Response to the Election Commission Consultation on Apportionment
A response to the Electoral Commission Consultation paper, Distribution between electoral regions of UK MEPs (April 2003)
David Butler, Steve Fisher, Ron Johnston, Iain McLean, Alistair McMillan, Roger Mortimore, and Peyton Young[1]
6 June, 2003
In the allocation of seats for the European Parliament across the regions of the UK, the Electoral Commission has been fortunate to have been given clear and simple criteria for making its recommendation, and to be dealing in an area in which the discipline of political science[2] has found a single best solution. The solution, known alternatively as the Webster or Sainte–Laguë method (they are identical in terms of outcome), is the only method which will achieve the Commission’s statutory mandate to ensure that ‘the ratio of electors to MEPs is as nearly as possible the same in each electoral region’. The method also copes in a straightforward and sensible way with changes in population relativities and is unaffected by adding or taking away new regions/countries from the apportionment process. This completely removes any dilemma relating to how to deal with the minima of three MEPs for any electoral region. Whether small regions[3] are included in the initial apportionment, or allocated the statutory minima at the start of the process has no effect on the distribution of seats between other regions under this method. The Webster/Sainte–Laguë method also has the benefit of being extremely simple to work out – it needs no more than a simple Excel Spreadsheet. It is, and has been rigorously proved to be, the best and most efficient of apportionment systems, with advantages over all other methods. The Electoral Commission should have no hesitation in recommending the Webster/Sainte–Laguë method for the apportionment of MEPs to the regions of the UK.
Unfortunately, the history of the allocation of seats to electoral regions has been bedevilled by misunderstanding and political manipulation. Balinski and Young (1982) trace the history of apportionment in the United States, showing how methods favouring large or small States, or Democrats or Republicans, were advocated and implemented according to changing circumstances. Without understanding the underlying properties of the apportionment process, decisions were made according to partisan assessments of the outcomes. In the United Kingdom, the apportionment process for the House of Commons has been carried out under self-contradictory rules, which have led to biases in the allocation of seats (see McLean and Mortimore 1992, McLean and Butler 1996). In the present instance, the Election Commission has the opportunity to learn from studies of the process of apportionment, which point to a single, non-biased, and optimal method.
This paper outlines the method known as Webster/Sainte–Laguë in a non-technical manner (for mathematical proofs see Balinski and Young 1982: Appendix A). It provides worked examples for both the theoretical examples used by the Electoral Commission and the actual regional electorates within the UK.
The worked examples which accompany the Electoral Commission Consultation Paper (2003) are all methods which lead to biased outcomes, and are in danger of suffering from well studied and avoidable paradoxes in allocation. A brief discussion of the problems associated with each method is provided.
The Webster/Sainte–Laguë method
The Webster/Sainte–Laguë method is extremely simple. In the US context it has been set out thus:
Choose the size of the house to be apportioned. Find a divisor x so that the whole numbers nearest to the quotients of the states sum to the required total. Give to each state its whole number (Balinski and Young 1982: 32)[4]
For the allocation of EU Parliamentary seats across the regions[5] of the UK, the number of seats to be apportioned (currently 87) will be set by the Treaty of Nice and subsequent EU Treaties. Given that the rules laid out (Electoral Commission 2003: 6) set a minimum allocation of three seats for each electoral region, the above method becomes:
Webster/Sainte–Laguë method:Take the number of seats to be allocated across the UK. Find a divisor x so that the whole numbers nearest to the quotients of the regions (given by the number of electors divided by x) sum to the required total. If no region is allocated less than three seats, give to each region its whole number. If any regions are allocated less than three seats, allocate them three seats. Excluding these regions, pick a new divisor so that the whole numbers nearest to the quotients of the remaining regions sum to the remaining number of seats. Give to each region its whole number.
The method set out by the French mathematician Sainte–Laguë operates in a slightly different way to the Webster method outlined here. However, the two are identical in terms of outcome[6].
Why is the Webster/Sainte–Laguë method seen by informed commentators as an optimal solution to the issue of the apportionment of seats amongst regions?[7]
1. It always apportions the right number of seats. This may sound slightly ridiculous, but some apparently intuitive methods end up allocating more/fewer seats than there are to fill[8].
