# On STV and proportionality

STV is not the proportional system many think it is

It is often asserted that there are two proportional systems on offer at this year’s referendum on New Zealand’s voting system. The statement makes me shudder every time I hear it. It’s true the single transferable vote (STV) normally gives minor parties some representation. But to call it proportional, in the way that the mixed member proportional (MMP) system is proportional, is not telling the whole story.

Starting at the beginning (skip this part if you’re a geek)
It is a mundane start, but it will help to establish what we mean by “proportional”. Proportionality is the idea that a political party’s representation in parliament should be proportional to their share of electors’ support. If a party receives 27 per cent of votes, they should, in principle, be entitled to about 27 per cent of the seats in parliament. Needless to say, proportionality has its advocates and opponents, but that debate’s for elsewhere. This is about whether STV fits that category.

STV is quite a complicated system, but the gist of it is this: Candidates stand for election to fill several (typically up to seven) vacancies. To be elected, a candidate needs to reach a number of votes known as the quota. On their ballot papers, electors rank candidates from first to last (though they don’t have to rank everyone). Each elector gets one vote. It goes to their first preference, unless that candidate has more votes than the quota or (if no candidate has an excess) they are coming last, in which case it goes to their next preference, and so on. The quota works so that just enough candidates are elected to fill the vacancies available.

If that was a bit over your head, the New Zealand Electoral Commission has a short video explaining the system. The Australian Electoral Commission also has a video (they use STV to elect their Senate), which I think gives a better (and longer) explanation, though our version would be a little different from the one they use. The AEC one is excellent; I know it’s Australian but I recommend that you watch it anyway.

Is STV proportional?
The first thing to note is that STV doesn’t even try to be proportional. The Sainte-Laguë method, which is used to allocate seats under MMP, is expressly designed to grant parties a share of seats in parliament proportional to their share of votes. The quota system of STV is designed to minimise wasted candidate votes and discourage tactical voting—admirable ends in themselves, but not the same as proportionality.

A case might then be made that STV has the effect of being proportional, even if it is a side effect. But note, first, that in STV electors vote for candidates, not parties. And, further, note that they vote not just for one candidate as in first-past-the-post (FPP), but they rank them in order from first to last. In FPP we can say an elector “voted” for the party their candidate stood for (though technically they didn’t). But the ranking in STV complicates that definition. If a voter ranked all of Party X’s candidates above all the rest, then we can easily say that he voted for Party X. But what if he ranked a Party X candidate first, then a Party Y one second, then another from Party Y, then Party Z, Party Y, Party X and finally Party W?

The ability not just to vote for a “party” when you like some of its candidates but dislike others is, indeed, an important advantage of STV. But it throws murkiness on to the concept of voting for a “party” that underlies the notion of proportionality.

The case where it is
There is a special case that is, in effect, proportional. This is the case where all electors cast their rankings to conform to a “party ticket”. For example, a Party X ticket would require a voter to rank all Party X candidates above all others and rank them all in the same order (X1, X2, X3, …). There are no deviations from party tickets, i.e. no-one chooses their own rankings, so they effectively just vote for a party.

Consider this basic example. There are five vacancies in an electorate. There are 400 who vote with the Party X ticket, 400 with the Party Y ticket and 200 with the Party Z ticket:

 Number of voters 400 voters 400 voters 200 voters Preference 1 Preference 2 Preference 3 Preference 4 etc. X1 X2 X3 Z1 etc. Y1 Y2 Y3 X1 etc. Z1 Z2 Z3 Y1 etc.

If we use the Hare quota, which is 200, then X1, Y1 and Z1 are elected in the first round. The excess votes from X1 and Y1, 200 each, go to X2 and Y2 respectively, who are elected. So the elected candidates are X1, X2, Y1, Y2 and Z1—each party picks up seats in proportion to the number of votes. If we use the Droop quota, we get the same result, except that the quota is 168, so X2 and Y2 pick up 232 transferred votes, not just 200.

We can generalise this. Say there are $N$ vacancies. The Party X ticket gets $v_x$ votes, the Party Y ticket $v_y$ votes, and so on. Everyone votes on a party ticket. We’ll call the quota $q$. Then if the Party X ticket just gets the quota, $v_x=q$, they’ll get one seat. If they get just over $n_x$ times the quota, $v_x = n_xq$then $(n_x-1)q$ votes are transferred to the second candidate, $(n_x-2)q$ to the third, and so on until $n_x$ candidates from Party X are elected. (There might be a few votes left over, but short of the quota.) Since $q$ is determined so that the number meeting the quota is the number of vacancies, that is, $n_x+n_y+n_z+\dots = N$, and since roughly speaking $n_P$ is proportional to $v_P$ for each party $P$ (since $v_P \approx n_Pq$), the effect is that the $N$ seats are divided in proportion to the number who voted by each party ticket.

