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Posts tagged ‘Sainte-Lague’

The minimum swing needed in the special votes for the Greens to steal that seat

A 4.6-point swing in specials relative to preliminary, about 0.6 points more than my projection, would get the Greens a 14th seat at National’s expense. But even less might be sufficient.

I think a projection, which I posted yesterday, is more useful information that the “minimum required swing” calculations that I did for the 2011 election. But if anyone is interested in knowing exactly what the Greens need to clear the bar, here is that spreadsheet I did in 2011, updated with 2014 data: Required changes in special votes to steal seats, 2014 election.

I won’t go through all the various possibilities. If you’re curious, feel free to do that yourself using the spreadsheet. The situation that is most interesting, because (other than no change) it’s most likely, is where the Greens would pick up a 14th seat in exchange for National’s 61st (and absolute majority).

The Greens would need to swing +4.60 points in the special votes alone (not total votes), relative to the preliminary count, to take a seat. I have them projected to swing +4.05 points, so this isn’t unrealistic.

Here’s the catch: In the preliminary results, Labour has the 120th quotient and National the 119th. This means that Labour would lose a seat before National. So they need to gain a little to make sure it’s National, not them, who loses the seat. This isn’t hard: they’re 561 votes away (relative to an assumption that specials are distributed identically to preliminary), or +0.19 points. My projection has them well ahead of this.

So the “minimum change required” situation has both Labour and the Greens taking from National. You’ll notice in the table below that the Greens don’t actually need National to lose as much as it did in my projection: -4.79 points, not as much as my projected loss of -5.58 points.

In terms of absolute votes, this is a swing of 1,648 relative to my projection. I said in my last post that 1,129 votes more would suffice. The assumptions behind these figures are different. The projection margin (1,129) assumes that the Greens only gain votes, and no-one loses any, i.e. it’s likely to be an overestimate (not by double though—the equivalent National loss is 5,058). The “minimum required” swing (1,648 relative to projection) assumes that all other parties perform in the specials identically to the preliminary result. Historically (and hence in my projection) most left-wing parties do better in the specials at National’s expense, and these help the Greens too. So this is also likely to be an overestimate, probably more so.

So really, the most important hope for the Greens to hang on to is that they will perform about 0.6 points, or maybe (depending on what else happens) even just 0.35 points, better in the special votes than the 2011 swings might imply.* As I said in my last post, it’s certainly possible, but it’ll be very close. If I were Steffan Browning or Maureen Pugh, I’d have a very nervous two weeks ahead.

Party Preliminary Projection Special required Change on prelim Change on projection Total required Change on prelim
National 48.06% 42.48% < 43.26% -4.79% +0.79% < 47.47% -0.59%
Labour 24.69% 28.10% > 24.88% +0.19% -3.22% > 24.71% +0.02%
Green 10.02% 14.07% > 14.63% +4.60% +0.56% > 10.59% +0.57%

* My projection applies the 2011 preliminary-to-special swings multiplicatively, not additively, to the 2014 preliminary result. A full method description is in the post describing the projection.

Projection of special votes for the 2014 election

On my projection for the 2014 special votes, based on the 2011 impact, the Greens will miss out on a 14th seat by a whisker and National will retain its outright majority. Maybe.

I crunched some numbers to project (I didn’t say “predict”) the impact of special votes. I suspect most of you will just want to know the answer, so I’ll cut to the chase first, then give a bit of analysis, then give a bit more details, and I’ll talk about my method fourth.

Special votes include, among other things, overseas votes and votes cast for a different electorate to the polling place location. They aren’t counted on election night; they’re just set aside for the full count released two weeks later. And they’re not normally representative of the vote as a whole. Historically, special votes have favoured the Greens significantly—they have often picked up an extra seat from it, as they did in 2011.

The most fundamental assumption I make is that you can use the 2011 impact of the special votes as a guide to the 2014 impact. Some other minor assumptions will become apparent in the method description at the end.

