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.

Preliminaries

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 ☐

[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.

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

• Clydebank and Milngavie (ballot paper)
• Cunninghame North (ballot paper)
• Cunninghame South (ballot paper)
• Dumbarton (ballot paper)
• Eastwood (ballot paper)
• Greenock and Inverclyde (ballot paper)
• Paisley (ballot paper)
• Renfrewshire North and West (ballot paper)
• Renfrewshire South (ballot paper)
• Strathkelvin and Bearsden (ballot paper)

Allocation of regional seats

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

	CON	GRN	LAB	LD	SNP	UKIP
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.

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.