The email we’ve had more than any other since the 8th of May is this one:

“Please can you explain how the Scottish election system works, and whether it’s a good idea for me as an SNP voter to give my list vote to someone else so as to ensure the maximum number of pro-Yes MSPs in Holyrood?”

We’d planned to leave that question until much nearer the relevant time, but to be honest we’re getting fed up of reading them, so let’s see if we can sort it out now.

The Scottish Parliament is determined by a process called the Additional Member System (AMS), and specifically a variant of it using the D’Hondt Method of counting.

It’s actually not complicated at all (certainly not compared to the mind-numbing maths horror of STV), but the calculations involved in allocating all the seats to ensure a roughly proportional represention of the various parties are lengthy, and most people’s attention spans are short. So we’ll keep this as brief as possible.

For the purposes of AMS, Scotland is divided into eight areas. Each has between eight and 10 constituency MSPs, and exactly seven “list” MSPs. The constituency MSPs are elected by First Past The Post (FPTP), just like in Westminster elections, and the list seats are then distributed in such a way as to try to make the total vote shares for each party more closely match the numbers of seats they end up with.

(If Holyrood elections were conducted by FPTP alone, then on May’s results there’d be roughly 123 SNP MSPs, and just seven from the other four parties put together.)

To demonstrate how the list-seat allocation works, we’re going to create an imaginary region, called (since it doesn’t really exist) Brigadoonshire.

Brigadoonshire has nine constituency seats, and in our hypothetical example the SNP take them all in a clean sweep. The list votes for the region, meanwhile, break down – and this is AN ILLUSTRATION NOT A PREDICTION – like this:

SNP: 100,000

Labour: 70,000

Conservatives: 60,000

Liberal Democrats: 25,000

Green: 15,000

What the D’Hondt method does is put those numbers through seven rounds of voting: one round for each list seat to be allocated. In each round, every party has its list vote divided by the number of seats it already has in the region – both constituency and list – plus 1. We call that number (total seats +1) the divisor.

(You have to add the 1 because otherwise when a party didn’t win any constituency seats you’d be starting off dividing by zero, which is mathematically impossible.)

So in Round 1, the SNP’s list vote is divided by 10 (because it has nine constituency seats and nine plus one is 10), and everyone else’s is divided by one, ie it stays the same. So for the first list seat, the adjusted vote becomes:

Labour: 70,000

Conservatives: 60,000

Liberal Democrats: 25,000

Green: 15,000

SNP: 10,000 (100,000 divided by 10)

That means Labour win Round 1 and get the first of the seven list seats. But because it now has a seat in the region, the party’s divisor goes up from one to two. So when we get to Round 2, the Labour vote is cut in half.

(In each round the process is that we take the party at the top of the table out, give them a seat, apply the new divisor to their original vote, and then slot the party back into the table at its new position. It might make it easier to understand if we mark the only party that’s changed position with an arrow.)

The Round 2 table is therefore:

Conservatives: 60,000

> Labour: 35,000 (70,000 divided by two)

Liberal Democrats: 25,000

Green: 15,000

SNP: 10,000

So now the Conservatives have a seat, and their divisor also goes up by one while everyone else’s numbers stay the same. Which means Round 3 comes out like this:

Labour: 35,000

> Conservatives: 30,000 (60,000 divided by two)

Liberal Democrats: 25,000

Green: 15,000

SNP: 10,000

Labour get another seat and their divisor goes up by one again, to three. (Divisors are always applied to the INITIAL number of votes – in Labour’s case 70,000 – NOT to the number carried over from preceding rounds.) So Round 4 runs:

Conservatives: 30,000

Liberal Democrats: 25,000

> Labour: 23,333 (70,000 divided by three)

Green: 15,000

SNP: 10,000

Another seat for the Tories, so their divisor goes up again, everyone else stays the same, and Round 5 plays out like this:

Liberal Democrats: 25,000

Labour: 23,333

> Conservatives: 20,000 (60,000 divided by three)

Green: 15,000

SNP: 10,000

Now the Lib Dems have a seat, and their divisor goes up. Round 6:

Labour: 23,333

Conservatives: 20,000

> Liberal Democrats: 12,500 (25,000 divided by two)

Green: 15,000

SNP: 10,000

Another seat for Labour, whose divisor now becomes four. So finally, the votes in Round 7 for the last of the list seats are:

Conservatives: 20,000

> Labour: 17,500 (70,000 divided by four)

Green: 15,000

Liberal Democrats: 12,500

SNP: 10,000

And the Tories get their third seat.

