Electoral Disproportionality: What Is It and How Should We Measure It?

Alan Renwick

Note: A shorter, less technical version of this post is available on the Constitution Unit blog. Readers seeking a quick answer to the question of whether the UK election was the most disproportional the UK has seen are encouraged to go there.

In a report published earlier this month, the Electoral Reform Society declares the 2015 general election “the most disproportional election to date in the UK”. The ERS’s website cranks up the rhetoric further: “It’s official: this election was the most disproportionate in UK history.” In marked contrast, my own first cut at analysing the election results, published a few weeks earlier, said that this was not the UK’s most disproportional election: that, indeed, it was the least disproportional since 1992.

So what is going on here? Which of us is right?

The simple answer to the first of those questions is that we have used different methods to measure proportionality. The simple answer to the second question is that there is no simple answer: it very much depends on what you mean by disproportionality. Surprisingly, this is a point that the existing literature – so far as I can tell – has not adequately dealt with. I am writing this post in the hope of sparking some discussion.


Measuring Disproportionality

A perfectly proportional election is one in which every party wins seats in exact proportion to its share of the votes. A party with 45 per cent of the votes wins 45 per cent of the seats, a party with 5 per cent of the votes wins 5 per cent of the seats, and so on. Disproportionality, then, refers to the degree to which the actual result deviates from this ideal. In order to compare levels of disproportionality across lots of elections, we need a way of measuring it.

The difficulty is that there are many – indeed, in principle, infinitely many – ways of measuring it. Michael Gallagher (1991), in his classic 1991 article on the subject, reviewed six indices. Two more recent reviews by Taagepera and Grofman (2003) and by Karpov (2008) each test out nineteen indices – and their lists of possibilities do not entirely overlap.

So how are we to go about working out which of these many possible measures we should use? The two recent reviews just cited both consider how the various indices measure up against a range of statistical axioms. This is clearly an important step. As Monroe (1994) argued in an earlier contribution taking a strongly axiom-driven approach, however, it should not be regarded as enough. As he wrote, “The statistical choices about the form of the measurement should not be confused with the normative choices about the object of the measurement, however. Even when the distributional standard is clear, the choice of which deviations are important may not be.” (Monroe: 1994: 146).

Indeed, several early contributions – including Gallagher (1991) and Cox and Shugart (1991) – emphasized that measures of disproportionality differ from each other in part because they get at different underlying conceptions of what disproportionality actually is. Borisyuk et al. (2004) make a related point.

So far as I can see, however, very little work has been done to consider what we actually mean by disproportionality and how it might best be measured. The index that has become the industry standard – the least-squares or Gallagher index, discussed by Gallagher in his 1991 article – seems to have been propelled to popularity at least in part by its endorsement in Arend Lijphart’s classic 1994 book Electoral Systems and Party Systems. There, Lijphart did justify his preference in large part through appeal to intuitions about what looks more or less disproportional (Lijphart 1994: 58–67). As I shall argue in a moment, however, Lijphart’s exploration of intuitions is very brief and does not consider one of the main contenders for the title of best index. We need to probe some intuitions a bit more deeply.


The Loosemore-Hanby Index

The Electoral Reform Society’s report measures disproportionality using what is sometimes called the DV (deviation from proportionality) score. In order to differentiate it clear from other measures, I shall follow standard practice among political scientists and refer to it as the Loosemore-Hanby index, named after the two authors who originally proposed it (Loosemore and Hanby 1971). This index looks at the deviation between each party’s vote share and its seat share: if a party obtains, say 25 per cent of the votes and 20 per cent of the seats, the deviation is 5. The index adds up the absolute values of these deviations across all parties running in the election and divides the total by two.

This index became the standard measure of disproportionality in the 1980s. At least in part, that was because it has a nice intuitive meaning: it represents the percentage of parliamentary seats that all of the over-represented parties combined hold above their proportional share (or, equivalently the deficit experienced by all the under-represented parties). In the UK election of 2015, for example, the value of the Loosemore-Hanby index is 24.0. That means that the parties whose seat shares exceed their vote shares (the Conservatives, Labour, the SNP, the DUP, and – fractionally – Sinn Féin and the SDLP) collectively hold 24 per cent of the seats in the House of Commons more than they would if they were represented in exact proportion to their votes – that is, about 156 seats more. Conversely, the under-represented parties (UKIP, the Liberal Democrats, the Greens, and other smaller parties) collectively hold about 156 seats fewer than they would in the case of perfect proportionality.

