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Fair Voting Rules in Committees

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Power, Voting, and Voting Power: 30 Years After

Abstract

In simple weighted committees with a finite number of members, fixed weights, and changing quota there exist a finite number of different quota intervals of stable power with the same sets of winning coalitions for all quotas from each of them. If in a committee the sets of winning coalitions for different quotas are the same, then the power indices based on pivots, swings, or minimal winning coalitions are also the same for those quotas. If the fair distribution of voting weights is defined, then the fair distribution of voting power means to find a quota that minimizes the distance between relative voting weights and relative voting power (optimal quota). The problem of the optimal quota has an exact solution via the finite number of quotas from different intervals of stable power.

Earlier version of this chapter has been published in Homo Oeconomicus 27(4); see Turnovec (2011).

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Notes

  1. 1.

    The square root rule is based on the following propositions: Let us assume n units with population p1, p2, …, pn, and the system of representation by a super-unit committee with voting weights w1, w2, …, wn. It can be rigorously proved that for sufficiently large pi the absolute Penrose-Banzhaf power of individual citizen of unit i in unit’s referendum is proportional to the square root of pi. If the relative Penrose-Banzhaf voting power of unit i representation is proportional to its voting weight, then indirect voting power of each individual citizen of unit i is proportional to the product of voting weight wi and square root of population pi. Based on the conjecture (not rigorously proved) that for n large enough the relative voting power is proportional to the voting weights, the square root rule concludes that the voting weights of the units’ representations in the super-unit committee, proportional to square roots of units’ population, lead to the same indirect voting power of each citizen independently of the unit she is affiliated with.

  2. 2.

    Supporters of the Penrose-Banzhaf power concept sometimes reject the Shapley-Shubik index as a measure of voting power. Their objections to the Shapley-Shubik power concept are based on the classification of power measures on so-called I-power (voter’s potential influence over the outcome of voting) and P power (expected relative share in a fixed prize available to the winning group of committee members, based on cooperative game theory) introduced by Felsenthal, Machover and Zwicker (1998). The Shapley-Shubik power index was declared to represent P-power and as such is unusable for measuring influence in voting. We tried to show in Turnovec (2007) and Turnovec, Mercik, Mazurkiewicz (2008) that objections against the Shapley-Shubik power index, based on its interpretation as a P-power concept, are not sufficiently justified. Both Shapley-Shubik and Penrose-Banzhaf measure could be successfully derived as cooperative game values, and at the same time both of them can be interpreted as probabilities of being in some decisive position (pivot, swing) without using cooperative game theory at all.

  3. 3.

    The definition of an absolute HP power index is provided by the author (a similar definition of absolute PB power can be found in Brueckner (2001), the only difference is that we relate the number of MWC positions of member i to the total number of coalitions, not to the number of coalitions of which i is a member).

  4. 4.

    For a discussion about the possible probabilistic interpretation of the relative PB and HP, see Widgrén (2001).

  5. 5.

    The index of fairness follows the same logic as measures of deviation from proportionality used in political science to evaluate the difference between results of an election and the composition of an elected body—e.g., the measure given in Loosemore and Hanby (1971) is based on the absolute values of the deviation metric, or Gallagher (1991) using a square roots metric.

References

  • Baldwin, R., & Widgrén, M. (2004). Winners and losers under various dual majority rules in the EU’s council of ministers. In M. Wiberg (Ed.), Reasoned choices (pp. 42–89). Turku: The Finnish Political Science Association.

    Google Scholar 

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

    Google Scholar 

  • Berg, S., & Holler, M. J. (1986). Randomized decision rules in voting games: A model for strict proportional power. Quality and Quantity, 20, 419–429.

    Article  Google Scholar 

  • Brueckner, M. (2001). Expected probabilistic characterization of power indices. In M.J. Holler & G. Owen (Eds.), Power indices and coalition formation, Boston-Dordrecht-London: Kluwer.

    Google Scholar 

  • Deegan, J., & Packel, E. W. (1978). A new index of power for simple n-person games. International Journal of Game Theory, 7, 113–123.

    Article  Google Scholar 

  • Felsenthal, D.S., & Machover, M. (1998). The Measurement of Voting Power, Theory and Practice, Cheltenham: Edward Elgar.

    Google Scholar 

  • Felsenthal, D.S., & Machover, M. (2004), Analysis of QM rule adopted by the EU inter-governmental conference, Brussels, 18 June 2004, Research report online, London: LSE. [http://eprints.lse.ac.uk/2531].

  • Felsenthal, D.S., & Machover, M. (2007). Analysis of QM rule adopted by the council of the European Union, Brussels, 23 June 2007, Research report online, London: LSE. [http://eprints.lse.ac.uk/archive0000431].

  • Felsenthal, D. S., Machover, M., & Zwicker, W. (1998). The bicameral postulates and indices of a priori voting power. Theory and Decision, 44, 83–116.

    Article  Google Scholar 

  • Gale, D. (1960). The theory of linear economic models. New York: McGraw-Hill.

    Google Scholar 

  • Gallagher, M. (1991). Proportionality, disproportionality and electoral systems. Electoral Studies, 10, 33–51.

