Skip to main content
SpringerLink
Log in

Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Complete games are also known in the literature of simple games as linear games or directed games.

  2. In Carreras and Freixas (1996) they are called δ-minimal winning vectors.

  3. More precisely, the notation wg(n,∗,r) stands for \(\sum_{t = 1}^{n}{wg}(n,t,r)\).

  4. Unfortunately we cannot apply Vandermonde’s identity \(\sum _{j=0}^{k}\binom{m}{j} \binom{n}{ k-j}=\binom{m+n}{ k}\) directly.

References

  • Alonso-Meijide, J. M., & Bowles, C. (2005). Generating functions for coalitional power indices: an application to the IMF. Annals of Operations Research, 137, 21–44.

    Article  Google Scholar 

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

    Google Scholar 

  • Brickell, E. F. (1989). Some ideal secret sharing schemes. Journal of Combinatorial Mathematics and Combinatorial Computing, 9, 105–113.

    Google Scholar 

  • Carreras, F., & Freixas, J. (1996). Complete simple games. Mathematical Social Sciences, 32, 139–155.

    Article  Google Scholar 

  • Carreras, F., & Freixas, J. (2008). On ordinal equivalence of power measures given by regular semivalues. Mathematical Social Sciences, 55, 221–234.

    Article  Google Scholar 

  • Carreras, F., Freixas, J., & Puente, M. A. (2003). Semivalues as power indices. European Journal of Operational Research, 149, 676–687.

    Article  Google Scholar 

  • 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 

  • Diffo Lambo, L., & Moulen, J. (2002). Ordinal equivalence of power notions in voting games. Theory and Decision, 53, 313–325.

    Article  Google Scholar 

  • Dubey, P., & Shapley, L. S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4, 99–131.

    Article  Google Scholar 

  • Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: theory and practice, problems and paradoxes. Cheltenham: Edward Elgar.

    Book  Google Scholar 

  • Freixas, J. (1997). Different ways to represent weighted majority games. Top, 5, 201–212.

    Article  Google Scholar 

  • Freixas, J. (2004). The dimension for the European Union Council under the Nice rules. European Journal of Operational Research, 156, 415–419.

    Article  Google Scholar 

  • Freixas, J. (2010). On ordinal equivalence of the Shapley and Banzhaf values for cooperative games. International Journal of Game Theory, 39, 513–527.

    Article  Google Scholar 

  • Freixas, J., & Molinero, X. (2010). Weighted games without a unique minimal representation in integers. Optimization Methods & Software, 25, 203–215.

    Article  Google Scholar 

  • Freixas, J., & Puente, M. A. (1998). Complete games with minimum. Annals of Operations Research, 84, 97–109.

    Article  Google Scholar 

  • Freixas, J., & Puente, M. A. (2008). Dimension of complete simple games with minimum. European Journal of Operational Research, 188, 555–568.

    Article  Google Scholar 

  • Freixas, J., Molinero, X., & Roura, S. (2012b). Complete voting systems with two types of voters: weightedness and counting. Annals of Operations Research, 193, 273–289.

    Article  Google Scholar 

  • Gvozdeva, T., & Slinko, A. (2011). Weighted and roughly weighted simple games. Mathematical Social Sciences, 61, 20–30.

    Article  Google Scholar 

  • Gvozdeva, T., Hameed, A., & Slinko, A. (2013). Weightedness and structural characterization of hierarchical simple games. Mathematical Social Sciences. doi:10.1016/j.mathsocsci.2012.11.007.

    Google Scholar 

  • Isbell, J. R. (1956). A class of majority games. Quarterly Journal of Mathematics, 7, 183–187.

    Article  Google Scholar 

  • Isbell, J. R. (1958). A class of simple games. Duke Mathematical Journal, 25, 423–439.

    Article  Google Scholar 

  • Kilgour, D. M. (1983). A formal analysis of the amending formula of Canada’s Constitution Act. Canadian Journal of Political Science, 16, 771–777.

    Article  Google Scholar 

  • Korshunov, A. D. (2003). Monotone Boolean functions. Uspekhi Matematičeskih Nauk, 58(5(353)), 89–162.