Basically, on my projection, the results would stay the same. The Greens will get closer to a 14th seat, but because on the preliminary results they only just got their 13th one, special votes won’t propel them far enough for a 14th. However, they will be tantalisingly close, moving their 14th seat to the 121st quotient.

Party Preliminary results Projected final results Gain/loss
Votes % Seats Vote % Seats
National Party 1,010,464 48.06% 61 1,136,155 47.37% 61 -0.69%
Labour Party 519,146 24.69% 32 602,304 25.11% 32 +0.42%
Green Party 210,764 10.02% 13 252,394 10.52% 13 +0.50%
New Zealand First Party 186,031 8.85% 11 204,919 8.54% 11 -0.30%
Māori Party 27,074 1.29% 2 33,134 1.38% 2 +0.09%
ACT New Zealand 14,510 0.69% 1 16,685 0.70% 1 +0.01%
United Future 4,533 0.22% 1 5,098 0.21% 1 -0.00%
Conservative 86,616 4.12% 0 94,357 3.93% 0 -0.19%
Internet MANA 26,539 1.26% 0 32,826 1.37% 0 +0.11%
Aotearoa Legalise Cannabis Party 8,539 0.41% 0 10,894 0.45% 0 +0.05%
Ban1080 4,368 0.21% 0 4,992 0.21% 0 +0.00%
Democrats for Social Credit 1,609 0.08% 0 1,983 0.08% 0 +0.01%
The Civilian Party 906 0.04% 0 1,035 0.04% 0 +0.00%
NZ Independent Coalition 895 0.04% 0 1,023 0.04% 0 +0.00%
Focus New Zealand 677 0.03% 0 774 0.03% 0 +0.00%

I project the Green gain to be 0.50%, and the National loss to be 0.69%. In a lot of cases, this can be enough to win and lose a seat, respectively: one seat is roughly 0.8% of the vote (0.83% if you ignore “wasted” votes), so it’s easy to imagine that another 0.5% might push you over.

But in this election, it wouldn’t be enough. The reason is that, in the preliminary results, the Greens only just made their 13th seat: it was the 118th quotient, i.e., the 118th seat to be allocated. National, on the other hand, has some room to slack before losing a seat—in fact, it would have picked up the 121st quotient, if there was one. In effect, the Greens would need to pick up to nearly a “whole seat” worth of votes to pick up seat number 14.

In my projection, the Greens nearly get there, but not quite. They move their 14th seat from the 127th quotient to the 121st. So if there was one more seat in Parliament, it would be theirs. And they’re very close: with 1,129 votes more (all other vote counts staying the same), they would steal the 120th quotient from National. Conversely, if National had had 5,058 votes less (all other counts staying the same), they would give up the 120th quotient to the Greens.

What does this mean?
In practice, what this tell us is not necessarily that Steffan Browning won’t make back in at Maureen Pugh’s expense. It’s that he might—but it’ll be very close. Certainly, the Greens shouldn’t be expecting another seat from specials, like they could in 2011.

There are, however, reasons to believe the wind might blow in the Greens’ favour. Firstly, advance votes were way up on previous elections. This was partly due to a concerted effort from parties to promote advance voting this election, and (I think) more so from the left than the right. Now, ordinary advance votes are counted on election night. But special advance votes—which include votes cast in advance outside the voter’s electorate—are not. And, as @annagrammatiste pointed out to me on Twitter, a lot of those special advance votes will have been cast at universities, some of which had advance voting booths on campus this year. University students, on average, lean left.

Secondly, estimated overseas votes doubled in this election, from 19,500 in 2011 to 38,500 in 2014. Overseas voters are known for being particularly Green-heavy.

Without those reasons, I would have said that it’s more likely that the seat allocations will stay put, but not enough to put money on it. But these weaken that likelihood. My instinct is that it probably won’t be enough, but objectively, I’m not really sure. It’ll be super close.