This is basically how “above the line” voting in the Australian Senate elections works: electors choose a party and their vote is then assumed to conform to that party ticket. Since over 95% of voters vote “above the line”, the effect is proportional. In fact, given that so few voters vote below the line, it’s hard to believe that voters who put more thought into their rankings than a party name have any substantial influence, which sort of renders the added complexity a bit pointless.

Disproportionality from small electorates
I doctored the numbers of that example to make the result neat. Obviously, the more seats there are in a single electorate, the closer we get to proportionality. Conversely, where there are only a few vacancies, the “margin of error” can be quite high. In the above case, if 499 voters had voted with the Party X ticket and 301 voters with the Party Y ticket, the result would have been the same (with either quota). So Parties X and Y get the same number of seats, despite having 49.9% and 30.1% respectively of votes—a difference in support of 19.8 percentage points. This distortion is unavoidable with low numbers of seats.

This can seem like a small issue, but when you aggregate this effect over dozens of electorates, each running separate contests, it can be a substantial effect. Sometimes, the “errors” in different electorates will cancel out. But other times, they will add up and all favour the same party.

That’s what happened in the Maltese general election of 1981. The Nationalist Party got a majority of votes (50.9%), but the Labour Party won a majority of seats and hence formed the government. This would have happened again in 1987, 1996 and 2008, but for the introduction of “bonus seats” after the 1981 crisis. So if you think STV’s “proportionality” will protect New Zealand from repeats of the infamous 1978 and 1981 (FPP) elections—where Labour got more votes than National, but National formed the government—you should think again. The only system on offer this referendum that protects against this anomaly is MMP.

Where proportionality gets murky
Even without the “errors” induced by a smaller number of vacancies, though, proportionality can still be murky. Consider this case. There are five vacancies, and at least three parties. The 1000 voters are divided as follows (for simplicity, we assume there are no other combinations):

 Number of voters 400 voters 400 voters 200 voters Preference 1 Preference 2 Preference 3 Preference 4 etc. X1 Z2 X2 X3 etc. Y1 Z3 Y2 Y3 etc. Z1 Z2 Z3 X1 etc.

If we use the Hare quota, which is 200, then X1, Y1 and Z1 reach the quota in the first round. X1 has 200 excess votes, which are all transferred to Z2, who now reaches the quota and is elected. Y2 also has 200 excess votes, which are transferred to Z3, who is elected. So the elected candidates are X1, Y1, Z1, Z2 and Z3. If we use the Droop quota (168), we get the same result, with Z2 and Z3 being elected on 264 and 232 transferred votes respectively. Note that Party Z has three of the five seats.

Whether this is proportional depends on how you look at it. If we say that a elector “voted” for the party of their first-choice candidate, then Parties X and Y should have won two seats each—but only won one. At the same time, Party X supporters clearly preferred Candidate Z2 over Party X’s own X2, and similarly for Party Y with respect to Z3. This is how STV’s supposed to work: voters choose candidates, not parties. But it seems contrary to proportionality that a party that had just 20% of the first-preference vote gained a majority (three of five) of vacancies for this electorate.

You could argue that this is really what the electorate asked for when everyone gave Party Z their second-preference vote. You could also argue that, if Party X supporters liked Z2 that much, it probably says something about Z2. This lack of proportionality could be, and for some is, an argument for STV. But we must conclude that either (a) STV is not proportional, or (b) the concept of proportionality makes no sense when looking at STV.

It’s not like MMP
In some ways, I like STV. I like the idea of minimising vote wastage, and in some contexts (mainly professional societies), the added complexity can be useful.

But STV offers no guarantee of proportionality between the parties on which national politics is based. There’s not even a guarantee that a party with a majority of votes will form the government. The Sainte-Laguë method of MMP, on the other hand, will never give fewer seats to a party with more votes.

There is, of course, more to voting systems than just proportionality. But if proportionality is a top priority for you, you should not view MMP and STV as on equal ground. The only truly proportional system in this referendum is MMP.