Theory and Decision

, Volume 83, Issue 4, pp 469–498 | Cite as

On the characterization of weighted simple games

  • Josep FreixasEmail author
  • Marc Freixas
  • Sascha Kurz


This paper has a twofold scope. The first one is to clarify and put in evidence the isomorphic character of two theories developed in quite different fields: on one side, threshold logic, on the other side, simple games. One of the main purposes in both theories is to determine when a simple game is representable as a weighted game, which allows a very compact and easily comprehensible representation. Deep results were found in threshold logic in the sixties and seventies for this problem. However, game theory has taken the lead and some new results have been obtained for the problem in the past two decades. The second and main goal of this paper is to provide some new results on this problem and propose several open questions and conjectures for future research. The results we obtain depend on two significant parameters of the game: the number of types of equivalent players and the number of types of shift-minimal winning coalitions.


Simple games Weighted games Characterization of weighted games Trade robustness Invariant-trade robustness 



This research was partially supported by funds from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER Funds) under Grant MTM2015-66818-P (MINECO/FEDER).


  1. Anthony, M., & Holden, S. (1994). Quantifying generalization in linearly weighted neural networks. Complex Systems, 8, 91–114.Google Scholar
  2. Bean, D., Friedman, J., & Parker, C. (2008). Simple majority achievable hierarchies. Theory and Decision, 65, 285–302.CrossRefGoogle Scholar
  3. Beimel, A., Tassa, T., & Weinreb, E. (2008). Characterizing ideal weighted threshold secret sharing. SIAM Journal on Discrete Mathematics, 22, 360–397.CrossRefGoogle Scholar
  4. Beimel, A., & Weinreb, E. (2006). Monotone circuits for monotone weighted threshold families. Information Processing Letters, 97, 12–18.CrossRefGoogle Scholar
  5. Bohossian, V., & Bruck, J. (2003). Algebraic techniques for constructing minimal weights threshold functions. SIAM Journal on Discrete Mathematics, 16, 114–126.CrossRefGoogle Scholar
  6. Carreras, F., & Freixas, J. (1996). Complete simple games. Mathematical Social Sciences, 32, 139–155.CrossRefGoogle Scholar
  7. Chvátal, V. (1983). Linear Programming. New York: W.H. Freeman.Google Scholar
  8. Chow, C. (1961). Boolean functions realizable with single threshold devices. In Proceedings of the Institute of Radio Engineers, Vol. 49 (pp. 370–371).Google Scholar
  9. Dedekind, R. (1897). Über Zerlegungen von Zahlen durch ihre größten gemeinsammen Teiler. Gesammelte Werke, 1, 103–148.Google Scholar
  10. de Keijzer, B., Klos, T., & Zhang, Y. (2014). Finding optimal solutions for voting game design problems. Journal of Artificial Intelligence, 50, 105–140.Google Scholar
  11. Dubey, P., & Shapley, L. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4, 99–131.CrossRefGoogle Scholar
  12. Einy, E., & Lehrer, E. (1989). Regular simple games. International Journal of Game Theory, 18, 195–207.CrossRefGoogle Scholar
  13. Elgot, C. (1961). Truth functions realizable by single threshold organs. In AIEE Conference Paper 60-1311 (October), revised November 1960; paper presented at IEEE Symposium on Switching Circuit Theory and Logical Design.Google Scholar
  14. Freixas, J., & Kurz, S. (2013). The golden number and Fibonacci sequences in the design of voting systems. European Journal of Operational Research, 226, 246–257.CrossRefGoogle Scholar
  15. Freixas, J., & Kurz, S. (2014). Enumerations of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum. Annals of Operations Research, 222, 317–339.CrossRefGoogle Scholar
