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The Nakamura numbers for computable simple games

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  • Kumabe, Masahiro
  • Mihara, H. Reiju

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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Bibliographic Info

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

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Date of creation: 23 Jun 2007
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Handle: RePEc:pra:mprapa:3684

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

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  1. Weber, Robert J., 1994. "Games in coalitional form," Handbook of Game Theory with Economic Applications, Elsevier, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 36, pages 1285-1303 Elsevier.
  2. Peleg, Bezalel, 2002. "Game-theoretic analysis of voting in committees," Handbook of Social Choice and Welfare, Elsevier, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 8, pages 395-423 Elsevier.
  3. Kumabe, Masahiro & Mihara, H. Reiju, 2006. "Computability of simple games: A characterization and application to the core," MPRA Paper 437, University Library of Munich, Germany.
  4. 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.
  5. Mihara, H. Reiju, 2004. "Nonanonymity and sensitivity of computable simple games," Mathematical Social Sciences, Elsevier, Elsevier, vol. 48(3), pages 329-341, November.
  6. Mihara, H.R., 1994. "Arrow's Theorem and Turing Computability," Papers, Minnesota - Center for Economic Research 276, Minnesota - Center for Economic Research.
  7. Truchon, M., 1993. "Voting Games and Acyclic Collective Choice Rules," Papers, Laval - Recherche en Politique Economique 9315, Laval - Recherche en Politique Economique.
  8. Kumabe, Masahiro & Mihara, H. Reiju, 2006. "Computability of simple games: A complete investigation of the sixty-four possibilities," MPRA Paper 440, University Library of Munich, Germany.
  9. Kelly, Jerry S., 1988. "Social choice and computational complexity," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 1-8, February.
  10. Lewis, Alain A., 1988. "An infinite version of arrow's theorem in the effective setting," Mathematical Social Sciences, Elsevier, Elsevier, vol. 16(1), pages 41-48, August.
  11. Rajat Deb, 2004. "Rights as alternative game forms," Social Choice and Welfare, Springer, Springer, vol. 22(1), pages 83-111, 02.
  12. Richter, Marcel K. & Wong, Kam-Chau, 1999. "Computable preference and utility," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 339-354, November.
  13. Kolpin, Van, 1990. "Equivalent game forms and coalitional power," Mathematical Social Sciences, Elsevier, Elsevier, vol. 20(3), pages 239-249, December.
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Cited by:
  1. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Computability of simple games: A characterization and application to the core," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 348-366, February.
  2. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Computability of simple games: A complete investigation of the sixty-four possibilities," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 150-158, March.
  3. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Preference aggregation theory without acyclicity: The core without majority dissatisfaction," Games and Economic Behavior, Elsevier, Elsevier, vol. 72(1), pages 187-201, May.

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