  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.