  16. Freixas, J., & Kurz, S. (2014). On minimum integer representations of weighted games. Mathematical Social Sciences, 67, 9–22.CrossRefGoogle Scholar
  17. Freixas, J., & Molinero, X. (2009). Simple games and weighted games: A theoretical and computational viewpoint. Discrete Applied Mathematics, 157, 1496–1508.CrossRefGoogle Scholar
  18. Freixas, J., & Molinero, X. (2010). Weighted games without a unique minimal representation in integers. Optimization Methods and Software, 25, 203–215.CrossRefGoogle Scholar
  19. Freixas, J., Molinero, X., & Roura, S. (2012). Complete voting systems with two types of voters: Weightedness and counting. Annals of Operations Research, 193, 273–289.CrossRefGoogle Scholar
  20. Freixas, J., & Pons, M. (2010). Hierarchies achievable in simple games. Theory and Decision, 68, 393–404.CrossRefGoogle Scholar
  21. Freixas, J., & Puente, M. A. (1998). Complete games with minimum. Annals of Operations Research, 84, 97–109.CrossRefGoogle Scholar
  22. Freixas, J., & Puente, M. A. (2008). Dimension of complete simple games with minimum. European Journal of Operational Research, 188, 555–568.CrossRefGoogle Scholar
  23. Freixas, J., & Zwicker, W. (2003). Weighted voting, abstention, and multiple levels of approval. Social Choice and Welfare, 21, 399–431.CrossRefGoogle Scholar
  24. Friedman, J., McGrath, L., & Parker, C. (2006). Achievable hierarchies in voting games. Theory and Decision, 61, 305–318.CrossRefGoogle Scholar
  25. Gabelman, I. (1961). The functional behavior of majority (threshold) elements, Ph.D. dissertation, Electrical Engineering Department, Syracuse University.Google Scholar
  26. Golomb, S. (1959). On the classification of Boolean functions. IRE Transactions on Circuit Theory, 6, 176–186.CrossRefGoogle Scholar
  27. Gvozdeva, T., & Slinko, A. (2011). Weighted and roughly weighted simple games. Mathematical Social Sciences, 61, 20–30.CrossRefGoogle Scholar
  28. Hammer, P., & Holzman, R. (1992). Approximations of pseudoboolean functions; applications to game theory. ZOR Methods and Models of Operations Research, 36, 3–21.CrossRefGoogle Scholar
  29. Hammer, P., Ibaraki, T., & Peled, U. (1981). Threshold numbers and threshold completions. Annals of Discrete Mathematics, 11, 125–145.Google Scholar
  30. Hammer, P., Kogan, A., & Rothblum, U. (2000). Evaluation, strength and relevance of Boolean functions. SIAM Journal on Discrete Mathematics, 13, 302–312.CrossRefGoogle Scholar
  31. Herranz, J. (2011). Any 2-asummable bipartite function is weighted threshold. Discrete Applied Mathematics, 159, 1079–1084.CrossRefGoogle Scholar
  32. Houy, N., & Zwicker, W. (2014). The geometry of voting power: Weighted voting and hyper-ellipsoids. Games and Economic Behavior, 84, 7–16.CrossRefGoogle Scholar
  33. Hu, S. (1965). Threshold Logic. Berkeley: University of California Press.Google Scholar
  34. Isbell, J. (1956). A class of majority games. Quarterly Journal of Mathematics Oxford Series, 7, 183–187.CrossRefGoogle Scholar
  35. Isbell, J. (1958). A class of simple games. Duke Mathematics Journal, 25, 423–439.CrossRefGoogle Scholar
  36. Kartak, V. M., Kurz, S., Ripatti, A. V., & Scheithauer, G. (2015). Minimal proper non-IRUP instances of the one-dimensional cutting stock problem. Discrete Applied Mathematics, 187, 120–129.CrossRefGoogle Scholar
  37. Kilgour, D. (1983). A formal analysis of the amending formula of Canada’s Constitution Act. Canadian Journal of Political Science, 16, 771–777.CrossRefGoogle Scholar
  38. Kurz, S. (2012). On minimum sum representations for weighted voting games. Annals of Operations Research, 196, 361–369.CrossRefGoogle Scholar
  39. Kurz, S., Molinero, X., & Olsen, M. (2016). On the construction of high-dimensional simple games. In Proceedings of the 22nd European Conference on Artificial Intelligence (pp. 1–13).Google Scholar
