Three Apportionment Problems, with Applications to the United Kingdom

Abstract

Generically, the problem of apportionment may be defined as the problem of assigning a vector of integer numbers to each of a number of entitled entities that comes as close as possible to giving each entity its proportionate share of representation. Within that, there are a number of sub-problems. The correct solution (if a uniquely best solution exists) to one may not be the correct solution to another.

An earlier version of this chapter was delivered at the “Workshop on Electoral Methods”, Department of Mathematics, KTH, Stockholm, May 2011. Many thanks for comments received then especially from Svante Linusson, Svante Janson, and Paul Edelman. The usual disclaimer applies.

Appendix. References to dH and S-L Apportionment in the UK Parliament

All cited from http://hansard.millbanksystems.com/

25.11.1997. Jack Straw [Labour minister (Home Secretary)]

Those who argue for Sainte-Lague say that it favours smaller parties and that d’Hondt discriminates in favour of larger parties. However, we do not believe that that is so, and we have calculated the likely effect of using all three divisors. The differences that they produce are minimal, and we have decided to use the d’Hondt divisor for four reasons. First, we believe that it will produce a fair result. Secondly, Sainte-Lague does not necessarily produce more proportional results. I have already introduced the proportionality index to the House—I noticed how hon. Members listened with bated breath. By calculating the index score for six regions using the 1994 European election vote in the United Kingdom, we found that, on average, d’Hondt scored higher than Sainte-Lague.

HC Deb 20 January 1998 vol 304 cc509-10W 509W

1.1 § Mr. Beith [Liberal Democrat]

To ask the Secretary of State for the Home Department, pursuant to his oral statement of 25 November 1997, Official Report, columns 812–13, whether he will publish the calculations on which his statement, regarding the proportionality of the Sainte-Lague and d’Hondt divisors, was based; if he will calculate the number of seats that would be won by each party in Scotland under the (a) d’Hondt and (b) Sainte-Lague divisors using the 1994 European Election results; and if he will make a statement.

1.2 § Mr. George Howarth [Labour junior minister]

The calculations were based on the votes cast in the 1994 elections to the European Parliament in six of the 11 regions for which the European Parliamentary Elections Bill provides.

In five of the six regions the choice of divisor made no difference to the final allocation of seats. In Scotland, the effect was as follows:

• d’Hondt:

• Four labour, one conservative,

• Three Scottish National Party

• Sainte-Lague:

• Three labour, one conservative,

• One Scottish Liberal Democrat,

• Three Scottish National Party.

During the Bill’s Second Reading debate, I gave the House figures which suggested that the two divisors produced different results in the London region.

Revised calculations show that both divisors produce the same result in London. I apologise for the original error.

1.3 § Mr. Beith

To ask the Secretary of State for the Home Department what divisors were used to calculate the allocation of seats between the regions of England in the European Parliamentary Elections Bill. [24043]

1.4 § Mr. Straw

[holding answer 19 January 1998]: No divisors were used. The allocation of seats to the English regions set out in Schedule 1 to the European Parliamentary Elections Bill was arrived at by dividing the total English electorate by the number of English seats (71) to produce an average figure. Seats were then allocated to regions in such a way as to ensure that the sum of the divergencies from this figure was as low as possible.

[James Clappison, (Conservative) 26.02.98]

Other people are more interested in this subject, and one of those is Professor Ian Maclean of Oxford university.

1.5 § Mr. Beith

A Liberal Democrat.

1.6 § Mr. Clappison

He may well be. He is certainly an expert on electoral systems, and he knows his stuff on these matters. The Home Secretary’s comments came to his attention and he suspected that something was wrong—in fact, it was a bit more than a suspicion. It was impossible for the Home Secretary to be right, because the Sainte-Lague system is never less proportional than the d’Hondt system and is frequently more proportional than it—contrary to what the Home Secretary told us on Second Reading. The good professor thought that there had been a mistake—I am glad that the Home Secretary admits it now—he carried out simulations and he found that he was correct. I am glad that the Home Secretary has realised his mistake.

1.7 § Mr. Straw

rose

1.8 § Mr. Clappison

Before the Home Secretary intervenes again, I must advise him that it would have been a good idea for the Government to answer my written question on that subject.

1.9 § Mr. Straw

rose—

1.10 § The Chairman

Order. Mr. Clappison has the Floor.

1.11 § Mr. Clappison

I shall certainly give way, but the Home Secretary might also like to respond to my next point. I asked the Government last week whether they had made a mistake, and I received the answer earlier this week that they were still thinking about it. The Home Secretary has obviously thought about it, so perhaps he can give us an answer.

1.12 § Mr. Straw

The hon. Gentleman’s implication is preposterous. He should know that, because I wrote to the shadow Home Secretary, his right hon. Friend the Member for North-West Cambridgeshire (Sir B. Mawhinney)—just as I wrote to the Liberal Democrat spokesman and other party leaders—as soon as we were aware that an error had been made. I apologised for that error and I have placed in the Library the details of the revised calculations. Furthermore, the Under—Secretary of State for the Home Department, my hon. Friend the Member for Knowsley, North and Sefton, East (Mr. Howarth), put on record in the Official Report of 20 January exactly what the situation was, and he repeated my unreserved apology to the House.

HC Deb 04 March 1998 vol 307 c627W

1.13 § Mr. Clappison [Con]

To ask the Secretary of State for the Home Department what assessment he has made of the paper of 17 December 1997 written by Professor Ian McLean of Oxford University on the effect of different proportional representation formulae on the allocation of seats to parties in the European elections with particular reference to his conclusions on the use of the d’Hondt and Saint-Lague divisions.

1.14 § Mr. George Howarth

[holding answer 23 February 1998]: Professor McLean’s paper has been read with interest. The Government remain of the view that the d’Hondt divisor is the most suitable one to use for European Parliamentary elections, a view which was endorsed in a division on 26 February when the Committee of the whole House considered the European Parliamentary Elections Bill.