Talk:Highest averages method

2003 comments

First, I can't tell you how happy that user:Ruhrjung added this page, and the sainte-lague one, and that we've had all these people working on these pages. I'm going to suggest that we replace the example with the following made-up one: apportioning seats to states for an 8-seat council of New England states. The reasons I'd want to make the change are: 1) the "preferences" (in this case, population) aren't made up, so nobody has to account for "is this a realistic spread of preferences or is this tailored", and 2) it emphasises the difference between d'H and S-L (that's why I chose not to use modified S-L).

However, I'm not going to be bold because Ruhrjung clearly put in a lot of work to make the existing example (whose table structure I have copied) and I don't want to offend him. Is this okay? Which example do you think is better?

	d'Hondt method							Sainte-Laguë method
States	MA	CT	ME	NH	RI	VT		MA	CT	ME	NH	RI	VT
Pop. (1000s)	6,349	3,405	1,274	1,235	1,048	608		6,349	3,405	1,274	1,235	1,048	608
mandate	quotient
1	6,349	3,405	1,274	1,235	1,048	608		6,349	3,405	1,274	1,235	1,048	608
2	3,175	1,703	637					2,116	1,135	425	412
3	2,116	1,135						1,270	681
4	1,587							907
5	1,270
seat	seat allocation
1	6,349							6,349
2		3,405							3,405
3	3,175							2,116
4	2,116									1,274
5		1,703						1,270
6	1,587										1,235
7			1,274						1,135
8	1,270											1,048

Also, whichever example we use, I want to use and explain a table like this:

	d'Hondt method							Sainte-Laguë method
States	MA	CT	ME	NH	RI	VT		MA	CT	ME	NH	RI	VT
Seats	5	2	1	0	0	0		3	2	1	1	1	0
Ratio	3.65	1.96	0.73	0.71	0.60	0.35		3.65	1.96	0.73	0.71	0.60	0.35
Diff.	+1.35	+0.04	+0.27	-0.71	-0.60	-0.35		-0.65	+0.04	+0.27	+0.29	+0.40	-0.35

• Seats=seats allocated under this system
• Ratio=Total Seats to allocate*State Population/N.E. Population
• Diff.= Seats-Ratio

-- User:DanKeshet

Go along. I'm busy at work for the next three days - probably without any time to "recover" in front of the computer.

Your proposed addition is quite in line with my thoughts on what's relevant and interesting. I had a thought with the made up number of votes, namely that they make percentage-comparisons easy for the untrained, as the total number of casted votes was set to 100.000. And as quite a few elections in the last 30 years (i.e. in my lifetime:-) has been done with these methods, it never occured to me that one of them ought to be taken as an example. By making up the numbers, the which-and-why question was avoided. :-) My idea was to add rows to the tables with approximately the following information:

const.                     result in percentage 
size         d'Hondt                              Sainte-Laguë
         party 1    party 2,   party 3...       party 1,  party 2,  party 3...
 1       100% (1)                               100% (1)
 2       100% (2)                               100% (2)
 3        67% (2)    33% (1)                     67% (2)   33% (1)
 4        50% (2)    25% (1)    25% (1)          50% (2)   25% (1)   25% (1)
 5
 6
 7
 8      
 9
10

I had further had the idea to have two sections in the article with two headings: One for comparing the unmodified Sainte-Laguë method, and one (as I had started) for the method with the first divisor set equal to 1.4 - and how unlikely it might now ever seem, I had actually finished that work when my computer locked, and the work was lost, ...and I soured.

Do as you like with the figures. I wouldn't advice you to use any New England example, as the whole wikipedia project already as it is is pretty much US-centered, which isn't always a good thing, and as it might seem odd to list half-a-dozen of non-US countries where the method is in uncontroversial use, and then give an example from USA. However, I'm glad someone more than me has found it relevant to work on these articles, and I'm sure the end result will become pretty good whatever you choose to do of your New England idea.

best regards!
-- Ruhrjung 17:52 28 Jul 2003 (UTC)

---

Uf! I'm sorry to hear about your computer lockup! I look forward to the text being regenerated. So, I agree about the US-centrism, and I'll drop the New England example, but I still think it's better to use a "real-life" made-up example (EU? AU?) than an out-of-the-blue made-up example.

