Arc of Prosperity

Scottish Independence within the EU – with a Scandinavian Slant


The 2016 Holyrood election according to the Danish Electoral System

christiansborg photo
Photo by fooishbar
When people in Scotland discuss an alternative to the Additional Member System currently used for Holyrood elections, they often assume the only real alternatives are FPTP (the system used for Westminster elections), STV (used for Scottish council elections) or d’Hondt with party lists (known from elections to the European Parliament.

However, a different system is used in Denmark (and similar ones are used in Norway and Sweden), and it is taken for granted there — and nobody ever suggests changing the system, so it’s definitely not a bad way to conduct elections.

It is basically Sainte-Laguë with top-up seats and personal votes instead of party lists (Sainte-Laguë is a variant of d’Hondt).

It has several attractive properties:

  1. All politicians need personal votes to get elected. There isn’t a party list where the person at the top of it can lean back in the knowledge that they’ll get elected no matter what.
  2. There is even competition amongst candidates from the same party, so that voters can elect the ones they like the best.
  3. It is reasonably fast to count (different from STV, which in practice has to be done computationally). In Denmark, they normally count the party votes on the night so that you know exactly how many seats each party has won, and then they count the personal votes the next day.
  4. Practically every vote counts: Because of the national top-up seats almost every vote counts — the only truly wasted votes are the ones cast for tiny parties that didn’t gain any representation (like UKIP or RISE).

To make the system more tangible, I have here tried to show what the 2016 Holyrood election would have looked like if this system had been used instead of AMS.

NB: I have simplified the system slightly in various ways. For instance, Denmark operates with a fourth layer between the regions and the national results, and the parties have several options to choose from with regard to how party votes should be distributed to the candidates. I don’t believe these differences are critical for the present purpose, but of course civil servants should look into the details if Holyrood ever decides to switch to this system.


The current constituencies would be kept, but they would change status to being nomination constituencies (“opstillingskredse” in Danish), which means that the local parties would be able to put up candidates for election, just like they do now. (Actually, Denmark has more nomination constituencies, so if Scotland adopts this system it might make sense to increase the number from 73. This would have the advantage of making politics more local.)

However, electing members of parliament would happen in larger units, electoral regions (“valgkredse” in Danish) — I’ve used the current Holyrood regions for this purpose, except that I’ve put Orkney & Shetland and Na h-Eileanan an Iar into separate regions. In Denmark, each electoral regional will elect as many members of parliament as the number of nomination constituencies within it, but I’ve kept the number of seats within each region unchanged.

Each electoral region consists of regional seats (Danish “kredsmandater”) and a few national top-up seats (Danish “tillægsmandater”). The regional seats are allocated locally, without any reference to events outside the electoral region, whereas the top-up ones are allocated nationally based on votes cast across the country. For instance, in this simulation the West of Scotland region containts 17 seats: 14 regional ones and 3 top-op ones.

Step 1: The election

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

As an example, here is the ballot paper for Eastwood in the West Scotland region. The local candidates here are Jackson Carlaw, John Duncan, Ken MacIntosh, Stewart Maxwell, [GRN candidate, Eastwood] and [UKIP candidate, Eastwood] (because the Greens and UKIP didn’t put up any candidates in most constituencies in the real election I’ve used this notation where necessary):

Ballot Paper for the Eastwood Constituency

Electoral Region: West Scotland

Mark exactly one box (☐) like this: ☒

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

Conservative Party ☐

Jackson Carlaw ☐
Ann Le Blond ☐
Graeme Brooks ☐
Maurice Corry ☐
Maurice Golden ☐
Jamie Greene ☐
Paul Masterton ☐
Billy McClure ☐
Andrew Polson ☐
David Wilson ☐

Green Party ☐

[GRN candidate, Eastwood] ☐
[GRN candidate, Dumbarton] ☐
Ross Greer ☐
[GRN candidate, Greenock and Inverclyde] ☐
[GRN candidate, Clydebank and Milngavie] ☐
[GRN candidate, Cunninghame North] ☐
[GRN candidate, Paisley] ☐
[GRN candidate, Cunninghame South] ☐
[GRN candidate, Renfrewshire South] ☐
[GRN candidate, Renfrewshire North and West] ☐

Labour ☐

Ken MacIntosh ☐
Jackie Baillie ☐
Johanna Baxter ☐
Neil Bibby ☐
Gail Casey ☐
Joe Cullinane ☐
Mary Fee ☐
Margaret McCarthy ☐
Siobhan McCready ☐
Paul O’Kane ☐