That’s basically the gist of this post. If you like voting systems, read on.


Some more detail
To dive in a little deeper, here’s a quick primer on how the Sainte-Laguë method works. This is the method used in New Zealand to proportionally allocate seats. The easiest way to think of it (in my opinion) is that you allocate the seats one by one, to the party with the highest “quotient” at the time. Every party’s initial quotient is their total number of votes, so the first seat effectively goes to the party with the most votes. Then, every time you allocate a seat, you divide that party’s total votes by their next divisor to get their new quotient (leaving the rest untouched). The first divisor for each party is 1, then it goes 3, 5, 7, and so on. You repeat till you’ve allocated all 120 seats.

How are quotients relevant? Well, here are the quotients near 120 for the preliminary results:

Quotient National Labour Greens NZ First Māori ACT UF
118 8350.9 8240.4 8430.6 8088.3 5414.8 4836.7 4533.0
119 8350.9 8240.4 7806.1 8088.3 5414.8 4836.7 4533.0
120 8215.2 8240.4 7806.1 8088.3 5414.8 4836.7 4533.0
121 8215.2 7986.9 7806.1 8088.3 5414.8 4836.7 4533.0
122 8083.7 7986.9 7806.1 8088.3 5414.8 4836.7 4533.0
123 8083.7 7986.9 7806.1 7441.2 5414.8 4836.7 4533.0

As I said, the Greens got the 118th quotient, and won’t pick up another one until 127. National’s getting quotients more often because it has more votes, and the gap between dividing by 121 and 123 is “smaller” than between 25 and 27 (Greens), so to speak.

Here is the same for my projected results:

Quotient National Labour Greens NZ First Māori ACT UF
118 9547.5 9560.4 9347.9 8909.5 6626.7 5561.7 5098.4
119 9547.5 9266.2 9347.9 8909.5 6626.7 5561.7 5098.4
120 9389.7 9266.2 9347.9 8909.5 6626.7 5561.7 5098.4
121 9237.0 9266.2 9347.9 8909.5 6626.7 5561.7 5098.4
122 9237.0 9266.2 8703.2 8909.5 6626.7 5561.7 5098.4
123 9237.0 8989.6 8703.2 8909.5 6626.7 5561.7 5098.4

The Green’s 14th quotient is just 41.8 away from National’s 61st. So we find a rudimentary “margin” for the Greens by multiplying this difference by the 14th divisor: 41.8 × 27 = 1,129. For National, we use the 61st divisor: 41.8 × 121 = 5,058.

Okay, now for the exciting part. (Heh.) There are lots of ways to do this projection; Graeme Edgeler has one that basically gives the same outcome. This is how mine works.

I took the preliminary results for 2011 and subtracted them from the final results to get the “special vote addition”. I use quote marks because it’s not actually all special votes: it also includes votes cast at polling places with fewer than 6 votes (these aren’t counted on election night) and any corrections. But anyway, I compare the preliminary percentages to the “special vote addition” percentages, by division, to get a “multiplier” for each party. This multiplier represents a relationship between the preliminary votes and the special votes for each party.

I apply this multiplier to the preliminary vote percentages for each party in 2014. If a party didn’t contest the 2011 election, I just use 1 (i.e., no adjustment). Now, after doing this, the percentages won’t add up to 100, so I scale them so that they do.

The next part is the complication. The preliminary total vote count includes informals (ballots where you can’t tell who they voted for); obviously the sum of parties’ votes don’t. And the preliminary special vote estimate (I take the figure that includes overseas votes and fewer-than-6 places) includes ballots that might later be ruled invalid, because the statutory declaration wasn’t completed correctly or something like that. So I use 2011 ratios of preliminary to final counts to project how many valid special votes there will be in 2014. It doesn’t end up making much of a difference (0.4%), but hey, I wouldn’t know unless I tried.

