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

Juho Laatu juho4880 at yahoo.co.uk
Sat Jul 14 19:08:33 PDT 2012

```Here are some responses to the requests and comments. I'm afraid this mail may not have much new content to others than Mike Ossipoff and me. This mail is more about the discussion than the actual content. But if you are interested, just read forward.

Juho

On 15.7.2012, at 2.21, Michael Ossipoff wrote:

>>
>> 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.
>
> [endquote]
>
>>>> 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?

s/q is one funcion that can be used to approximate the "n% of the votes, n% of the seats" principle.

>
> 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?

In some other places in my mail, yes.

>
> Vt = total votes. St = total seats. V1 = my votes. S1 = my seats. V2 =
>
> Because of the correct percentages specified above, in keeping with
>
> 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
> equation).
>
> 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
> wording.
>
> (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.

I disagree with nothing so far, except that I believe you are describing the way you think.

>
> 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.

I didn't see any links to SL and LR yet (as far as I understood, so far you said that ideal proportional representation aims at making s/q values equal).

>
> 3. In my most recent posting, I showed the following: Your objection
> to SL,

I don't object 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.

Number of Hare quotas (St*Vn/Vt) and ideal s/q describe the same ideal proportional result if we assume that the ideal state can be reached with discrete seats or if we allow fractional seats. Different methods differ mostly in how they handle cases where the ideal results can not be reached.

> I then showed that if you want that closeness,
> then you really want SL.

SL is one way to allocate the seats when one must deviate from ideal proportionality. I don't see why that approach would be ideal. (I think you discussed above only ideal proportions, not how SL deals with deviations from the ideal proportions. Only word "closeness" was maybe intended to refer to how SL hadles deviations from the ideal proportions.)

> 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
> specific.

I think you jumped to conclusion that SL must be the correct method after describing ideal proportionality (that is not SL specific) in general. You did not mention how SL or other methods deviate from the ideal proportions. I know how it works, but I think you just did not include any concrete claims about that in your arguments.

>
> 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.

Maybe you can point out those questions or requests that I did not answer (or maybe rewrite those questions) so that I can answer them.

>
>>>
>>> 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.

I think I already answered those questions. I believe we agree on what the basic formula for ideal (accurate) representation is, as expressed as s/q or as number of Hare quotas or any other way.

>
> 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.

This is the ideal case (accurate proportionality) with no roundings, deviations or anything like that.

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

Neither of those methods can guarantee equal s/q if we allocate discrete seats in the usual way.

>
>
>>>>> . 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.
>
> [endquote]
>
> No,there were a total of 49 seats in Raph's example.

I wrote about the 46 seat variant of Raph Frank's example. I generated that variant in order to point out maximum fluctuation. There is no major difference in content.

>
>
> 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.
>
> [endquote]
>
> 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.

I don't think I forgot those values. I just didn't write about the feelings of those small parties. In the 46 seat example all small parties got 1 seat with 0.69 quotas. In the quoted text I talked about the large party that "lost" 8 seats.

>
> 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.

Both cases are the same as ideal targets. Methods may differ on how they measure the "as equal as possible" criterion and how they allocate the seats based on that.

> 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 want the results be close to the ideal (fractional) value (if we want to implement accurate proportionality).

> 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.

You seem to be generating some new method here. I don't object to that.

> (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
> to_.
>
> 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
> house-size.
>
> 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.

That's an interesting approach. Let's see how this will be linked to the earlier discussion.

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

I don't think I described yet in what specific way I want the nuber of seats to be close to ideal proportionality.

>
>> 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.

That's the ideal proportionality case. I used word "approximately" above to refer to possible deviations from this ideal.

>
>>
>>> 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.

Yes, those are good descritions on how systematic discussion may proceed.

>
> Mike Ossipoff

```