Social Choice and Welfare

, Volume 31, Issue 4, pp 621–640 | Cite as

The Nakamura numbers for computable simple games

  • Masahiro Kumabe
  • H. Reiju Mihara
Original Paper

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

355_2008_300_MOESM1_ESM.pdf (27 kb)
ESM (PDF 28 KB)

References

  1. Andjiga NG and Mbih B (2000). A note on the core of voting games. J Math Econ 33: 367–372 CrossRefGoogle Scholar
  2. Arrow KJ (1963). Social choice and individual values, 2nd edn. Yale University Press, New Haven Google Scholar
  3. Austen-Smith D and Banks JS (1999). Positive political theory I: collective preference. University of Michigan Press, Ann Arbor Google Scholar
  4. Banks JS (1995). Acyclic social choice from finite sets. Soc Choice Welf 12: 293–310 CrossRefGoogle Scholar
  5. 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: 157–165 CrossRefGoogle Scholar
  6. Bartholdi III JJ, Tovey CA and Trick MA (1989b). The computational difficulty of manipulating an election. Soc Choice Welf 6: 227–241 CrossRefGoogle Scholar
  7. Deb R (2004). Rights as alternative game forms. Soc Choice Welf 22: 83–111 CrossRefGoogle Scholar
  8. Kelly JS (1988). Social choice and computational complexity. J Math Econ 17: 1–8 CrossRefGoogle Scholar
  9. Kolpin V (1990). Equivalent game forms and coalitional power. Math Soc Sci 20: 239–249 CrossRefGoogle Scholar
  10. Kumabe M, Mihara HR (2007) Computability of simple games: a complete investigation of the sixty-four possibilities. MPRA Paper 4405, Munich University LibraryGoogle Scholar
  11. Kumabe M and Mihara HR (2008). Computability of simple games: a characterization and application to the core. J Math Econ 44: 348–366 CrossRefGoogle Scholar
  12. Lewis AA (1988). An infinite version of Arrow’s Theorem in the effective setting. Math Soc Sci 16: 41–48 CrossRefGoogle Scholar
  13. Mihara HR (1997). Arrow’s Theorem and Turing computability. Econ Theory 10: 257–76 CrossRefGoogle Scholar
  14. Mihara HR (1999). Arrow’s theorem, countably many agents and more visible invisible dictators. J Math Econ 32: 267–287 CrossRefGoogle Scholar
  15. Mihara HR (2004). Nonanonymity and sensitivity of computable simple games. Math Soc Sci 48: 329–341 CrossRefGoogle Scholar
  16. Nakamura K (1979). The vetoers in a simple game with ordinal preferences. Int J Game Theory 8: 55–61 CrossRefGoogle Scholar
  17. Odifreddi P (1992). Classical recursion theory: the theory of functions and sets of natural numbers. Elsevier, Amsterdam Google Scholar
  18. 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 395–423. Elsevier, Amsterdam Google Scholar
  19. Richter MK and Wong K-C (1999). Computable preference and utility. J Math Econ 32: 339–354 CrossRefGoogle Scholar
  20. Shapley LS (1962). Simple games: an outline of the descriptive theory. Behav Sci 7: 59–66 CrossRefGoogle Scholar
  21. Soare RI (1987). Recursively enumerable sets and degrees: a study of computable functions and computably generated sets. Springer, Berlin Google Scholar
  22. Tanaka Y (2007). Type two computability of social choice functions and the Gibbard-Satterthwaite theorem in an infinite society. Appl Math Comput 192: 168–174 CrossRefGoogle Scholar
  23. Truchon M (1995). Voting games and acyclic collective choice rules. Math Soc Sci 29: 165–179 CrossRefGoogle Scholar
  24. van Hees M (1999). Liberalism, efficiency and stability: some possibility results. J Econ Theory 88: 294–309 CrossRefGoogle Scholar
  25. Weber RJ (1994). Games in coalitional form. In: Aumann, RJ and Hart, S (eds) Handbook of game theory, vol 2, chap 36, pp 1285–1303. Elsevier, Amsterdam Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Masahiro Kumabe
    • 1
  • H. Reiju Mihara
    • 2
  1. 1.Kanagawa Study CenterThe University of the AirYokohamaJapan
  2. 2.Graduate School of ManagementKagawa UniversityTakamatsuJapan

Personalised recommendations