Liberal Democrats ☐

John Duncan ☐
Rod Ackland ☐
Frank Bowles ☐
Katy Gordon ☐
Tristan Gray ☐
Ruby Kirkwood ☐
Eileen McCartin ☐
Aileen Morton ☐
Charity Pierce ☐
John Watson ☐

Scottish National Party ☐

Stewart Maxwell ☐
George Adam ☐
Tom Arthur ☐
Kenneth Gibson ☐
Rona MacKay ☐
Derek MacKay ☐
Ruth Maguire ☐
Stuart McMillan ☐
Gil Paterson ☐
Gail Robertson ☐


[UKIP candidate, Eastwood] ☐
[UKIP candidate, Strathkelvin and Bearsden] ☐
[UKIP candidate, Dumbarton] ☐
[UKIP candidate, Greenock and Inverclyde] ☐
[UKIP candidate, Clydebank and Milngavie] ☐
[UKIP candidate, Cunninghame North] ☐
[UKIP candidate, Paisley] ☐
[UKIP candidate, Renfrewshire South] ☐
[UKIP candidate, Cunninghame South] ☐
[UKIP candidate, Renfrewshire North and West] ☐

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.

Sådan kan en stemmeseddel se ud. #FV15 #magentalove #aarhus
Sådan kan en stemmeseddel se ud. #FV15 #magentalove #aarhus.
And yes, ballot papers can be really long in Denmark. I think I’ve once seen one that was more than a metre long. In Sweden, where they have a similar system, they have separate ballot papers for each party, and the voter picks one and puts it inside an envelope. I’m not sure that’s a better solution, though.

(For the purpose of this simulation, I have used the constituency votes for the large parties and the list votes for the small parties. For simplicity I’ve also ignored all parties smaller than UKIP. Furthermore I’ve assumed that everybody will vote for their local candidate. 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 very different.)

All the ballot papers can be found here.

Step 2: Allocation of regional seats

Once counting starts, the first thing to do is to allocate the regional seats.

Please see the local results (there are links in the ballot papers) for this, but here are the results for the West Scotland region as an example:

Nomination constituencies

Allocation of regional seats

538203 voters: 17 seats, of which 14 regional seats. Turnout was 63%.

1 64732 (3) 17219 (10) 90468 (2) 12106 148659 (1) 5854
3 21577 (7) 5739 30156 (5) 4035 49553 (4) 1951
5 12946 (13) 3443 18093 (9) 2421 29731 (6) 1170
7 9247 2459 12924 (14) 1729 21237 (8) 836
9 7192 1913 10052 1345 16517 (11) 650
11 5884 1565 8224 1100 13514 (12) 532
13 4979 1324 6959 931 11435 450

What this shows is first of all that there are 14 regional seats and three top-up seats.

The table lists all parties that put up candidates in this electoral region. The first line (marked with 1) shows the actual number of votes received for each party, i.e., the SNP got 148,659 votes, Labour 90,468, etc.

The next line shows the number of votes divided by 3, and the last one the number of votes divided by 13. For larger regions, one would produce more rows, dividing the number of votes by 15, 17, 19 and so forth.

Once the table has been produced, one looks for the largest number in it. In this case, it’s the 148,659 votes cast for the SNP. This means that the first regional seats goes to this party, and this is marked in the table by highlighting the number in blue and putting “(1)” after the number.

Now one has to find the second-largest figure, which is 90,468, and the second seat therefore goes to Labour. Similarly, the third seat is allocated to the Conservatives.

When we get to allocating the fourth seat, the SNP’s number of votes divided by 3 (49,553) is larger than any other remaining figure, and the fourth seat thus goes to the SNP.

We proceed in this way until all the 14 regional seats have been allocated.

Step 3: Allocation of the top-up seats to the parties

After allocating regional 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 win at least one regional seat or get at least 2% of the votes. For this reason, UKIP gets excluded (getting 1.9% is not enough).

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 regional 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.)

Here are the country-wide results:

Party Votes Percent Total seats Regional seats Top-Up seats
SNP 1059897 44.2% 58 49 9
LAB 514261 21.4% 28 22 6
CON 501844 20.9% 28 22 6
GRN 150429 6.3% 8 8 0
LD 126414 5.3% 7 5 2
UKIP 46426 1.9% 0 0 0
2399271 129 106 23

Step 4: Allocating top-up seats to specific regions

Now that the top-up seats have been allocated to the parties, they need to be placed in specific electoral regions.

