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


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.


Social Choice Initial Segment Recursive Function Game Form Simple Game 
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Supplementary material

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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

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