# 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*/(2*N* − *k* + 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(2*N* − k)/7 + 1). More generally, if the first divisor is changed to *m*, then the effective threshold is *V*/((2*N* − *k*)/*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*/2*N*. 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*/2*N*. 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.

**Proof**