Written by Colin+ in ask uncle colin, psephology.

*Ask Uncle Colin* is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

Up here in Scotland, we've got an election tomorrow. It's not as simple as the stupid first past the post elections you have down there in England, but even with our superior Scottish intelligence, some people are still struggling to understand how the system works. Do you think you could explain?

Jings! Enough FPTP Foolishness; Electoral Reform Should Occur Now

Hi, JEFFERSON! Gladly.

The first thing to note is, there is no perfect electoral system, but some are less imperfect than others. The one in use in most English and UK elections is First Past The Post: the country (or county or city) is split up into zones that will have one representative; the electorate of that zone then votes, and whoever gets more votes than anyone else, wins.

That seems reasonable, except for one major problem: the proportion of representatives belonging to any given party is far removed from the proportion of votes they received. For example, at the last election, the party with the most votes won the most seats, which seems right; however, the Conservatives polled a little short of 37% of the national vote and received a little more than half of the seats -- about 40% more than they would have under a proportional system. They weren't the most over-represented1, though: the SNP polled under 5% nationwide and received 8.6% of the seats, practically every MP in Scotland2. At the other end of the scale, UKIP, for whom I normally have no sympathy, received only one seat despite getting nearly one in eight votes nationwide. (The third and fourth parties in terms of seats -- the SNP and Liberal Democrats -- received fewer votes between them than UKIP alone, but won 64 seats.) The geographical distribution of your votes, in many cases matters at least as much as the number of them3.

It's clear to nearly anyone who gives it a moment's thought that this is ridiculous. When the Scottish Parliament was created in the late 1990s, a more sensible system was adopted: the Additional Member System (AMS), which is rather more representative -- although still imperfect (the SNP polled around 45% and won 53% of the seats; the other major parties were within a few percentage points of what they 'deserved').

AMS combines elements of FPTP (so everyone has a local representative, known as an MSP) with elements of a regional list system: each of the eight regions elects around nine constituency MSPs and seven for the region as a whole. While the constituency MSPs are elected exactly as you'd expect, the regional MSPs... well, it's (comparatively) complicated.

The aim in each region is to add extra MSPs for each qualifying party so that the proportions (including the constituency MSPs) match the number of regional votes as closely as possible. That would be complicated enough even if there wasn't a likelihood that some parties would have already picked up more than their 'fair' share of MSPs from the constituency ballot. Still with me? Anyway, there's a method for it, known as the D'Hondt method (named for Victor D'Hondt, although a similar method was first used by Thomas Jefferson4 .) I'm going to run through an example (Lothian region, 2011, but with rounded numbers), first without the constituency members first, and then add them in.

Start by listing the parties and their totals:

SNP: 111

Labour: 71

Conservative: 33

Green: 22

Independent: 19

Liberal Democrat: 16

If it were a one-seat region, it would be natural to award it to the SNP, and that's exactly what the method does: nobody has any seats yet, so there's no complication. However, how about the second seat? Now the SNP has one seat, in the next round, each of its votes counts as half a vote. So, in the next round, the totals are:

LAB: 71

SNP (1): 55.5

CON: 33

GRN: 22

IND: 19

LD: 16

At this point, Labour have the highest effective total, so they get the next seat, and their votes count as half in the next round:

SNP (1): 55.5

LAB (1): 35.5

CON: 33

GRN: 22

IND: 19

LD: 16

Now the SNP are back in the lead, and they take the next seat; in the following round, their votes count for a third instead of a half. (The rule is that in each round, your votes count for $\frac{1}{s+1}$, where $s$ is the number of seats you have.)

SNP (2): 37

LAB (1): 35.5

CON: 33

GRN: 22

IND: 19

LD: 16

The SNP take the fourth seat as well, with their votes in the next round counting for a quarter5 .

LAB (1): 35.5

CON: 33

SNP (3): 27.75

GRN: 22

IND: 19

LD: 16

This process continues for as long as there are seats to distribute. The next three seats would go to Labour, to the Conservatives, and again to the SNP. If there were only seven seats for grabs, the SNP would have four, Labour two and the Conservatives one. That's broadly in line with their ratio of votes, if you ignore the smaller parties; if you had originally set the rule that a party got one seat for every 27.75 votes, the SNP would have had 4, Labour 2-and-a-bit, and the Conservatives 1-and-a-bit, but you get nothing for the and-a-bits.

In the actual election, the SNP won eight of the nine constituency seats, with Labour holding the other, and so their vote numbers had to be adjusted before allocating the regional seats. With eight seats already, each SNP regional vote counted for a ninth, while Labour's one seat meant each of their votes counted for a half in the first round, like this:

LAB (1): 35.5

CON: 33

GRN: 22

IND: 19

LD: 16

SNP (8): 12.333

After Labour took the first seat, their votes counted for a third:

CON: 33

LAB (2): 23.667

GRN: 22

IND: 19

LD: 16

SNP (8): 12.333

The remaining seats went to the Conservatives (whose vote became halved to 16.5), Labour again (now to 17.75), the Greens (to 11), the Independent ticket (to 9.5), Labour (to 14.2) and the Conservatives (to 11). Overall, the seats ended up as:

SNP: 8

LAB: 4

CON: 2

GRN: 1

IND: 1

That's remarkably close to the result for the full D'Hondt method (the only difference being that the SNP took a seat the Liberal Democrats would otherwise have won.)

So, why does D'Hondt work? It's not really about different votes counting for different amounts: it's about finding a fair threshold for the number of votes to win a seat. Dividing the vote tally by successive amounts finds the most efficient threshold at each level. For one seat, the threshold is "the number of votes the biggest party got." For two, it's "either the number of votes the second party got, or half the number the first party got, whichever is bigger." That is, if the first party has more than twice as many votes as the second, it deserves both of the seats, or else they deserve one each.

In honesty, I was skeptical about it when I started writing this -- it seemed convoluted and over-engineered -- but the D'Hondt method looks like a pretty practical way to divide seats up fairly fairly, especially as we have Machines for This Kind Of Thing. Explaining how it works to the average voter... that's another matter. But if I had to, I'd summarise it as:

The D'Hondt method finds out the right number of votes per seat, so that the seats are distributed pretty evenly, and there aren't too many MSPs

Because, if there's one thing that nobody wants, it's any more politicians.

-- Uncle Colin

* Thanks to @textuallimits for inadvertently suggesting the question. I should emphasise that the question does not, to the best of my knowledge or belief, represent her attitude towards Anglo-Scottish intelligence. Or my attitude, for that matter. It does reflect my attitude towards FPTP, though.

- ignoring the Northern Irish parties [↩]
- Even restricting the analysis to Scotland, the SNP won 56 out of 59 seats -- 95% -- with 50% of the vote. [↩]
- Don't get me started on the Alliance in 1983: 7.7 million votes gave them 23 seats; Labour, with 8.5 million, won 209 [↩]
- Hey! That's YOUR name! [↩]
- It seems a little unfair that the SNP would take the fourth seat -- they'd have 75% of the seats having only beaten Labour by about 60-40. It turns out that the actual score was 61-39; the SNP won the fourth seat very narrowly. [↩]