Social Choice and Welfare

, Volume 31, Issue 4, pp 621–640

The Nakamura numbers for computable simple games

  • Masahiro Kumabe
  • H. Reiju Mihara
Original Paper

DOI: 10.1007/s00355-008-0300-5

Cite this article as:
Kumabe, M. & Mihara, H.R. Soc Choice Welfare (2008) 31: 621. doi:10.1007/s00355-008-0300-5


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.

Supplementary material

355_2008_300_MOESM1_ESM.pdf (27 kb)

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