The last step is to use the projected special vote percentages and the projected valid special vote count, to project the “special vote addition” for each party, add those to the preliminary count and that’s the projected result.

If you get excited by numbers like I do, here’s the spreadsheet I used to figure this out: Special vote impact projection 2014. The actual Sainte-Laguë stuff is in a different spreadsheet, but I’ve pasted the borderline quotients into that spreadsheet.

Effective thresholds in MMP when there is no threshold

Abolishing the 5% threshold in MMP (as I advocate) doesn’t mean that a party getting just one vote picks up one in 120 seats.  It’s fairly intuitive that there is still an “effective threshold”: a number of votes that parties must get to earn their first seat.  That then begs the question: How many votes is enough?

The answer depends on the method used to translate the party vote to seats in Parliament.  New Zealand (and a number of other countries) uses a method called the Sainte-Laguë method.  Another common method is the d’Hondt method.  In this post I’ll assume you’re familiar with at least one of them (they are very similar); if you’re not, Wikipedia explains them reasonably well.

The Sainte-Laguë method is more sympathetic to smaller parties than the d’Hondt method, so we expect the Sainte-Laguë effective threshold to be lower.  The report of the 1986 Royal Commission on the Electoral System lists the thresholds in Addendum 2.1, on page 74.  The threshold for an N-member House, when there are k parties other than the one-seat party, is V/(2Nk + 1) for the Sainte-Laguë method and V/(N + 1) for the d’Hondt method.

I couldn’t find proofs of these effective thresholds, so I derived those results myself.  That proof is in a PDF file here.

That then helps me to find the effective threshold of a modified Sainte-Laguë method that I support, which is the same one that they use in Norway and Sweden.  In this method, the first divisor is changed to 1.4 (instead of 1).  The threshold is then V/(5(2N − k)/7 + 1).  More generally, if the first divisor is changed to m, then the effective threshold is V/((2Nk)/m + 1).

What does that even mean?
Those formulae don’t really mean much at first glance.  The best way to find meaning is to compare them to V/N.  That is, in a 120-seat house, how does the “effective threshold” compare to 1/120 of all party votes, or 0.83%?

To make life easier, we’ll make an approximation: we’ll assume that N is much larger than k, i.e. there are many more seats than parties, which is generally true.  We’ll also use the fact that N >> 1.

Then the Sainte-Laguë effective threshold is approximately V/2N.  That means, in order to get one seat out of 120, you need roughly half of 1/120th of the party vote, or 1/240th of the party vote, or about 0.42%.

For the modified Sainte-Laguë method, it’s roughly mV/2N.  Basically that means you take 1/120th of the vote and multiply it by m/2.  For example, if m = 1.4, then you need about 70% of 1/120th of the vote, which is about 0.58%.

The d’Hondt threshold, roughly V/N, is just 1/120th of the party vote (or marginally less), or about 0.83%.

It seems fair to me that a party falling just short of 1/120 of the party vote should get one seat in Parliament.  But awarding them a seat for achieving just half of that seems a bit unfair—and disproportional—to me.  The effective threshold should be enough to be “close-ish” to 1/120.  I would put “close-ish” at about 70% of 1/120 of the party vote, which is why the modified Sainte-Laguë method used by Norway and Sweden seems sensible to me.


If there was no threshold

If the five per cent threshold didn’t apply, the Conservative Party and the Aotearoa Legalise Cannabis Party would have polled high enough to gain seats:

% Party





National Party
Labour Party
Green Party
NZ First Party
Conservative Party
Maori Party

ACT New Zealand
United Future
Aotearoa Legalise Cannabis Party
Other parties

* The Maori Party got 2 seats but won 3 electorates, so they keep their third electorate as an overhang seat
† There are actually 70 electorates, but Christchurch Central is currently tied before special votes, so for this count we’re just pretending it doesn’t exist.  It’s tied between National and Labour, so it doesn’t actually affect the total number of seats.