2. It is not biased towards large or small regions. Other methods (such as the d’Hondt/Jefferson method) tend (on balance, rather than in every case), to favour either large or small regions. As Balinski and Young (1982: 77) show, using simple simulations, ‘Webster’s method is perfectly unbiased for any distribution of seats’.
3. It does not suffer from paradoxes associated with shifts in relative regional populations, or the inclusion or exclusion of regions. Under some methods of apportionment, a region which has a rapidly growing electorate can lose out to one in which the relative growth in electorate is much slower – known as the population paradox. Similarly, if the total number of seats is increased, whilst regional electorates remain the same, some methods will lead to regions losing seats, known as the Alabama paradox (after its first known occurrence). Further, some methods are sensitive to the inclusion of regions which are apparently irrelevant to the apportionment decision. Including a new region A with (say) three seats, and increasing the total number of seats by three, can still lead to redistribution of seats between regions B and C. This is known as the new States paradox. Sensitivity to these factors is important when (as in the case under examination) there are issues of minimum numbers of seats, and whether to include small regions in initial apportionments. The Webster/Sainte–Laguë method (along with other divisor methods) avoids these paradoxes.
4. It is the best method for apportioning seats close to the overall average UK electorate. Amongst the class of apportionment methods known as ‘divisor’ methods (the only ones which avoid over/under allocation of seats, and which avoid the paradoxes outlined in point 3 above), the Webster/Sainte–Laguë method is the one most likely to track the average electorate per seat (or quota). Since the ratio of electors to MEPs is the fundamental criterion laid down for the Election Commission, it should carry particular weight.
Theoretical problems with the Election Commission proposals
The Election Commission outlines four methods of redistributing seats, and gives two scenarios under which each are tested. It has to be made clear that all four of the methods suggested in the Consultation paper presented by the Election Commission areextremely unsatisfactory.
The first method is termed the Divisor method, but in actual fact appears to be what is known as a quotamethod. It suffers from the fundamental flaw that it may not allocate the correct total number of seats. As such, it violates a key principle which should guide this method of apportionment, particularly since the total number of UK seats will be fixed under European Treaty. It is easy to illustrate the problem with this method, using a contrived example. Take a situation where there are 100 seats to be distributed between 10 regions, and the total electorate is 100,000,000, with electorates as distributed in Table 1 below. The average electorate (i.e. the quota) is 1,000,000. It is not necessary to involve any minimum requirement of seats in order to confuse this system sufficiently for it to become unworkable. The example in Table 1 shows that allocating each region a rounded quota figure, leads to a total allocation of 101 seats. Some method must be devised for reducing the allocation by one. However, tinkering with the method, in order to correct this is likely to introduce bias, and susceptibility to counter-intuitive paradoxes of apportionment. It is clear that the first method suggested by the Election Commission is fundamentally flawed. Whilst quota allocation has some intuitive appeal, it can only act as a guide to the apportionment of a fixed number of seats. It may work in some circumstances, but in many cases it will not allocate the correct number of seats.
It could be argued that this is a carefully constructed theoretical example of a mathematical anomaly, and that this sort of thing is unlikely to happen in practice. However, this is not the case – quota method apportionment frequently leads to the wrong number of seats being allocated. If the quota method were applied to the regional allocation of seats using the actual regional electorates for 2001 (shown in Table 4 below), the quota method apportions the wrong number of seats for both the 87 seat case and the 72 seat case[9]. In the first situation, the rounded quota allocations only sum to 86, and in the second the rounded quota allocations only sum to 71[10]. The indeterminacy of quota methods is a serious and practical problem; it fundamentally undermines any case for basing the apportionment of seats on this procedure.
