In party list PR, as PR in general, seats are awarded in proportion to
votes. That is to say a seat is awarded for each 1/N of the vote, in
an N-seat election. But there are different ways of dealing with fractions,
different "allocation formulas". Most are adequate, but they differ in
how closely they carry out PR's goal of equal representation per person.

I'm going to describe 3 allocation formulas: Sainte-Lague, Largest Remainder, & d'Hondt. These 3 methods were actually first proposed as ways of apportioning the U.S. House of Representatives, and they correspond, respectively, to the Webster, Hamilton, & Jefferson apportionment methods. This relationship with the apportionmnent methods is important, because it, in some cases, shows the principles upon which the PR methods are based. Also, the relative acceptance & success of these methods for apportionment may be relevant to their likely success for PR, at least in the U.S.

Where the PR & apportionment implementations differ, this discussion will be about the simpler apportionment implementation, because it's directly based on principle. Another article will show how the PR procedure derives from the simpler definition from principle.

But the methods will be referred to here by both their PR & apportionment names. I emphasize that, for example, Webster's method & the Sainte-Lague procedure give the same seat allocation; the Sainte-Lague procedure is 1 way of implementing Webster's method.

As I said, PR gives a seat for every 1/N of the total vote, in an
N-seat election. Say there are 100 seats, then parties (& independent
lists) get a seat for each 1/100 of the total vote, and independents
are seated if they have 1/100.

But suppose a party has 3.4/100 of the total vote. The obvious thing to do is to round to the nearest whole number, and give that party 3 seats. That's what Webster's method does.

The only problem occurs when sometimes, due to rounding, when this is done for each of the parties, the total seats awarded might add up to 99 or 101, instead of 100. Only rarely would it differ from 100 by more than that.

The simplest solution would be to just let there be 99 or 101 seats instead of 100. If there's concern about the cost of the 101th seat, that cost would average out over time. And the very popular German district system doesn't guarantee an exactly constant house-size anyway, regardless of whether the allocation rule does. So the simlest solution is simply to round off and leave it as-is. This is called "Simple Rounding".

But suppose we insist on awarding exactly 100 seats? Say that, using 1 seat per 1/100 of the vote, this awards only 99 seats. That 1/100 of the vote is called the "quota". In practice, it would be a certain number of votes--for instance, if the total votes is a million, then the 1/100 quota amounts to 10,000 votes. When we put each party's seats as close as possible to 1 seat per quota, by dividing each party's votes by the quota & rounding off, we got only 99 seats.

Well, we don't want to give up the idea of a uniform quota, a common ratio between seats & votes, because that's the goal of PR. So if we want to award seats based on a common ratio, or quota, and the 10,000 vote quota awards only 99 seats, then what would you suggest we do, to get another seat?

Change the quota. We lower the quota slightly, until it gives us 100 seats. We lower it till, by dividing it into each party's votes, & rounding off, to determine that party's seats, we award a total of 100 seats.

That's Webster's method.

I emphasize that the common quota is a common ratio between seats & votes, and rounding off puts each party's seats as close as possible to what that common quota calls for. The goal is equal seats per vote, resulting in equal representation per person.

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That was the quota definition of Webster's method. Webster actually worded his own definition slightly differrently, in a way that's very brief: To determine each party's seats:

Divide each party's votes by the same number & round off.

This common divisor is chosen so that the total number of seats awarded equals the desired house (or district) size.

Again, this common divisor is a common ratio between seats & votes, and rounding off puts each party's seats as close as possible to what that common ratio calls for.

Some object that the divisor definition is unclear because it orders division by an as-yet unspecified number, unlike the quota definition. But Webster's divisor definition has the best brevity.

