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Probabilistic power indices for voting rules with abstention

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  • Freixas, Josep

Abstract

In this paper, we introduce eight power indices that admit a probabilistic interpretation for voting rules with abstention or with three levels of approval in the input, briefly (3, 2) games. We analyze the analogies and discrepancies between standard known indices for simple games and the proposed extensions for this more general context. A remarkable difference is that for (3, 2) games the proposed extensions of the Banzhaf index, Coleman index to prevent action and Coleman index to initiate action become non-proportional notions, contrarily to what succeeds for simple games. We conclude the work by providing procedures based on generating functions for weighted (3, 2) games, and extensible to (j,k) games, to efficiently compute them.

Suggested Citation

  • Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
  • Handle: RePEc:eee:matsoc:v:64:y:2012:i:1:p:89-99 DOI: 10.1016/j.mathsocsci.2012.01.005
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    References listed on IDEAS

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    1. Annick Laruelle & Federico Valenciano, 2005. "Assessing success and decisiveness in voting situations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, pages 171-197.
    2. Josep Freixas, 2010. "On ordinal equivalence of the Shapley and Banzhaf values for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 513-527, October.
    3. Laruelle,Annick & Valenciano,Federico, 2011. "Voting and Collective Decision-Making," Cambridge Books, Cambridge University Press, number 9780521182638, December.
    4. Roland Pongou & Bertrand Tchantcho & Lawrence Diffo Lambo, 2011. "Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity," Theory and Decision, Springer, pages 157-178.
    5. Josep Freixas & William S. Zwicker, 2003. "Weighted voting, abstention, and multiple levels of approval," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 399-431, December.
    6. Carreras, Francesc & Freixas, Josep, 2008. "On ordinal equivalence of power measures given by regular semivalues," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 221-234, March.
    7. Rubinstein, Ariel, 1980. "Stability of decision systems under majority rule," Journal of Economic Theory, Elsevier, vol. 23(2), pages 150-159, October.
    8. Freixas, Josep & Zwicker, William S., 2009. "Anonymous yes-no voting with abstention and multiple levels of approval," Games and Economic Behavior, Elsevier, vol. 67(2), pages 428-444, November.
    9. Tchantcho, Bertrand & Lambo, Lawrence Diffo & Pongou, Roland & Engoulou, Bertrand Mbama, 2008. "Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories," Games and Economic Behavior, Elsevier, vol. 64(1), pages 335-350, September.
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    Cited by:

    1. Matthew Gould & Matthew D. Rablen, 2017. "Reform of the United Nations Security Council: equity and efficiency," Public Choice, Springer, pages 145-168.
    2. Matthew Gould & Matthew Rablen, 2013. "Equitable Representation in the Councils of the United Nations: Theory and Application," CESifo Working Paper Series 4519, CESifo Group Munich.
    3. Freixas, Josep & Tchantcho, Bertrand & Tedjeugang, Narcisse, 2014. "Achievable hierarchies in voting games with abstention," European Journal of Operational Research, Elsevier, vol. 236(1), pages 254-260.
    4. Sascha Kurz, 2014. "Measuring Voting Power in Convex Policy Spaces," Economies, MDPI, Open Access Journal, vol. 2(1), pages 1-33, March.

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