The Number of Parties: A Problem and a Solution

The mean-square-based measures, in contrast, are designed with only one dimension in mind and are not able to provide an estimate of political divergences across dimensions, although it is possible to aggregate the separate polarisations for each dimension.

Other measures of polarisation

In addition to these two most widely used measures – the pairwise measure and the standard deviation/variation-based measure – there are several other operationalisations of party system po-larisation that have been suggested and used in party system research, reflecting other nuances of the concept. One version is the maximum distance among parties in a system, referring to the idea that polarisation is defined by the existence of extreme, anti-system parties on both sides of the ideological spectrum. This measure is used, for example, by Dejaeghere and Dassonneville (2015), Matakos, Troumpounis, and Xefteris (2015) and Andrews and Money (2009). Another measure, which reflects a similar idea, is the proportion of extremist parties in the party system. This has been used among others by King et al. (1990) and Warwick (1994). A tangent measure, which has found application in closely related research into voting behaviour, is the measure of party system compactness suggested by Alvarez and Nagler (2004) that takes also the dispersion of voters into account (and is thus suitable for only such cases, where comparable information about the political positions of voters is available, severely restricting the range of application). This measure has been used in party system polarisation research among others by Ezrow (2008) and Dow (2011).

two most common measures of polarisation, and simulate their association to fragmentation, we can better understand the problem. For the following, hypothetical party systems of 2 to 10 parties were generated with party positions and weights randomly drawn from separate uniform distributions 10,000 times. I use the Esteban and Ray measure (equation 5.6) and the ideological standard deviation measure (equation 5.2) to calculate the amount of polarisation for each configuration and the effective number of parties (Laakso and Taagepera 1979) to determine fragmentation. The results of the simulation are shown on Figure 5.1.

0 1 2 3 4

2.5 5.0 7.5 10.0

Fragmentation

Polarisation, ideological SD

0 1 2 3 4 5

2.5 5.0 7.5 10.0

Fragmentation

Polarisation, Esteban and Ray

Figure 5.1: Simulation of Association between Fragmentation and Polarisation. The figure shows 10,000 replica-tions of a party system withN={2...10}parties and weights and positions randomly drawn from uniform distributions.

We can see that both the Esteban and Ray measure and the ideological standard deviation based measure are related to the number of parties in the system (fragmentation). As fragmentation rises, there seems to be a very clear increasing lower bound to the values of polarisation and thus the expected value of polarisation increases as the number of parties increases.

Another way to think of this mechanical association is to ask what would happen for a given party system if we added a number of parties with random ideological locations to the system. Let’s assume we have a party system with 4 parties with certain fixed relative sizes and fixed ideological positions. How would the polarisation of the system increase if we added 1 to 4 parties to the systems with randomly varying sizes and ideological locations? The results of this thought experiment are shown on Figure 5.2.

We can clearly see that adding additional parties is very likely to result in higher polarisation, regardless of the positions of the parties. If there is a mechanical relationship between polarisation and party system fragmentation, then this is a problem, because the two phenomena can then no longer be separately studied. If polarisation is on the left hand side of the equation and the number

CEUeTDCollection

0 1 2

1 2 3 4

Additional parties

Change in polarisation, ideological SD

0 1 2 3

1 2 3 4

Additional parties

Change in polarisation, Esteban and Ray

Figure 5.2: Polarisation and Additional Parties in the System. The figure shows the distributions of 10,000 repli-cations of a party system with N = 4 parties to which 1 to 4 parties are added with random positions and relative sizes. The box-plot shows the inter-quartile range (median in the middle) and the whiskers the largest/smallest value no further than 1.5 of the inter-quartile range from the edge of the box.

of parties or fragmentation is added to the right hand side, the model will show an association and model fit will improve regardless of the actual association between fragmentation as polarisation.

Therefore, we need an alternative measure of polarisation, one not dependent on the fragmen-tation of the party system. As we would prefer a general measure that could aggregate distances along one or many dimensions, a possibility that is based on pairwise differences in any number of dimensions is considered here. For each party we can calculate the distance from every other party and thus as a first step we could characterise each party by the average distance it has from every other party. Taking the weighted average of such average pairwise distances would result in a pairwise polarisation measure that can be expressed as follows:¹

Ppw =

n

X

i n

X

j

wi

1

n−1dij (5.7)

where w_i is the weight of partyi,d_ij is the distance between parties i andj andn is the number of parties. The fundamental difference from the Esteban and Ray measure is that instead of weighting each pairwise distance by the product of w_i and w_j, the same weight _n−1¹ is used for all parties, which effectively applies only for parties i 6=j in d_ij, because the distance of a party from itself is by definition 0. This measure is uncorrelated with the number of parties or party system fragmentation and the results of the above simulations for this measure can be seen on Figure 5.3.

Instead of the Esteban and Ray measure for polarisation, this version will thus be used, as it

1 A slightly different solution is suggested by Schmitt (2016).

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0.0 2.5 5.0 7.5

2.5 5.0 7.5 10.0

Fragmentation

Polarisation, alternative PW measure

−1 0 1 2

1 2 3 4

Additional parties

Change in polarisation, alternative PW measure

Figure 5.3: Polarisation, Fragmentation and an Alternative Pairwise Measure.The figure shows how an alternative pairwise measure that is suggested here performs in the above mentioned simulations. The box-plot shows the inter-quartile range (median in the middle) and the whiskers the largest/smallest value no further than 1.5 of the inter-inter-quartile range from the edge of the box.

clearly has more desirable properties than the alternatives. The estimate of polarisation does seem to go up as additional parties are added to the system, but this increase is much weaker than for the other two indices. And it should be noted that it does have a problem with heteroskedasticity – the possible values of polarisation have a much higher variance, especially if fragmentation is at or below two parties. This is also a problem for the other two measures. However, such low fragmentation rarely exists in the party systems that we are looking and thus this cannot be a major problem here. Furthermore, unlike the ideological standard deviation and the Esteban and Ray measures, this proposed measure has a rather simple interpretation – it is the weighted average of the mean pairwise distance of each party from all other parties.²