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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.

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  • 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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    1. Laruelle,Annick & Valenciano,Federico, 2011. "Voting and Collective Decision-Making," Cambridge Books, Cambridge University Press, number 9780521182638, January.
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    9. 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.
    10. 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.
    11. 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.
    12. William S. Zwicker, 2009. "Anonymous Voting Rules with Abstention: Weighted Voting," Studies in Choice and Welfare, in: Steven J. Brams & William V. Gehrlein & Fred S. Roberts (ed.), The Mathematics of Preference, Choice and Order, pages 239-258, Springer.
    13. 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.
    14. 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.
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    Cited by:

    1. M. Musegaas & P. E. M. Borm & M. Quant, 2018. "Three-valued simple games," Theory and Decision, Springer, vol. 85(2), pages 201-224, August.
    2. Matthew Gould & Matthew D. Rablen, 2017. "Reform of the United Nations Security Council: equity and efficiency," Public Choice, Springer, vol. 173(1), pages 145-168, October.
    3. Josep Freixas & Montserrat Pons, 2022. "A critical analysis on the notion of power," Annals of Operations Research, Springer, vol. 318(2), pages 911-933, November.
    4. 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.
    5. Matthew Gould & Matthew D. Rablen, 2013. "Equitable Representation in the Councils of the United Nations: Theory and Application," CEDI Discussion Paper Series 13-07, Centre for Economic Development and Institutions(CEDI), Brunel University.
    6. Sascha Kurz, 2014. "Measuring Voting Power in Convex Policy Spaces," Economies, MDPI, vol. 2(1), pages 1-33, March.
    7. Izabella Stach, 2022. "Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions," Group Decision and Negotiation, Springer, vol. 31(2), pages 317-334, April.
    8. Matthew Gould & Matthew D. Rablen, 2016. "Equitable representation in councils: theory and an application to the United Nations Security Council," Public Choice, Springer, vol. 169(1), pages 19-51, October.
    9. Josep Freixas & Montserrat Pons, 2021. "An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval," Group Decision and Negotiation, Springer, vol. 30(2), pages 447-462, April.
    10. Joseph Armel Momo Kenfack & Bertrand Tchantcho & Bill Proces Tsague, 2019. "On the ordinal equivalence of the Jonhston, Banzhaf and Shapley–Shubik power indices for voting games with abstention," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 647-671, June.
    11. Michela Chessa & Vito Fragnelli, 2022. "The Italian referendum: what can we get from game theory?," Annals of Operations Research, Springer, vol. 318(2), pages 849-869, November.
    12. Friedel Bolle, 2022. "Voting with abstention," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 24(1), pages 30-57, February.
    13. Josep Freixas, 2020. "The Banzhaf Value for Cooperative and Simple Multichoice Games," Group Decision and Negotiation, Springer, vol. 29(1), pages 61-74, February.
    14. Bertrand Mbama Engoulou & Pierre Wambo & Lawrence Diffo Lambo, 2023. "Banzhaf–Coleman–Dubey–Shapley sensitivity index for simple multichoice voting games," Annals of Operations Research, Springer, vol. 328(2), pages 1349-1364, September.
    15. Bertrand Mbama Engoulou & Pierre Wambo & Lawrence Diffo Lambo, 2023. "A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input," Group Decision and Negotiation, Springer, vol. 32(4), pages 871-888, August.
    16. Josep Freixas & Roberto Lucchetti, 2016. "Power in voting rules with abstention: an axiomatization of a two components power index," Annals of Operations Research, Springer, vol. 244(2), pages 455-474, September.

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