Table 1 below shows the value of the Loosemore-Hanby index for each UK general election since 1945. As can be seen, the 2015 election does indeed have a strong claim to having the highest – or at least equal highest – level of disproportionality in the series. The previous record holder was the election of 1983 – when, infamously, the SDP–Liberal Alliance secured 3.5 per cent of the seats on 25.4 per cent of the votes while Labour won 32.2 per cent of the seats on 27.6 per cent of the votes. Overall disproportionality then, according to the Loosemore-Hanby index, was 24.1. That is fractionally higher than the 2015 figure. But the difference is actually only 0.05, and it is fair therefore to declare this a dead heat. According to Loosemore-Hanby, therefore, the 2015 election ranks in joint first place in terms of disproportionality.


Table 1. Disproportionality in UK General Elections since 1945: The Loosemore-Hanby Index

1945 1950 1951 1955 1959 1964 1966 1970 1974F 1974O
14.9 8.6 3.5 5.0 8.6 11.0 9.7 8.7 19.7 18.9
1979 1983 1987 1992 1997 2001 2005 2010 2015
15.2 24.1 20.8 18.0 21.2 22.0 20.5 22.8 24.0

Sources: Calculated by the author from data in Nohlen and Stöver (2010) and the websites of UK Political Info and the Electoral Commission.


Or Maybe Gallagher

The trouble with this, however, is that, much as the Loosemore-Hanby index has intuitive appeal, it has now fallen significantly out of favour among political scientists. The most widely used measure of disproportionality today is, as I have indicated, the Gallagher index, discussed by Gallagher in 1991. This was the measure that I used in my initial post-election analysis, leading me to the conclusion that this was the most proportional election since 1992. I used it, however, not because I had thought about the issue, but just because it’s the index that we political scientists habitually now use.

I shall explain the Gallagher index for the uninitiated in a moment, but the key point for now is that, according to this measure, the 2015 election was very far from being the UK’s most disproportional ever. In fact, as Table 2, below, shows, it was only the eighth most disproportional. Since the post-war Conservative–Labour duopoly began to break down in 1974, only two of the eleven elections have shown clearly lower levels of disproportionality.


Table 2. Disproportionality in UK General Elections since 1945: The Gallagher Index

1945 1950 1951 1955 1959 1964 1966 1970 1974F 1974O
11.7 6.9 2.9 4.1 7.3 8.9 8.4 6.6 15.5 15.0
1979 1983 1987 1992 1997 2001 2005 2010 2015
11.6 20.6 17.8 13.6 16.6 17.8 16.6 15.1 15.0

Sources: As Table 1.


So we have our first point of investigation. Why do the two indices produce such different verdicts? And are there any clear reasons for favouring one of these measures over the other?


What’s Wrong with Loosemore-Hanby?

The Loosemore-Hanby index has fallen from favour for two main reasons. The first is an indisputable problem: the index fails what is generally known in statistics as Dalton’s principle of transfers (those wanting detailed discussion might best go Monroe 1994). This is one of the statistical axioms that are standardly applied in assessments of disproportionality measures, and it is one of those that have relevance to the kinds of normative question that we encounter when considering disproportionality in real-world elections.

In respect of disproportionality measures, the principle of transfers asserts that if you take some seats from a party that is relatively better off and give them to a party that is relatively worse off, the value of your measure should fall. The Loosemore-Hanby index, however, doesn’t always do that. In fact, it is completely insensitive to transfers between over-represented parties – even if one is more over-represented than the other – or between under-represented parties. As we have already seen, it measures the total amount of over-representation and the total amount of under-representation, but it is entirely unresponsive to shifts within either side of this divide.

