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On the ordinal equivalence of the Jonhston, Banzhaf and Shapley–Shubik power indices for voting games with abstention

Author

Listed:
  • Joseph Armel Momo Kenfack

    (The University of Yaoundé I (MASS)
    THEMA Laboratory, The University of Cergy-Pontoise)

  • Bertrand Tchantcho

    (The University of Yaoundé I (MASS)
    THEMA Laboratory, The University of Cergy-Pontoise)

  • Bill Proces Tsague

    (The University of Yaoundé I (MASS))

Abstract

The aim of this paper is twofold. We extend the well known Johnston power index usually defined for simple voting games, to voting games with abstention and we provide a full characterization of this extension. On the other hand, we conduct an ordinal comparison of three power indices: the Shapley–Shubik, Banzhaf and newly defined Johnston power indices. We provide a huge class of voting games with abstention in which these three power indices are ordinally equivalent. This is clearly a generalization of the work by Freixas et al. (Eur J Oper Res 216:367–375, 2012) and a twofold extension of Parker (Games Econ Behav 75:867–881, 2012) in the sense that, the ordinal equivalence emerges for three power indices (not just for the Shapley–Shubik and the Banzhaf indices), and it holds for a class of games strictly larger than the class of I-complete (3,2) games namely semi I-complete (3,2) games.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jogath:v:48:y:2019:i:2:d:10.1007_s00182-018-0650-x
    DOI: 10.1007/s00182-018-0650-x
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    References listed on IDEAS

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    1. Sébastien Courtin & Bertrand Tchantcho, 2015. "A note on the ordinal equivalence of power indices in games with coalition structure," Theory and Decision, Springer, vol. 78(4), pages 617-628, April.
    2. Laruelle,Annick & Valenciano,Federico, 2011. "Voting and Collective Decision-Making," Cambridge Books, Cambridge University Press, number 9780521182638, September.
    3. Lorenzo-Freire, S. & Alonso-Meijide, J.M. & Casas-Mendez, B. & Fiestras-Janeiro, M.G., 2007. "Characterizations of the Deegan-Packel and Johnston power indices," European Journal of Operational Research, Elsevier, vol. 177(1), pages 431-444, February.
    4. Sébastien Courtin & Bertrand Tchantcho, 2015. "A note on the ordinal equivalence of power indices in games with coalition structure," Post-Print hal-00914910, HAL.
    5. Dominique Lepelley & N. Andjiga & F. Chantreuil, 2003. "La mesure du pouvoir de vote," Post-Print halshs-00069255, HAL.
    6. Roland Pongou & Bertrand Tchantcho & Lawrence Diffo Lambo, 2011. "Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity," Theory and Decision, Springer, vol. 70(2), pages 157-178, February.
    7. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
    8. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    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.
    10. 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.
    11. Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
    12. R J Johnston, 1978. "On the Measurement of Power: Some Reactions to Laver," Environment and Planning A, , vol. 10(8), pages 907-914, August.
    13. Freixas, Josep & Marciniak, Dorota & Pons, Montserrat, 2012. "On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices," European Journal of Operational Research, Elsevier, vol. 216(2), pages 367-375.
    14. 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.
    15. 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.
    16. Rubinstein, Ariel, 1980. "Stability of decision systems under majority rule," Journal of Economic Theory, Elsevier, vol. 23(2), pages 150-159, October.
    17. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    18. Lawrence Diffo Lambo & Joël Moulen, 2002. "Ordinal equivalence of power notions in voting games," Theory and Decision, Springer, vol. 53(4), pages 313-325, December.
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    Cited by:

    1. 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.
    2. 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.

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