MMP maths: How party vote percentages become seats in parliament

Your indispensable guide to counting the numbers on election night. Simon Wilson explains the official way to do it and offers a cheat’s alternative.

You vote for a party that doesn’t make it into parliament and your party vote is just discarded? Sorry, it’s true. Your vote is set to one side and not considered when they determine the final number of MPs for each party. But in a ghostly kind of way, it does live on in the final determination, and not in the way you may have intended…

There are two ways for a party to get into parliament. One is to reach the “threshold” of 5% of the total number of party votes cast. The other is to win a simple majority of electorate votes in an electorate. In other words, your party vote will be discarded if your party does not achieve either of those things.

But that’s not the end of it. How exactly do they allocate the seats to each party and what happens to the seats that might have corresponded to the votes of all those parties that didn’t make it?

Take this example: The Lion Party gets 40%, the Tiger Party gets 35%, the other parties that do get into parliament get 15% among them, and the parties that don’t make it get a total of 10%.

In a parliament of 120 seats, that would give the Lions 48 seats, the Tigers 42 seats and the others 18 seats between them. But that’s only 108 seats. How do they fill the remaining 12 seats?

The answer is, they don’t calculate them like that. In New Zealand we use something called the Sainte-Laguë formula, invented in 1910 by French mathematician André Sainte-Laguë. It’s very similar to an American system, invented 78 years earlier, called the Webster formula. The French, then as now, were not aware the Americans had led the way.

Essentially, under Sainte-Laguë, the Electoral Commission starts with 120 seats, looks at the votes, and sees that the Lions won the most seats, so allocates them one seat. For the next seat, it sees that the Tigers won almost as many, so it gives them one. The third seat might go to the Lions, the fourth to a minor party, the fifth to the Lions again, the sixth to the Tigers, the seventh to another minor party, and so on. With each seat, the calculation is based on deciding which party has the best claim to get that seat, in order to maintain the relativities of all the parties.

By the time they get to 120, each party will have the number of seats that correspond proportionately to its share of the vote, relative to all the other parties’ shares of the vote.

They don’t literally sit down and discuss it, of course. There’s a calculator. It’s publicly available here and you can test it by entering whatever percentages you want to look at, to see what the outcome would be.

Actual Electoral Commission calculator not shown

However, just in case you’re one of the veritable handfuls of people who find themselves near a calculator but not a computer, there’s a cheat’s way to do it that produces the same result.

Think of it this way: those 12 seats that were empty in the example above have to be shared out, pro rata, among the parties that did get into parliament. So the Lions, with 40% of the vote, get 40% of the 12 seats. The Tigers get 35% and the others get 15% to share.

For the Lions, 40% of 12 is 4.8, call it 5. That means they get 48+5 seats, 53 in all. The Tigers’ 35% is 4.2, giving them 42+4 for a total of 46. And for the other parties, 15% of 12 seats is 1.8, call it 2. That gives them 18+2 for a total of 20.

But 53+46+20 is only 119. That last seat would go to the Tigers, on a straight Sainte-Lague principle: which party is closest at this point to deserving the next seat?

So, in this parliament, the Lions, with 40% of the vote would end up with 53 seats, which is 44% of the seats. The Tigers, with 35% of the vote, would get 47. As I said, this is a cheat’s way. It won’t be exactly right in all possible circumstances. But if you tap these vote percentages (40/35/15/10) into the Electoral Commission calculator you’ll see it produces the same result. It’s a good rule-of-thumb guide.

Who forms a government? In this example, the Lions and Tigers will need to make friends with at least some of the smaller parties, the Antelopes and Hyenas and Meerkats.

But what if there was an Elephant Party that won, say, 46% of the vote? That would give it 55 seats. If the discarded votes were 10%, as above, the Elephants would get another 5.5% of the seats, which rounds to 6, for a total of 61 seats. One more than half of the 120 seats in parliament.

This is how a party with less than half the party vote can still win a majority in parliament.

But what about the overhang?

An “overhang” occurs when a party wins more electorate seats than its party vote allows for. Say there’s a Giraffe Party that wins three electorates but only 1% of the total party vote. According to that 1% party vote, it should have only 1% of the seats, which is one seat. But that doesn’t mean the Giraffes have to give up two of their seats. They won the electorates and the three winning candidates will all be MPs sitting in parliament. The system copes by adding them to the total, which means for this term of parliament it will have 122 seats.

If an overhang happens, it affects the overall percentages. Our Lions, for example, would get 40% of 122, plus 40% of the 10% vote for failed parties, which is 12.2 with rounding, that’s 49+5, total of 54 seats. One more seat than if the parliament did not have an overhang.

If the Elephants won 46% of a 122-seat parliament, they would probably end up with 62 seats, which is still over half, but it’s such a narrow advantage they will need those Meerkats or they won’t be able to govern.

Yes, it’s complicated. The Electoral Commission’s calculator makes it easy, honest. But some of us like getting out the calculator. Anyway, here’s a video of some giraffes swimming in a river.

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