Shugart and Taagepera (2017) have a nice schematic in Votes from Seats that shows how the various logical models that they build link up, shown in the left-hand panel of Figure 1. At the top is the seat-product, which is determined by the district magnitude, \(M\), and the total number of seats, \(S\). After that, comes the number of seat winning parties, \(N_{S0}\), and the number of vote winning parties, \(N_{V0}\). Next, they model the seat and vote shares for the first placed party, \(s_1\) and \(v_1\), and then use these shares to model other useful measures like the effective number of vote- and seat-winning parties, \(N_{S2}\) and \(N_{V2}\), and the disproportionality, \(D_2\), which is also known as the Gallagher index.
My current project is to see if we can simplify the schematic on the left so that it looks more like the one on the right. In particular, I would like to see if we can do away with the simplifying step in layer 3 where we assume that the first-placed party’s share are a good proxy for all parties’ shares. In a recent paper, I show that we can model the effective number of parties, \(N_{2}\), as a function of the actual number of parties, \(N_{0}\), without this simplifying step. The resulting equation—which I have called the “square root rule”—holds that \(N_{2} = N_0^{1/2}\). In doing so, I also show that we can model the number of vote-winning parties, \(N_{V0}\), as a function of the number of seat-winning parties, \(N_{S0}\), and the number of parties that contest the election, \(N_{C}\), such that \(N_{V0} = \left( N_{S0} \times N_C \right)^{1/2}\). We are going to use both of these equations below, so keep them in mind.
The problem that we face, however, is that we still need the first-party vote and seat shares to estimate disproportionality, \(D_{2}\). I have some ideas about how we might overcome this problem, which I am going to spell out in this post. However, I am only about 80% of the way there, so a complete solution still eludes me. Hopefully, though, by spelling out my reasoning I might make it easier for someone else to chart a way forward. Though the intention is to model disproportionality as a function of actual numbers of parties, we’re going to start with the effective number of parties for reasons that will soon become clear.
The Effective Number of Parties
Laakso and Taagepera (1979) define the effective number of parties, \(N_{2}\), as follows:
\[ N_{2} = \frac{1}{\sum_{i=1}^{N_{0}} p_{i}^{2}} \]
Where \(p\) is some distribution of party shares (whether of seats, votes, or anything else) and \(N_{0}\) is the actual number of share-winning parties. It can be shown1 that we can rewrite this equation such that:
\[ N_{2} = \frac{N_{0}}{1 + N_{0}\left(N_{0}-1\right)\mathrm{Var}\left(p\right)} \]
Where \(\mathrm{Var}\left(p\right)\) is the variance of the distribution of party shares. Thanks to the square root rule, we know that we can replace \(N_2\) with \(N_0^{1/2}\). After making this substitution and rearranging to solve for \(\mathrm{Var}\left(p\right)\), we arrive at a logical model of the variance of party shares as a function of the number of share winning parties:
\[ \mathrm{Var}\left(p\right) = \frac{N_{0}^{1/2} - 1}{N_{0}\left(N_{0}-1\right)} \]
This, I think, is a useful discovery in its own right and could have many potential applications. For now though, keep it in mind along with the other two equations that I discuss in the initial paragraphs.
Disproportionality
We express the Gallagher index of disproportionality (Gallagher 1991), which we denote \(D_{2}\) , as follows:
\[ D_{2} = \sqrt{\frac{1}{2} \sum_{i=1}^{N_{V0}} \left(v_i -s_i \right)^2} \]
Where \(v\) is a distribution of vote shares, \(s\) is a distribution of seat shares, and \(N_{V0}\) is the number of vote winning parties. As before, it can be shown that we can rewrite this equation such that:
\[ D_{2} = \sqrt{\frac{N_{V0}-1}{2} \left( \mathrm{Var}\left(v\right) + \mathrm{Var}\left(s\right) - 2\mathrm{Cov}\left(v,s\right) \right)} \]
By definition, the covariance of any two variables is also equal to their correlation multiplied by the product of their standard deviations, such that \(\mathrm{Cov}\left(v,s\right) = \mathrm{Corr}\left(v,s\right) \times \mathrm{Var}\left(v\right)^{1/2} \times \mathrm{Var}\left(s\right)^{1/2}\). This gives:
\[ D_{2} = \sqrt{\frac{N_{V0}-1}{2} \left( \mathrm{Var}\left(v\right) + \mathrm{Var}\left(s\right) - 2 \times \mathrm{Corr}\left(v,s\right) \times \mathrm{Var}\left(v\right)^{1/2} \times \mathrm{Var}\left(s\right)^{1/2}\right)} \]
The equation is now growing in size. However, note that we have logical models for all of the terms that it contains save for the correlation, \(\mathrm{Corr}\left(v, s\right)\): \(N_{V0} = \left(N_{S0} \times N_{C} \right)^{1/2}\) and \(\mathrm{Var}\left(p\right) = \frac{N_{0}^{1/2} - 1}{N_{0}\left(N_{0}-1\right)}\). Ordinarily, we would determine a suitable value for the correlation by computing the mean of its two extremes, -1 and +1. However, I’m not so sure that that’s the best way forward in this case. If we compute the arithmetic mean we get \(\left(-1 + 1\right)/2 = 0\), implying no correlation. But electoral systems rarely produce seat distributions that share no relationship with the vote distributions that inform them. Likewise, if we take the geometric mean, we get into even trickier territory, since \(\sqrt{-1 \times 1} = \sqrt{-1} = i\) an imaginary number. Personally, I’m not against these models straying into imaginary and complex numbers, but it would certainly make things a lot more complicated.
Future Steps
So far, I have modelled disproportionality by using the equation above, substituting out the appropriate terms, and assuming that the correlation between vote and seat shares is 1. But perhaps there is some smart way forward that I have missed. Likewise, other approaches might be more fruitful. Two that spring to mind are:
To note that \(\frac{1}{N_{2}}\) represents the share for each effective party. Using these instead of the first party shares might better approximate disproportionality since they reflect the entire distribution.
To model proportionality using the Rényi divergence instead of the Gallagher index. As I show in this paper, the effective number of parties shares an exact equivalence with the Rényi entropy. So it might be that there is some way to get at proportionality if we operationalise it in the same way that we operationalise the effective number of parties.
References
Gallagher, Michael. 1991. “Proportionality, Disproportionality and Electoral Systems.” Electoral Studies 10 (1): 33–51. https://doi.org/10.1016/0261-3794(91)90004-C.
Laakso, Markku, and Rein Taagepera. 1979. “‘Effective’ Number of Parties: A Measure with Application to West Europe.” Comparative Political Studies 12 (1): 3–27. https://doi.org/10.1177/001041407901200101.
Shugart, Matthew S., and Rein Taagepera. 2017. Votes from Seats: Logical Models of Electoral Systems. Cambridge, UK: Cambridge University Press.
Footnotes
I hate it when people say this, then don’t actually show how, but this is a post not a paper and want to keep things short, so please take this on trust!↩︎