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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andjiga NG, Chantreuil F, Lepelley D (2003) Lames ure du pouvoir de vote. Math Sci Hum 163:111–145
Banzhaf J (1965) Weighted voting doesn’t work : a mathematical analysis. Rutgers Law Rev 19:317–343
Banzhaf JF (1966) Multi-member electoral districts-do they violate the one man, one vote principle. Yale Law J 75:1309–1338
Banzhaf JF (1968) One man, 3.312 votes: a mathematical analysis of the electoral college. Vilanova Law Rev 13:304–332
Bernardi G (2018) A new axiomatization of Banzhaf index for games with abstention. Group Decis Negot 1:165–177
Diffo Lambo L, Moulen J (2001) Quel pouvoir mesure-t-on dans un jeu de vote. Math Sci Hum 152:27–47
Diffo Lambo L, Moulen J (2002) Ordinal equivalence of power notions in voting games. Theory Dec 53:313–325
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4:99–131
Felsenthal DS, Machover M (1997) Ternary voting games. Int J Game Theory 26:335–351
Freixas J (2005) The Banzhaf power index for games with several levels of approval in the input and output. Ann Oper Res 137:45–66
Freixas J (2012) Probabilistic power indices for voting rules with abstention. MASS 64:89–99
Freixas J (2020) The Banzhaf value for cooperative and simple multichoice games. Group. Decis Negot 29:61–74
Freixas J, Lucchetti R (2016) Power in voting rules with abstention: an axiomatization of two components power index. Ann Oper Res 2:455–474
Freixas J, Pons M (2021) An appropriate way to extend the Banzhaf index for multiple levels of approval. Group Decis Negot 30:447–462
Freixas J, Tchantcho B, Tedjeugang N (2014) Voting games with abstention: linking completeness and weightedness. Decis Support Syst 57:172–177
Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21:399–431
Haimanko O (2017) The axiom of equivalence to individual power and the Banzhaf index. Behav Games Econ. https://doi.org/10.1016/j.geb.2017.05.003
Isbell R (1958) A class of simple games. Duke Math J 25:423–439
Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Oper Res 1:89–104
Parker C (2012) The influence relation for ternary voting games. Games Econ Behav 75:867–81
Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57
Pongou R, Tchantcho B, Tedjeugang N (2014) Power theories for multi-choice organizations and political rules: rank-order equivalence. Oper Res Perspect 1:42–49
Shapley LS, Shubik M (1954) A model for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792
Taylor AD, Zwicker WS (1993) Weighted voting, multicameral representation, and power. Games Econ Behav 5:170–181
Taylor AD, Zwicker WS (1999) Simple Games. Princeton Univ. Press, Princeton
Tchantcho B, Diffo Lambo L, Pongou R, Mbama Engoulou B (2008) Voters’ power in voting games with abstention: influence relation and ordinal equivalence of power theories. Games Econ Behav 64:335–350
Tomiyama Y (1987) Simple game, voting representation and ordinal power equivalence. Int J Polit Inf 11:67–75
Acknowledgements
The authors want to thank anonymous reviewers for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mbama Engoulou, B., Wambo, P. & Diffo Lambo, L. A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input. Group Decis Negot 32, 871–888 (2023). https://doi.org/10.1007/s10726-023-09831-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10726-023-09831-3