[EM] Sainte-Laguë/Webster v Huntington-Hill

Sun Apr 7 11:21:15 PDT 2024

 Since Michael Ossipoff has moved the argument to here from the Voting Theory Forum, I think I have a right of reply (whether he personally reads it or not), and will try to make this post mostly about the actual discussion point rather than arguing about the argument (though I will mention some things for context). Michael said he was done with the conversation previously, but in his post he'd put things in public that I felt needed rebutting, so it was reasonable for me to reply for the benefit of anyone else reading, if nothing else.
With vote-for-one party-list PR (equivalently apportionment, but I'll speak in terms of PR), you have exact proportionality when the number of seats a party has divided by its number of voters is the same for every party. We can say that an individual voter's representation level is s/v, where s is the number of seats their party has won and v the number of voters of that party. A higher s/v is better for a voter. Perfect PR is when s/v is the same for everyone. This was not in dispute.
Sainte-Laguë (equivalently Webster for apportionment) elects the set of candidates that minimises the variance of s/v (looking at absolute differences). This variance of s/v is what I consider to be the most appropriate measure of disproportionality. Huntington-Hill minimises the ratio of voters' representation from exact PR.

I would argue that it makes most sense to use the arithmetic variance because it is essentially an arithmetic distribution. If you total s/v for all voters, then you get the same number (s) under any result. If you were instead looking at v/s, you would use harmonic statistics. If you were using data that multiplied to a set number, you would use geometric statistics. And so on.
And because Huntington-Hill uses geometric statistics for arithmetic data, it breaks if any party gets 0 seats (it is important to note that this is a causal thing, not a coincidence). If, for example, the number of seats was equal to the number of parties with at least one vote, Huntington-Hill must award each party exactly one seat each (even if one party has 99% of the vote and several have just one vote). Michael questioned whether breaking in one case made Huntington-Hill the wrong measure in general, and I answered yes. If we want something to be mathematically correct, it should work in all cases. It might give good results in many cases, but you can't trust it to reliably do so, and to be a contender for objectively correct, it must work in all cases. Analogously, something is not a law of physics if it breaks under certain conditions.
These were the main arguments I put in favour of Sainte-Laguë over Huntington-Hill. I also pointed out that in "Fair Representation: meeting the ideal of one man", Balinski and Peyton Young conclude that Webster (equivalently Sainte-Laguë) is "the unique unbiased divisor method". Michael simply doubted that they said that, but it was indeed a quote. I later expanded on the quote (which I found on Google books online - https://www.google.co.uk/books/edition/_/ZsSHDwAAQBAJ?hl=en&gbpv=1 ) "Webster's is the unique unbiased divisor method. It seems amazing therefore that Hill's method could have been chosen in 1941 on precisely the ground that it was the unbiased method, and that Webster's method was discarded." This is a quote, not a paraphrasing.
Michael mentioned Huntington's paper to me, yes. He did not provide a reference or quote, and upon finding it myself I had a look through it and did not find anything that challenged the positive arguments that I had put forward in favour of Sainte-Laguë. It should also be pointed out that Michael at no point made any real attempt to argue against my specific points, or indeed address the Balinski and Peyton Young quote and I did not block him for that. It is also poor form in an argument to simply say "I am posting this paper as my argument. You must rebut that to win", especially without highlighting the specific, relevant parts of the paper. In any case, when pressed, I went back to the paper (available here: https://www.ams.org/journals/tran/1928-030-01/S0002-9947-1928-1501423-0/S0002-9947-1928-1501423-0.pdf - The Apportionment of Representatives in Congress by E.V. Huntington) and found what I considered to be the crux of the argument:
"The rather vague concept of the inequality between two states is thus reduced to the more definite concept of the inequality between two numbers.
The question then comes down to this: what shall be meant by the inequality between these two numbers? Shall we mean the absolute difference between the two numbers, or the relative difference between them? If the size of the congressional districts is large, say 250,000 in one state and 250,005 in the other, then the difference of five people is of little consequence in so large a number. But if the districts were themselves very small, say 10 and 15, then the same difference of five people becomes important; 15, we say, is larger than 10 by fifty per cent, while 250,005 is larger than 250,000 by only (1/500) th of 1 per cent. In the present problem it is clearly the relative or percentage difference, rather than the mere absolute difference, which is significant."
I pointed out in reply to this that any (sensible) method will attempt to give the district with 15 people 50% more representatives than the one with 10 and the one with 250,005 a fraction of a per cent more than the one with 250,000. It doesn't take a method that specifically looks at percentage differences in individual representation to do that. We would need to look at how methods actually behaved. But I also pointed out (perhaps more relevantly) that the variance we are looking at is the variance of s/v, not variance of total number of seats, and therefore the big numbers simply cancel out and we are dealing with numbers of the same order of magnitude as each other. If you look at s/v and use ratios as well, you are double-compensating. If there's something else Michael wants me to refute, it's up to him to explicitly point it out.
That's largely it. But I should also point out for context that this was initially a discussion about Michael's "Bias-Free" method. I argued that while it might be unbiased under a certain definition of bias and under certain assumptions, it is biased towards small parties when using the most sensible measure of proportionality (the Sainte-Laguë measure). This is where Huntington-Hill came in, as Michael argued that Sainte-Laguë wasn't the only sensible measure. However, he went further and argued that Huntington-Hill made more sense than Sainte-Laguë. Michael did also separately say that Sainte-Laguë is nearly unbiased by his measure, and Huntington-Hill is twice as biased as Sainte-Laguë, so I'm not sure how that squares with Huntington-Hill making more sense as a method.
Anyway, Michael might or might not read this, but maybe I should wear it as a badge of honour that I'm the only member on here that he won't unblock.
Toby

    On Sunday, 7 April 2024 at 03:26:42 BST, Michael Ossipoff <email9648742 at gmail.com> wrote:  

I'm reading from the Spam-Folder to find out what nominations I've missed from blocked individuaos.
But there's no guarantee that I'll find everything that I missed, & so it would be great if someone would post a few additional updates of the nominations-list, at least until I get the remaining 2 persons( for a total of 3 of the 4) provisionally unblocked.
On Sat, Apr 6, 2024 at 3:15 AM Toby Pereira <tdp201b at yahoo.co.uk> wrote:

 I notice you haven't included my nominations. I'm guessing you blocked me after you threw your toys out of the pram over your "Bias Free" method on the Voting Theory Forum. I

I told you that I was done with that conversation, & then you posted "I know that Michael said he was done with this conversation, but..."
...& then proceeded to continue your repetition of your unusual personal definition of bias & your interpretation of "most proportional"., without saying where Professor Huntington went wrong with his justification of ratio as a measure of variation in s/v. You said that you found Huntington's paper, but evidently either didn't read it or didn't understand it. You didn't share with us what you believe was the error in his argument, or why your interpretation is better than his.
I certainly had the right to block you instead of continuing to receive your messages with your repetition. You didn't give me a choice, when you continued it.

 notice Kristofer suggested some methods that aren't in the list either. I'm not sure this can really count as a poll of EM members if this person has decided to selectively ignore people.

I haven't decided to ignore anyone. I've requested that nominations from the 4 blocked individuals be re-posted by someone else. Additionally, I've already provisionally unblocked Garman, & will next unblock Kristofer & Bristow, provisionally.
Anyway, there's no need to say that anyone is "running" the poll. The nominations count, & are distributed to all of us. Someone other than me should post a few more nomination-list updates, because obviously mine can't be reliably-complete until I've unblocked the 2 remaining persons whom I intend to provisionally unblock, for the purpose of my participation in the poll 

Toby

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