Referring to the divisor definition of Webster [link to that definition] suppose that we start with a divisor so large that no party gets a seat. Then let's start lowering the divisor so that parties will begin getting seats. As we keep lowering the divisor, more parties will get seats, more seats will be awarded. The object is to lower the divisor till (if 100 seats are desired) 100 seats are awarded.
So start with a divisor so large that no party gets a seat. As we lower the divisor, some party will get the 1st seat. Which party? This depends on which party can get a seat with the largest divisor. When every party has 0 seats, we want a party to qualify for 1 seat instead of 0. This means that, when we divide its votes by the divisor, we want to get .5, since that can be rounded off to 1 instead of 0.
For a particular party, what divisor, when divided into its votes, will give us .5? Find out by dividing .5 into that party's votes. This will tell you what divisor will round that party up to 1. So we do this for each party, dividing its votes by .5 The party with the largest result of that division is the party which will be the 1st one rounded up as the divisor is lowered. It gets the 1st seat.
We keep doing this: Say a party has 1 seat. It qualifies for 2 seats if dividing its votes by the divisor results in 1.5 What divisor, divided into its votes, will result in 1.5? We find out by dividing 1.5 into its votes.
So, in general: Divide each party's votes by .5 more than its current number of seats. We can call this division result that party's "score". The party with largest score gets the next seat. Then we again divide each party's votes by .5 more than its current number of seats, to determine which party gets the next seat, etc.
To clarify where it comes from, I've written that wordily. Here's a briefly stated electoral law for this systematic procedure for Webster:
All parties initially have 0 seats. Repeatedly, each party's votes are divided by .5 more than its current number of seats, and the party with largest division-result receives the next seat. This procedure is repeated till the desired number of seats has been awarded. The advantage of this systematic procedure rule is that it avoids the trial-&-error that would be needed to carry out Webster's method, directly from Webster's definition. The countries using Sainte-Lague must have felt that this is important, because they use a systematic procedure.
Actually, though, the systematic procedure that they use differs slightly from the above one: If you double all those numbers ending in .5, by which you divide the parties' votes, you get odd numbers. _That_ is the version of the Webster systematic procedure that is actually used in Sainte-Lague countries. It's the procedure that's given in books, under the name "Sainte-Lague".
The reason, most likely, why Sainte-Lague, as used, uses odd numbers instead of the numbers ending in .5, is that, in pre-computer days, doubling those numbers ending in .5, to get odd numbers, would reduce the number of digits, and get rid of fractions, which would make a non-computerized calculation easier.
But the way I worded it before makes for a much briefer & clearer wording. The odd-numbers procedure could be worded:
"Repeatedly, each party's votes are divided by the odd number equal to one more than twice that party's current number of seats, and the party with the highest division result gets the next seat. This is repeated till the desired number of seats have been awarded"
Obviously a more complicted wording.
The _World Atlas of Elecions_, available in most university libraries, refers to the above procedure as "pure Sainte-Lague". Another version of this is referred to in that book as "Modified Sainte-Lague". Modified Sainte-Lague has a provision to make it more difficult for a small partyk to get its 1st seat. Though some proportionalists are proposing pure Sainte-Lague, no one is proposing modified Sainte-Lague. But it's worth mentioning because some of the Sainte-Lague countries use it:
Where pure Sainte-Lague (odd numbers procedure), when a party has 0 seats, would divide that party's votes by the 1st odd number, 1, modified Sainte-Lague instead divides by 1.4, just to make it more difficult to get a 1st seat, in order to discourage small parties.
But if one isn't using a computer, that can amount to a lot of work, actually using that procedure. There's a quicker way for doing a systematic procedure without a computer:
You'd start as in the quota definition of Webster: If there are 100 seats to be allocated, then the 1st quota is 1/100 of the total vote. So you determine how many votes this quota is, and divide it into each party's votes, to determine how many quotas it has.
If, when you round each party to the nearest whole number and give it that number of seats, let's say this only awards 99 seats:
We have to award one more seat, so we simply do what we did before: Divide each party's votes by .5 more than its current number of seats to find out which party would be the 1st to round up if we were to begin lowering the quota. We give the 100th seat to the party with largest division-result.
Of course it we instead had 98 as the initial number of seats, instead of 99, we'd simply have to repeat the procedure, as described before.
But what if the initial quota awards 101 seats. Then we have to find out which party should lose a seat. An argument analogous to the previous one, this time involving raising the quota until a party loses a seat, leads to the following: Divide each party's votes by .5 less than its current number of seats. The party with lowest division-result is the one that loses a seat.
Why this is--if a party has 3 seats, we want to find out what divior or quota it would take to round it down to 2. So we want to find out what divisor, when divided into that party's votes, would give 2.5 And we find that out by dividing 2.5 into the party's votes. The 1st party to lose a seat as we raise the divisor will be the one with the lowest result of dividing its votes by .5 less than its current number of seats.
This is just essentially the same as the previous arguments, except that it's about finding out who should lose a seat, when the initial quota has awarded too many.
I emphasize that all these procedures (Websters definition, *The systematic procedure starting with 0 seats, dividing by numbers ending in .5 * The systematic procedure dividing by odd numbers, with all parties initially having 0 seats * The systematic procedure I've just described)--All these procedures give the same identical seat allocation.
It may be that a sytematic procdeure makes for a better electoral law, but Webster's definition, either his divisor definition, or the quota definition, is the one that is based directly on the principle of putting each party's seats as close as possible to what is called-for by a common ratio between seats & votes.
Most books on PR give, for Sainte-Lague, the odd numbers formula, without any word on where it comes from, leaving the impression that Sainte-Lague is an arbitrary unexplained procedure. I've approached Sainte-Lague from principle, deriving the systematic procedures from the definition based on principle.
An argument analogous to the ones used above leads to the following
systematic procedure for Jefferson's method:
All parties initially have 0 seats. Repeatedly divide each party's
by 1 more than its current number of seats, and give the next seat
to the party with the largest division-result. Repeat this till
the desired number of seats has been awarded.
All parties initially have 0 seats. Repeatedly divide each party's by 1 more than its current number of seats, and give the next seat to the party with the largest division-result. Repeat this till the desired number of seats has been awarded.
The reason why it's "1 more than the current number of seats, instead of ".5 more than the current number of seats" is that with Jefferson there's no such thing as rounding up, and so to for a party with 2 seats to qualify instead for 3 seats, the result of dividing its votes by the common divisor has to actually equal 3. But the argument leading to the above systematic procedure for Jefferson's method is the same as that for Webster, except for that one difference in rounding.
As I said, this systematic procedure for Jefferson's method is called "d'Hondt". Most countries use it in the form I've written, although it has also been used in an implementation similar to the quicker systematic procedure I described for Webster, for when a computer isn't being used. In that implementation, the parties aren't started with 0 seats, but rather a 1st quota is used, to get an initial result closer to the desired result. The country that was doing this is no longer using d'Hondt, and (as of 1995) it may be that no country is using this quick implementation.