Abstract
Multi-criteria simple games constitute an extension of the basic framework of voting systems and collective decision-making. The study of power index plays an important role in the theory of multi-criteria simple games. Thus, in this paper, we propose the extended Banzhaf index for these games, as the natural generalization of this index in conventional simple games. This approach allows us to compare various criteria simultaneously. An axiomatic characterization of this power index is established. The Banzhaf index is computed by taking into account the minimal winning coalitions of each class. Since this index depends on the number of ways in which each player can effect a swing, one of the main difficulties for finding this index is that it involves a large number of computations. We propose a combinatorial procedure, based on generating functions, to obtain the Banzhaf index more efficiently for weighted multi-criteria simple games. As an application, the distribution of voting power in the European Union is calculated.
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Notes
For x,y∈R k we denote x≥y⇔x i ≥y i , x≠y.
References
Algaba, E., Bilbao, J. M., Fernández, J. R., & López, J. J. (2003). Computing power indices in weighted multiple majority games. Mathematical Social Sciences, 46, 63–80.
Alonso-Mejide, J. M., & Bowles, C. (2005). Generating functions for coalitional power indices: an application to the IMF. Annals of Operations Research, 137, 21–44.
Amer, R., Carreras, F., & Magaña, M. (1998). Extension of values to games with multiple alternatives. Annals of Operations Research, 84, 63–78.
Banzhaf, J. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 317–343.
Bolger, E. M. (1993). A value for games with n players and r alternatives. International Journal of Game Theory, 22, 319–334.
Brams, S. J., & Affuso, P. J. (1976). Power and size. A new paradox. Theory and Decision, 7, 29–56.
Dubey, P., & Shapley, L. S. (1979). Mathematical properties of the Banzhaf index. Mathematics of Operations Research, 4, 99–131.
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. London: Edward Elgar.
Fernández, J. R., Algaba, E., Bilbao, J. M., Jiménez, A., Jiménez, N., & López, J. J. (2002). Generating functions for computing the Myerson value. Annals of Operations Research, 109, 143–158.
Fishburn, P. C. (1973). The theory of social choice. Princeton: Princeton University Press.
Freixas, J. (2005). Banzhaf measures for games with several levels of approval in the input and output. Annals of Operations Research, 137, 45–65.
Freixas, J., & Zwicker, W. S. (2003). Weighted voting, abstention, and multiple levels of approval. Social Choice and Welfare, 21, 399–431.
Grilli di Cortona, P., Manzi, C., Pennisi, A., Ricca, F., & Simeone, B. (1999). SIAM monographs on discrete mathematics and applications. Evaluation and optimization of electoral systems. Philadelphia: SIAM.
Hsiao, C. R., & Raghavan, T. E. S. (1993). Shapley value for multichoice cooperative games, I. Games and Economic Behavior, 5, 240–256.
Monroy, L., & Fernández, F. R. (2007). Weighted multi-criteria simple games and voting systems. Foundations of Computing and Decision Sciences, 32, 295–313.
Monroy, L., & Fernández, F. R. (2009). A general model for voting systems with multiple alternatives. Applied Mathematics and Computation, 215, 1537–1547.
Monroy, L., & Fernández, F. R. (2011). The Shapley–Shubik index for multi-criteria simple games. European Journal of Operational Research, 209, 122–128.
Pongou, R., Tchantcho, B., & Diffo Lambo, L. (2011). Political influence in multi-choice institutions: cyclicity, anonymity and transitivity. Theory and Decision, 70, 157–178.
Rubinstein, A. (1980). Stability of decision systems under majority rule. Journal of Economic Theory, 23, 150–159.
Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. The American Political Science Review, 48, 787–792.
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The research of the authors is partially supported by the Andalusian Ministry of Economics, Innovation and Science project P09-SEJ-4903, the Spanish Ministry of Science and Innovation projects ECO2011-29801-C02-01, and the Spanish Ministry of Science and Technology project MTM2010-19576-C02-01.
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Monroy, L., Fernández, F.R. Banzhaf index for multiple voting systems. An application to the European Union. Ann Oper Res 215, 215–230 (2014). https://doi.org/10.1007/s10479-013-1374-8
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DOI: https://doi.org/10.1007/s10479-013-1374-8