Links to sections:
The 'Ross quota' is here named for J F S Ross. In his 1955 book,
Elections and Electors, he identified a 'natural law' governing the
permissible variations, in the number of constituents per constituency in a
single member system, with regard to local community boundaries.
The Ross quota generalises his rule for one member systems to two-member
systems, or three-member systems, etc. In other words, the Ross quota applies
to any uniform member system.
In a democracy, voters are supposed to be equally represented. Each member of parliament ( MP ) represents an equal number of voters. A parliament of 600 MPs might represent 36 million voters. So, each MP should be responsible for 36 million voters divided by 600. That is 60 thousand voters per MP.
This total voters, V, divided by total number of representatives, R ( or
the 'seats' they take in parliament ) is called the 'Hare quota'. That is V/R.
This quota is named after Thomas Hare. In mid-nineteenth century Britain, he
proposed proportional representation for general elections. His scheme treated
the country as a single constituency, in which his quota of votes was enough
to elect a candidate as an MP.
In the example, any candidate, who gathered 60 thousand votes from anywhere in
the country, would be elected to parliament. Hare's system, which
proportionally counts ballot papers giving ranked choices of candidates,
virtually maximises the voters' choice of representation.
At the other extreme to Hare's system, there is the single member system,
the most local system. This minimises choice, or the electoral principle, in
favor of the locality principle. But the Hare quota is still needed to give
the required number of constituents per single member constituency. That is 60
thousand, in the above example.
In fact, this required average of voters per representative is a starting
point for practically any system of representative democracy. That includes
multi-member systems. A two-member system would require the Hare quota to be
doubled, to estimate the required number of constituents per two-member
constituency. That is 2 x 60 = 120,000. A three-member system would require
constituencies to be three times the Hare quota or 180 thousand
constituents.
When Hare introduced his system in 1857, the Hare quota was meant to be an
elective quota. But it certainly does not serve that purpose in a single
member constituency. One candidate must win half the votes not to be defeated
by another. The Hare quota would require a candidate to win all the votes, if
it were used as the elective quota.
The proper elective quota is not the Hare quota of V/R but V/(R+1). This
latter quota is named the Droop quota, after H R Droop, who explained it in
1869. When there is one representative to be elected ( R = 1 ) the Hare quota
would give total votes divided by one ( V/1). But the Droop quota gives total
votes divided by one, for the number of representatives, plus one:
V/(1+1).
If both candidates got half the votes, a random choice would have to be made, to break the tie.
In a two-member constituency, the Droop quota, or elective minimum votes a
candidate needs to be elected, is one third of the votes. Thus, two elected
candidates guarantee a minimum proportional representation of two-thirds the
voters in that two-member constituency.
In a three-member constituency, the Droop quota is one-quarter the votes,
giving a 'PR' of three-quarters the votes. And so on.
Thus, for a 600-member constituency of the whole nation, like that proposed by
Hare, the strictly correct elective quota would not be V/R = 36m./600 =
60,000.
Instead, the Droop quota gives V/(R+1) = 36m./(600+1) = 59,901 ( rounded-up to
the nearest whole number to ensure no candidate is wrongly elected on less
than the minimum ).
The Hare quota decides how many voters each representative will serve. But the Droop quota is needed to decide between voters who the reps will be. There is another quota that follows directly on from these two. Ive called it the Ross quota because it is implicit in his explanation of the limit to allowed variations in the number of constituents, with regard to natural boundaries, in a single member system.
Suppose the above example that requires each single member constituency be of 60,000 constituents. These are to give local representation. That argues their boundaries must be true to the localities. The question is how much bigger or smaller than 60 thousand should be allowed, with respect to natural variations in the population sizes of local communities?
J F S Ross answered this. A constituency should not be bigger than 80 thousand nor smaller than 40 thousand. Why? Because, a constituency of 80,001 is 20,001 bigger than the equal constituencies' benchmark of 60,000. If you divide the over-size constituency into two single member constituencies of 40,000 and 40,001, they are both closer to the required 60,000.
There was no practical reason for Ross to go on to ask what are the permissible variations for a two-member system, which was the English general election system, that had been gradually abolished over the nineteenth century and earlier part of the twentieth century.
However, the question of permissible variations in any uniform member system is of more than theoretical interest. Using the previous example, a two member system would mean constituencies had two seats representing 60,000 constituents each. The total constituents per two-member constituency is 120,000.
