Abstract
In order to study voting situations when voters can also abstain and the output is binary, i.e., either approval or rejection, a new extended model of voting rule was defined. Accordingly, indices of power, in particular Banzhaf’s index, were considered. In this paper we argue that in this context a power index should be a pair of real numbers, since this better highlights the power of a voter in two different cases, i.e., her being crucial when switching from being in favor to abstain, and from abstain to be contrary. We also provide an axiomatization for both indices, and from this a characterization as well of the standard Banzhaf index (the sum of the former two) is obtained. Some examples are provided to show how the indices behave.
Similar content being viewed by others
Notes
If we modelize this example as a weighted game the weights for yes are the same, the details are exposed in Example 1.3-(iii).
In the Sect. 6 we shall say something about axiomatization of \(I^{YN}_p\).
This shows that the null-axiom and the reduced axiom at the highest level imply YA-equal treatment on unanimity games, which by transfer can be extended to all games. YA-equal treatment for the (3, 2) game V and players \(p,r \in N\) means that \(P_p^{YA}[V] = P_r^{YA}[V]\) whenever \(V(S)-V(S_{p\downarrow })=V(S)-V(S_{r\downarrow })\) for all tripartitions \(S \in 3^N\) with \(p,r \in S_1\). Analogously, one may consider YN- and AN-equal treatment with the corresponding implications.
References
Albizuri, M. J., & Ruiz, L. M. (2001). A new axiomatization of the Banzhaf semivalue. Spanish Economic Review, 3, 97–109.
Alonso-Meijide, J. M., & Bowles, C. (2005). Generating functions for coalitional power indices: An application to the IMF. Annals of Operations Research, 137, 21–44.
Alonso-Meijide, J. M., Bilbao, J. M., Casas-Méndez, B., & Fernández, J. R. (2009). Weighted multiple majority games with unions: Generating functions and applications to the European Union. European Journal of Operational Research, 198, 530–544.
Amer, R., Carreras, F., & Magaña, A. (1998). The Banzhaf–Coleman index for games with r alternatives. Optimization, 44, 175–198.
Amer, R., Carreras, F., & Magaña, A. (1998). Extension of values to games with multiple alternatives. Annals of Operations Research, 84, 63–78.
Banzhaf, J. F. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19, 317–343.
Barua, R., Chakravarty, S. R., & Roy, S. (2005). A new characterization of the Banzhaf index of power. International Game Theory Review, 7, 545–553.
Bishnu, M., & Roy, S. (2012). Hierarchy of players in swap robust voting games. Social Choice and Welfare, 38, 11–22.
Bolger, E. M. (1986). Power indices for multicandidate voting games. International Journal of Game Theory, 15, 175–186.
Bolger, E. M. (1993). A value for games with \(n\) players and \(r\) alternatives. International Journal of Game Theory, 22, 319–334.
Bolger, E. M. (2000). A consistent value for games with \(n\) players and \(r\) alternatives. International Journal of Game Theory, 29, 93–99.
Braham, M., & Steffen, F. (2008). M. Braham & F. Steffen (Eds), Power, freedom, and voting (essays in honor of Manfred J. Holler). Berlin: Springer.
Cook, W. D. (2006). Distance-based and ad hoc consensus models in ordinal preference ranking. European Journal of Operational Research, 172, 369–385.
Côrte-Real, P. P., & Pereira, P. T. (2004). The voter who wasn’t there: Referenda, representation and abstention. Social Choice and Welfare, 22, 349–369.
Diffo Lambo, L., & Moulen, J. (2002). Ordinal equivalence of power notions in voting games. Theory and Decision, 53, 313–325.
Feltkamp, V. (1995). Alternative axiomatic characterizations of the Shapley and Banzhaf values. International Journal of Game Theory, 24, 179–186.
Felsenthal, D. S., & Machover, M. (1997). Ternary voting games. International Journal of Game Theory, 26, 335–351.
Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Theory and practice, problems and paradoxes. Cheltenham: Edward Elgar.
Freixas, J. (2005). Banzhaf measures for games with several levels of approval in the input and output. Annals of Operations Research, 137, 45–66.
