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

Abstract

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.

Appendix: Generating functions

Generating functions are a widely used tool in probability calculus. Given a random variable x that can assume non-negative integer values ℓ with probability p _ℓ , its generating G(z) function is defined as G(z)=∑p _ℓ z ^ℓ, with 0≤z<1. The generating function possesses a set of useful properties and in several cases makes calculations easier. Among the main properties there are

Given two variables identically distributed, x ₁ and x ₂, it is easily seen that the generating function F(z) for the sum variable, y=x ₁+x ₂, is F(z)=G ²(z). Since F(z)=∑ _ℓ q _ℓ z ^ℓ, it is straightforward to derive the probabilities for y as

$$q_\ell= \frac{1}{\ell!}\frac{\partial F^\ell}{\partial^\ell z} \bigg|_{z=0}. $$

In a similar fashion if G ₁(z),G ₂(z),…,G _N (z) are the generating functions of N (differently distributed) random variables x ₁,x ₂,…,x _N , the generating function for y=x ₁+x ₂+⋯+x _N will be given by the product F(z)=G ₁(z)⋅G ₂(z)…G _N (z).