The Nakamura numbers for computable simple games
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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.
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- Arrow KJ (1963). Social choice and individual values, 2nd edn. Yale University Press, New Haven Google Scholar
- Austen-Smith D and Banks JS (1999). Positive political theory I: collective preference. University of Michigan Press, Ann Arbor Google Scholar
- Kumabe M, Mihara HR (2007) Computability of simple games: a complete investigation of the sixty-four possibilities. MPRA Paper 4405, Munich University LibraryGoogle Scholar
- Odifreddi P (1992). Classical recursion theory: the theory of functions and sets of natural numbers. Elsevier, Amsterdam Google Scholar
- 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
- Soare RI (1987). Recursively enumerable sets and degrees: a study of computable functions and computably generated sets. Springer, Berlin Google Scholar
- 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