By the way, do you know about Wikipedia:WikiProject Voting Systems? You don't need to know anything about the project in order to participate, but if we want project-wide standards, then that could be a place we work them out.

See you,

DanKeshet 04:35, 29 Jul 2003 (UTC)

One more step taken, on the outlined road, but the weather is nice, and the summer short up here in the North, why I refuse to hide indoors more. :))
-- Ruhrjung 09:05, 5 Aug 2003 (UTC)

No worries. Relax, take your time, it will still be here when you get back. I have added a section on the other way of conceiving of the highest averages methods. I understand that nobody who didn't already understand what I wrote will probably gain an understanding, but I meant it as starter text which can be ruthlessly rewritten until it actually explains the reasoning behind the procedures. DanKeshet 17:49, 6 Aug 2003 (UTC)

I saw that. The colour is nice. Your attempt to explain the method as if it was largest remainder method is maybe not quite simple to understand, that's true, but if so, it can surely be mended in due time. I'm for instance pretty fond of the following sparse wording, quoted from http://www.barnsdle.demon.co.uk/vote/appor.html:

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.

-- Ruhrjung 13:58, 7 Aug 2003 (UTC)

Would it be useful to mention minor methods like Hill's method? (Hill's method is used for assigning representatives to US states, and has the property that every "party" gets at least one representative.) Rob Speer 08:04, Jul 17, 2004 (UTC)

why?

Im having trouble understanding the exact rationale behind these methods. Why do seemingly provisionaly choosen divisors result in a proportional result? Also, why are such complicated ways of allocation used at all; it would seem that simply dividing the number of votes each party won with the total votes and multiplying this with the number of available seets, rounded down, and then largest remaining fractions up to the remaining number of seets chosen (if you write this out, it is mathematically equivalent to Hare quota) would by definition result in the most proportional results, and its the most intuitive - mathematically most evident - way of doing this. --195.29.116.30 04:54, 19 July 2006 (UTC)

• Thaat is a largest remainder method and is not mathematically equivalent.--Rumping (talk) 10:17, 5 June 2009 (UTC)

You’re describing Largest-Remainder. Its departures from proportionality are universally agreed to be much greater than those of Webster/Sainte-Lague. …& that’s not even counting Largest-Remainder’s nonmonoticity (negative response).

But you started right: X% of the vote should get X% of the seats. So yes, multiply each party’s vote % by the desired house-size, to get the parties’ “ Rightful seats”.

Rightful seats typically ends in a fraction, so round to the nearest whole number. Assign to the party that number of seats.

Due to vagaries of rounding, the total assigned seats might not sum to the desired house-size. No problem:

Before rounding, multiply each party’s rightful-seats by the same number (the “multiplier”).

If the total allocation was too low, the multiplier should be a bit greater than 1. If the total allocation was too high, the multiplier should be a by less then 1.

Then, as before, round to the nearest whole number, & give the party that number of seats.

So, the house-size is adjusted by multiplying each party’s rightful seats by some same number before rounding.

By trial-&-error, find the multiplier that gives the desired house-size.

That gives the Webster/Sainte-Lague allocation.

If, instead of rounding to the nearest whole number, you round everyone up, that’s Adams.

…or if you round everyone down, Jefferson/d’Hondt.

2600:6C55:7900:2B8:5514:AAA9:18DB:87CB (talk) 22:03, 23 December 2024 (UTC)

For the (necessary) SL modification, the 1st rounding-point is changed from.5 to .7 2600:6C55:7900:2B8:5514:AAA9:18DB:87CB (talk) 22:15, 23 December 2024 (UTC)

____________

Let’s suppose those figures:

A: 340,000 B: 280,000 C: 160,000 D: 60,000 E: 15,000

The result using Sainte-Laguë method would be: 3, 2, 1, 1

If we use D'Hond method, then: 2, 2, 2, 1.