The 2015 election result provides us with clear examples of this problem. UKIP won 12.6 per cent of the votes but only one seat. The Liberal Democrats won just 7.9 per cent of the vote, but nevertheless held on to eight seats. Clearly, both of these parties fell well short of their proportional seat shares overall. But, just comparing between the two of them, UKIP are much worse off than the Lib Dems. If we accept that there are only nine seats to share between them, there is no way to deny that the outcome would have been more proportional had UKIP won five or six of these and the Liberal Democrats three or four. The Loosemore-Hanby index, however, stays exactly the same whatever distribution of those nine seats across the two parties we try.

Similarly, Labour and the Conservatives were both over-represented in 2015, but the Conservatives were over-represented by a great deal more. It is clear, therefore, that seat transfers from the Conservatives to Labour (up to a point, of course) would, on any reasonable definition of disproportionality, reduce that disproportionality. According to Loosemore-Hanby, however, things remain exactly the same.

The Gallagher index, by contrast, does not suffer from this problem. Here, transferring a seat from a better represented to a worse represented party always reduces disproportionality.


Small and Large Deviations from Proportionality

The second reason usage of Loosemore-Hanby has declined is concern over the fact that it treats lots of very small deviations from proportionality as equivalent to a few large deviations. Imagine one election in which ten parties run and each wins a seat share that is one percentage point higher or lower than its vote share. Then imagine another election in which there are two parties and the seat share secured by each is five percentage points above or below its vote share. We might intuitively think that the deviations from proportionality in the first case are too small to be worth bothering about too much, while the deviations in the second case are big and potentially very important. Yet the Loosemore-Hanby index produces the same disproportionality score (5) for both.

The Gallagher index responds to the intuition that a few big deviations matter more than many small ones by giving bigger deviations greater weight. Specifically, it starts, like the Loosemore-Hanby index, with the difference between each party’s vote share and its seat share, but then it squares each of these differences before summing them. It then divides the sum by two and takes the square root in order to leave us on roughly the scale that we started with. Squaring the differences gives extra weight to bigger deviations: a deviation of five percentage points becomes 25, while a deviation of one stays at one.

The question to ask here is whether we really all share the intuition that a few big deviations always matter more than many small ones. The case for this intuition tends to be made using examples such as the one I have just offered, where very many very small deviations are compared to very few much larger deviations: in fact, the one comparison of cases that Lijphart uses to explain why he favours Gallagher over Loosemore-Hanby is exactly the one I have described, with ten deviations of one and two of five. But do we still have the same intuitions when we compare real world examples that are less extreme?

Take, for example, how our two indices of disproportionality treat the Liberal Democrats, UKIP, and the Green Party at the last two UK general elections. The relevant numbers are shown in Table 3. In 2010, there was one large deviation (affecting the Liberal Democrats) and there were also two much smaller deviations. In 2015, there were three more middling deviations. The Loosemore-Hanby index doesn’t care about the size of these deviations individually: it just adds them up and comes to the conclusion that the total amount of disproportionality affecting these three parties was somewhat higher in 2015 than in 2010. (The table doesn’t show the full calculation of the index, as that would require us to include all the other parties as well. But the column marked “│V – S│” – showing the absolute values of the vote–seat differences – gives the first step in the calculation of the index as it affects these parties.)


Table 3. How Loosemore-Hanby and Gallagher treat disproportionalities in 2010 and 2015

2010 2015
Votes (%) Seats (%) │V – S│ (V – S)2 Votes (%) Seats (%) │V – S│ (V – S)2
Lib Dem 23.03 8.77 14.26 203.29 7.87 1.23 6.64 44.08
UKIP 3.01 0 3.01 9.59 12.64 0.15 12.49 155.98
Green 0.96 0.15 0.81 0.65 3.77 0.15 3.62 13.08
Total 18.16 213.54 22.75 213.14
Aggregate 27.09 8.92 24.28 1.54


For the Gallagher index, by contrast, the individual size of the deviations does matter. The large deviation faced by the Liberal Democrats in 2010 is given extra weight. As a result, the total amount of deviation for these parties registered in the calculation of the index – shown by the total of the squared vote–seat differences – is virtually identical for both elections – just over 213 in each case.

