The Nakamura numbers for computable simple games

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. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. We would like to thank an anonymous referee for useful suggestions. The discussion in footnote 3 and Remark 4, among other things, would not have been possible without his/her suggestion.