It seems to contradict the idea of a 'more proportional' system.

Thanks. —Preceding unsigned comment added by 81.35.196.9 (talk) 15:03, 13 November 2008 (UTC)

• You may have made an error with your D'Hondt calculations: I make it 3,3,1,0 as 340000/3 > 280000/3 > 160000/2 > 60000/1. As for which is more proportional, I have my doubts about any system which does not guarantee that a party with more than half the votes will win at least half the seats. --Rumping (talk) 10:17, 5 June 2009 (UTC)

Modified S-L

The table calculates modified S-L with 1.4, 3.4, 5.4,.... This results in a top row with a divisor of 1.4, which is incomparable with all the other tables. Why not use the mathematically equivalent series 1, 2 3/7 (2.429..), 3 6/7,...? This would lead to more comparable numbers (although it would take a footnote to explain it.) 187.143.7.74 (talk) 14:49, 12 February 2010 (UTC)

No, it doesn't. Modified S-L (and the table) uses 1.4, 3, 5, .... --Roentgenium111 (talk) 18:05, 24 January 2013 (UTC)

The table has been modified to allow for direct comparisons across rows, by using 0.5, 1.5... for Sainte-Lague (which is the "more correct" fencepost sequence; using 1, 3, ... is just a convenience to avoid dealing with fractions). Closed Limelike Curves (talk) 21:02, 2 February 2024 (UTC)

Sainte-Laguë and splits

I have radically changed the part which previously said: "D'Hondt favors the merging of parties, while Sainte-Laguë favours neither merging nor splitting parties which expect to gain more than 1 or 2 seats. (It does favor splitting of very small parties – expecting to gain only 1-2 seats – into still smaller ones). Modified Saint-Laguë prevents this splitting advantage for small parties, while remaining impartial towards party size for all larger parties" on the grounds that it is not true. Just to illustrate from the current example, if the Yellow party split its 47,000 votes into 6 new parties with about 7,800 votes each, it would win 6 seats in both unmodified Sainte-Laguë and modified Sainte-Laguë (with unmodified Sainte-Laguë it might even successfully do a split into 7 parties with 6,700 votes each so going from 4 to 7 seats). The reality is that Sainte-Laguë often rewards splits, even for large parties, rather like the Hare quotas in Hong Kong. --Rumping (talk) 16:45, 22 May 2015 (UTC)

Fill gaps from the German Wikipedia

For missing references and other useful material (including mathematical properties of the apportionments achieved and other algorithms always yielding the same apportionment) see http://de.wikipedia.org/wiki/Sitzzuteilungsverfahren . -- Wegner8 07:13, 19 October 2017 (UTC) — Preceding unsigned comment added by Wegner8 (talk • contribs)

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is huntington-hill example correct

Hi

not sure on this but I'm not sure the huntington hill example is correct.

because the divisor is n(n+1) the first number in the series is infinity - hence all who meet the threshold get at least one seat. so the first round should both eliminate two of the parties, and assign four seats to the other parties leaving six to be allocated. these will go four to yellow, one to white, one to red, giving overall allocation of 5,2,2,1. In this example it is the same seat allocation as d'hondt and modified S-L.