So the question, again, is whether Gallagher really captures our intuitions about the importance of these various deviations better than Loosemore-Hanby. My feeling – now that I have thought about the matter – is that it does not. My feeling is that, if what we are concerned about is simply how proportional or disproportional the election result is, all of these deviations matter in proportion to their size. In 2010, as the bottom row of Table 3 shows, the three parties won 27 per cent of the vote and 9 per cent of the seats. In 2015, their vote share fell slightly, but their collective seat share – already far below their proportional share – collapsed to well below 2 per cent. That looks to me like an increase in disproportionality.

The Loosemore-Hanby index is sometimes criticized for, as Lijphart puts it, “exaggerating the disproportionality of systems with many parties” (Lijphart 1994: 60). In fact, however, this index is entirely neutral as to how any given amount of under- or over-representation is divided up among parties. By contrast, the Gallagher index downplays disproportionality the more the votes are fragmented across parties such than any given party’s vote–seat deviation is smaller. As the fragmentation of the UK party system has increased over recent years, therefore, the standard measure of disproportionality has, it would appear, increasingly understated the true level of disproportionality.

I will be interested to hear what readers think about this. If my hunch is shared by others, there will be a good case for saying that Loosemore-Hanby better captures our perceptions of proportionality in these elections than does Gallagher.


But We Haven’t Finished Quite Yet: Absolute versus Relative Measures of Disproportionality

That already gives us some reason to doubt whether the established wisdom among political scientists about how to measure electoral disproportionality adequately captures changes in disproportionality in recent UK general elections. But there is more for us to think about here just than that.

In the 2010 election, the Liberal Democrats secured, as we have seen, 8.8 per cent of the seats from 23.0 per cent of the votes. In 2015, UKIP won just 0.15 per cent of the seats from 12.6 per cent of the votes. Which of these is the more strikingly disproportional result? Whether we use the Loosemore-Hanby index or the Gallagher index, the answer is that the Lib Dems in 2010 suffered the greater under-representation: their vote–seat gap, at 14.3 per cent, was greater than UKIP’s, at 12.5.

But many people (abstracting from any feelings they might have for or against either party) seem more troubled by the UKIP result: the Liberal Democrats might have been under-represented in 2010, but they did at least secure 38.1 per cent of their proportional share of the seats; UKIP in 2015, by contrast, won just 1.2 per cent of its proportional share. When I did a quick – entirely unscientific – test of opinion among my Facebook friends in the process of preparing this post, the substantial majority (though not all) of those who responded came back with the view that a UKIP-type result was more “unfair” than a Lib Dem-type result (I say “type” here because I actually made the numbers a bit less extreme in order to make the test harder).

The intuition here seems to be that what matters is not the absolute deviation of seat shares from vote shares – the starting point of both the Loosemore-Hanby and Gallagher indices – but rather the relative deviation: the deviation in proportion to the party’s support. To take a further example, Loosemore-Hanby and Gallagher both see as much disproportionality if a party with 30 per cent of the vote gets 25 per cent of the seats as if a party with 5.1 per cent of the vote win 0.1 per cent of the seats. Most people find that odd.

So far as I am aware, the first person to give detailed consideration to relative measures of electoral disproportionality was Michael Gallagher in the 1991 article that kicked off so much of this discussion. In fact, though that article became the basis upon which the Gallagher index rose to replace the Loosemore-Hanby index as the standard measure of disproportionality, its conclusion actually favoured a different index – one based on relative deviations. This was what Gallagher called the Sainte-Laguë index and what some others label the χ2 index. Gallagher said of the Sainte-Laguë index that there “is a strong argument in favour of its adoption as the standard measure of disproportionality” (Gallagher 1991: 49).

The Sainte-Laguë index simply takes the square of the vote–seat deviation for each party, divides it by that party’s vote share, and then sums these values across all the parties. The part of the calculation for the 2010 general election relating to the Lib Dems, therefore, is 14.3 squared and then divided by 23.0 – which comes out at 8.8 – while the element for UKIP in 2015 is 12.5 squared and divided by 12.6 – which is 12.3. For Sainte-Laguë, therefore, UKIP in 2015 experienced more marked under-representation than did the Lib Dems in 2010.