Under the Sainte-Laguë method with the threshold, National had the 117th, 118th and 120th seats, and Labour had the 119th seat.  Those last four seats would be lost.  The Conservatives, at 2.76%—higher than the Maori and ACT Parties combined—polled high enough to get the 18th seat in the allocations, as well as the 56th and 91st.  The ALCP had a close shave polling at 0.48% and picking up seat number 104 with 1,185 votes to spare (above the new 120th quotient, 8,331).‡

List candidates who would have made it List candidates who wouldn’t have made it
Colin Craig (Conservatives)—18th seat
Kathy Sheldrake (Conservatives)—56th seat
Larry Baldock (Conservatives)—91st seat
Michael Appleby (ALCP)—104th seat
Aaron Gilmore (National)—120th seat
Raymond Huo or Rajen Prasad (Labour)§—119th seat
Cam Calder (National)—118th seat
Jackie Blue (National)—117th seat

‡ In a 120-seat Parliament with no threshold, you generally need roughly 0.42% to get one seat.  The way I think of it is that you need more than half of 1/120th of the total votes, which works out to 0.42%.
§ If Brendon Burns wins Christchurch Central, he displaces Mr Huo out of Parliament and Mr Prasad into the last list position, so it would be Mr Prasad.  If Mr Burns loses, it would be Mr Huo.

This would obviously have seen the National-ACT-United Future combination fall short of a majority at 59 seats.  But there’s a solid chance that National would have considered the Conservatives a suitable coalition partner.  (They don’t agree on everything, but then, no two parties do.)  That would then have seen the right-wing group on 62 seats, still enough to govern.

Of course, even if the Conservatives were excluded here, the Maori Party’s three seats would also have been enough to give National a majority.  But that would have made life a lot harder for National’s legislative programme (as ACT and the Maori Party don’t always vote together).

Labour would have had a much, much harder time stitching up a majority, given that they would need NZ First (who said they won’t work with anyone), Mana (who Labour said they won’t touch) and ALCP (who knows?), and even with all of them (plus the Greens and Maori Party) they’d still fall one seat short.

So, no change in government and no radical change in its form, unless you count the entry of the Conservatives into a centre-right government as a big change.  (That said, at 2.76% they probably deserve those three seats.)

As for the ALCP getting into Parliament, it might scare a few people but the ALCP’s done okay before: in 1996, they got 1.66% of the party vote.  Nonetheless, 0.48% can seem like a short mandate to get 1/121st of the seats in Parliament.  If we used the modified Sainte-Laguë method, where the first divisor is changed from 1 to 1.4 but all subsequent divisors are the same (3, 5, 7, …) (as is used in Norway and Sweden) then that seat would have gone to National instead.

I used to do this exercise for fun (well, once, in 2008). It matters a bit more to me now because I’ve come to support abolishing the threshold altogether.  It matters even more with the review of MMP that will be coming up if MMP (as predicted) wins the referendum.  The change to this election result would have been relatively insignificant and I don’t like either of the parties that didn’t make it, but it makes little sense to me that a party receiving four times the vote of a party that gets one seat, should get no representation in Parliament at all.

What it would take to steal a seat in the special votes

Update: I’ve uploaded the spreadsheet I used to make these calculations—a link is at the bottom of the post.

While the two million ordinary votes have been counted, there are still around 220,000 special votes left. We won’t know the effects of these votes until final results are released on 10 December. In the past, they’ve changed the provisional results: last election, they saw the Greens take a seat from National. What would it take for that to happen again in 2011?


  • Greens need a 2.23-point swing on National in the special votes (alone) to take a seat from them.
  • Greens need a 2.27-point swing on Labour (in the special votes alone) to take a seat from them.
  • To gain two more seats, the Greens would need to almost double their vote in the specials (i.e. get 20.23%).
  • NZ First is closest to picking up another seat. They only need a 0.43-point swing on National, or a 0.54-point swing on Labour (though the latter is more complicated).
  • It’s impossible for NZ First to lose a seat.
  • National would take a seat off Labour with a 3.73-point swing. Labour would do the same to National with a 4.5-point swing. Both situations require NZ First to lose a little as well.
  • Maori, ACT, Mana and United Future would need to quintuple, double, triple and decuple their votes in the specials respectively, so their seat counts won’t be budging.