Largest Remainder starts out like Webster's quota definition.
As in the previous example, we start out with the quota of 1/100
of the total vote, or 10,000 votes. But we only give each party
a seat for each _whole_ quota. In other words, we round each
party down. So that party with 3.4 quotas gets 3 seats because
we round it down. If it had 3.9 quotas we'd still round it down
to 3 seats. This rounding down of everyone means that we're going
to have seats left over. These seats are given to the parties
with the largest fractional parts. For instance if .9 is the largest
fraction, of the various parties, then that party with 3.9 would
get an additional seat. If there are 2 seats leftover, and the party
with 3.4, with its fraction of .4, has the 2nd biggest fraction, then
it gets the other leftover seat, giving it 4 seats. For 3.4 quotas.

Largest Remainder's advocates say it makes sense to give seats to the biggest fractions ("remainders"). But such advocates, at that point, are parting ways with their original goal, when wanting PR: equal representation per person. Largest Remainder's standard makes a lot of sense for an auction. But I showed, above, that equal representation per person, equal seats per vote, as nearly as possible is gotten by putting each party as close as possible to what is indicated by a common ratio between seats & votes.

Jefferson is like Webster, except that instead of rounding off to
the nearest whole number, it rounds down. Other than that, it's
identical to the definitions I've given for Webster's method.

While Webster/Sainte-Lague is unbiased with respect to party size, and Hamilton/Largest-Remainder is also unbiased if done right, Jefferson/d'Hondt favors large parties. For this reason it's never proposed by U.S. PR advocates. For that reason also, because it favors the large, it was rejected for apportionment.

Webster/Sainte-Lague is the optimally proportional PR allocation
rule, being directly based on a common seats/votes ratio.

If it's permissible for the district-size to vary by 1 seat sometimes, then Webster's very simplest version, simple rounding can be used. Largest Remainder advocates have claimed that Webster is less simple or less obvious than Largest Remainder when an exactly constant number of seats is necessary, and when Webster has to adjust its quota (divisor).

True, Largest Remainder is more obvious, because we're more familiar with auctions than with PR. But Webster isn't complicated, and its adjustment of the quota to slightly change the number of seats is also obvious, when we've decided that we want to come as close as possible to the results of a common quota.

Though unbiased, Largest Remainder is erratic, compared to Webster. But, though this means that it isn't really as good as Webster, in carrying out PR's purpose, Largest Remainder is still perfectly adequate. When comparing the merits of allocation rules, it's easy to lose perspective of the fact that even less-accurate PR is still PR. Some countries use Largest Remainder for PR, and nothing here is intended to imply that countries using Largest Remainder should change their allocation method.

But in a country where PR is new, a proposal for the least criticizable allocation rule might have the best chance for success. And Largest Remainder has a long history of criticism in the U.S. George Washington's 1st Presidential veto was a veto of a bill to apportion the House by Hamilton's method (Largest Remainder), because it "has no common divisor or proportion". Jefferson's method was adopted instead, but during the 19th century it bothered people that it was favoring the large states, and so another bill for Largest Remainder was passed by Congress, and this time not vetoed. But Congress soon had a problem with some paradoxical behavior of Largest Remainder: It could take a seat from a state that's gained population and give it to a state that's lost population. (of course in PR it could take a seat from a party that's gained votes & give it to a party that's lost votes). Though Largest Remainder was the law, Congress avoided it by choosing the House size so that Largest Remainder & Webster would give the same results. Around 1900 Webster's method became law.

d'Hondt's bias toward large parties is of course less pronounced in large districts. d'Hondt doesn't have Largest Remainder's paradoxical possibilities. Though Largest Remainder is preferable to d'Hondt, based on bias, and though Webster/Sainte-Lague is the ideal best it's important to emphasize that the differences between the allocation methods have less effect than district size, and that all 3 allocation methods are actually quite adequate. My advocacy of Webster/Sainte-Lague over Largest-Remainder is based on the need to minimize criticizability of a PR proposal for a country where PR is new.

Also, since this discussion only deals with party list allocation methods, it seems important to emphasize that party list PR isn't the only kind--another form of PR called "STV" is also a popular proposal.