Skip to main content
SpringerLink
Log in

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The Lisbon voting system of the Council of the European Union, which became effective in November 2014, cannot be represented as the intersection of six or fewer weighted games, i.e., its dimension is at least 7. This sets a new record for real-world voting bodies. A heuristic combination of different discrete optimization methods yields a representation as the intersection of 13,368 weighted games. Determination of the exact dimension is posed as a challenge to the community. The system’s Boolean dimension is proven to be 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We remark that rounding populations to, say, thousands is common in applied work because this simplifies computations, e.g., of the voting power distribution in the EU Council. Rounding, however, leads to a different set of winning coalitions, i.e., is analyzing ‘wrong’ rules.

  2. For example, every 16-member winning coalition is minimal but few are also shift-minimal.

  3. The example is the smallest possible: all complete simple games with \(n\le 5\) are weighted.

  4. Just to give an example, \(\big \langle \{4,\ldots ,28\},\{1,3,4,5,7,8, 10,12,15,\ldots ,28\}; \{1,4,5,7,\ldots ,12, 14,\ldots , 28\}, \{3,\ldots ,8,10,12, 13,15,\ldots ,28\}\big \rangle \) is a 2-trade for the first two losing coalitions.

  5. For the general ILP modeling of weighted games we refer to [11].

  6. In order to check this intuition, we have computed the difference \(\left\| \mathcal {P}(v_{\text {EU28}})-\mathcal {P}(v_1\wedge v_2)\right\| _1\) for four different power measures \(\mathcal {P}\) (cf. [1]): it is only around \(7 \times 10^{-7}\) for the least square nucleolus and \(9 \times 10^{-7}\) for the normalized Banzhaf index, but 0.00537 for the Shapley-Shubik index and 0.167 for the nucleolus.

References

  1. Alonso-Meijide, J., Álvarez-Mozos, M., Fiestras-Janeiro, M.: The least square nucleolus is a normalized Banzhaf value. Optim. Lett. 1–7 (to appear). doi:10.1007/s11590-014-0840-9

  2. Cheung, W.-S., Ng, T.-W.: A three-dimensional coting system in Hong Kong. Eur. J. Oper. Res. 236(1), 292–297 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. De\({\check{\i }}\)neko, V.G., Woeginger, G.J.: On the dimension of simple monotonic games. Eur. J. Oper. Res. 170(1), 315–318 (2006)

  4. Faliszewski, P., Elkind, E., Wooldridge, M.: Boolean combinations of weighted voting games. In: Proceedings of the 8th International Conference on Autonomous Agents and Multiagent Systems, pp. 185–192. International Foundation for Autonomous Agents and Multiagent Systems (2009)

  5. Freixas, J.: The dimension for the European Union Council under Nice rules. Eur. J. Oper. Res. 156(2), 415–419 (2004)

    Article  MATH  Google Scholar 

  6. Freixas, J., Marciniak, D.: On the notion of dimension and codimension. In: Petrosyan, L. A., Zenkevich, N. A. (eds) Proceedings of the 3rd International Conference Game Theory and Management, pp. 67–81. The International Society of Dynamic Games (Russian Chapter) (2010)

  7. Freixas, J., Puente, M.A.: Dimension of complete simple games with minimum. Eur. J. Oper. Res. 188(2), 555–568 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gvozdeva, T., Slinko, A.: Weighted and roughly weighted simple games. Math. Social Sci. 61(1), 20–30 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  10. Kurz, S.: On minimum sum representations for weighted voting games. Ann. Oper. Res. 196(1), 361–369 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kurz, S.: On the inverse power index problem. Optimization 61(8), 989–1011 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Taylor, A.D.: Mathematics and Politics. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  13. Taylor, A.D., Zwicker, W.S.: Weighted voting, multicameral representation, and power. Games Econ. Behav. 5(1), 170–181 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgments

The authors thank two anonymous referees for constructive suggestions. The usual caveat applies.

Author information

Authors and Affiliations

  1. Department of Mathematics, Physics, and Computer Science, University of Bayreuth, 95440, Bayreuth, Germany

    Sascha Kurz

  2. Department of Economics, University of Bayreuth, Bayreuth, Germany

    Stefan Napel

  3. PCRC, University of Turku, Turku, Finland

    Stefan Napel

Authors
  1. Sascha Kurz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Napel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sascha Kurz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kurz, S., Napel, S. Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record. Optim Lett 10, 1245–1256 (2016). https://doi.org/10.1007/s11590-015-0917-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0917-0

Keywords

Mathematics Subject Classification

Search

Navigation