    Article  Google Scholar 

  • Holler, M. J. (1982a). An introduction to the analysis of power, voting, and voting power. In M. J. Holler (Ed.), Power, voting and voting power (pp. 15–30). Würzburg, Wien: Physica Verlag.

    Chapter  Google Scholar 

  • Holler, M. J. (1982b). Forming coalitions and measuring voting power. Political Studies, 30, 262–271.

    Article  Google Scholar 

  • Holler, M. J. (1985). Strict proportional power in voting bodies. Theory and Decision, 19, 249–258.

    Article  Google Scholar 

  • Holler, M. J. (1987). Paradox proof decision rules in weighted voting. In M. J. Holler (Ed.), The logic of multiparty systems (pp. 425–436). Dordrecht: Martinus Nijhoff Publishers.

    Chapter  Google Scholar 

  • Holler, M. J., & Packel, E. (1983). Power, luck and the right index. Zeitschrift für Nationalőkonomie (Journal of Economics), 43, 21–29.

    Article  Google Scholar 

  • Hosli, M. O. (2008). Council decision rules and european union constitutional design. AUCO Czech Economic Review, 2(1), 76–96.

    Google Scholar 

  • Johnston, R. J. (1978). On the measurement of power: Some reaction to laver. Environment and Planning, A10, 907–914.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Leech, D., & Aziz, H. (2008). The double majority voting rule of the EU reform treaty as a democratic ideal for an enlarging union: An appraisal using voting power analysis, Presented at the 2008 EPCS meeting in Jena.

    Google Scholar 

  • Loosemore, J., & Hanby, J. V. (1971). The theoretical limits of maximum some analytical expression for electoral systems. British Journal of Political Science, 1, 467–477.

    Article  Google Scholar 

  • Nurmi, H. (1982). The problem of the right distribution of voting power. In M. J. Holler (Ed.), Power, voting and voting power (pp. 203–212). Würzburg, Wien: Physica Verlag.

    Google Scholar 

  • Nurmi, H. (1997a). The representation of voter groups in the European parliament. A Penrose-Banzhaf index analysis. Electoral Studies, 16, 317–322.

    Article  Google Scholar 

  • Nurmi, H. (1997b). On power indices and minimal winning coalitions. Control and Cybernetics, 26(4), 609–611.

    Google Scholar 

  • Owen, G. (1972). Multilinear extensions of games. Management Science, 18, 64–79.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2006). Penrose voting system and optimal quota. Acta Physica Polonica, B37(11), 3133–3143.

    Google Scholar 

  • Słomczyński, W., & Życzkowski, K. (2007). From a toy model to double square root system. Homo Oeconomicus, 24(3–4), 381–400.

    Google Scholar 

  • Turnovec, F. (1997). Power, power indices and intuition. Control and Cybernetics, 26(4), 613–615.

    Google Scholar 

  • Turnovec, F. (2007). New measure of voting power. AUCO Czech Economic Review, 1(1), 4–14.

    Google Scholar 

  • Turnovec, F. (2008a). National, political and institutional influence in European Union decision making. AUCO Czech Economic Review, 2(2), 154–173.

    Google Scholar 

  • Turnovec, F. (2008b). Efficiency of fairness in voting systems. Homo Oeconomicus, 25(3–4), 335–348.

    Google Scholar 

  • Turnovec, F. (2009). Fairness and squareness: Fair decision making rules in the EU council. Badania Operacyjne i Decyzje, 4, 109–124.

    Google Scholar 

  • Turnovec, F. (2011). Fair voting rules in committees, strict proportional power and optimal quota. Homo Oeconomicus, 27(4), 463–479.

    Google Scholar 

  • Turnovec, F., Mercik, J. W., & Mazurkiewicz, M. (2008). Power indices methodology: Decisiveness, pivots and swings. In M. Braham & F. Steffen (Eds.), Power, freedom and voting, essays in honor of Manfred J. Holler (pp. 23–37). Berlin, Heidelberg: Springer.

    Chapter  Google Scholar 

  • Widgrén, M. (2001). On the probabilistic relationship between the public good index and the normalized Banzhaf index. In M. J. Holler & G. Owen (Eds.), Power indices and coalition formation. Boston-Dordrecht-London: Kluwer.

    Google Scholar 

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Acknowledgments

This research was supported by the Czech Science Foundation, project No. 402/09/1066 “Political economy of voting behavior, rational voters’ theory and models of strategic voting” and by the Max Planck Institute of Economics in Jena. The author would like to thank Manfred J. Holler, Andreas Nohn and an anonymous referee for valuable comments to an earlier version of the chapter.

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  1. Institute of Economic Studies, Charles University in Prague, Opletalova 26, 110 00, Prague 1, Czech Republic

    František Turnovec

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  1. Institute of SocioEconomics, Department of Economics, University of Hamburg, Von-Melle-Park 5, Hamburg, 10146, Germany

    Manfred J Holler

  2. Department of Political Science and, Contemporary History, University of Turku, Assistentinkatu 7, Turku, 20014, Finland

    Hannu Nurmi

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Turnovec, F. (2013). Fair Voting Rules in Committees. 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_19

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  • DOI: https://doi.org/10.1007/978-3-642-35929-3_19

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