Table 1Example showing inconclusivity of quota method
Population / hypothetical quota allocation / hypothetical quotasettlement rounded
Region 1 / 9600000 / 9.6 / 10
Region 2 / 14700000 / 14.7 / 15
Region 3 / 5800000 / 5.8 / 6
Region 4 / 8700000 / 8.7 / 9
Region 5 / 6500000 / 6.5 / 7
Region 6 / 12100000 / 12.1 / 12
Region 7 / 7100000 / 7.1 / 7
Region 8 / 15100000 / 15.1 / 15
Region 9 / 6200000 / 6.2 / 6
Region 10 / 14200000 / 14.2 / 14
Total / 100000000 / 100 / 101
The second Outlier method is the same as the first, but excluding any small regions that have an automatic entitlement to three seats. This system suffers from the same fundamental flaws as the first method.
The third Iterative method, using an scheme based on the d’Hondt formula, avoids the problem with the first two methods. It is a scheme that will manage to allocate the correct number of seats. The d’Hondt method of seat allocation is equivalent to the Jefferson divisor method (see McLean and Mortimore 1992: 295, Balinski and Young 1982: 18). The problem with this method is that it tends to be biased towards large regions. As such, it tends to violate the principle of equalizing the ratio of electors to MEPs across regions. It is similar, but inferior, to the Webster/Sainte–Laguë method.
The Regression method suggested by Prof. Paul Whiteley is based on a central and flawed assumption, that quota methods will produce ‘an ideal distribution of seats’, when (as detailed above) it does no such thing. However, given that small regions have to have a minimum number of seats, the method goes on to use regression analysis to find the best fit for alternative allocations. The rationale behind this method is abstruse, and there is no theoretical justification for using the size of residual error in order to reallocate seats.
There appear to be three key problems with this model:
- The simple regression model assumes a linear relationship between seats and votes, but this assumption is violated by the fact that setting a threshold or minimal number of seats for small regions skews any distribution. The presence of (one or more) points which (in the hypothetical example) do not deserve three seats, but are allocated three anyway, will skew the distribution. Given that any resulting linear estimate will be a bad estimator of the other points, it is not clear why this should be used to reallocate seats. It seems almost certain to penalize large regions, simply because they suffer from the bad predictive quality of the model through being further away from the point at which it is skewed away from a better fit.
- The proximity of any allocation of seats under this method will not be directly related to the ratio of electors to MEPs. The residual measures the shortest distance between a suggested distribution and some hypothetical predicted best-fit, rather than the deviance from any actual alternative distribution. As such, it violates the central criterion that the ratio of electors to MEPs should be as near as possible the same in each electoral region.
- Such a model is liable to suffer from each of the paradoxes of apportionment outlined above. It is very sensitive to (what should be) irrelevant effects.
Simulations using the Webster/Sainte–Laguë method and comparisons with the Election Commission proposals
Table 2 replicates the Election Commission figures for Scenario 1, and Table 3 for Scenario 2 (Election Commission 2003: 19). To these are added extra columns, showing the distribution of seats under the Webster/Sainte–Laguë method, and electoral quotas for all regions, and the electoral quota for all regions except Region B (which is given 3 seats according to the minima criterion under all of the methods).
Table 2Scenario 1 and different methods of apportionment
Region / Electorate / Divisor / Outlier / Iterative / Regression / Webster/ Sainte–Laguë / Quotas (including Region B) / Quotas (excluding Region B)A / 3,456,789 / 5 / 5 / 5 / 5 / 5 / 5.2 / 5.0
B / 1,234,567 / 3 / 3 / 3 / 3 / 3 / 1.9 / –
C / 5,678,901 / 8 / 8 / 8 / 9 / 8 / 8.6 / 8.3
D / 4,567,890 / 7 / 7 / 7 / 7 / 7 / 6.9 / 6.7
E / 8,901,234 / 13 / 13 / 13 / 12 / 13 / 13.4 / 13.0
Total / 23,839,381 / 36 / 36 / 36 / 36 / 36 / 36 / 33
For Scenario 1, each of the Divisor, Outlier, Iterative, and Webster/Sainte–Laguë methods produce the same distribution of seats, after Region B has been allocated the minimum of 3. This allocation corresponds to the nearest number of quota entitlements for regions A, C, D, and E, after Region B (and its three seats) have been discounted. However, the Regression method deviates from the number of quota seats each region deserves. Region E contains 13.0 ‘quota’ seats, but is allocated only 12 seats under the Regression method. It is clear that the distribution under this method has greater deviation from the principle that the ratio of electors to MEPs should be as nearly as possible the same.
