The UK General Election in 2005 according to the Danish Electoral System

This page shows how the UK General Election in 2005 would have been counted according to Danish electoral law (link in Danish).

Introduction

When British people discuss an alternative to FPTP, they often take for granted that the only options are AV (as in Australia), STV (as in Ireland), or perhaps AMS (as in Scotland and Germany).

However, a different system is used in Denmark (and similar systems are used in Norway and Sweden), and they are taken for granted there.

It is basically Sainte-Laguë with top-up seats.

However, instead of explaining the maths behind the system, I have here tried to show what the results of the General Election in 2005 would have been if this system had been used instead.

Set up

The current constituencies would be kept, but they would change status to being nomination constituencies (“opstillingskredse” in Danish), which means that the constituency associations would be able to put up candidates for election, just as they do now.

However, electing members of parliament would happen in larger units, electoral constituencies (“valgkredse” in Danish), that will typically consist of around 10 nomination constituencies.

Each electoral constituency will elect as many members of parliament as the number of nomination constituencies within it, so the number of MPs will be the same as before.

Each electoral constituency consists of a large number of constituency seats (Danish “kredsmandater”) and a small number of top-up seats (Danish “tillægsmandater”). The constituency seats are allocated locally, without any reference to events outside the electoral constituency, whereas the top-up ones are allocated nationally and regionally.

Step 1: The election

Each voter will be given a ballot paper listing all candidates in the entire election constituency, but with the one from their own nomination constituency listed before the other candidates.

As an example, here is the ballot paper for Argyll & Bute. The local candidates here are McGrigor, Reid, Manson, Strong and Henderson:

Ballot Paper for Nomination Constituency Argyll & Bute

Electoral Constituency: Argyll, Bute & Highland
Electoral Region: West Scotland

Mark exactly one box (☐) like this: ☒

You can choose to vote for either a party or a candidate.

Cons ☐

J.A.R.N. McGrigor ☐
J.B.P. Hodgson ☐
P.A. Ross ☐
R.I. Rowantree ☐

Grn ☐

D. Jardine ☐
D. MacLeod ☐

LibDem ☐

A. Reid ☐
D.G. Alexander ☐
C.P. Kennedy ☐
J.A. Thurso ☐

Labour ☐

C. Manson ☐
C.M. Conniff ☐
A.W. Jamieson ☐
D.J. Stewart ☐

PubP ☐

D.S. Lawson ☐

SNP ☐

I.C. Strong ☐
K.A. Shirron ☐
D.G. Thompson ☐
M.E. Will ☐

SSP ☐

D. Henderson ☐
L. Ivory ☐
G. MacDonald ☐
A.E. Macleod ☐

UKIP ☐

P.A. Anderson ☐

Ind (J.G.P. Campbell) ☐

Ind (M. Grant) ☐

Each voter has to tick exactly one box. If they vote for a candidate, it is a vote both for the party that this candidate represents and for the actual candidate. If they vote for a party, it is a vote for the party only.

(For the purpose of producing this, I have assumed all voters voted for the candidate they voted for at the FPTP election. In reality, given the greater choice of candidates, and given the option of voting for just the party, of course the results from an actual election under this system would have been different.)

All the ballot papers can be found here.

Step 2: Allocating constituency seats

Once counting starts, the first thing to do is to allocate the constituency seats. Please see the local results (there are links in the ballot papers) for this, but here is an example:

Ayrshire & Lanark

511545 voters: 7 seats, of which 6 constituency seats. Turnout was 61%.

	Cons	Grn	LibDem	Labour	OCV	SL	SNP	SSCP	SSP	UKIP	Ind (D. McFarlane)	Ind (J. Houston)	Ind (R. Gentle)
1	47038 (5)	1575	49402 (3)	148121 (1)	415	1166	53422 (2)	592	4953	2317	416	160	1513
3	15679	525	16467	49373 (4)	138	388	17807	197	1651	772	138	53	504
5	9407	315	9880	29624 (6)	83	233	10684	118	990	463	83	32	302
7	6719	225	7057	21160	59	166	7631	84	707	331	59	22	216

What this shows is first of all that there are seven nomination constituencies in Ayrshire & Lanark, and there are six constituency seats (and therefore one top-up seat).

The table lists all parties that put up candidates in this electoral constituency (note that independents are listed as if they were parties in their own right). The first line (marked with 1) shows the actual number of votes received for each party. That is, the Conservatives got 47,038 votes, the Green Party got 1,575 votes, etc.

The next line shows the number of votes divided by 3, and the last one the number of votes divided by 5. For larger constituencies, one would produce more rows, dividing the number of votes by 7, 9, 11 and so forth.

Once the table has been produced, one looks for the largest number in it. In this case, it's the 148,121 votes cast for Labour. This means that the first constituency seats goes to this party, and this is marked in the table by highlighting the number in bold and putting “(1)” after the number.

Now one has to find the second-largest figure, which is 53,422, and the second seat therefore goes to the SNP. Similarly, the third seat is allocated to the Liberal Democrats.

When we get to allocating the fourth seat, Labour's number of votes divided by 3 (49,373) is larger than the number of votes that the Tories got (47,038), and the fourth seat thus goes to Labour.

The Conservatives then get the fifth seat, and finally, Labour get the last constituency seat here.

Step 3: Allocating top-up seats

After allocating constituency seats in all electoral constituencies in the country, the next step is to allocate the top-up seats.

To do this, all the votes cast for all parties in the entire country are added up.