  40. Kurz, S., & Napel, S. (2016). Dimension of the Lisbon voting rules in the EU Council: A challenge and new world record. Optimization Letters, 10, 1245–1256.CrossRefGoogle Scholar
  41. Kurz, S., & Tautenhahn, N. (2013). On Dedekind’s problem for complete simple games. International Journal of Game Theory, 42, 411–437.CrossRefGoogle Scholar
  42. Littlestone, N. (1988). Learning when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2, 285–318.Google Scholar
  43. May, K. (1952). A set of independent, necessary and sufficient conditions for simple majority decision. Econometrica, 20, 680–684.CrossRefGoogle Scholar
  44. Muroga, S. (1971). Threshold Logic and Its Applications. New York: Wiley-Interscience.Google Scholar
  45. Muroga, S., Toda, I., & Kondo, M. (1962). Majority decision functions of up to six variables. Mathematics Computation, 16, 459–472.CrossRefGoogle Scholar
  46. Muroga, S., Toda, I., & Takasu, S. (1961). Theory of majority decision elements. Journal Franklin Institute, 271, 376–418.CrossRefGoogle Scholar
  47. Muroga, S., Tsuboi, T., & Baugh, R. (1970). Enumeration of threshold functions of eight variables. IEEE Transactions on Computers C-19(9), 818–825.Google Scholar
  48. Neumann, J. V., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.Google Scholar
  49. Parberry, I. (1994). Circuit Complexity and Neural Networks. Cambridge: The M.I.T. Press.Google Scholar
  50. Peled, U., & Simeone, B. (1985). Polynomial-time algorithms for regular set-covering and threshold synthesis. Discrete Applied Mathematics, 12, 57–69.CrossRefGoogle Scholar
  51. Peleg, B. (1968). On weight of constant sum majority games. SIAM Journal of Applied Mathematics, 16, 527–532.CrossRefGoogle Scholar
  52. Picton, P. (2000). Neural Networks (2nd ed.). Great Britain: The Macmillan Press, Ltd.Google Scholar
  53. Ramamurthy, K. (1990). Coherent Structures and Simple Games. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  54. Reiterman, J., Rödl, V., Sinajova, E., & Tuma, M. (1985). Threshold hypergraphs. Discrete Applied Mathematics, 54, 193–200.CrossRefGoogle Scholar
  55. Roychowdhury, V., Siu, K., & Orlitsky, A. (Eds.). (1994). Theoretical Advances in Neural Computation and Learning. Stanford, USA: Kluwer Academic Publishers.Google Scholar
  56. Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17, 1163–1170.CrossRefGoogle Scholar
  57. Simmons, G. (1990). How to (really) share a secret. In Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology. Springer, London (pp. 390–448).Google Scholar
  58. Siu, K., Roychowdhury, V., & Kailath, T. (1995). Discrete Neural Computation: A Theoretical Foundation. New Jersey: Prentice Hall.Google Scholar
  59. Tassa, T. (2007). Hierarchical threshold secret sharing. Journal of Cryptology, 20, 237–264.CrossRefGoogle Scholar
  60. Taylor, A. D., & Pacelli, A. (2008). Mathematics and Politics (2nd ed.). New York: Springer.CrossRefGoogle Scholar
  61. Taylor, A. D., & Zwicker, W. S. (1992). A characterization of weighted voting. Proceedings of the American Mathematical Society, 115, 1089–1094.CrossRefGoogle Scholar
  62. Taylor, A. D., & Zwicker, W. S. (1995). Simple games and magic squares. Journal of Combinatorial Theory Series A, 71, 67–88.CrossRefGoogle Scholar
  63. Taylor, A. D., & Zwicker, W. S. (1999). Simple Games: Desirability Relations, Trading, and Pseudoweightings. New Jersey: Princeton University Press.Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversitat Politècnica de Catalunya (Campus Manresa)ManresaSpain
  2. 2.CirprotecTerrassaSpain
  3. 3.Department of MathematicsUniversity of BayreuthBayreuthGermany

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