We can try a formula that fits the case of single member systems, and see
if it works for two member systems. The votes per constituency divided by the
number of seats plus two, or V/(R+2), gives 60,000/(1+2) = 20,000. This gives
the permitted variation of constituents in the single member system.
This formula also looks promising as the Ross quota, because it follows
directly on from the Hare quota, V/R, and the Droop quota, V/(R+1).
Now to apply V/(R+2) to a two member system: 120,000/(2+2) = 30,000.
One considers that the variations in a single member system were determined in terms of variations in the permitted number of constituents per representative. We hold to this, rather than constituents per constituency: it would be wrong to think that the permitted variation in two member constituencies was 120,000 plus or minus 30,000 constituents.
What is wrong with the 150,000 to 90,000 variation? The smallest permissible constituency is allowed to go down to two seats representing 45,000 each. But the largest permissible constituency would only be allowed to run down to two seats representing 50,000 each, after a third portion of 50,001 was transfered towards making-up another two-member constituency.
Only the lower limit is correct: 90,000. This means that the two seats in
the double member constituency must not be more than 15,000 each below the
required 60,000 constituents per representative.
The upper limit on two-member constituencies is not 150,000 but 135,000.
Consider that a community numbers 135,001. It would then be more equitable, in
terms of the 120,000 'benchmark' of equal constituencies, to make this two
member constituency into three seats, helping to make up other two-member
constituencies, say, a minimal constituency of 90,000 plus a seat of
45,001.
Compare going over a 150,000 limit. Such a two-member constituency would be changed into three seats to be fit conveniently into the two-member system. The constituency lower limit would be 100,000, leaving, say, 50,001 for a seat in another constituency. This means a 50,000 variation between maximum and minimum permitted variation. Whereas the 135,000 to 90,000 limits show only a 45,000 variation.
One might argue, a 135 thousand constituency implies two seats each
representing sixty-seven and a half thousand. They are both only seven and a
half thousand from the required average voters per seat. This suggests making
a corresponding lower limit of two seats representing fifty-two and a half
thousand, for a constituency of 105,000. But then the upper limit would have
to be three times 52 1/2 or one hundred and fifty seven and a half thousand.
In other words, too big a variation ( 52 1/2 thou. ) is being made again.
Beyond the single member system, the equal variation about a constituency average does not work. For instance, the 45,000 variation between 90,000 and 135,000 could be adjusted to an equal twenty-two and a half thousand variation about 120,000. That is ninety seven and a half thousand to one hundred and forty two and a half thousand. But this would be inequitable, because it would imply a lower limit of two times forty eight and three-quarter thousand compared to an upper limit of three times forty seven and a half thousand, before a transfer of one of those three portions.
One might also ask why not have a smaller variation than 45,000? It cannot be smaller than 40,000, because that would put the limits of variation between two and three times 40,000, or 80,000 and 120,000. That would make the required average as also the upper limit of variation. That fails the condition of allowing a variation about the average for natural variations in local communities.
Whereas, it is not till 45,000 variation is reached that we arrive at a formula consistent with the case for the single member system. Namely, V/(R+2) = 60,000/(1+2) = 20,000. This formula happens to give the variation for the upper limit, as well as the lower limit. But the formula we can generally use for the upper limit in multi-member systems also works for single member systems.
Thus, for a two member system, the upper variation may be given by the
formula: V/(R+2)R = 120,000/(2+2)2 = 15,000. Added to 120,000 this gives the
maximum constituency of 135,000.
The formula also works for single member systems: 60,000/(1+2)1 = 20,000 for
the upper variation, as well as the lower. But this is just an accident of two
formulas for upper and lower variations giving the same result for single
member systems.
Is the unequal variation from 90,000 to 135,000 about 120,000 inequitable? It may be assumed that under-representation in the constituency system will be matched by a corresponding over-representation in another part of the system. This is the inequitable consequence of having to allow for natural variations in the size of local communities. I believe its implication for the lop-sided variation of ninety to one hundred and thirty five thousand is as follows:
The over-size constituencies are not allowed to be as over-size as the under-size constituencies are allowed to be under-size. Therefore, voters, in over-size constituencies, cannot be as under-represented, voter for voter, as their counter-parts can be over-represented in under-size constituencies. This means that the system balance of under-representation to over-representation must be met by there being more voters with their lesser degree of under-representation than voters, in under-size constituencies, allowed a greater degree of over-representation.