Given Gallagher’s advocacy of the Sainte-Laguë index and the fact that his article has become by far the most cited piece of work relating to the measurement of electoral disproportionality, it is remarkable that this index has subsequently been almost entirely ignored. It is included by Taagepera and Grofman and by Karpov in their reviews. So far as I can tell, however, no one since Gallagher in 1991 has suggested that this index should actually be used.

Lijphart, in his 1994 book, did discuss at some length the issue of whether absolute or relative measures of proportionality should be preferred, and he concluded in favour of the former. But he did so because the relative measure that he tested was yet another index – the d’Hondt index – which does indeed produce some very counter-intuitive results and which Gallagher has already concluded in 1991 was clearly inferior to the Sainte-Laguë index. Lijphart opted for Gallagher over d’Hondt by examining intuitions in several hypothetical cases. He began by acknowledging a case where the idea of relative proportionality does appear more intuitive than absolute proportionality:

“I believe that most people would agree that, in these illustrative cases of v1 = 40%/s1 = 41% and v2 = 10%/s2 = 11%, the larger party’s deviation from proportionality is indeed not as serious as the smaller party’s because it is a smaller relative deviation.” (Lijphart 1994: 63)

But he then added, “I would also suggest that this criterion becomes operative … only when the absolute deviations are the same.” (Lijphart 1994: 63). He justified this, saying

“I think that few people would agree that a 4 per cent deviation is the case of v1 = 40%/s1 = 44% is normatively on a par with v2 = 10%/s2 = 11%, v3 = 5%/s3 = 5.5% and v4 = 1%/s4 = 1.1%, as the d’Hondt notion of proportionality would maintain.” (Lijphart 1994: 63).

He then adds a further example: a four-seat district contested by two parties, one of which wins 41,000 votes, the other 10,000 votes. The d’Hondt index (which, as the name suggests, mirrors the logic of the d’Hondt proportional allocation method) is minimized if the larger party wins all four seats, rather than if that party receives three seats and the smaller party one. But Lijphart remarks, “I would submit that the most widely held view of proportionality would consider the second result to be the more just and equitable” (Lijphart 1994: 64).

Lijphart’s intuitions seem to me spot on here. As I have said, however, he is applying his intuitions to the wrong index. Had he instead tested the Sainte-Laguë index, he would have found that it did not violate his intuitions in either of the scenarios where d’Hondt falls down: Sainte-Laguë agrees that v1 = 40%/s1 = 44% is more disproportional than v2 = 10%/s2 = 11% and that giving one of the four seats to the party on 10,000 votes is more proportional than giving them all to the party on 41,000.

As I have said, the d’Hondt index is closely related to the d’Hondt formula for the proportional allocation of seats. The value of the index is, by definition, minimized by the allocation achieved through the d’Hondt formula (Gallagher 1991). The Sainte-Laguë index is related to the Sainte-Laguë electoral formula in the same way. Even when Lijphart wrote, it was well known that the d’Hondt formula violates one of our core intuitions about proportionality – that an electoral system, in order to be fully proportional, should not show systematic bias in favour of large or small units – while Sainte-Laguë is scrupulously neutral (Balinski and Young 1982; for more recent analysis, see also Schuster et al. 2003). It should therefore have been expected that the d’Hondt index of disproportionality would similarly violate our intuitions, while the Sainte-Laguë index, at least in this respect, would fit them. This makes it all the more puzzling that Lijphart did not consider whether the Sainte-Laguë index might be a runner.

Nevertheless, we have all subsequently simply followed Lijphart’s conclusion and used the Gallagher index without giving serious attention to the alternative.


How Does the Sainte-Laguë Index Perform?

Of course, one reason Sainte-Laguë has been ignored could be that it fails the test of the statistical axioms. Before going further, therefore, this needs to be checked.

