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Square Root Voting System, Optimal Threshold and \( \uppi \)

Abstract

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favor of the square root voting system, in which voting weights of member states are proportional to the square root of their population. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold \(q\) for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic ‘union’ of \(M\) states and derive in this case an explicit approximate formula for the level of the optimal threshold: \(q\simeq 1/2+1/\sqrt{\pi M}\).

Keywords

  • European Union
  • Member State
  • Vote System
  • Small State
  • Large State

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter was presented to the Voting Power in Practice Symposium at the London School of Economics, 20–22 March 2011, sponsored by the Leverhulme Trust.

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References

  • Ade, F. (2006). Decision making in Europe: Were Spain and Poland right to stop the constitution in December 2003? (Preprint 2006), from http://congress.utu.fi/epcs2006/docs/D3_ade.pdf

  • Algaba, E., Bilbao, J. R., & Fernandez, J. R. (2007). The distribution of power in the European constitution. European Journal of Operational Research, 176, 1752–1766.

    CrossRef  Google Scholar 

  • Andjiga, N.-G., Chantreuil, F., & Leppeley, D. (2003). La mesure du pouvoir de vote. Mathematical Social Sciences, 163, 111–145.

    Google Scholar 

  • Baldwin, R. E., & Widgrén, M. (2004). Winners and losers under various dual majority rules for the EU Council of Ministers. CEPR Discussion Paper No. 4450, Centre for European Policy Studies, Brussels 2004, from http://www.cepr.org/pubs/dps/DP4450.asp

  • Banzhaf, J. F. (1965). Weighted voting does not work: A mathematical analysis. Rutgers Law Review, 19, 317–343.

    Google Scholar 

  • Bârsan-Pipu, N., & Tache, I. (2009). An analysis of EU voting procedures in the enlargement context. International Advances in Economic Research, 15, 393–408.

    CrossRef  Google Scholar 

  • Beisbart, C., Bovens, L., & Hartmann, S. (2005). A utilitarian assessment of alternative decision rules in the Council of Ministers. European Union Politics, 6, 395–419.

    CrossRef  Google Scholar 

  • Bengtsson, I., & Życzkowski, K. (2006). Geometry of quantum states. Cambridge: Cambridge UP.

    CrossRef  Google Scholar 

  • Chang, P.-L., Chua, V. C. H., & Machover, M. (2006). LS Penrose’s limit theorem: tests by simulation. Mathematical Social Sciences, 51, 90–106.

    CrossRef  Google Scholar 

  • Feix, M. R., Lepelley, D., Merlin, V., & Rouet, J. L. (2007). On the voting power of an alliance and the subsequent power of its members. Social Choice and Welfare, 28, 181–207.

    CrossRef  Google Scholar 

  • Felsenthal, D. S., & Machover, M. (1997). The weighted voting rule in the EUs Council of Ministers, 1958–95: Intentions and outcomes. Electoral Studies, 16, 33–47.

    CrossRef  Google Scholar 

  • Felsenthal, D. S., & Machover, M. (1998). Measurement of voting power: Theory and practice, problems and paradoxes. Cheltenham: Edward Elgar.

    Google Scholar 

  • Felsenthal, D. S., & Machover, M. (1999). Minimizing the mean majority deficit: The second square-root rule. Mathematical Social Science, 37, 25–37.

    CrossRef  Google Scholar 

  • Felsenthal, D. S., & Machover, M. (2001). Treaty of nice and qualified majority voting. Social Choice and Welfare, 18, 431–464.

    CrossRef  Google Scholar 

  • Gelman, A., Katz, J. M., & Tuerlinckx, F. (2002). The mathematics and statitics of voting power. Statistical Science, 17, 420–435.

    CrossRef  Google Scholar 

  • Gelman, A., Katz, J. M., & Bafumi, J. (2004). Standard voting power indexes do not work: an empirical analysis. British Journal of Political Science, 34, 657–674.

    CrossRef  Google Scholar 

  • Hosli, M. O. (2008). Council decision rules and European Union constitutional design. AUCO Czech Economic Review, 2, 76–96.

    Google Scholar 

  • Jones, K. R. W. (1991). Riemann-Liouville fractional integration and reduced distributions on hyperspheres. Journal of Physics A, 24, 1237–1244.

    CrossRef  Google Scholar 

  • Kirsch, W., Słomczyński, W., & Życzkowski, K. (2007). Getting the votes right. European Voice, 3–9, 12.

    Google Scholar 

  • Kirsch, W. (2007). On Penrose’s square-root law and beyond. Homo Oeconomicus, 24, 357–380.

    Google Scholar 

  • Kirsch, W. (2010). The distribution of power in the Council of Ministers of the European Union. In M. Cichocki & K. Życzkowski (Eds.), Institutional design and voting power in the European Union (pp. 93–107). Farnham: Ashgate Publishing Group.