Then one excludes small parties. These are the ones that didn't either receive at least one constituency seat or get at least 2% of the votes. For instance, UKIP got only one constituency seat, but even if hadn't got any, they would still have received top-up seats because they got more than 2% of the votes. Similarly, Respect did not get even close to 2% of the votes, but they managed to get one constituency seat, so they will get a top-up seat, too.

Quite a few small parties, such as the Green Party and the BNP, have not managed to get elected because of these restrictions. However, the Green Party did not put up candidates everywhere, but if they did, I would consider it likely they would be able to get at least 2% of the votes cast, even if the still couldn't get a single constituency seat. It is more unlikely that the BNP would manage to get in, given that their appeal seems to be concentrated in certain areas, so getting 2% nationally would be quite difficult for them, and getting one constituency seat would require even more support in their hotspots.

To allocate the top-up seats, one calculates the share of the vote and then tops up with top-up seats to make the share of seats the same.

That is, if a party got 10% of the votes, it should get 10% of the seats in parliament, i.e., 65 seats, so if it only got 50 constituency seats, it will get a top-up of 15 seats.

(The actual calculations are slightly more complex than this, but this is the principle. The figures below have been done according to the actual rules.)

To get as many seats as possible, parties can enter into an electoral pact (Danish “valgforbund”) with other like-minded parties. I have here assumed that such pacts would exist between Labour and the SDLP, between the Conservatives and the Ulster Unionists, between the LibDems and the APNI, and between the SNP and Plaid Cymru.

Here are the country-wide results, listing the number of votes, constituency seats, top-up seats and total seats:

Step 4: Allocating top-up seats nationally

Now the top-up seats need to be placed in the most appropriate places. For this purpose, the electoral constituencies are grouped into regions.

The calculations are similar to the ones for allocating seats in the electoral constituencies, except that the whole country is being looked at.

In the following table, the constituency seats already allocated above are marked with an X, and each party's votes in each region have been divided by 1, 3, 5, etc.

To allocate the first top-up seat, the largest number in the entire table is found (computers are much better at this than humans), in this case it's the 53,367 votes that UKIP got in East Anglia. The second-largest number is the 52,982 votes that UKIP got in region South, so the next top-up seat is allocated here.

The procedure is repeated many times. Once a region has got all the top-up seats it is entitled to, it can get no more, even if the largest number left in the table is in this region. This is why top-up seat number 53 is given to the LibDems in region South (24,005) instead of the Conservatives in East Anglia (24,011).

The same applies to parties, so seat 110 goes to Labour in Greater Manchester (19,364) instead of Northern Ireland's Sinn Féin (19,392) that has already got the one top-up seat it is entitled to.

This procedure is repeated until all 130 top-up seats have been allocated. The last top-up seat is won by Respect in Wales with only 643 votes.

Step 5: Allocating top-up seats within the regions

Now that the top-up seats have been allocated to the regions, within each of these they need to be allocated to the electoral constituencies within it.

This is done in exactly the same way, except that the votes are divided by 1, 4, 7, etc. instead of 1, 3, 5, etc.

Here are the calculations for West Scotland as an example. (To see the results from the other regions, please click on the links in the big table above.)

The first top-up seat goes to the LibDems in Dumfries, Galloway & Borders, the second one to the SNP in Glasgow suburbs, and so on.

		1	4	7	10	13	16
Ayrshire & Lanark	Cons	X	11759	6719	4703	3618	2939
	LibDem	X	12350	7057	4940	3800	3087
	Labour	X	X	X	14812 (4)	11393	9257
	SNP	X	13355	7631	5342	4109	3338
	UKIP	2317	579	331	231	178	144
Dumfries, Galloway & Borders	Cons	X	11808	6747	4723	3633	2952
	LibDem	32298 (1)	8074	4614	3229	2484	2018
	Labour	X	10633	6076	4253	3271	2658
	SNP	14142	3535	2020	1414	1087	883
	UKIP	1031	257	147	103	79	64
Glasgow	Cons	X	4054 (5)	2316	1621	1247	1013
	LibDem	X	8471	4840	3388	2606	2117
	Labour	X	X	X	11200	8615	7000
	SNP	X	8085	4620	3234	2487	2021
Glasgow suburbs	Cons	X	12298	7027	4919	3784	3074
	LibDem	X	X	10446	7312	5625	4570
	Labour	X	X	X	X	15436 (3)	12542
	SNP	X	15475 (2)	8842	6190	4761	3868
	UKIP	726	181	103	72	55	45

Step 6: Determining MPs

Now that we have established exactly how many seats each party gets in each electoral constituency, we need to determine which of the candidates standing that have been elected.

To do this, we simply count the number of votes cast for each candidate, and the candidate with the most votes gets the first seat, the one with the second-most votes gets the next seat, and so on.

As an example, let us look at the Tories in Oxfordshire.

This party got two seats here (both of them constituency seats). The first one goes to David Cameron with 26,571 votes, and the second one to Tony Baldry.

The last four candidates on the list are not elected. However, Boris Johnson becomes the first reserve in case Cameron or Baldry has to step down, and Ed Vaizey second reserve. In this way, by-elections are not needed.

Cons (2)

D.W.D. Cameron MP	26571
A.B. Baldry MP	26382
A.B. Johnson MP	24894
E.H.B. Vaizey MP	22354
A.C. McLean	16653
V. Morris	6992

This completes the election.

Appendix A: MPs elected under FPTP that would also have been elected under the Danish system

Appendix B: MPs elected under FPTP that would not have been elected under the Danish system