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A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input

Author

Listed:
  • Bertrand Mbama Engoulou

    (The University of Douala)

  • Pierre Wambo

    (Ecole Normale Superieure)

  • Lawrence Diffo Lambo

    (Ecole Normale Superieure)

Abstract

In this paper, we propose an axiomatization of the totally critical raw Banzhaf index (TCRBI). In the literature, the TCRBI was introduced as a generalization of the Banzhaf power index on the class of yes–no voting games with several ordered levels of approval in input ((j, 2)-simple games). Its principal strength, compared to earlier generalizations of the Banzhaf power index, is the fact that it preserves the desirability relation regardless of the number of input levels of approval. Our main result is that, up to a multiplicative positive real number, the TCRBI is the only power index on the class of (j, 2)-simple games, satisfying the following set of axioms : Equal treatment, Positivity, Null Player, and Linear Transfer.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:grdene:v:32:y:2023:i:4:d:10.1007_s10726-023-09831-3
    DOI: 10.1007/s10726-023-09831-3
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    References listed on IDEAS

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    1. 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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    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. Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
    7. Josep Freixas, 2020. "The Banzhaf Value for Cooperative and Simple Multichoice Games," Group Decision and Negotiation, Springer, vol. 29(1), pages 61-74, February.
    8. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
    9. Dominique Lepelley & N. Andjiga & F. Chantreuil, 2003. "La mesure du pouvoir de vote," Post-Print halshs-00069255, HAL.
    10. Pongou, Roland & Tchantcho, Bertrand & Tedjeugang, Narcisse, 2014. "Power theories for multi-choice organizations and political rules: Rank-order equivalence," Operations Research Perspectives, Elsevier, vol. 1(1), pages 42-49.
    11. Giulia Bernardi, 2018. "A New Axiomatization of the Banzhaf Index for Games with Abstention," Group Decision and Negotiation, Springer, vol. 27(1), pages 165-177, February.
    12. 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.
    13. 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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