Decisions in Economics and Finance

, Volume 41, Issue 2, pp 489–529 | Cite as

Proper strong-Fibonacci games

  • Flavio Pressacco
  • Laura ZianiEmail author


We define proper strong-Fibonacci (PSF) games as the subset of proper homogeneous weighted majority games which admit a Fibonacci representation. This is a homogeneous, type-preserving representation whose ordered sequence of type weights and winning quota is the initial string of Fibonacci numbers of the one-step delayed Fibonacci sequence. We show that for a PSF game, the Fibonacci representation coincides with the natural representation of the game. A characterization of PSF games is given in terms of their profile. This opens the way up to a straightforward formula which gives the number \(\varPsi (t)\) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of \(\varPsi (t)\) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio \(\varPsi (t+2)/\varPsi (t)\) converges toward the golden ratio \({\varPhi }\).


Weighted majority games Natural representation Homogeneous representation Profile vector Fibonacci numbers Golden ratio 

JEL Classification




We wish to thank the editor and three anonymous reviewers for their helpful comments and suggestions.


  1. Baron, D.P., Ferejohn, J.A.: Bargaining in legislatures. Am. Political Sci. Rev. 83, 1181–1206 (1989)CrossRefGoogle Scholar
  2. Fragnelli, V., Gambarelli, G., Gnocchi, N., Pressacco, F., Ziani, L.: Fibonacci representations of homogeneous weighted majority games. In: Nguyen, N.T., Kowalczyk, R., Mercik, J. (eds.) Transactions on computational collective intelligence XXIII. Lecture Notes in Computer Science, vol. 9760, pp. 162–171. Springer, Berlin (2016)CrossRefGoogle Scholar
  3. Freixas, J., Kurz, S.: The golden number and Fibonacci sequences in the design of voting structures. Eur. J. Oper. Res. 226, 246–257 (2013)CrossRefGoogle Scholar
  4. Freixas, J., Kurz, S.: On minimum integer representations of weighted games. Math. Soc. Sci. 67, 9–22 (2014)CrossRefGoogle Scholar
  5. Freixas, J., Molinero, X.: Weighted games without a unique minimal representation in integers. Optim. Methods Softw. 25, 203–215 (2010)CrossRefGoogle Scholar
  6. Freixas, J., Molinero, X., Roura, S.: Complete voting systems with two classes of voters: weightedness and counting. Ann. Oper. Res. 193–1, 273–289 (2012)CrossRefGoogle Scholar
  7. Gurk, H.M., Isbell, J.R.: Simple Solutions, Contributions to the Theory of Games, Vol. IV, Annals of Mathematics Studies, vol. 40, pp. 247–265. Princeton University Press, Princeton (1959)Google Scholar
  8. Isbell, R.: A class of majority games. Q J Math 7, 183–187 (1956)CrossRefGoogle Scholar
  9. Isbell, R.: On the enumeration of majority games. Math. Tables Aids Comput. 13, 21–28 (1959)CrossRefGoogle Scholar
  10. Kalandrakis, T.: Proposal rights and political power. Am. J. Political Sci. 50–2, 441–448 (2006)CrossRefGoogle Scholar
  11. Krohn, I., Sudhölter, P.: Directed and weighted majority games. ZOR - Math. Methods Oper. Res. 42, 189–216 (1995)CrossRefGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. Maschler, M., Peleg, B.: A characterization, existence proof and dimension bounds for the kernel of a game. Pacific J. Math. 18–2, 289–328 (1966)CrossRefGoogle Scholar
  14. Montero, M.: Proportional Payoffs in Majority Games (2008). Available at SSRN:
  15. Ostmann, A.: On the minimal representation of homogeneous games. Int. J. Game Theory 16, 69–81 (1987)CrossRefGoogle Scholar
  16. Peleg, B.: On weights of constant-sum majority games. SIAM J. Appl. Math. 16–3, 527–532 (1968)CrossRefGoogle Scholar
  17. Philippou, A., Bergum, G.E., Horadam, A.F.: Fibonacci Numbers and Their Applications, Mathematics and its Applications, vol. 18. Springer, Dordrecht (1986)CrossRefGoogle Scholar
  18. Pressacco, F., Plazzotta, G., Ziani, L.: Bilateral symmetry and modified Pascal triangles in Parsimonious games. (2013). with validation no.: hal-00948123
  19. Pressacco, F., Ziani, L.: A Fibonacci approach to weighted majority games. J Game Theory 4, 36–44 (2015)Google Scholar
  20. Rosenmüller, J.: Weighted majority games and the matrix of homogeneity. Oper. Res. 28, 123–141 (1984)Google Scholar
  21. Rosenmüller, J.: Homogeneous games: recursive structure and computation. Math. Oper. Res. 12–2, 309–330 (1987)CrossRefGoogle Scholar
  22. Rosenmüller, J., Sudhölter, P.: The nucleolus of homogeneous games with steps. Discrete Appl. Math. 50, 53–76 (1994)CrossRefGoogle Scholar
  23. Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17–6, 1163–1170 (1969)CrossRefGoogle Scholar
  24. Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, Princeton (1944)Google Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2018

Authors and Affiliations

  1. 1.Department of Economics and Statistics D.I.E.S.Udine UniversityUdineItaly

Personalised recommendations