[EM] Sainte-Lague vs d'Hondt for party list PR

Michael Ossipoff email9648742 at gmail.com
Sat Jul 14 16:21:25 PDT 2012

> In Finland the government is currently planning to join some districts. I believe they are planning that partially for strategic reasons (larger D'Hondt districts vs. country level proportionality). In the U.S. state splitting is probably not easy, but to my understanding gerrymandering of the smaller districts could be possible. You know better.


>>> Yes, some people might look at the s/q values
>> Some people?   But not you. :-)
> I would check the s/q values right after I have checked that each party got at least close to the number of seats that their proportion of the votes entitles them to. :-)

Do you deny that the parties would have "at least close to the number
of seats that their proportion of the votes entitles them to", to the
greatest extent possible if their s/q are equal?

Say my party has 10 percent of the votes and 10 percent of the seats.
Your party has 15 percent of the votes and 15 percent of the seats.
That's what you're referring to, correct?

Vt = total votes. St = total seats. V1 = my votes. S1 = my seats. V2 =
your votes. S2 = your seats.

Because of the correct percentages specified above, in keeping with
your goal,, :

S1/St = V1/Vt.

 S2/St = V2/Vt

For  each of the above equations, multiply both sides by St, and
divide both sides by V1 (in the first equation) or V2 (in the 2nd

You get:

S1/V1 = St/Vt

S1/V2 = St/Vt

Therefore, if you achieve the goal that you say is your goal, then the
two parties s/q are equal.

In other words, your goal is achievable only by making the s/q equal.

Your goal is another way of saying that s/q should, as nearly as
possible, be equal. Equal s/q is your goal. Same goal, different

(I use "q" as a unit measure of votes or population. Some quota. Some
quotient of votes or population divided by a common divisor. I use
that "q" in order that what I say is applicable to PR (which involves
votes) and apportionment (which involves population). But, if you
prefer, we can replace "q" with "p", standing for "person".

If you disagree, say so and say why. Tell me which part of the above
demonstration you disagree with.

If you don't disagree, then you also don't disagree that making s/q
equal would achieve your own favorite goal.

I have made three  arguments for SL over LR:

1. Above, I showed that your expressed goal is identical to making s/q equal.

2. Immediately below, I quote my argument that equal representation
for everyone means equal s/q.

If you disagree with my justification of one statement based on
previous statements, then tell me which justification you disagree
with,and why. Be specific.

3. In my most recent posting, I showed the following: Your objection
to SL, is that it puts parties' seats too far from their numbers of
Hare quotas. Your standard for rightness and fairness is that seats be
close to Hare quotas. I then showed that if you want that closeness,
then you really want SL. If you don't agree with that conclusion, then
tell which of the following you disagree with: a) One of the
justifications that I gave (in my previous posting) for a statement,
based on previous statements; or b) One of the assumptions that I
made, regarding what you like or want in an allocation--Again, be

I've posted those 3 arguments for SL vs LR. You haven't answered any
of those 3 arguments. By "answering" an argument, I mean; If you
disagree with the argument's conclusion, then you need to say exactly
what statement in the argument you disagree with, and why.

>> Remember, I tried to agree to disagree about whether people have a
>> right to equal representation. You refused to disagree about that. You
>> said that you agree that people have a right to equal representation.
>> Equal representation means the same representation for everyone. The
>> same representation for everyone means the same representation per
>> person. The same representation per person means equal s/q.
> But is that what people usually mean when they talk about equal representation?
>> So, do you or do you not agree that people have a right to equal
>> representation?.
> Equal representation, yes. But s/q is to me just one mathematical formula (out of many) that can be used to measure different properties of the results of an election.

Nonsense. Above, I showed that equal representation for everyone means
equal s/q. In that argument, I made a series of statements. If you
disagree with one of those statements, then tell us which one you
disagree with, and why.

Near the beginning of this post, I showed that giving to each party a
percentage of the seats equal to its percentage of its votes (the goal
that you espoused), can only be achieved by making the parties' s/q
equal. If you disagree with a statement in that argument, don't
hesitate to say so, and specify which statement, and why you disagree
with it.

>> What unfair use of the quotas/votes?
> Here I referred to the 8 quotas and 8 seats of one party in the bad example (no strategies assumed). People who think that n% of the votes should entitle them to n% of the seats are the ones that may find deviations from this principle unfair.

As I showed above, "people who think that n% of the votes should
entitle them to n% of the seats" should also want equal s/q.
...because your goal can't be achieved without equal s/q.

You said:

In Sainte-Laguë / Webster the problem is that in principle (not
usually in practice) it can deviate from this target not only in how
the fractional seats are allocated but by multiple seats.


No. As I showed above, your goal is achievable only by equal s/q.
That's what SL achieves. LR does not.

