Annals of Operations Research

, Volume 196, Issue 1, pp 361–369 | Cite as

On minimum sum representations for weighted voting games

  • Sascha KurzEmail author


A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.


Simple games Weighted voting games Minimum realizations Realizations with minimum sum 



The author thanks the anonymous referees for carefully reading a preliminary version of this article and giving useful comments to improve the presentation.


  1. Alon, N., & Edelman, P. H. (2010). The inverse Banzhaf problem. Social Choice and Welfare, 34(3), 371–377. CrossRefGoogle Scholar
  2. Carreras, F., & Freixas, J. (1996). Complete simple games. Mathematical Social Sciences, 32, 139–155. CrossRefGoogle Scholar
  3. de Keijzer, B. (2009). On the design and synthesis of voting games. Master’s thesis, Delft University of Technology. Google Scholar
  4. de Keijzer, B., 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
  5. Freixas, J., & Molinero, X. (2009). On the existence of a minimum integer representation for weighted voting games. Annals of Operations Research, 166, 243–260. CrossRefGoogle Scholar
  6. Freixas, J., & Molinero, X. (2010). Weighted games without a unique minimal representation in integers. Optimization Methods & Software, 25(2), 203–215. CrossRefGoogle Scholar
  7. Isbell, J. (1958). A class of simple games. Duke Mathematical Journal, 25, 423–439. CrossRefGoogle Scholar
  8. Isbell, J. (1959). On the enumeration of majority games. Mathematical Tables and Other Aids To Computation, 13, 21–28. CrossRefGoogle Scholar
  9. Krohn, I., & Sudhölter, P. (1995). Directed and weighted majority games. ZOR. Zeitschrift Für Operations-Research, 42(2), 189–216. Google Scholar
  10. Kurz, S. (2012). On the inverse power index problem. Optimization. doi: 10.1080/02331934.2011.587008. 21 pp. Google Scholar
  11. Kurz, S., & Tautenhahn, N. (2012, accepted). On Dedekind’s problem for complete simple games. International Journal on Game Theory, 25 p. Google Scholar
  12. Muroga, S., Toda, I., & Kondo, M. (1962). Majority decision functions of up to six variables. Mathematics of Computation, 16, 459–472. CrossRefGoogle Scholar
  13. Muroga, S., Tsuboi, T., & Baugh, C. R. (1970). Enumeration of threshold functions of eight variables. IEEE Transactions on Computers, 19, 818–825. CrossRefGoogle Scholar
  14. Niskanen, S., & Östergård, P. (2003). Cliquer user’s guide, version 1.0 (Tech. Rep. T48). Communications Laboratory, Helsinki University of Technology. Google Scholar
  15. Östergård, P. R. J. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120(1–3), 197–207. CrossRefGoogle Scholar
  16. Peled, U. N., & Simeone, B. (1985). Polynomial-time algorithms for regular set-covering and threshold synthesis. Discrete Applied Mathematics, 12, 57–69. CrossRefGoogle Scholar
  17. Read, R. C. (1978). Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations. Annals of Discrete Mathematics, 2, 107–120. CrossRefGoogle Scholar
  18. Sudhölter, P. (1996). The modified nucleolus as canonical representation of weighted majority games. Mathematics of Operations Research, 21(3), 734–756. CrossRefGoogle Scholar
  19. Tautenhahn, N. (2008). Enumeration einfacher Spiele mit Anwendungen in der Stimmgewichtsverteilung. Master’s thesis, Bayreuth, 269 pages, in German. Google Scholar
  20. Taylor, A. D., & Zwicker, W. S. (1999). Simple games. Desirability relations, trading, pseudoweightings. New Jersey: Princeton University Press. 246 p. Google Scholar
  21. Winder, R. O. (1965). Enumeration of seven-argument threshold functions. IEEE Transactions on Electronic Computers, 14, 315–325. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Physics, and Computer ScienceUniversity of BayreuthBayreuthGermany

Personalised recommendations