The most comprehensive axiom-based review is that of Taagepera and Grofman (2003), who assess their nineteen possible indices in terms of twelve criteria. The Sainte-Laguë index (they called it the χ2 index) comes out of this comparison as one of the top four indices. It falls short of the two best performing indices, which are our old friends, Loosemore-Hanby and Gallagher. But the three criteria it falls short on are of limited relevance. One – the symmetry of how the terms are used in the index equation – matters for the measurement of some other forms of difference that Taagepera and Grofman are also interested in, but not for the measurement of disproportionality. Another – the fact that the Sainte-Laguë index becomes indeterminate if a party wins seats but no votes – is irrelevant to democratic elections, where such a result is impossible. The third – that the index is quite sensitive to the lumping of parties winning a substantial share of the votes into an “others” category – could be more serious in some cases. The data used here for UK elections, however, have “others” categories representing less than 1 per cent of the vote for all elections except 1945, where it is 1.8 per cent. So, at least in this case, it is not a big concern.

So far as the established axioms go, therefore, the Sainte-Laguë index seems to be at least as good as the main alternatives.

That, however, is as far as Taagepera and Grofman take us, for they do not give detailed consideration to whether we should prefer an absolute or relative measure. They do discuss the very similar contrast between difference and ratio measures – where they take what looks to me like the odd step of defining the Sainte-Laguë index as a difference measure because it starts not with the ratio of seats to votes but the ratio of vote–seat deviations to votes. As far as evaluating these options, however, they offer only an admission that their hunches are not clear.

We have seen already that the Sainte-Laguë index fits with Lijphart’s intuitions about hypothetical examples and with my and most of my Facebook friends’ intuitions about the last two UK general elections.

Furthermore, this index appears to be a rather more subtle reflection of our intuitions than I have suggested so far. I have said that it offers a measure of relative proportionality. But a pure relative measure – one that was interested only in vote–seat differences as a proportion of a party’s size – would certainly not fit our intuitions, as it would attach as much importance to a tiny as to a much larger party. Such a pure relative measure would start not with the parties’ squared vote–seat deviations, but with the unsquared deviations – dividing these by party vote shares and then summing. This index would score a maximum possible value of 1 for every party that won votes (no matter how few) but no seats, and would therefore in essence turn into little more than a count of all the micro-parties that ran somewhere in the country. The squaring of differences in the Sainte-Laguë index essentially removes this problem. Micro-parties do still weigh more heavily in the Sainte-Laguë index than in either of the other indices I have been discussing: to be precise, parties that win no seats contribute exactly twice as much to the Sainte-Laguë index as to the Loosemore-Hanby index and much less than that to Gallagher. But their impact remains small (in fact, it is precisely their vote share).

How, then, does the Sainte-Laguë index fare when we look at it in relation to disproportionality in a longer series of recent UK general elections. Table 4, below, shows the numbers. Like the other indices I have discussed, it shows a marked step up in 1974: all the indices agree that this was the key turning point for British elections in terms of disproportionality. And what is most notable is that, on this measure, the 2015 election unambiguously emerges as the most disproportional election in post-war history, comfortably surpassing the previous high, reached in 1983.


Table 4. Disproportionality in UK General Elections Since 1945: The Sainte-Laguë Index

1945 1950 1951 1955 1959 1964 1966 1970 1974F 1974O
12.3 8.0 1.6 2.5 6.5 11.2 8.5 8.5 22.0 20.9
1979 1983 1987 1992 1997 2001 2005 2010 2015
16.1 30.2 24.3 19.5 21.8 23.3 24.2 25.1 32.8

Sources: As Table 1.


The key final intuition to consider here concerns the comparison between 1983 and 2015. Did disproportionality this year really surpass that of 1983, as the Sainte-Laguë index wants us to believe?

Table 5 shows the key numbers for these two elections. In addition to vote and seat shares, it includes the absolute value of the difference between vote and seat shares (the building block of the Loosemore-Hanby index and, after squaring, of the Gallagher index) and the squared vote–seat difference divided by vote shares (the basis of the Sainte-Laguë index). It should be remembered that these two terms are measured on different scales, so we can’t straightforwardly compare a figure in the │V – S│ column to a figure in the (V –S)2/V column. Rather, we should look at whether the relative contributions of different figures to each column as a whole fit with our intuitions.