>>>> . In the S1+N seats case the large party gets 43.48% of the seats with 61%
>>>> of the votes. Or in other words, all 20 seats with only 12.2 quotas (7.8
>>>> extra seats), or only 20 seats with 28.06 quotas (8.06 seats too little).
>>> [endquote]
>>> That's ok if the parties are genuine, natural and not the result of
>>> splitting strategy. As I described above.

>>> I'm afraid some people might get upset if they think they were entitled to 8
>>> seats more but will get none
>> Undoubtedly. But thinking that they're entitled to 8 seats, and being
>> entitled to 8 seats aren't quite the same thing.
> In that example there was a party that got 61% of the voters. Its supporters may feel that they are entitled to approximately 61% of the seats (assuming good proportionality). The total number of seats was 46.


No,there were a total of 49 seats in Raph's example.

You said:

If we want to give them 61% of the 46 seats, that would make 28.06
seats. But they got only 20 seats. Maybe they expected 28.06 to be
rounded to 28, or maybe to 27 due to some possible additional noise in
the calculations, but not to 20. One may have different kind of
agreements on what each party is entitled to, but even without any
such agreements people might get upset with if they get only 20 seats
in a system that is supposed to be proportional.


You're forgetting about how far off the small parties' percent of the
seats would be from their percentage of the votes, if they didn't get
the seats that SL gives to them.

1. I showed that N% of the votes getting N% of the seats is achievable
only if the parties' s/q are equal. You want the seat percentage to
match the vote percentages. Therefore, you want the s/q to be as equal
as possible.

2. You're saying that you want the parties' seats to be close to their
numbers of Hare quotas. So tell me this: Do you object to making their
seats _as close as possible_ to their Hare quotas? If you object to
that, then tell why. Now, let's say that, though we prefer a certain
house-size, we don't require it.

Each party's number of Hare quotas will be somewhere between two
integers. You want the party to get a number of seats equal to one of
those two integers. I suggest let's make the party's number of seats
equal to whichever of those integers it's number of Hare quotas is
closer to. If you object to that, then tell me how it conflicts with
your desire for the number of seats to be close to the number of Hare
quotas. (Remember, we're assuming that there's a new law saying that
the house size needn't remain the preferred one that the Hare quota is
calculated from).

We've gone you one better: Not only did we give to each party a number
of seats equal to one of the bounding integers, but we gave it a
number of seats equal to whichever bounding integer it was _closer

That might give us a number of seats that is different from the
initially-preferred house size from which the Hare quota was
calculated. We've pretended that a new law says that that is
permitted, and that our only goal is your goal of making the seats
close to the number of Hare quotas.

But suppose that we want a number of seats different from the one that
we get when we carry out the instruction I described 3 paragraphs ago.

Well, we could start with a different "Hare quota", couldn't we.
Pretend that some different number of seats was the "preferred"
House-size, giving us a different "Hare quota". Choose that "preferred
House-size", and resulting "Hare quota" so that when we carry out the
instruction I described 4 paragraphs ago, we get the really desired

Surely you're not going to say that the procedure described  5
paragraphs back is only fair and proportional for some Hare quotas but
not for other Hare quotas. But if you are, say so and say why.

The procedure I've described immediately above, the one that best
attains your goal of seats close to Hare quotas, is Sainte-Lague.

> Maybe the idea of getting approximately n% of the seats with n% of the votes in a good proportional system is close to what I was thinking of.

As I've said, that goal ia achievable only by making the parties' s/q equal.

>> Dividing the parties' votes by the same divisor, any common
>> divisor, and rounding off the quotients to the nearest whole number,
>> will put the parties s/q as close as possible to the ideal equal s/q.
>> If you use the Hare quota as the divisor, for that procedure, you'll
>> often get a total number of seats different from the desired
>> house-size. So you use a different divisor. Don't be wedded to the
>> Hare quota.
>> If we allow a variable house size, then we could say: Divide each
>> party's votes by the Hare quota (based on some most preferred
>> house-size), and round off the quotients to the nearest whole number.
>> That rounded off quotient is the number of seats to assign to each
>> party. That would be a fine method.
>> But the fact that that divisor is a "Hare quota" based on some
>> preferred (but not required) house-size doesn't make it special or
>> privileged. How can you think that is somehow fairer to use that
>> divisor instead of some other divisor?
> I think that was your theory, not mine.

Wrong. It isn't a theory.

It's an argument. That's different from a theory. It's an argument
that I expressed in better detail in my next posting (the one before
this one), and then repeated above, in this posting.

Here's how it works:

I make a series of statements. I tell how each statement is justified
by previous statements,and by suppositions about what you believe.

Then it's your turn  If you want to disagree with the argument, that
means that you want to tell which of my justifications for a
statement, based on previous statements, is incorrect--but don't
forget to tell why. Or else tell which of my suppositions about what
you believe is incorrect. If the latter, then you've specified that on
which we must agree to disagree.

Mike Ossipoff

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