Annals of Operations Research

, Volume 215, Issue 1, pp 245–256 | Cite as

The electoral system for the Italian Senate: an analogy with deterministic chaos?

An analysis via characteristic polynomials
  • G. PontualeEmail author
  • F. Dalton
  • S. Genovese
  • E. La Nave
  • A. Petri


The electoral system adopted for the allocation of seats in the Italian Senate utilizes a complex mechanism of awards at a regional level with the aim of strengthening, when necessary, the winning coalition and so improve overall government stability. The results presented here demonstrate that in a significant number of cases, the effect of the mechanism is opposite to that desired, to wit, weakening the resultant government by awarding more seats to the minority coalition. Indeed the award to the minority can even be such that the minority coalition becomes the majority and wins the election. The application of the award mechanism is strongly unpredictable as it depends crucially on the precise number of seats independently obtained in each region, and that each adjustment thereof can be positive, zero or negative; a characteristic that closely resembles the behaviour of a chaotic dynamical system whose trajectory, although purely deterministic, depends on infinitely precise details and is therefore unpredictable. To perform the systematic numerical analysis of the award effectiveness, we introduce characteristic polynomials, one for each electoral district, which carry information about all possible outcomes and award applications. Their product yields a polynomial containing the dependence of the result at national level on each of the regional awards.


Electoral systems Generating functions Chaotic systems 



A.P., F.D. and G.P. commemorate Bruno Simeone and his profound thoughts and teachings on the science of electoral systems.


  1. Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., & Vattay, G. (2012). Chaos: classical and quantum. Copenhagen: Niels Bohr Institute. Google Scholar
  2. Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1, 3rd ed.). New York: Wiley. Google Scholar
  3. Fragnelli, F., & Ortona, G. (2006). In B. Simeone & F. Pukelsheim (Eds.), Mathematics and democracy (pp. 65–81). Berlin and Heidelberg: Springer. CrossRefGoogle Scholar
  4. Decreto del Presidente della Repubblica 11/02/2006. Gazzetta Ufficiale 36, 13/02/2006. Google Scholar
  5. Gleick, J. (2011). Chaos: making a new science. Open road iconic ebooks. Google Scholar
  6. Grilli di Cortona, P., Manzi, C., Pennisi, A., Ricca, F., & Simeone, B. (1999). SIAM monographs on discrete mathematics and its applications: Evaluation and optimization of electoral systems. Philadelphia: SIAM. CrossRefGoogle Scholar
  7. Penrose, L. S. (1946). Journal of the Royal Statistical Society, 109(1), 53. CrossRefGoogle Scholar
  8. Penrose, L. S. (1952). On the objective study of crowd behaviour. London: H.K. Lewis. Google Scholar
  9. Saari, D. G. (2001). Chaotic elections! A mathematician looks at voting. Providence: Am. Math. Soc. (Here the word chaos is used in the general common sense.) Google Scholar
  10. Strogatz, S. H. (1994). Nonlinear dynamics and chaos: with applications in physics, biology, chemistry, and engineering. Reading: Addison-Wesley. Google Scholar
  11. Zyczkowski, K., & Slomczynski, W. (2004). arXiv:cond-mat/0405396v2.
  12. Zyczkowski, K., & Slomczynski, W. (2012). arXiv:1104.5213v2.
  13. Zyczkowski, K., Slomczynski, W., & Zastawniak, T. (2006). Physics World, 19, 35–37. Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • G. Pontuale
    • 1
    Email author
  • F. Dalton
    • 1
  • S. Genovese
    • 2
  • E. La Nave
    • 3
  • A. Petri
    • 1
  1. 1.CNR—Istituto dei Sistemi ComplessiRomeItaly
  2. 2.Ufficio legislativo Gruppo PDSenato della Repubblica ItalianaRomeItaly
  3. 3.CNR—Istituto dei Sistemi Complessi, Dipartimento di FisicaUniversità SapienzaRomeItaly

Personalised recommendations