Explanation of model

This analysis works as follows. As a starting point, we assume that special votes follow the same proportions as ordinary votes. We then transfer votes from a “losing party” to a “gaining party”. When this happens, their quotients (under the Sainte-Laguë method, see Elections NZ, Wikipedia) fall and rise respectively. The idea is to find the point where the last quotients of the parties gaining and losing seats are equal. This is the minimum vote transfer needed to steal the seat.

When I talk about “losing”, “gaining” or “transferring” votes, I mean relative to the assumption that the special votes are in proportion to the ordinary votes. A “swing” means the difference in percentage points between the special votes (not the total votes) and the ordinary votes.

We try not to touch the votes of other parties. This means that this analysis is a bit simplistic but shows the absolute minimum vote gains necessary, like a best-case scenario. (You need fewer votes if you’re only stealing from one party.) In some cases, though, a vote transfer between two parties results in a seat being picked up by a third party, so we adjust that party’s votes accordingly too.

Greens to take a seat from National

Party Provisional result Special required Swing Total required Change
National 47.99% < 45.76% -2.23% < 47.77% -0.22%
Greens 10.62% > 12.84% +2.22% > 10.84% +0.22%

Given the Greens’ history with special votes, this is the most likely scenario. For the Greens to take a seat from National, they need to get 12.84% of the special votes. (That is, of the 220,720 special votes, 12.84% need to be cast for the Greens, all at National’s expense—a “vote transfer” of 4,863 votes or 2.23 percentage points.) This would take the Greens’ total party vote from 10.62% to 10.84%, enough to see Mojo Mathers (14th on Greens list) replace Aaron Gilmore (53rd on National list).

This will probably happen: the Green co-leaders think they will hit 11 per cent. Based on a comparison to the special vote effect of 2008, Graeme Edgeler on the Legal Beagle finds similarly. I think it’s a good bet that, even if they don’t hit 11 per cent, they’ll pick up enough to get seat number 14.

Because National was allocated the 120th seat (using the Sainte-Laguë method), it’s the easiest to steal seats from. It’s theoretically possible for the Greens to steal from Labour though.

Greens to take a seat from Labour

  Provisional result Special required Swing Total required Change
Labour 27.13% < 24.86% -2.27% < 26.91% -0.22%
Greens 10.62% > 12.89% +2.27% > 10.84% +0.22%

If the Greens take their extra votes from Labour rather than National, they can see Labour’s last quotient (8,967) slip past National’s (8,930) to be lost to the Greens at 8,893. In this case, they would need 12.89% of the special votes, or a transfer of 4,971 votes, slightly more than what they would need stealing from National. At time of writing, iPredict’s live parliament projection had this scenario happening.

The current deadlock in Christchurch Central means that the exiting Labour MP could be either Raymond Huo (21st on list) or Rajen Prasad (20th), depending on whether Brendon Burns wins.

Note that if the Greens pull that same number of votes (just under 5,000) from both National and Labour, but split between them, they would not gain an extra seat. This is because National’s and Labour’s last quotients would both go down, but neither by as much—so the increase in the Greens’ last quotient won’t meet either. It works best for the Greens if their extra support in specials comes “from” a single major party.

It’s not possible for the Greens to steal from any other party: that would require the losing party to get a negative number of special votes.