The calculations are similar to the ones for allocating seats in the electoral constituencies, except that the whole country is being looked at, and we use the divisors 1, 4, 7, etc., instead of 1, 3, 5, etc.

In the following table, the regional seats already allocated above are marked with an X.

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 12,106 votes the Lib Dems got in West Scotland; because the Lib Dems are indeed due a top-up seat, the first top-up seat gets allocated to them there.

The second-largest number is the 11,784 votes the Lib Dems got in South Scotland, so the next top-up seat is allocated here.

The procedure is repeated many times. Once a party 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 belongs to this party. For instance, the third-largest number in the table is the 8,637 belonging to the Greens in the Lothians, but the Greens aren’t due any top-up seats, so instead the third seats goes to the SNP in the Lothians.

The same applies to regions. For instance, let’s look at the 16th top-up seat. It can’t go to the SNP in South Scotland (6792) because this region isn’t due any more seats; it can’t go to the Tories in Lothian (6783) for the same reason; and neither can the SNP in West Scotland get it (6757). It therefore goes to the SNP in North East Scotland (6746).

The last seat goes to Labour in the Highlands and Islands (4978). All the seats have now been allocated.

1 4 7 10 13 16 19 22 25
Central Scotland CON X X 19: 6065 4245 3265 2653 2234 1929 1698
GRN X 3181 1817 1272 978 795 669 578 508
LAB X X X 10: 7609 5853 4756 4005 3458 3043
LD 5024 1256 717 502 386 314 264 228 200
SNP X X X X X X X 18: 6481 5703
Glasgow CON X 12: 7226 4129 2890 2223 1806 1521 1313 1156
GRN X 5849 3342 2339 1799 1462 1231 1063 935
LAB X X X X 21: 5413 4398 3704 3199 2815
LD 5860 1465 837 586 450 366 308 266 234
SNP X X X X X X X 20: 5838 5137
Highlands and Islands CON X X 22: 5307 3715 2858 2322 1955 1688 1486
GRN X 3096 1769 1238 952 774 651 563 495
LAB X 23: 4978 2844 1991 1531 1244 1048 905 796
LD X 5086 2906 2034 1565 1271 1070 924 813
SNP X X X X X 4944 4163 3595 3164
Lothian CON X X X 6783 5218 4239 3570 3083 2713
GRN X 8637 4935 3455 2657 2159 1818 1570 1382
LAB X X X 4: 8497 6536 5310 4472 3862 3399
LD X 4736 2706 1894 1457 1184 997 861 757
SNP X X X X X 3: 8624 11: 7262 6272 5519
Mid Scotland and Fife CON X X X 15: 6827 5251 4267 3593 3103 2730
GRN X 4465 2551 1786 1373 1116 940 811 714
LAB X X 5: 8420 5894 4534 3684 3102 2679 2357
LD X 5102 2915 2040 1569 1275 1074 927 816
SNP X X X X X X 13: 7033 6074 5345
Na h-Eileanan an Iar CON 1499 374 214 149 115 93 78 68 59
GRN 919 229 131 91 70 57 48 41 36
LAB 3378 844 482 337 259 211 177 153 135
LD 965 241 137 96 74 60 50 43 38
SNP X 1718 982 687 528 429 361 312 274
North East Scotland CON X X X X 17: 6564 5333 4491 3878 3413
GRN X 3781 2160 1512 1163 945 796 687 604
LAB X X 6359 4451 3424 2782 2342 2023 1780
LD X 4613 2636 1845 1419 1153 971 838 738
SNP X X X X X X 9: 7811 16: 6746 5936
Orkney and Shetland CON 840 210 120 84 64 52 44 38 33
GRN 1474 368 210 147 113 92 77 67 58
LAB 955 238 136 95 73 59 50 43 38
LD X 3129 1788 1251 963 782 658 569 500
SNP X 1276 729 510 392 319 268 232 204
South Scotland CON X X X X 7: 8062 6551 5516 4764 4192
GRN X 3693 2110 1477 1136 923 777 671 590
LAB X X X 6463 4972 4039 3402 2938 2585
LD 2: 11784 2946 1683 1178 906 736 620 535 471
SNP X X X X X 6: 8066 6792 5866 5162
West Scotland CON X X X 6473 4979 4045 3406 2942 2589
GRN X 4304 2459 1721 1324 1076 906 782 688
LAB X X X X 14: 6959 5654 4761 4112 3618
LD 1: 12106 3026 1729 1210 931 756 637 550 484
SNP X X X X X X 8: 7824 6757 5946

Step 6: Determining MPs

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

Danish election posters.
Danish election posters.
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 thesecond-most votes gets the next seat, and so on. (Political parties in Denmark can choose between different systems, but this needn’t concern us here.)

