Proportional representation, Lorenz diagram and Gini measure

The former two weblog texts discussed British and Scottish incomprehension of democracy. Our discussion used numbers though. A picture says more than one thousand words. Lorenz curves are a nice way to display inequality. The data, calculations and following charts are in this excel sheet: 2015-2017-UK-Holland-NIreland-Lorenz-Gini

The UK General Elections in 2015 and 2017

The following two charts show the results in the UK General Elections of 2015 and 2017.

The horizontal axis gives the cumulative percentage of the popular vote. The vertical axis gives the cumulative percentage of the seats. If there is proportional allocation of seats, then the blue line of the seats would cover the pink diagonal.

The parties in the line-up have been ordered by mismatch. Conforming to the Sainte Laguë / Webster criterion (see this discussion by Alan Renwick), the mismatch is determined by ((the ratio of % seats to the % of votes) minus 1). For example in 2015, UKIP with 12.5% of the votes got only 0.2% of the seats, namely 1 seat for Nigel Farage himself. Their mismatch is 0.2 / 12.5 – 1 = 0.012 – 1. LibDem got 7.8% of the votes but only 1.2% of the seats, a mismatch of 0.158 – 1 (with rounding). Another mismatch are the parties that got no seats: the “Others” still got 2.1% of the votes, which means a mismatch of 0 / 2.1 – 1 = -1.

One might argue that the wasted votes should be omitted from the graph and Gini, yet, they rather stand out as a sore spot in current representation. A proportional representation of the wasted vote w in total n is possible by leaving seats empty or by filling the seats and taking a qualified majority f = 1/2 / (1 – w / n). In 2017 the wasted vote was 3.5% and then f = 50% / 96.5% = 51.8%. A representative majority in a full House of 650 seats then requires 337 seats, and not 325. See here.

The situation in 2017 has improved mainly because UKIP no longer really participated. The LibDem still got 7.3% of the vote and 1.8% of the seats, which is a marginal improvement.

The Gini coefficients are 29.7% in 2015 and 15.6% in 2017.

The graphs and coefficients are inaccurate because of strategic or tactical voting. A voter who favours a Conservative candidate but sees a loss against a Labour candidate might vote for the LibDem, reflecting a {Conservative, LibDem, Labour} preference ordering. There will be some averaging out, but the official votes will likely not reflect the true proportions of the first choices.

The nice thing of these graphs also is that one can recognise some of the parties. In both graphs the Conservatives are on the right hand side, with 36.4% of the votes and 50.8% of the seats in 2015, and 42.2% of the votes and 48.8% of the seats in 2017.

(These percentage take the wasted votes including the invalid votes as part of the denominator. Other sources may report that e.g. the Conservatives got 43.8 of the vote, looking only at the parties that got seats.)

The Dutch General Elections of 2017

The PR Gini for the UK shows that it is strikingly disproportional. Holland provides a useful point of reference. Holland had general elections in 2017 too, and its PR Gini is 3.6%. The major impact in Holland are the 2% of voters for small parties who got no seats. The Dutch qualified majority is f = 50% / 98% = 51%. In a PR system there will be strategic voting too, for example w.r.t. the coalition government. This however is no good reason to adjust the PR Gini coefficient, since such considerations are not quite those of proportionality, see also the discussion in the Appendix.


Conclusions are:

  1. The UK is at an alarming distance from proportional representation. This is detrimental for: (a) The ability to work together, compromise, form coalitions, and respect the opposition, (b) The possibility for smaller parties to partake in government and responsibility. (c) The entry and exit of new parties. (d) The notion among the electorate that they are represented.
  2. The Lorenz graph is a useful tool to show proportionality. The graph and Gini coefficient are not difficult to make. The ordering via the Sainte Laguë / Webster criterion gives (slightly) higher Gini coefficients (less proportionality) than ordering by the difference between % seats and % votes.

Technical appendix

The calculation of the Gini is straightforward. Each step, from one party to the next one, generates a small trapezium, with the area h (a + b) / 2. The height h is in this case the horizontal distance, given by the vote share of the next party. The sides a and b are the differences (on the left and right) between the diagonal and the cumulated seat curve. Summation of all these areas gives a total A. The Gini is equal to 2A, since the whole area of the square is 1. (The formulas for the Gini in wikipedia are more complex than needed for this piecewise linear application.)

Once I had decided to use the Lorenz graph, a google generated some predecessors.

Orit Kedar, Liran Harsgor and Raz A. Sheinerman (2013) refer on page 5 to Taagepera and Shugart (1989) Seats and Votes. Let me reproduce the quote from the first authors quoting the second authors:

“They note that ‘an alternative [to the measure of deviation from PR which they use] is the Gini index of inequality, which has theoretical advantages but is more complex to calculate’ (p. 204). They add that ‘the Gini index is the most widespread index of inequality, and it does satisfy Dalton’s principle [of transfers]. The Gini index is useful for many purposes other than electoral studies (where it has been little used)’ (p.263).”

The calculation above has been straightforward. It must be mentioned that Kedar et al. have a more complex analysis with districts though.

Anish Tailor and Nicolas Veron (2014) look at inequality in the European Parliament. Their problem is that Germany has 700,000 votes per seat while Malta has 70,000 votes per seat. They find a Gini of the UK of 6.3%, but this thus concerns another research question. If one would look at representation by parties then the EU Parliament might be less disproportional.

Kestelman (2005) also considers measures of apportionment and proportionality, and also refers to Taagepera and Shugart for the Gini (p14). He states that the Gini would be complicated to explain and calculate, while it is rather simple, see above Excel sheet. Thus, curiously:

“Fortunately highly correlated with LHI [Loosmore-Hanby Index], the Gini Disproportionality Index (GnI) is rather complicated to explain and calculate (virtually necessitating computerisation).” (p16)

Kestelman also suggests that STV would be a proportional method, but then he might neglect that an application to districts causes disproportionality over the whole nation (see this example).

Alexander Karpov (2008) gives a more analytical overview of the various measures.  He sums over the ratio of shares (%seats / %votes) but I do not see the rationale for this yet. The calculation above uses the ratio only for ordering, and uses levels with their intuitive interpretation.

Karpov’s article got a comment from Michela Chessa (2012) who points to indices that look at power. Indeed, if a party has 49.9% of the vote and the mere technique of apportionment would generate a majority of 50.1% in the seats, then one might wonder whether this is merely technique, and not a major decision on content.

Update June 23 2017: (1) Originally I sorted the parties on the difference between the % seats and the % votes, but the ratio is better, and indeed gives slightly higher Gini coefficients. One can easily check this by sorting differently in the excel sheet. (2) Alan Renwick repeats the useful distinction between the measure of proportionality and the measure on impact (like power). (3) If seats are allocated using one particular criterion (like the Sainte Laguë criterion), then it doesn’t seem so much useful to see what it means in terms of another criterion, for, if the other criterion really is better, then use this to assign the seats. Thus the issue is intellectually rather dead in continental Europe, that already applies proportionality. The issue only comes up here because of the situation in the UK.

Update June 30 2017: (1) Northern Ireland included, see this discussion. (2) The excel sheet for a country/year now contains an unsorted section and a section that sorts automatically. (3) Uniform chart sizes. (4) With a small correction, the Dutch PR Gini is 3.6 instead of 3.5.

Apparently I cannot find a picture of Max Lorenz (1876-1959) but there are some of Corrado Gini (1884-1965).

Corrado Gini (1884-1965) (Source: SIS, Instituto Centrale di Statistica)


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