Greens to take two extra seats from National or Labour

Since this would get James Shaw (15th on Greens list) into Parliament, I feel compelled to include this. The easiest way for the Greens to gain two seats is to take one from each of National (who got the 120th seat) and Labour (who got the 119th seat):

  Provisional result Special required Swing Total required Change
National 47.99% < 42.73% -5.26% < 47.47% -0.52%
Labour 27.13% < 23.03% -4.10% < 26.73% -0.40%
Greens 10.62% > 20.23% +9.61% > 11.57% +0.95%
NZ First 6.81% < 6.51% -0.27% < 6.78% -0.03%

This would also require NZ First to lose a little to prevent them getting one of the vacated seats first. Unfortunately, though, this situation does require the Greens to almost double their party vote (10.62% to 20.23%) in the specials, which for supporters of Mr Shaw is rather wishful thinking.

The Greens could also gain two seats off National or two off Labour with just a bit more (20.97% or 21.02% respectively).

New Zealand First to take a seat from National

  Provisional result Special required Swing Total required Change
National 47.99% < 47.56% -0.43% < 47.94% -0.05%
NZ First 6.81% > 7.23% +0.42% > 6.85% +0.04%

It’s actually easier for NZ First to take a seat from National than it is for the Greens to. That’s because NZ First, in the provisional results, is “next in line” for a seat: its next quotient of 8,867 is the highest of the non-qualifying quotients. So it just needs to catch up to National’s last quotient of 8,930, which it can do with a 0.43-point swing.

This isn’t likely to happen. NZ First is more likely to suffer from the special votes than pull 7.23% of them. But it’s not a foregone conclusion.

New Zealand First to take a seat from Labour

  Provisional result Special required Swing Total required Change
National 47.99% > 48.37% +0.38% > 48.02% +0.03%
Labour 27.13% < 26.20% -0.93% < 27.04% -0.09%
NZ First 6.81% > 7.35% +0.54% > 6.86% +0.05%

For this to happen, both National and NZ First have to gain votes at the expense of Labour. This is because when we transfer votes from Labour to NZ First, their quotients (originally 8,967 and 8,867) meet at 8,947, which is higher than National’s 8,930. Then NZ First wins the seat off Labour, who in turn wins it back off National—hence the need for National to gain votes too. The number of votes switching hands is still just 2,025, less than what the Greens need. But for NZ First it probably won’t happen.

New Zealand First to lose a seat to anyone: NZ First’s Asenati Taylor (8th on list) is safe. NZ First’s eighth seat sits at a quotient of 10,050, well ahead of the 120th quotient of 8,930. For them to lose a seat they would effectively need to get less than zero special votes. Sorry to disappoint you.

National to take from Labour or Labour to take from National: For either of these to happen, NZ First would need to lose 299 votes to the gaining party to stop it from getting the seat first. Including those votes, a 3.73-point swing towards National would do the trick for them, as would a 4.5-point swing towards Labour.

Maori Party to gain a third seat: Note that this wouldn’t actually gain the Maori Party a seat—they have three electorates but are only allocated two seats, so a third seat just means no overhang for Parliament. They would need at least 7.87% of the special votes to take the 120th seat from National and prevent NZ First from getting there first. They weren’t far over the bar for two seats with ordinary votes (1.35%) so their next quotient is down on 5,966, well short of 8,930. It’s also impossible for the Maori Party to lose a seat, but only just.

The one-seat parties (ACT, Mana and United Future): The ACT Party would need to get 2.41%, the Mana Party 3.10% and United Future 6.54% to benefit from special votes. Since this is doubling, tripling and decupling their votes respectively, it’s fair to say these parties won’t be affected by special votes. They can’t lose their party vote seat allocation (to create an overhang) either, even if they get zero special votes.

Update: The spreadsheet I did to figure out these “minimum vote transfers” can be found here (XLSX format). It’s a little esoteric but hopefully it’s followable.
Update 2: It appears that that spreadsheet doesn’t work in Excel 2007 and earlier, so you need Excel 2010, because I used the FLOOR() function with negative numbers but positive significance.
Update 3: Uploaded new version, that spreadsheet should now work in Excel 2007 and earlier.