As an example, let us look at the SNP in West Scotland.

Scottish National Party (7)

Rona MacKay 17060
Stuart McMillan 17032
Kenneth Gibson 16587
Gil Paterson 16158
Derek MacKay 14718
George Adam 14682
Tom Arthur 14272
Ruth Maguire 13416
Gail Robertson 13413
Stewart Maxwell 11321

This party got seven seats here (six regional seats and one national top-up seat). The first one goes to Rona MacKay with 17,060 votes, the second one to Stuart McMillan with 17,032 votes, the third one to Kenneth Gibson with 16,587 votes, and so on.

The last three candidates on the list are not elected. However, Ruth Maguire becomes the first reserve in case any of the seven elected members has to step down, with Gail Robertson being the second reserve. In this way, by-elections are not needed.

(Please note that the results are quite misleading because I’ve taken the figures from an AMS election. At the moment only people in Eastwood were able to vote for Stewart Maxwell, and his personal vote got squeezed last week because it was a three-way race, but if people all over the West Scotland region had been able to vote for him, I’m certain he would have been much higher up the list, given his high media profile.)

This completes the election.

Final result

Scottish National Party (58 MPs)

George Adam, Clare Adamson, Alasdair Allan, Tom Arthur, Colin Beattie, Keith Brown, Aileen Campbell, Willie Coffey, Angela Constance, Bruce Crawford, Roseanna Cunningham, Ash Denham, Graeme Dey, Bob Doris, James Dornan, Jennifer Dunn, Mairi Evans, Fergus Ewing, Linda Fabiani, Joe Fitzpatrick, Kate Forbes, Jeane Freeman, Kenneth Gibson, Jenny Gilruth, Toni Giugliano, Christine Grahame, Clare Haughey, Donna Heddle, Jamie Hepburn, Fiona Hyslop, DJ Johnston-Smith, Bill Kidd, Richard Lochhead, Richard Lyle, Gordon MacDonald, Angus MacDonald, Derek MacKay, Rona MacKay, Ben Macpherson, Gillian Martin, John Mason, Michael Matheson, Mark McDonald, Ivan McKee, Christina McKelvie, Stuart McMillan, Alex Neil, Gil Paterson, Shona Robison, Gail Ross, Michael Russell, Shirley-Anne Somerville, Stewart Stevenson, Nicola Sturgeon, John Swinney, David Torrance, Maureen Watt, Humza Yousaf.

Conservative Party (28 MPs)

Michelle Ballantyne, Miles Briggs, Alexander Burnett, Jackson Carlaw, Finlay Carson, Colin Clark, Ruth Davidson, Murdo Fraser, Jamie Greene, Kirstene Hair, Alison Harris, Alex Johnstone, Callum Laidlaw, John Lamont, Gordon Lindhurst, Dean Lockhart, Edward Mountain, Oliver Mundell, Robbie Munro, Andrew Polson, Douglas Ross, John Scott, Graham Simpson, Liz Smith, Alexander Stewart, Ross Thomson, Kyle Thornton, Adam Tomkins.

Labour (28 MPs)

Jackie Baillie, Claire Baker, Neil Bibby, Bill Butler, Kezia Dugdale, Patricia Ferguson, Neil Findlay, Iain Gray, Cara Hilton, Lesley Hinds, Daniel Johnson, James Kelly, Johann Lamont, Lewis MacDonald, Ken MacIntosh, Jenny Marra, Paul Martin, Siobhan McCready, Margaret McCulloch, Michael McMahon, Carol Mochan, Elaine Murray, Paul O’Kane, John Pentland, Alex Rowley, Elaine Smith, Linda Stewart, David Stewart.

Green Party (8 MPs)

Maggie Chapman, John Finnie, Ross Greer, Patrick Harvie, Alison Johnstone, Kirsten Robb, Mark Ruskell, John Wilson.

Liberal Democrats (7 MPs)

Kris Chapman, Alex Cole-Hamilton, Katy Gordon, Willie Rennie, Mike Rumbles, Tavish Scott, Jamie Stone.

One thought on “The 2016 Holyrood election according to the Danish Electoral System

  • This is absolutely fascinating, and is a better solution than what we already have. Surprised it’s not had more coverage!


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