Fragmentation, Concentration, Proliferation

Almost all research on party-system fragmentation operationalises the phenomenon using the effective number of parties. A well-conducted recent example of this is Vicente Valentim and Elias Dinas’ paper in the British Journal of Political Science. It also cites all sorts of papers that take a similar approach.

However, there are two ways to alter fragmentation when measured using the effective number of parties. First, we can increase or decrease the actual number of parties. In other words, we can adjust how many parties proliferate. Second, we can increase or decrease how much support is focussed on a small number of parties. That is, we can adjust the concentration of support.

We can see this in action if we consider the simplest model that links effective parties, $N_2$, and actual parties, $N_0$:

$$N_2 = N_0 ^ \theta$$

Here, the effective number of parties, $N_2$, measures fragmentation; the actual number of parties, $N_0$, measures the proliferation of actual political parties; and the model’s sole parameter, $\theta$, measures the concentration of support across actual parties, ranging from $\theta = 0$, where $N_2 = 1$, to $\theta = 1$, where $N_2 = N_0$.

Overview

The simplest model linking effective parties (N₂) and actual parties (N₀) is:

N₂ = N₀^θ

The parameter θ determines the concentration of political support across parties. When θ = 1, there are as many effective as actual parties. As θ moves from 1 to 0, the effective number of parties falls towards 1.

Actual parties (N₀)5

Concentration parameter (θ)0.65

Effective parties (N₂)2.85

Higher values of θ keep the effective count closer to the actual count, so influence is spread more evenly across parties.

The interactive above shows how party-system fragmentation as measured using the effective number of parties changes as we adjust $N_0$ and $\theta$. The key takeaway here is that there are two parameters, not one. So modelling the effective number of parties does not tell us much about the specific mechanism at play: is it that the predictor in question causes parties to proliferate (or not) or is it that it adjusts the extent to which support is concentrated in a handful of parties?

One final point: while I have treated $\theta$ as a parameter here, we could arguably treat any of the elements in the equation as parameters if we wanted. To see why, note that we can rearrange the equation above to get:

$$\theta = \frac{\log \left( N_2 \right)}{\log \left( N_0 \right)}$$

And, incidentally, this is the equation for efficiency (i.e. normalised entropy) in information theory, which makes sense given that the effective number of parties is an information-theoretic quantity in its own right.