So at the end of seven rounds, Brigadoonshire is now represented by these MSPs:

SNP: 9 (all constituency)

Labour: 3 (all list)

Conservative: 3 (all list)

Liberal Democrat: 1 (list)

And that’s how AMS works.

So now we have to deal with the tactical-voting issue. The whole point of AMS is that by redressing the unfairness that tends to result from FPTP, it’s designed to make tactical voting both unnecessary and pointless. Indeed, because you can’t know the constituency results in advance, it’s basically impossible.

For example, if the SNP only win eight of the constituency seats in Brigadoonshire instead of nine and the Tories pip them to the post for the other one – we’ll assume that Brigadoonshire is in the Borders – the seats on the vote numbers above then come out at SNP 8, Labour 4, Conservative 3, Lib Dem 1.

So because of/despite that Tory constituency win, the Tories still have three seats overall (one constituency and two list), but now Labour have effectively pinched one from the SNP. See what we mean about it being unpredictable?

We see from that scenario that trying to vote tactically on the list for a second party is a risky business, because you can’t take the FPTP element for granted – even slight changes in the constituency outcome can affect the list results in unforseeable ways. But is it worth the gamble?

We took a look at a real result – the Mid Scotland And Fife (MS&F) region from the 2011 Holyrood election, in which the SNP won eight out of nine constituency seats – to see how it would have turned out if one-third of SNP voters had given their list vote to the Greens instead.

Now, that’s an ENORMOUSLY generous figure for a tactical vote. The large majority of voters on all sides will vote for their preferred party in both ballots, and even those who do want to make a tactical Yes/left-wing vote will find themselves with a dilemma of which party to choose – the Greens, Tommy Sheridan’s Solidarity, the new “RISE” party seeking to replace the SSP, and possibly others.

(You only need to look at the hilarious results the “SNPout” campaign got in May’s general election – with the benefit of massive press hype and explicit backing from the Daily Mail and Telegraph – to see how few people actually tend to vote tactically, even in an FPTP election where it has a much better chance of working.)

So the Greens actually getting a third of the SNP list vote is basically a completely farcical notion. But let’s assume for the purposes of illustration that it happens, which on the 2011 MS&F numbers would mean the SNP vote falling from 116,691 down to 78,183 and the Greens rocketing from 10,914 to 49,811.

If you do the sums on that, the list result changes from Labour 3, Conservative 2, Lib Dem 1, SNP 1 to Labour 3, Conservative 2, Green 2, with the Greens capturing one seat each from the SNP and the Lib Dems.

In other words the Yes/left alliance would make a net gain of just one seat in the region for even an absolutely inconceivably massive tactical vote.

To beef up the data a bit we then applied the same formula to the Glasgow region, home to Patrick Harvie, which returned the Greens’ second-strongest performance in 2011. Again, the net Yes/left gain with a third of the SNP vote switching is just one seat – the Greens go from one to four, but two of them are taken from the SNP.

(The strongest Green region, Lothian, is a special case due to the presence in 2011 of the now tragically-deceased Margo McDonald, so it’s not really possible to even make a sensible extrapolation because we don’t know where her vote will go.)

Finally, we tested the Greens’ weakest region, Central Scotland, where on a one-third switch from SNP to Green there was no net gain at all – the Greens pick up three list seats, but all three of them are taken from the SNP.

These figures are all illustrations, not predictions. We have no way of knowing how many votes each party will get next May – eight months is a long time in politics – and no way of knowing how many constituency seats those votes will translate into.

But what we’ve learned is that even an unimaginably huge tactical list vote will likely at best produce a net gain of a couple of seats for Yes/left parties, and at worst could cost the SNP some seats and perhaps even their majority.

(AMS, we must remember, was also specifically designed to make winning majorities very difficult, and the Nats currently technically don’t have one at all, holding 64 seats out of 128. On our illustrative calculations here they lose six list seats in just three regions, and we can’t be sure how many constituencies they might win to compensate and regain a majority. The most they can gain is 20, as they already have 53 of 73.)

It’s not this site’s business to tell anyone how to vote. What these numbers strongly suggest, though, is that tactical voting – of any sort and for anyone’s benefit – in an AMS election is a mug’s game. You should vote for the party or parties that you most want to see form the government, rather than trying to second-guess the system. Because if you try, chances are it’ll make a chump out of you.

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Subsequent articles on this subject: Part Two, Part Three, Part Four, Part Five.