Skip to main content

A characterization of weighted simple games based on pseudoweightings


The paper provides a new characterization of weighted games within the class of simple games. It is based on a stronger form of the point-set-additive pseudoweighting property of simple games. The characterization obtained is of interest in various research fields such as game theory, coherent structures, logic gates, operations research and Boolean algebra. A (monotonic) simple game corresponds to an inequivalent (monotonic) function in Boolean algebra and a weighted game corresponds to a threshold function. The characterization obtained provides a better understanding of these mathematical structures while opening new prospects for solving numerous open problems in these areas.

This is a preview of subscription content, access via your institution.


  1. 1.

    Carreras, F., Freixas, J.: Compete simple games. Math. Soc. Sci. 32, 139–155 (1996)

    Article  Google Scholar 

  2. 2.

    Cheung, W.S., Ng, T.W.: A three-dimensional voting system in Hong Kong. Eur. J. Oper. Res. 236, 292–297 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Freixas, J.: The dimension for the European Union Council under the Nice rules. Eur. J. Oper. Res. 156, 415–419 (2004)

    Article  Google Scholar 

  4. 4.

    Freixas, J., Molinero, X.: Simple games and weighted games: a theoretical and computational viewpoint. Discrete Appl. Math. 157, 1496–1508 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Freixas, J., Freixas, M., Kurz, S.: On the characterization of weighted simple games. Theor. Decis. 83(4), 469–498 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Isbell, J.R.: A class of simple games. Duke Math. J. 25, 423–439 (1958)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kilgour, D.M.: A formal analysis of the amending formula of Canada’s Constitution Act. Can. J. Polit. Sci. 16, 771–777 (1983)

    Article  Google Scholar 

  8. 8.

    Kurz, S., Napel, S.: A Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record. Optim. Lett. 10(6), 1245–1256 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Le Breton, M., Montero, M., Zaporozhets, V.: Voting power in the EU council of ministers and fair decision making in distributive politics. Math. Soc. Sci. 63, 159–173 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Leech, D.: Voting power in the governance of the International Monetary fund. Ann. Oper. Res. 109, 375–397 (2002)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Taylor, A.D., Pacelli, A.M.: Mathematics and Politics: Strategy, Voting, Power, and Proof. Springer, New York (2008)

    Book  Google Scholar 

  12. 12.

    Taylor, A.D., Zwicker, W.S.: A characterization of weighted voting. Proc. Am. Math. Soc. 115, 1089–1094 (1992)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Taylor, A.D., Zwicker, W.S.: Weighted voting, multicameral representation, and power. Games Econ. Behav. 5, 170–181 (1993)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Taylor, A.D., Zwicker, W.S.: Simple games and magic squares. J. Combin. Theory Ser. A 71, 67–88 (1995)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Taylor, A.D., Zwicker, W.S.: Simple Games: Desirability Relations, Trading, and Pseudo Weightings. Princeton University Press, New Jersey (1999)

    MATH  Google Scholar 

  16. 16.

    Zhu, Z., Fang, C., Katzgraber, H.G.: borealis—a generalized global update algorithm for Boolean optimization problems. Optim. Lett. (2020).

    MathSciNet  Article  MATH  Google Scholar 

Download references


This research is partially supported by funds from the Spanish Ministry of Science and Innovation Grants MTM2015-66818-P, PID2019-I04987GB-I00. The author wishes to thank two anonymous referees for their helpful comments.

Author information



Corresponding author

Correspondence to Josep Freixas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Freixas, J. A characterization of weighted simple games based on pseudoweightings. Optim Lett 15, 1371–1383 (2021).

Download citation


  • Boolean functions
  • Threshold functions
  • Simple games
  • Weighted games
  • Pseudoweightings
  • Optimization