Table 5. Results of the General Elections of 1983 and 2015

1983 2015
Votes (%) Seats (%) │V – S│ (V –S)2/V Votes (%) Seats (%) │V – S│ (V –S)2/V
Conservative 42.4 61.1 18.7 8.2 36.8 50.8 14.0 5.3
Labour 27.6 32.2 4.6 0.8 30.4 35.7 5.2 0.9
Alliance/LD 25.4 3.5 21.8 18.8 7.9 1.2 6.6 5.6
UKIP 12.6 0.2 12.5 12.3
Greens 0.2 0 0.2 0.2 3.8 0.2 3.6 3.5
SNP 1.1 0.3 0.8 0.6 4.7 8.6 3.9 3.2
Plaid 0.4 0.3 0.1 0.0 0.6 0.5 0.1 0.0

Sources: As Table 1.


The first notable difference between the indices is in the importance they attach to disproportionalities affecting smaller and larger parties. For Loosemore-Hanby, the over-representation of the Conservatives is a very large component – indeed, in 2015, it is a larger part of disproportionality even than the under-representation of UKIP. The same applies, by extension, to Gallagher: remember that the effect of the squaring in the Gallagher index is to increase differences between smaller and bigger vote–seat gaps. For Sainte-Laguë, by contrast, the over-representation of the Conservatives makes up a smaller part of overall disproportionality: much more striking, from this index’s perspective, is the under-representation of the SDP–Liberal Alliance in 1983 and of UKIP in 2015. Similarly, Labour’s over-representation is in both elections given much more weight by Loosemore-Hanby (and somewhat more by Gallagher) than by Sainte-Laguë. Indeed, Loosemore-Hanby and Gallagher see Labour’s over-representation as a larger component of the disproportionality of the 2015 election than either the over-representation of the SNP or the under-representation of the Greens.

Sainte-Laguë’s treatment of Labour vs. the small parties certainly seems more intuitive than the treatment offered by either of the other indices. That Labour was over-represented in 2015 has barely been noticed, whereas the disproportionalities affecting the SNP and the Greens have been topics of widespread discussion.

Where our (at least, my) intuitions lie regarding the Conservatives is less clear. It is certainly true that complaints about disproportionality mainly focused in 1983 on the under-representation of the Alliance and in 2015 on the marked disparities among the four smaller Great Britain-wide parties, not on the over-representation of the Conservatives. But that may be not because the over-representation of the Conservatives was perceived not to matter, but because it was placed in a different box: it belonged to the discourse over manufactured majorities rather than the discourse over disproportionalities. So here it would appear that the indices are getting at different underlying notions of electoral fairness: Sainte-Laguë appears to capture very clearly whether the electoral system has been fair to particular parties and those who vote for those parties; Loosemore-Hanby and Gallagher relate more to the overall impact of these disproportionalities – a point that I will return to below.

Returning to the comparison between 1983 and 2015, the Gallagher index sees the 1983 outcome as more disproportional because it is very exercised by the two big vote–seat deviations in 1983 and rather less concerned about the more numerous but smaller deviations in 2015. The Loosemore-Hanby index sees the two elections as more-or-less equal in terms of disproportionality because it is equally exercised by vote–seat deviations that add up to about the same amount whether they are concentrated in a few parties or spread across many. And the Sainte-Laguë index places the 2015 election first in terms of disproportionality because it is most concerned about the fact that several parties this year secured such a small proportion of the seats to which they were proportionally “entitled”.

As I indicated previously, it is not clear to me why we should be less concerned about many small deviations than a few large ones – at least when the deviations are not tiny, as in the examples used by Lijphart, but rather involve significant disparities affecting parties with a significant voice, as here. As to the choice between Loosemore-Hanby and Sainte-Laguë, however, that seems to depend on which underlying concept we are interested in.



I started by asking who was right on the question of whether the 2015 election was the UK’s most disproportional: the Electoral Reform Society or me in my earlier analysis. That question is in danger of supposing that our task is to go to the maths in order to measure disproportionality correctly and see whether any hunches that we might have started off with were right or wrong.

