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Computability of simple games: A complete investigation of the sixty-four possibilities

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  • Kumabe, Masahiro
  • Mihara, H. Reiju

Abstract

Classify simple games into sixteen "types" in terms of the four conventional axioms: monotonicity, properness, strongness, and nonweakness. Further classify them into sixty-four classes in terms of finiteness (existence of a finite carrier) and computability. For each such class, we either show that it is empty or give an example of a game belonging to it. We observe that if a type contains an infinite game, then it contains both computable infinitegames and noncomputable ones. This strongly suggests that computability is logically, as well as conceptually, unrelated to the conventional axioms.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 440.

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Date of creation: Oct 2006
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Handle: RePEc:pra:mprapa:440

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Keywords: Voting games; infinitely many players; axiomatic method; complete independence; algorithms; Turing computability; recursion theory;

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References

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  1. Nabil I. Al-Najjar & Luca Anderlini & Leonardo Felli, 2006. "Undescribable Events," Review of Economic Studies, Oxford University Press, vol. 73(4), pages 849-868.
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  4. Mihara, H. Reiju, 1999. "Arrow's theorem, countably many agents, and more visible invisible dictators1," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 267-287, November.
  5. Mihara, H.R., 1994. "Arrow's Theorem and Turing Computability," Papers, Minnesota - Center for Economic Research 276, Minnesota - Center for Economic Research.
  6. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Preference aggregation theory without acyclicity: The core without majority dissatisfaction," MPRA Paper 11728, University Library of Munich, Germany.
  7. Banks, Jeffrey & Duggan, John & Le Breton, Michel, 2003. "Social Choice and Electoral Competition in the General Spatial Model," IDEI Working Papers 188, Institut d'Économie Industrielle (IDEI), Toulouse.
  8. Shanfeng Zhu & Xiaotie Deng & Maocheng Cai & Qizhi Fang, 2002. "On computational complexity of membership test in flow games and linear production games," International Journal of Game Theory, Springer, vol. 31(1), pages 39-45.
  9. 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.
  10. Kumabe, Masahiro & Mihara, H. Reiju, 2007. "The Nakamura numbers for computable simple games," MPRA Paper 3684, University Library of Munich, Germany.
  11. 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.
  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. H. Reiju Mihara, 1997. "Arrow's Theorem, countably many agents, and more visible invisible dictators," Public Economics, EconWPA 9705001, EconWPA, revised 07 May 1997.
  14. Anthony Downs, 1957. "An Economic Theory of Political Action in a Democracy," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 65, pages 135.
  15. 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.
  16. H. Reiju Mihara, 2003. "Nonanonymity and sensitivity of computable simple games," Game Theory and Information, EconWPA 0310006, EconWPA, revised 01 Jun 2004.
  17. Kelly, Jerry S., 1988. "Social choice and computational complexity," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 1-8, February.
  18. William Thomson, 2001. "On the axiomatic method and its recent applications to game theory and resource allocation," Social Choice and Welfare, Springer, vol. 18(2), pages 327-386.
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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, 2007. "The Nakamura numbers for computable simple games," MPRA Paper 3684, University Library of Munich, Germany.

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