In other words, in the lop-sided 90 to 135 thousand system, under-representation should be shared more between voters in over-size constituencies. Increasing the over-size limit to 150 thousand would only mean an equally great spread of the voters' under-representation, to their over-representation. But it wouldnt, I guess, change the over-all amount of under-representation.
However, this isnt the same as seeing working models including the Ross quota. See the section, after next.
Briefly, the above example can be shown for a three-member system. The
required average three member constituency become three times 60,000 equals
180,000. The Ross quota gives the variation to the least size constituency, as
V/(R+2) = 180,000/(3+2) = 36,000.
So, the least size constituency is 180,000 - 36,000 = 144,000.
To get the upper variation, generally divide the variation again by the
number of seats per constituency. Or, V/(R+2)R = 180,000/(3+2)3 = 12,000.
So, the maximum constituency is 192,000.
The Ross quota, V/(R+2) doesnt just give the lower limit of variation. In terms of the required average vote, it also gives the elective quota in the smallest permitted constituency. It derives the same value as the Droop quota in terms of the minimum permitted voters per constituency. ( Call the smallest vote: V,n where n stands for minimum. ) That is to say: V/(R+2) = V,n/(R+1).
Using the example, this is: 180,000/(3+2) = 144,000/(3+1) = 36,000.
In this respect, the relation between the Ross quota and the Droop quota is, in turn, similar to the relation between the Droop quota and the Hare quota, considered both in terms of quotas or portions of representation and variations in representation. This may be shown as follows:
The Hare quota, V/R gives the required average of voters per constituency. The Droop quota, V/(R+1) gives the elective quota for representatives in such a constituency. The proportion of representation in that constituency is: VR/(R+1). Thus, in a three member constituency the proportional representation ( 'PR' ) is R/(R+1) = 3/(3+1) or three-quarters of the votes.
Assuming all the constituents vote ( which of course they normally dont ) the permitted variation in number of votes from number of elective votes is: V - VR/(R+1) = V/(R+1). That is, the Droop quota is both the elective quota and the variation in votes from elective votes. This compares to the Ross quota as the elective quota in a minimal constituency and the variation from the average constituency to that minimal constituency.
For further symmetry, compare the Droop quota's proportional representation within constituencies to the Ross quota's proportional representation across constituencies. The Droop quota's PR is the number of seats times the quota: RV/(R+1). The Ross quota PR is the ratio of the minimal constituency to the maximal constituency, say, V,n/V,x.
From above, V/(R+2) = V,n/(R+1). So, V,n = (R+1)V/(R+2).
Also from above, V,x = V + V/(R+2)R = V( (R+2)R + 1)/(R+2)R ) =
V(R+1)²/(R+2)R.
After cancelling, V,n/V,x = R/(R+1). This is the same ratio of
representation across constituencies with the Ross quota, as the Droop quota
gives within a constituency.
In any constituency, the elective vote is the PR of the total vote. Across
constituencies, the minimal constituency is analgous to the elective vote and
the maximal constituency analgous to the total vote of any given constituency,
in an across-constituency proportional representation, whereby V,n =
V,x.R/(R+1).
The theoretical distribution of natural constituencies, based on local communities with varying electorates, in a uniform-member system, has to be within limits imposed by the need for equal representation.
From the previous section, the minimum permitted constituency has a greater variation from the required average constituency than the maximum permitted constituency: the difference is in the ratio of R to one, respectively.
Therefore there are R times as many voters over-represented in the minimum permitted constituency as there are voters under-represented in the maximum constituency. To compensate for this imbalance, the distribution of constituencies might be weighted, with R times as many maximum-size constituencies as minimum constituencies.
With regard to the whole range of constituencies between those two upper and lower limits, the binomial distribution models for the approximate number of over-size to under-size constituencies about a required average. See table one, where weight R equals the maximum weight five, in the weighting row, for five times as many maximum constituencies ( shown in the far right column ) as minimum constituencies ( shown in the far left column of numbers ).
In terms of the previous section's discussion of R, for representatives, it is simplest to assume that the uniform-member system, in this example is a five-member system.
