I left a few points hanging in **my previous post**. One was that AMS, being a proportional system, makes tactical voting impossible. Tactical voting, under a first-past-the-post system, is where your preferred party is polling third, with the two despised enemies neck-and-neck at the top: you vote for the enemy you despise less, because at least it will keep out the other scum. But under AMS, where parties with even 4% of the vote get seats, voting for the less-despised enemy simply gives them a better chance of a seat, and reduces your own party’s chances of winning one.

Another complication is how we should properly refer to the two votes that you have under the AMS system, and I sidestepped this issue by calling one the ‘constituency’ vote (which is not controversial) and the other the ‘second vote’. However, even ‘second vote’ won’t do, because the ballot paper could well ask for your constituency vote in its right-hand column, and what I’ve called your ‘second vote’ in its left-hand column. This would make my terminology disastrously counter-intuitive. So a better term for the second vote is the ‘list vote’, because you cast your vote not for an individual member but for a party list.

But even ‘list vote’ isn’t the term used in official descriptions, and that’s because of a complication that I haven’t told you about. The complication is that list votes aren’t totalled and seats allocated for the country as a whole; instead the country is divided into 8 regions of 7 list seats each, and those 7 seats are allocated in proportion to the list-vote shares *for that region only*. Each constituency is also within a region, and it is the constituency seats *for that region* that are topped up. So the more widely used (but less informative) term for the list vote is the ‘regional vote’. The regional basis of the top-up also makes the election result hard to predict: you can’t just say, “X party has Y percent of the votes, so they’ll get Z seats,” because different parties can be stronger in different regions.

The third and almost last complication is both interesting and infuriating, and arises from two inherent constraints in the seat-allocation process. One is that to allocate seats in exact proportions, you would have to allow fractions – one-third each of 10 seats is 3.3333 seats per party – and that won’t do. The other is that if you avoid the fractional problem by not allocating all the seats – give the three parties 3 seats each and leave the 10th seat unallocated – that won’t do either.

The formula used to get round these problems is the d’Hondt formula, which operates like a bidding system, but with the bids rigged by the formula. In the first round, each party bids its full number of list-votes, and the party with the most list-votes wins the first seat; but for the next round, that party’s list-votes are divided by the number of seats it now has plus 1 – i.e. it’s now divided by 2 – so that party can only bid half its list-votes for the second seat, which will therefore go to a party whose list-votes are more than half those of the leader. And so it proceeds: at each round, what you can bid are your list-votes divided by the number of seats you now have plus one; algebraically

Q = V / (S + 1)

where Q is what you’re allowed to bid, V is your list-votes and S is the number of seats you currently have.

To take a simple example, if you’re the Big Party and I’m the Tiny Party, and there are 7 seats to be allocated, your bids round by round will be all of your list-votes, then 1/2 your list-votes, then 1/3 of your list-votes, then 1/4 of your list-votes, and so on, and as long as that fraction of your list-votes is more than the whole of my list-votes you’ll be allocated that seat. If my list-votes are more than 1/7 of yours, but not more than 1/6, I’ll get the last seat.

The immediate importance of d’Hondt is that you can’t convert votes into seats by simple arithmetic: if you want to know how many seats a party would get if it won a given percentage of the vote, you have to specify an assumed vote-share for each party and then run d’Hondt: anything else is mere guesswork. For a Holyrood election you have to do this for each of the 8 regions separately. You also have to specify, for each party, the number of constituency seats – d’Hondt sets the value of S in the first round to each party’s constituency seats. Running a d’Hondt calculation is an easy and mechanical task, but not all commentators do it. There’ll be a d’Hondt calculator on my website soon, and when it’s ready I’ll post the link here.

It’s the last complication of all, however, that’s the most interesting. The aim of the system is to arrive at a proportional parliament by adding to the constituency seats a top-dressing of list seats. But what if a party’s constituency seats already exceed its list-vote share? Supposing, for example, in a 100-seat parliament with 60 contituency seats and 40 list seats, the party vote-shares and constituency seats are as follows:

VOTE-SHARES: SLOBS 40%, TOADS 50%, EARWIGS 10%

CONSTITUENCY SEATS: SLOBS 50, TOADS 10, EARWIGS 0

The Slobs are due 40 parliamentary seats, but they already have 50, and you can’t take those away; the Toads deserve 50 parliamentary seats, which are available, but then the Earwigs would get none, and they deserve 10 parliamentary seats. The problem is of course logically insoluble, because the terms of the system don’t allow it: AMS assumes that each party’s share of constituency seats will fall short of its list-vote share, giving room for a top-up.

I’ll deal in a later post with how d’Hondt resolves this conundrum – it’s a fair result in fact, but it has unexpected effects. And it will lead us, as some readers will already have guessed from the way my argument here is going, directly on to the Wings Over Scotland Devastating Electoral Initiative, which is looking as though it will play a crucial role in the 2021 election.