    Google Scholar 

  • Kurth, M. (2007). Square root voting in the Council of the European Union: Rounding effects and the jagiellonian compromise, (Preprint math.GM 0712.2699).

    Google Scholar 

  • Laruelle, A., & Widgrén, M. (1998). Is the allocation of voting power among the EU states fair? Public Choice, 94, 317–339.

    CrossRef  Google Scholar 

  • Laruelle, A., & Valenciano, F. (2002). Inequality among EU citizens in the EU’s Council decision procedure. European Journal of Political Economy, 18, 475–498.

    CrossRef  Google Scholar 

  • Laruelle, A., & Valenciano, F. (2008). Voting and collective decision-making. Bargaining and power. Cambridge: Cambridge UP.

    CrossRef  Google Scholar 

  • Leech, D. (2002). Designing the voting system for the Council of the EU. Public Choice, 113, 437–464.

    CrossRef  Google Scholar 

  • Leech, D., & Machover, M. (2003). Qualified majority voting: The effect of the quota. In M. Holler, et al. (Eds.), European Governance, Jahrbuch für Neue Politische Ökonomie (pp. 127–143). Mohr Siebeck: Tübingen.

    Google Scholar 

  • Leech, D., & Aziz, H. (2010). The double majority voting rule of the EU reform treaty as a democratic ideal for an enlarging union: An appraisal using voting power analysis. In M. Cichocki & K. Życzkowski (Eds.), Institutional design and voting power in the European Union (pp. 59–73). Farnham: Ashgate Publishing Group.

    Google Scholar 

  • Lindner, I., & Machover, M. (2004). LS Penrose’s limit theorem: proof of some special cases. Mathematical Social Sciences, 47, 37–49.

    CrossRef  Google Scholar 

  • Machover, M. (2010). Penrose’s square root rule and the EU Council of the Ministers: Significance of the quota. In M. Cichocki & K. Życzkowski (Eds.), Institutional design and voting power in the European Union (pp. 35–42). Farnham: Ashgate Publishing Group.

    Google Scholar 

  • Moberg, A. (2010). Is the double majority really double? The voting rules in the Lisbon Treaty. In M. Cichocki & K. Życzkowski (Eds.), Institutional design and voting power in the European Union (pp. 19–34). Farnham: Ashgate Publishing Group.

    Google Scholar 

  • Morriss, P. (1987) Power: A philosophical analysis (2nd ed. 2002). Manchester UP: Manchester.

    Google Scholar 

  • Owen, G. (1975). Multilinear extensions and the Banzhaf value. Naval Research Logistics Quaterly, 22, 741–750.

    CrossRef  Google Scholar 

  • Pajala, A., & Widgrén, M. (2004). A priori versus empirical voting power in the EU Council of Ministers. European Union Politics, 5, 73–97.

    CrossRef  Google Scholar 

  • Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109, 53–57.

    CrossRef  Google Scholar 

  • Penrose, L. S. (1952). On the objective study of crowd behaviour. London: H.K. Lewis & Co.

    Google Scholar 

  • Pöppe, Ch. Die Quadratwurzel, das Irrationale und der Tod, Spektrum der Wissenschaft, August 2007, pp. 102–105.

    Google Scholar 

  • Pukelsheim, F. (2007). Der Jagiellonische Kompromiss. Neue Züricher Zeitung, 20 Juni 2007.

    Google Scholar 

  • Pukelsheim, F. (2010). Putting citizens first: Representation and power in the European Union in institutional design and voting power in the European Union. In M. Cichocki, & K. Życzkowski (Eds.), (pp. 235–253). Farnham: Ashgate Publishing Group

    Google Scholar 

  • Ramaley, J. F. (1969). Buffon’s noodle problem. American Mathematical Monthly, 76, 916–918.

    CrossRef  Google Scholar 

  • Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Science Review, 48, 787–792.

    CrossRef  Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2004) Voting in the European Union: The square root system of penrose and a critical point. preprint cond-mat.0405396.

    Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2006). Penrose voting system and optimal quota. Acta Physica Polonica B, 37, 3133–3143.

    Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2007). From a toy model to the double square root voting system. Homo Oeconomicus, 24, 381–399.

    Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2010). Jagiellonian compromise—an alternative voting system for the Council of the European Union. In M. Cichocki & K. Życzkowski (Eds.), Institutional design and voting power in the European Union (pp. 43–57). Farnham: Ashgate Publishing Group.

    Google Scholar 

  • Życzkowski, K., Słomczyński, W., & Zastawniak, T. (2006). Physics for fairer voting. Physics World, 19, 35–37.

    Google Scholar 

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Correspondence to Karol Życzkowski .

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Życzkowski, K., Słomczyński, W. (2013). Square Root Voting System, Optimal Threshold and \( \uppi \) . In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_30

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