( Note that a simple adaptation of table one also shows how a non-uniform system of constituencies is distributed. The weighting row could be re-designated as the number of representatives per constituency. The binomial distribution then shows how sixteen constituencies would be either one-, two-, three-, four- or five-member constituencies. The average number of representatives per constituency is three. )
| Number of constituencies classed from under-size to over-size: |
1 | 4 | 6 | 4 | 1 |
| Weighting to make-up for constituencies being more under-size than over-size: |
1 | 2 | 3 | 4 | 5 |
| Totals of under-size to over-size ( about 18 average ) constituencies: |
1 | 8 | 18 | 16 | 5 |
J F S Ross pointed out that electoral legislation ignored the natural law ( which I call the Ross quota ) for constituency variations in a single member system. This should be a variation of one third or 33 1/3 per cent about the norm. I believe Canada allows a 30 per cent variation about the required average size single member constituency. That may not be too far out. But Ross said that Britain's legislation, in 1944, for a 25 per cent variation, was found to be unworkable.
In 1979, the victorious Tories tried to impose as equal constituencies as possible. As a result, limits even below the 25 per cent limit were imposed on unwilling localities, after being over-ruled in a hearing. Succesful local opposition no doubt depended on how much clout the local community had, rather than on the inherent justice of the case.
Take the example of two Yorkshire constituencies. The Boundary Commission decided to take two wards out of York, rather than leave the city as one constituency. These two wards were added to a rural constituency including the small town of Selby. York's local Labour and Tory associations were both opposed to this taking out a slice of the York 'cake'. The people of York successfully opposed the plan.
But the Boundary Commission got away with a similar sort of plan for the Yorkshire town of Scarborough and district. There two wards, that shared the town's seaside interests, were put in the agricultural constituency of Ryedale. The judge admitted the local opposition's case was just but said he had to defer to the government's wish to equalise constituencies as much as possible.
The government had not set a definite limit to constituency variations. So,
with respect to the boundary changes, some communities were bound to be
treated as 'more equal than others'. This 'Orwellian' government gave
inequitable guidelines, in its command of equal constituencies.
One must remember that by equalising the single member constituencies, the
Tory party in power increased its majority by about a hundred seats without
increasing its vote. Thus more apparent equality was a stalking horse for even
more inequality.
The single member system is the most prone to gerrymandering. For example, the Boundary Commission proposed that Colchester's two single member constituencies should have boundaries with one sub-urban constituency shaped like a do-nut and an other urban constituency shaped like the hole in the middle. The latter would most likely have gone to Labour.
The Town Clerk was asked if he would have objected to this, if the Tories
had not controled the local council. He would not answer the question till he
had retired. The hearing found against the do-nut as splitting town and
country.
Beyond splitting a community on inner versus outer city lines, the more basic
objection is to splitting a town at all with single member constituencies.
Dr J F S Ross, MC, BSc, PhD, civil engineer, army officer and college principal, turned freelance writer on democratic elections. Parliamentary Representation, in 1943, was the first book to reveal the backgrounds of MPs. Nowadays such work gives academics a job. Ross did the research in his own time and found considerable difficulties in his way. Indeed, a copy of the book found its way into the House of Commons library, as essential reading.
Ross' motivation was not an income. He showed how unrepresentative
supposedly representative democracy was. ( Ross' book included a Tory MP's
document on parliament as a plutocracy. ) This served as evidence for a
representative electoral system, namely the single transferable vote ( STV ).
Later statistics on MPs routinely still show striking deficiencies, such as in
science and technology -- Ross' own competance.
In 1955, another substantial work, Elections and Electors, continued Ross's aims. ( This was the book that converted this writer, when a student. Later, I asked for a copy from the secretary of the Electoral Reform Society, in charge of its vital ballot services. Like Ross, he had been a military man, Major Frank Britton. He had a military MBE. )
In 1959, Ross published a short book, The Irish Electoral System, to present the case for retaining STV in the Irish referendum.
In 1955, Ross wrote a pointed preface to the first edition of Voting in Democracies, by J Lambert and Enid Lakeman. In later editions this became Lakeman's standard work, How Democracies Vote. Like Ross, she became recognised as a leading expert on voting method.
Ross, Lakeman and Britton belonged to what might be called the old guard of electoral reform, who believed that the single transferable vote was rightly the democratic method and wouldnt support expedients like the alternative vote or party lists.
Richard Lung.
25 november; 14 december 2002.