Table 3Scenario 2 and different methods of apportionment
Region / Electorate / Divisor / Outlier / Iterative / Regression / Webster/ Sainte–Laguë / Quotas (including Region B) / Quotas (excluding Region B)A / 3,456,789 / 4 / 4 / 5 / 5 / 5 / 4.8 / 4.6
B / 1,234,567 / 3 / 3 / 3 / 3 / 3 / 1.7 / –
C / 5,678,901 / 8 / 8 / 8 / 7 / 7 / 7.8 / 7.5
D / 6,789,012 / 9 / 9 / 9 / 9 / 9 / 9.4 / 9.0
E / 8,901,234 / 12 / 12 / 11 / 12 / 12 / 12.3 / 11.8
Total / 26,060,503 / 36 / 36 / 36 / 36 / 36 / 36 / 33
In Scenario 2 the Webster/Sainte–Laguë apportionment corresponds to the Regression method, and differs from the Divisor, Outlier, and Iterative examples. This case is interesting, because when Region B is excluded from the analysis (it is allocated the minimum of 3 seats) and the quota allocations for the remaining four are worked out, a simple rounding solution allocates the wrong number of seats. This is another example of the indeterminacy of quota methods, as highlighted by the case given in Table 1. If the Quotas excluding region B are rounded and added up, region A should have 5 seats, Region C should have 8, Region D should have 9, and Region E 12. These add up to 34, rather than the 33 seats which remain after Region B has been allocated 3 of the 36 seats. In this case, it is not clear how the Election Commission came up with the Divisor or Outlier allocation, since both methods are indeterminate for this example. These two methods deviate from the quota by giving Region A fewer seats than it should have given its quota allocation of 4.6, whereas Region C with 7.5 quotas gets 8 seats. This, again, clearly deviates from the principle that the ratio of electors to MEPs should be as nearly as possible the same. The Iterative method removes the ‘extra’ seat allocated according to a quota method from Region E, which is allocated 11 seats, when it contains 11.8 quotas.
For both Scenarios 1 and 2, the Webster/Sainte–Laguë method is most consistently close to the quota allocation. This is not surprising, since it has been shown to be the mathematically most likely to do so. This corresponds to the criterion of allocating seats so that the ratio of electors to MEPs is as nearly as possible the same in each electoral region, subject to a minimal allocation of 3 seats. It has been shown, above, that the Webster/Sainte–Laguë method is technically superior to any other method for fairly apportioning seats. These two Scenarios give practical examples of the superior nature of seat allocations under the Webster/Sainte–Laguë method.
Finally, a worked example (with spreadsheet), is included for the UK, using figures for the Electorate in 2001[11]. The results are shown in Table 4, the working are shown in the accompanying Excel Worksheet (Webster method.xls). To illustrate the relative share of the electorate, the percentage of each electorate in each region is shown (column 2), along with the number of quota entitlements for each region, given total UK allocation of 87 and 72 seats (columns 3 and 6). The average population per seat (i.e. the quota) is 509,478 for the 87 seat example, and 615,619 for the 72 seat example.
With no minima, the Webster/Sainte–Laguë allocation for 87 seats is shown in column 4. It is calculated using a divisor of 504,000. This is smaller than the quota, because using the quota and rounding off the entitlements only leads to the allocation of 86 seats (and hence the Electoral Commission ‘Divisor’ (i.e. quota) method would be inconclusive). The Webster/Sainte–Laguë method allocates the extra seat to the South West region – receiving 8 seats with a quota entitlement of just 7.4. If Northern Ireland is given an extra seat, in order to ensure it receives the minimum of 3 seats, a new divisor has to be used to reallocate the remaining 84 seats (column 5). If a quota system were used, this would raise issues of whether to continue to include the Northern Ireland population to determine the quota, or not. Under the Webster/Sainte–Laguë method, this is not an issue. All that is required is a divisor which distributes the right number of seats. In this case, a divisor of 509,000 allocates the right number of seats. The extra seat given to Northern Ireland is taken from the South West, which in the earlier allocation had been given an above quota number of seats.