Freixas, J. (2005). The Shapley–Shubik power index for games with several levels of approval in the input and output. Decision Support Systems, 39, 185–195.
Freixas, J. (2012). Probabilistic power indices for voting rules with abstention. Mathematical Social Sciences, 64, 89–99.
Freixas, J., & Parker, C. (2015). Manipulation in games with multiple levels of output. Journal of Mathematical Economics, 61, 144–151.
Freixas, J., & Zwicker, W. S. (2003). Weighted voting, abstention, and multiple levels of approval. Social Choice and Welfare, 21, 399–431.
Freixas, J., Marciniak, D., & Pons, M. (2012). On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices. European Journal of Operational Research, 216, 367–375.
Freixas, J., Tchantcho, B., & Tedjeugang, N. (2014). Achievable hierarchies in voting games with abstention. European Journal of Operational Research, 236(1), 254–260.
Freixas, J., Tchantcho, B., & Tedjeugang, N. (2014). Voting games with abstention: Linking completeness and weightedness. Decision Support Systems, 57, 172–177.
Haller, H. (1994). Collusion properties of values. International Journal of Game Theory, 23, 261–281.
Holler, M. J., Ono, R., & Steffen, F. (2001). Constrained monotonicity and the measurement of power. Theory and Decision, 50, 385–397.
Holler, M. J., & Napel, S. (2004). Monotonicity of power indices and power measures. Theory and Decision, 56, 93–111.
Jones, M., & Wilson, J. (2010). Multilinear extensions and values for multichoice games. Computational Statistics, 72(1), 145–169.
Leech, D. (2002). Voting power in the governance of the International Monetary Fund. Annals of Operations Research, 109, 375–397.
Lehrer, E. (1988). An axiomatization of the Banzhaf value. International Journal of Game Theory, 17, 89–99.
Levitin, G. (2003). Optimal allocation of multi-state elements in linear consecutively connected systems with vulnerable nodes. European Journal of Operational Research, 150, 406–419.
Monroy, L., & Fernández, F. R. (2014). Banzhaf index for multiple voting systems. An application to the European Union. Annals of Operations Research, 215, 215–230.
Morriss, P. (2002). Power: A philosophical analysis (2nd ed.). Manchester: Manchester University Press.
Obata, W., & Ishii, H. (2003). A method for discriminating efficient candidates with ranked voting data. European Journal of Operational Research, 151, 233–237.
Ono, R. (2001). Values for multialternative games and multilinear extensions. In M. J. Holler & G. Owen (Eds.), Power indices and coalition formation (pp. 63–86). Kluwer Dordrecht: Academic Publishers.
Owen, G. (1978). Characterization of the Banzhaf–Coleman index. SIAM Journal of Applied Mathematics, 35, 315–327.
Parker, C. (2012). The influence relation for ternary voting games. Games and Economic Behavior, 75, 867–881.
Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109, 53–57.
Straffin, P. D. (1982). Power indices in politics. In S. J. Brams, W. F. Lucas, & P. D. Straffin (Eds.), Political and Related Models (Vol. 2, pp. 256–321). New York: Springer.
Tchantcho, B., Diffo Lambo, L., Pongou, R., & Mbama Engoulou, B. (2008). Voters’ power in voting games with abstention. Games and Economic Behavior, 64, 335–350.
Acknowledgments
The ideas of the paper were discussed, and the paper itself was prepared mostly during some exchange visits of the two authors. Both are grateful to the hosting departments for their warm hospitality. They also acknowledge a grant from GNAMPA, CNR, supporting the visit in Italy of the first author and Grant MTM2012-34426/FEDER of the Ministry of Economy and Competitiveness. The research of the second author was partially supported by Ministero dell’Istruzione, dell’Universitá e della Ricerca Scientifica (COFIN 2009). We also thank the three referees of the paper and Giulia Bernardi for their helpful comments, which allowed to improve the final version of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Freixas, J., Lucchetti, R. Power in voting rules with abstention: an axiomatization of a two components power index. Ann Oper Res 244, 455–474 (2016). https://doi.org/10.1007/s10479-016-2124-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-016-2124-5