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Heuristic and exact solutions to the inverse power index problem for small voting bodies

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  • Sascha Kurz

    ()
    (University of Bayreuth)

  • Stefan Napel

    ()
    (University of Bayreuth)

Abstract

Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. The paper considers approximations to this inverse problem for the Penrose-Banzhaf index by hill-climbing algorithms and exact solutions which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.

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Bibliographic Info

Paper provided by Friedrich-Schiller-University Jena, Max-Planck-Institute of Economics in its series Jena Economic Research Papers with number 2012-045.

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Date of creation: 23 Jul 2012
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Handle: RePEc:jrp:jrpwrp:2012-045

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Keywords: electoral systems; simple games; weighted voting games; square root rule; Penrose limit theorem; Penrose-Banzhaf index; institutional design;

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  1. Noga Alon & Paul Edelman, 2010. "The inverse Banzhaf problem," Social Choice and Welfare, Springer, vol. 34(3), pages 371-377, March.
  2. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
  3. Lindner, Ines & Owen, Guillermo, 2007. "Cases where the Penrose limit theorem does not hold," Mathematical Social Sciences, Elsevier, vol. 53(3), pages 232-238, May.
  4. Deineko, Vladimir G. & Woeginger, Gerhard J., 2006. "On the dimension of simple monotonic games," European Journal of Operational Research, Elsevier, vol. 170(1), pages 315-318, April.
  5. Lindner, Ines & Machover, Moshe, 2004. "L.S. Penrose's limit theorem: proof of some special cases," Mathematical Social Sciences, Elsevier, vol. 47(1), pages 37-49, January.
  6. Laruelle,Annick & Valenciano,Federico, 2008. "Voting and Collective Decision-Making," Cambridge Books, Cambridge University Press, number 9780521873871, October.
  7. Annick Laruelle & Mika Widgrén, 1998. "Is the allocation of voting power among EU states fair?," Public Choice, Springer, vol. 94(3), pages 317-339, March.
  8. Leech, Dennis, 2002. "Power Indices As An Aid To Institutional Design : The Generalised Apportionment Problem," The Warwick Economics Research Paper Series (TWERPS) 648, University of Warwick, Department of Economics.
  9. Serguei Kaniovski, 2008. "The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent," Social Choice and Welfare, Springer, vol. 31(2), pages 281-300, August.
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