lewis, Alain A., 1988. "An infinite version of arrow's theorem in the effective setting," Mathematical Social Sciences, Elsevier, vol. 16(1), pages 41-48, August.
  7. Kumabe, Masahiro & Mihara, H. Reiju, 2006. "Computability of simple games: A characterization and application to the core," MPRA Paper 437, University Library of Munich, Germany.
  8. Peleg,Bezalel, 2008. "Game Theoretic Analysis of Voting in Committees," Cambridge Books, Cambridge University Press, number 9780521074650, October.
  9. Kumabe, Masahiro & Mihara, H. Reiju, 2006. "Computability of simple games: A complete investigation of the sixty-four possibilities," MPRA Paper 440, University Library of Munich, Germany.
  10. Weber, Robert J., 1994. "Games in coalitional form," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 36, pages 1285-1303 Elsevier.
  11. Kolpin, Van, 1990. "Equivalent game forms and coalitional power," Mathematical Social Sciences, Elsevier, vol. 20(3), pages 239-249, December.
  12. Richter, Marcel K. & Wong, Kam-Chau, 1999. "Computable preference and utility," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 339-354, November.
  13. Kelly, Jerry S., 1988. "Social choice and computational complexity," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 1-8, February.
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Cited by:
  1. Kumabe, Masahiro & Mihara, H. Reiju, 2006. "Computability of simple games: A complete investigation of the sixty-four possibilities," MPRA Paper 440, University Library of Munich, Germany.
  2. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Computability of simple games: A characterization and application to the core," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 348-366, February.
  3. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Preference aggregation theory without acyclicity: The core without majority dissatisfaction," Games and Economic Behavior, Elsevier, vol. 72(1), pages 187-201, May.

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  1. 利用者:Theorist2/Archived in Wikipedia (Japanese)
  2. Nakamura number in Wikipedia (English)
  3. Txantiloi:Cite doi/10.1007.2Fs00355-008-0300-5 in Wikipedia (Basque)
  4. ライスの定理 in Wikipedia (Japanese)
  5. User:Theorist2/Unused stuffs in Wikipedia (English)
  6. Template:Cite doi/10.1007.2Fs00355-008-0300-5 in Wikipedia (English)
  7. Cooperative game in Wikipedia (English)
  8. Rice's theorem in Wikipedia (English)
  9. الگو:Cite doi/10.1007.2Fs00355-008-0300-5 in Wikipedia (Persian)
  10. 中村ナンバー in Wikipedia (Japanese)
  11. 協力ゲーム in Wikipedia (Japanese)
  12. Template:Cite doi/10.1007.2Fs00355-008-0300-5 in Wikipedia (Japanese)
  13. User:Theorist2/Archived in Wikipedia (English)
  14. قالب:Cite doi/10.1007.2Fs00355-008-0300-5 in Wikipedia (Arabic)

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