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The Nakamura numbers for computable simple games

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Author Info
Kumabe, Masahiro
Mihara, H. Reiju

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Abstract

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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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 3684.

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Date of creation: 21 Nov 2007
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Handle: RePEc:pra:mprapa:3684

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Related research
Keywords: Nakamura number voting games core Turing computability axiomatic method multi-criterion decision-making

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Find related papers by JEL classification:
C69 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Other
D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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References listed on IDEAS
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  1. Andjiga, Nicolas Gabriel & Mbih, Boniface, 2000. "A note on the core of voting games," Journal of Mathematical Economics, Elsevier, vol. 33(3), pages 367-372, April. [Downloadable!] (restricted)
  2. H. Reiju Mihara, 1997. "Arrow's Theorem and Turing computability," Economic Theory, Springer, vol. 10(2), pages 257-276. [Downloadable!] (restricted)
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  3. Richter, Marcel K. & Wong, Kam-Chau, 1999. "Computable preference and utility," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 339-354, November. [Downloadable!] (restricted)
  4. Lewis, Alain A., 1988. "An infinite version of arrow's theorem in the effective setting," Mathematical Social Sciences, Elsevier, vol. 16(1), pages 41-48, August. [Downloadable!] (restricted)
  5. Kumabe, Masahiro & Mihara, H. Reiju, 2007. "Computability of simple games: A complete investigation of the sixty-four possibilities," MPRA Paper 440, University Library of Munich, Germany. [Downloadable!]
  6. Truchon, Michel, 1995. "Voting games and acyclic collective choice rules," Mathematical Social Sciences, Elsevier, vol. 29(2), pages 165-179, April. [Downloadable!] (restricted)
  7. Mihara, H. Reiju, 2004. "Nonanonymity and sensitivity of computable simple games," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 329-341, November. [Downloadable!] (restricted)
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  8. Kelly, Jerry S., 1988. "Social choice and computational complexity," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 1-8, February. [Downloadable!] (restricted)
  9. Kumabe, Masahiro & Mihara, H. Reiju, 2007. "Computability of simple games: A characterization and application to the core," MPRA Paper 437, University Library of Munich, Germany. [Downloadable!]
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  10. Kolpin, Van, 1990. "Equivalent game forms and coalitional power," Mathematical Social Sciences, Elsevier, vol. 20(3), pages 239-249, December. [Downloadable!] (restricted)
  11. Rajat Deb, 2004. "Rights as alternative game forms," Social Choice and Welfare, Springer, vol. 22(1), pages 83-111, 02. [Downloadable!] (restricted)
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