An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players

  • Antônio Francisco NetoEmail author
  • Carolina Rodrigues Fonseca
Original Research


We introduce a new generating function based method to compute the Banzhaf, Deegan–Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph \(G=\left( V,E\right) \) with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG’s) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377–387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG’s the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG’s by computing the PG power index of the associated voting game.


Banzhaf power index Deegan–Packel power index Public good power index Shapley–Shubik power index Multiple weighted voting games Incompatible players Generating function Commutative algebra Partition analysis Graph 



AFN would like to thank the financial support of CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) under grant 307211/2015-0. CRF would like to thank FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais) for a graduate scholarship.


  1. Algaba, E., Bilbao, J. M., Fernández García, J., & López, J. J. (2003). Computing power indices in weighted multiple majority games. Mathematical Social Sciences, 46(1), 63–80.Google Scholar
  2. Alonso-Meijide, J. M., & Bowles, C. (2005). Generating functions for coalitional power indices: An application to the IMF. Annals of Operations Research, 137(1), 21–44.Google Scholar
  3. Alonso-Meijide, J. M., & Casas-Méndez, B. (2007). The public good index when some voters are incompatible. Homo Oeconomicus, 24(3–4), 449–468.Google Scholar
  4. Alonso-Meijide, J. M., Casas-Méndez, B., Holler, M. J., & Lorenzo-Freire, S. (2008). Computing power indices: Multilinear extensions and new characterizations. European Journal of Operational Research, 188(2), 540–554.Google Scholar
  5. Alonso-Meijide, J. M., Bilbao, J. M., Casas-Méndez, B., & Fernández, J. R. (2009). Weighted multiple majority games with unions: Generating functions and applications to the European Union. European Journal of Operational Research, 198(2), 530–544.Google Scholar
  6. Alonso-Meijide, J. M., Casas-Méndez, B., & Fiestras-Janeiro, M. G. (2015). Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players. Applied Mathematics and Computation, 252(1), 377–387.Google Scholar
  7. Alonso-Meijide, J. M., Freixas, J., & Molinero, X. (2012). Computation of several power indices by generating functions. Applied Mathematics and Computation, 219(8), 3395–3402.Google Scholar
  8. Andrews, G. E., Paule, P., & Riese, A. (2001). MacMahon’s partition analysis: The omega package. European Journal of Combinatorics, 22(7), 887–904.Google Scholar
  9. Banzhaf, J. F, I. I. I. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19, 317–343.Google Scholar
  10. Bilbao, J. M., Fernández, J. R., Jiménez Losada, A., & López, J. J. (2000). Generating functions for computing power indices efficiently. TOP, 8(2), 191–213.Google Scholar
  11. Bondy, J. A., & Murty, U. S. R. (1982). Graph theory with applications. Amsterdam: Elsevier.Google Scholar
  12. Carreras, F. (1991). Restriction of simple games. Mathematical Social Sciences, 21(3), 245–260.Google Scholar
  13. Chessa, M. (2014). A generating functions approach for computing the public good index efficiently. TOP, 22(2), 658–673.Google Scholar
  14. Coleman, J. S. (1971). Control of collectives and the power of a collectivity to act. In B. Lieberman (Ed.), Social choice (pp. 269–300). New York: Gordon and Breach.Google Scholar
  15. Cox, A. D., Little, J., & O’Shea, D. (2007). Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. Berlin: Springer.Google Scholar
  16. Datta, R. S. (2010). Finding all Nash equilibria of a finite game using polynomial algebra. Economic Theory, 42(1), 55–96.Google Scholar
  17. Deegan, J, Jr., & Packel, E. W. (1978). A new index of power for simple n-person games. International Journal of Game Theory, 7(2), 113–123.Google Scholar
  18. Dubey, P., & Shapley, L. S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4(2), 99–131.Google Scholar
  19. Egorychev, G. P. (1984). Integral representation and the computation of combinatorial sums. Providence: American Mathematical Society.Google Scholar
  20. Einy, E. (1988). The Shapley value on some lattices of monotonic games. Mathematical Social Sciences, 15(1), 1–10.Google Scholar
  21. Faaland, B. (1972). On the number of solutions to a diophantine equation. Journal of Combinatorial Theory (A), 13(2), 170–175.Google Scholar
  22. Fernández, J. R., Algaba, E., Bilbao, J. M., Jiménez, A., Jiménez, N., & López, J. J. (2002). Generating functions for computing the Myerson value. Annals of Operations Research, 109(1–4), 143–158.Google Scholar
  23. Gould, H. W. (1972). Combinatorial Identities. A standardized set of tables listing 500 binomial coefficient summations. Henry W. Gould, Morgantown, W. Va.Google Scholar
  24. Holler, M. (1982). Forming coalitions and measuring voting power. Political Studies, 30(2), 262–271.Google Scholar
  25. Kirsch, W., & Langner, J. (2010). Power indices and minimal winning coalitions. Social Choice and Welfare, 34(1), 33–46.Google Scholar
  26. Lange, F., & Kóczy, L. (2013). Power indices expressed in terms of minimal winning coalitions. Social Choice and Welfare, 41(2), 281–292.Google Scholar
  27. Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2(3), 225–229.Google Scholar
  28. Neto, A. F. (2019). Generating functions of weighted voting games, MacMahon’s partition analysis, and Clifford algebras. Mathematics of Operations Research, 44(1), 74–101.Google Scholar
  29. Neto, A. F., & dos Anjos, P. H. R. (2014). Zeon algebra and combinatorial identities. SIAM Review, 56(2), 353–370.Google Scholar
  30. Owen, G. (1972). Multilinear extensions of games. Management Science, 18(5), 64–79.Google Scholar
  31. Owen, G. (1975). Multilinear extensions and the Banzhaf value. Naval Research Logistic Quarterly, 22(4), 741–750.Google Scholar
  32. Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109(1), 53–57.Google Scholar
  33. Rodríguez-Veiga, J., Novoa-Flores, G. I., & Casas-Méndez, B. (2016). Implementing generating functions to obtain power indices with coalition configuration. Discrete Applied Mathematics, 214(11), 1–15.Google Scholar
  34. Schott, R., & Staples, G. S. (2012). Operator calculus on graphs theory and applications in computer science. London: Imperial College Press.Google Scholar
  35. Shapley, L. S. (1953). A value for n-person games. In: Contributions to the theory of games II (Vol. 28, pp. 307–317). Princeton: Princeton University Press.Google Scholar
  36. Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distributions of power in a committee system. The American Political Science Review, 48(3), 787–792.Google Scholar
  37. Wilf, H. S. (1990). Generatingfunctionology. New York: Academic Press.Google Scholar
  38. Yakuba, V. (2008). Evaluation of Banzhaf index with restrictions on coalitions formation. Mathematical and Computer Modelling, 48(9–10), 1602–1610.Google Scholar

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Authors and Affiliations

  1. 1.DEPRO, Escola de MinasUFOPOuro PretoBrazil
  2. 2.PPEA, DEECO, UFOPMarianaBrazil

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