I hope it is now clear that this gets things the wrong way round. Rather than going to the maths to check our hunches, we need to go to our hunches to work out which bit of maths is most appropriate.

Working on the basis of the examples I have explored here, it seems to me that the Sainte-Laguë is getting at what I would call “disproportionality in itself”. It looks at how similar each party’s seat share is to its vote share. And it embodies the notion that “similarity” here can only sensibly be thought of in relative terms: vote and seat shares are more out of kilter if a party wins 0.1 per cent of the seats on 5.1 per cent of the votes (or vice versa) than if those shares are 40.1 and 45.1.

The Loosemore-Hanby and Gallagher indices, meanwhile, appear to be closer to getting at the impact of disproportionality-in-itself on how the country is actually governed. Because it weights by party size, Sainte-Laguë, compared to the other indices, underplays the disproportionalities that affect larger parties. But the larger parties may be particularly important for the nature of government: the fact that the Conservatives secured an overall majority of seats in May on the basis of only 37 per cent of the votes clearly matters a great deal for how we are governed – and, indeed, perhaps has a bigger impact than the under-representation of UKIP.

The trouble with this is that how disproportionalities affect the result clearly has to do with many more things than the Loosemore-Hanby or Gallagher indices can capture in themselves. If we imagine an election in which one party gets 47 per cent of the votes and 52 per cent of the seats while another party wins 25 per cent of the votes and 40 per cent of the seats, any index will regard the second disproportionality as greater than the first; but it may well be that the first has the bigger impact on governance. In trying to do more than measure proportionality-in-itself, therefore, the Loosemore-Hanby and Gallagher indices may be trying to do more than can reasonably be expected of any single index.

The question of whether Gallagher should be preferred over Loosemore-Hanby as a measure of disproportionality-as-it-affects-governance illustrates the same difficulty. It might be that lots of small deviations matter less than a few large ones. Equally, however, it might not. If, say, all the under-represented parties belong to one likely coalition while all the over-represented parties belong to another, then the multiple small disproportionalities could add up to make a big difference to the overall result. Again, trying to get at how much various disproportionalities matter not just in normative terms for disproportionality-in-itself, but rather in consequential terms for the impact of disproportionalities on governance seems to be trying to do more than one index can manage.

It seems to me, therefore, that there is a strong case to be made for the Sainte-Laguë index as our standard measure of disproportionality – a case that, so far as I am aware, has not been properly explored before. If we are interested not just in disproportionality-in-itself, but also in the impact of the electoral system on governance, then we need to investigate that separately and additionally – by looking, for example, at the incidence and size of manufactured majorities.

Returning, finally, to the debate between the Electoral Reform Society and me with which I began, it appears fair to conclude, therefore, that they were right and I was wrong – the 2015 election is indeed the most disproportional that the UK has seen, at least since 1945. But the reasons they were right are not those they had in mind.

As I indicated at the start, I have written this post in the hope of sparking some discussion. Do you have intuitions that diverge from those I have suggested above? Are there scenarios in which the Sainte-Laguë index generates its own counter-intuitive results? Are there contributions to the existing literature that I have not mentioned that shed important light? Above all, is analysis of the Sainte-Laguë index something that you think I should pursue further, or should I drop it and go back to something less esoteric?

I look forward to hearing your responses.



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Borisyuk, Galina, Colin Rallings, and Michael Thrasher (2004). “Selecting Indices of Electoral Proportionality: General Properties and Relationships.” Quality and Quantity 38, no. 1 (February), 51–74.

Cox, Gary W., and Matthew Soberg Shugart (1991). “Comment on Gallagher’s ‘Proportionality, Disproportionality and Electoral Systems’.” Electoral Studies 10, no. 4 (December), 348–52.

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Schuster, Karsten, Friedrich Pukelsheim, Mathias Drton, and Norman R. Draper (2003). “Seat Biases of Apportionment Methods for Proportional Representation.” Electoral Studies 22, no. 4 (December), 651–76.

Taagepera, Rein, and Bernard Grofman (2003). “Mapping the Indices of Seats–Votes Disproportionality and Inter-Election Volatility.” Party Politics 9, no. 6 (November), 659–77.



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