    Article  Google Scholar 

  • Krohn, I., & Sudhölter, P. (1995). Directed and weighted majority games. Mathematical Methods of Operations Research (ZOR), 42, 189–216.

    Article  Google Scholar 

  • Kurz, S. (2012). On minimum sum representations for weighted voting games. Annals of Operations Research, 196, 361–369.

    Article  Google Scholar 

  • Kurz, S., & Tautenhahn, N. (2012). On Dedekind’s problem for complete simple games. International Journal of Game Theory. doi:10.1007/s00182-012-0327-9.

    Google Scholar 

  • Leech, D. (2002a). Designing the voting system for the council of the European Union. Public Choice, 113(3–4), 437–464.

    Article  Google Scholar 

  • Leech, D. (2002b). Voting power in the governance of the International Monetary fund. Annals of Operations Research, 109, 375–397.

    Article  Google Scholar 

  • Maschler, M., & Peleg, B. (1966). A characterization, existence proof, and dimension bounds for the kernel of a game. Pacific Journal of Mathematics, 18, 289–328.

    Article  Google Scholar 

  • May, K. (1952). A set of independent, necessary and sufficient conditions for simple majority decision. Econometrica, 20, 680–684.

    Article  Google Scholar 

  • Morriss, P. (2002). Power: a philosophical analysis (2nd ed.). Manchester: Manchester University Press.

    Google Scholar 

  • Muroga, S., Toda, I., & Kondo, M. (1962). Majority decision functions of up to six variables. Mathematics of Computation, 16, 459–472.

    Article  Google Scholar 

  • Owen, G. (1978). Characterization of the Banzhaf–Coleman index. SIAM Journal on Applied Mathematics, 35, 315–327.

    Article  Google Scholar 

  • Peled, U. N., & Simeone, B. (1985). Polynomial-time algorithms for regular set-covering and threshold synthesis. Discrete Applied Mathematics, 12, 57–69.

    Article  Google Scholar 

  • Shapley, L. S. (1953). A value for n-person games. In A. W. Tucker & H. W. Kuhn (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press.

    Google Scholar 

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

    Article  Google Scholar 

  • Simmons, G. J. (1990). How to (really) share a secret. In Proceedings of the 8th annual international cryptology conference on advances in cryptology (pp. 390–448). London: Springer.

    Google Scholar 

  • Stinson, D. R. (1992). An explanation of secret sharing schemes. Designs, Codes and Cryptography, 2, 357–390.

    Article  Google Scholar 

  • Straffin, P. D. (1982). Power indices in politics. In S. J. Brahms, S. J. Lucas, & P. D. Straffin (Eds.), Political and related models (Vol. 2). New York: Springer.

    Google Scholar 

  • Tassa, T. (2007). Hierarchical threshold secret sharing. Journal of Cryptology, 20, 237–264.

    Article  Google Scholar 

  • Taylor, A. D., & Pacelli, A. (2008). Mathematics and politics (2nd ed.). New York: Springer.

    Book  Google Scholar 

  • Taylor, A. D., & Zwicker, W. S. (1999). Simple games: desirability relations, trading, and pseudoweightings. New Jersey: Princeton University Press.

    Google Scholar 

  • Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press.

    Google Scholar 

  • Young, J. M. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72.

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the two referees of this paper for their interesting comments and also for their exhaustive reports that contributed to improve the original submitted version.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics III and High Engineering School (Manresa Campus), Technical University of Catalonia, Barcelona, Spain

    Josep Freixas

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

    Sascha Kurz

Authors
  1. Josep Freixas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sascha Kurz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josep Freixas.

Additional information

Research of the first author partially funded by Grants SGR 2009–1029 of Generalitat de Catalunya and MTM 2012–34426 from the Spanish Economy and Competitiveness Ministry, from the Spanish Science and Innovation Ministry.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freixas, J., Kurz, S. Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum. Ann Oper Res 222, 317–339 (2014). https://doi.org/10.1007/s10479-013-1348-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-013-1348-x

Keywords

Search

Navigation