Annals of Operations Research

, Volume 215, Issue 1, pp 137–163 | Cite as

Heuristic and exact solutions to the inverse power index problem for small voting bodies

  • Sascha Kurz
  • Stefan Napel


Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. The paper considers approximations to this inverse problem for the Penrose-Banzhaf index by hill-climbing algorithms and exact solutions which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.


Electoral systems Simple games Weighted voting games Square root rule Penrose limit theorem Penrose-Banzhaf index Institutional design 


  1. Alon, N., & Edelman, P. H. (2010). The inverse Banzhaf problem. Social Choice and Welfare, 34(3), 371–377. CrossRefGoogle Scholar
  2. Aziz, H., Paterson, M., & Leech, D. (2007). Efficient algorithm for designing weighted voting games. In IEEE international, multitopic conference, 2007, INMIC 2007 (pp. 1–6). Available at Google Scholar
  3. Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19(2), 317–343. Google Scholar
  4. Carreras, F., & Freixas, J. (1996). Complete simple games. Mathematical Social Sciences, 32(2), 139–155. CrossRefGoogle Scholar
  5. Chang, P.-L., Chua, V. C., & Machover, M. (2006). L.S. Penrose’s limit theorem: tests by simulation. Mathematical Social Sciences, 51(1), 90–106. CrossRefGoogle Scholar
  6. De Nijs, F., & Wilmer, D. (2012). Evaluation and improvement of Laruelle-Widgrén inverse Banzhaf approximation. Mimeo. Available at
  7. Deĭneko, V. G., & Woeginger, G. J. (2006). On the dimension of simple monotonic games. European Journal of Operational Research, 170(1), 315–318. CrossRefGoogle Scholar
  8. Dubey, P., & Shapley, L. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4(2), 99–131. CrossRefGoogle Scholar
  9. Fatima, S., Wooldridge, M., & Jennings, N. (2008). An anytime approximation method for the inverse Shapley value problem. In Proceedings of the 7th international conference on autonomous agents and multi-agent systems, Estoril, Portugal (pp. 935–942). Available at
  10. Felsenthal, D., & Machover, M. (1998). The measurement of voting power—theory and practice, problems and paradoxes. Cheltenham: Edward Elgar. Google Scholar
  11. Freixas, J., & Kurz, S. (2011). On minimal integer representations of weighted games. Mimeo. Available at
  12. Freixas, J., & Molinero, X. (2009). On the existence of a minimum integer representation for weighted voting systems. Annals of Operations Research, 166, 243–260. CrossRefGoogle Scholar
  13. Freixas, J., & Molinero, X. (2010). Weighted games without a unique minimal representation in integers. Optimization Methods & Software, 25(2), 203–215. CrossRefGoogle Scholar
  14. 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. CrossRefGoogle Scholar
  15. Isbell, J. R. (1956). A class of majority games. Quarterly Journal of Mathematics, 7(1), 183–187. CrossRefGoogle Scholar
  16. Kaniovski, S. (2008). The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent. Social Choice and Welfare, 31(2), 281–300. CrossRefGoogle Scholar
  17. Keijzer, B. d. (2009). On the design and synthesis of voting games. Master’s thesis, Delft University of Technology. Google Scholar
  18. Keijzer, B. d., Klos, T., & Zhang, Y. (2010). Enumeration and exact design of weighted voting games. In Proceedings of the 9th international conference on autonomous agents and multiagent systems (Vol. 1, pp. 391–398). Google Scholar
  19. Kirsch, W., Słomczyński, W., & Życzkowski, K. (2007). Getting the votes right. European Voice, (3–9 May), 12. Google Scholar
  20. Kurth, M. (2008). Square root voting in the Council of the European Union: rounding effects and the Jagiellonian compromise. Available at
  21. Kurz, S. (2012a). On minimum sum representations for weighted voting games. Annals of Operations Research, 196(1), 361–369. CrossRefGoogle Scholar
  22. Kurz, S. (2012b). On the inverse power index problem. Optimization, 61(8), 989–1011. CrossRefGoogle Scholar
  23. Kurz, S., Maaser, N., & Napel, S. (2012). On the egalitarian weights of nations. Mimeo, University of Bayreuth. Google Scholar
  24. Laruelle, A., & Valenciano, F. (2008). Voting and collective decision-making. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  25. Laruelle, A., & Widgrén, M. (1998). Is the allocation of power among EU states fair? Public Choice, 94(3–4), 317–340. CrossRefGoogle Scholar
  26. Leech, D. (2002a). Designing the voting system for the EU Council of Ministers. Public Choice, 113(3–4), 437–464. CrossRefGoogle Scholar
  27. Leech, D. (2002b). Voting power in the governance of the international monetary fund. Annals of Operations Research, 109(1), 375–397. CrossRefGoogle Scholar
  28. Leech, D. (2003). Power indices as an aid to institutional design: the generalised apportionment problem. In M. J. Holler, H. Kliemt, D. Schmidtchen, & M. E. Streit (Eds.), Jahrbuch für Neue Politische Ökonomie (Vol. 22). Tübingen: Mohr Siebeck. Google Scholar
  29. Lindner, I., & Machover, M. (2004). L.S. Penrose’s limit theorem: proof of some special cases. Mathematical Social Sciences, 47(1), 37–49. CrossRefGoogle Scholar
  30. Lindner, I., & Owen, G. (2007). Cases where the Penrose limit theorem does not hold. Mathematical Social Sciences, 53(3), 232–238. CrossRefGoogle Scholar
  31. Lucas, W. F. (1992). HistoMAP module: Vol. 19. Fair voting: weighted votes for unequal constituencies. Lexington: COMAP. Google Scholar
  32. Neyman, A. (1982). Renewal theory without replacements. Annals of Probability, 10(2), 464–481. CrossRefGoogle Scholar
  33. Nurmi, H. (1982). The problem of the right distribution of voting power. In M. J. Holler (Ed.), Power, voting, and voting power. Würzburg: Physica-Verlag. Google Scholar
  34. Pennisi, A., Ricca, F., Serafini, P., & Simeone, B. (2007). Amending and enhancing electoral laws through mixed integer programming in the case of Italy. In E. Yashin (Ed.), Proceedings of the 8th international conference on economic modernization and social development. Moscow: HSE. Google Scholar
  35. Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109(1), 53–57. CrossRefGoogle Scholar
  36. Penrose, L. S. (1952). On the objective study of crowd behaviour. London: Lewis Google Scholar
  37. Ricca, F., Scozzari, A., Serafini, P., & Simeone, B. (2012). Error minimization methods in biproportional apportionment. Top, 20(3), 547–577. CrossRefGoogle Scholar
  38. Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. The American Political Science Review, 48(3), 787–792. CrossRefGoogle Scholar
  39. Słomczyński, W., & Życzkowski, K. (2006). Penrose voting system and optimal quota. Acta Physica Polonica B, 37, 3133–3143. Google Scholar
  40. Słomczyński, W., & Życzkowski, K. (2007). From a toy model to the double square root system. Homo Oeconomicus, 24(3/4), 381–399. Google Scholar
  41. Słomczyński, W., & Życzkowski, K. (2011). Square root voting system, optimal threshold and π. Available at
  42. Taylor, A. D., & Zwicker, W. S. (1999). Simple games. Princeton: Princeton University Press. Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of BayreuthBayreuthGermany
  2. 2.Public Choice Research CentreTurkuFinland

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