--Dirtyrottenscoundrel (talk) 09:32, 23 October 2020 (UTC) Dirtyrottenscoundrel (talk) 09:32, 23 October 2020 (UTC)

The example was incorrect and has been corrected. Closed Limelike Curves (talk) 20:58, 2 February 2024 (UTC)

Unexplained deletion of content

@Closed Limelike Curves you are removing the Imperiali quota (post(k) = k + 2) from the divisor table (and elsewhere) without explanation in your edit comments. Please explain that here. Thank you —Quantling (talk | contribs) 20:52, 26 January 2024 (UTC)

Did I? I'll fix that; I dropped it from the table intentionally because it's:

1. Not technically a proper divisor method (signposts must be k+1) at most, and

2. Subsumed by the stationary divisor methods.

But I'll add a comment describing it as part of the stationary family of divisor methods. Closed Limelike Curves (talk) 21:00, 26 January 2024 (UTC)

1. Yes, I've seen that you changed the text to support that "Not technically a proper divisor method (signposts must be k+1) at most", but what is your source for this restriction of at most k + 1? The restriction appears false in light of the existence of the Imperiali quota.
2. Yes, but especially because it has a name, we should include it, yes? As in, we don't include k + π because no one uses it!

—Quantling (talk | contribs) 21:06, 26 January 2024 (UTC)

Seeing no objections, I am going to restore equal footing to the Imperiali quota, including the possibility that post(k) could be greater than k + 1. —Quantling (talk | contribs) 14:48, 31 January 2024 (UTC)

The restriction is provided in Balinski and Young, as well as Pukelsheim (the two texts that form the basis for this article). fenceposts that violate the restriction k <= post(k) <= k+1 fail the exactness/idempotence axiom of proportional representation systems--a proportional representation algorithm M*(p) must be idempotent, i.e. M*(p) = M*(M*(p)). In other words, every whole number must round to itself. Closed Limelike Curves (talk) 20:58, 2 February 2024 (UTC)

Best names for the discussed methods

@Closed Limelike Curves you've renamed several of the methods discussed so that the wikilinks take us to pages that are named differently. As evidenced by the choice of the names of these linked articles, the new names appear to be inferior to the names that they have replaced. But likely you have a reason for preferring the names that you are using ... what is it? Thank you —Quantling (talk | contribs) 21:11, 26 January 2024 (UTC)

I’m not super wedded to either set of names, but my reasoning in general was that:

1. I’d prefer to be consistent in using one set of names (American) or the other (European) throughout this article; unfortunately it’s a bit more difficult to try and make the European names fit. Adams’ method is only occasionally associated with Cambridge; the Dean and Hill methods and have no European names whatsoever.
2. As a rule of thumb, I think it’s reasonable to go by priority (and Jefferson/Webster have priority, as they came up with the methods first).
3. The English-language literature on this seems to lean towards using the American names more often.

Closed Limelike Curves (talk) 00:30, 27 January 2024 (UTC)

The European names are more well-known. In addition, in English-language literature, the American names are more likely to be encountered in older articles (especially ones that deal with U.S. congressional apportionment). In the talk page for Sainte-Laguë method, a user noted that in literature published since 2015, the European name "Sainte-Laguë" was used nearly three times more than the American name "Webster". Glide08 (talk) 16:29, 27 October 2024 (UTC)

Allows zeros

@Closed Limelike Curves et al: The new column in the table "Allows zeros" might need some re-thinking. Any of the methods that gives post(k=0) = 0 can be said to not allow zeros in that every alternative will get one representative before any alternative gets a second alternative. However, if giving one representative to every alternative makes too many representatives then a floor number of votes is established and only those alternatives with that many votes will get a representative. In this case, some alternatives will get zero representatives even though "Allows zeros" is listed as "no". —Quantling (talk | contribs) 21:26, 26 January 2024 (UTC)

Seeing no objections, I am going to remove the column "Allows zeros" from the table, so that it does not mislead the reader into believing that some systems disallow zeros in all cases. —Quantling (talk | contribs) 14:50, 31 January 2024 (UTC)

@Closed Limelike Curves:. Your recent edit to restore the "Allows zeros" column to the table didn't quite work / appears incomplete. I look forward to seeing your ideas for this. Thank you —Quantling (talk | contribs) 14:31, 2 February 2024 (UTC)

Corrected, thanks. Closed Limelike Curves (talk) 20:54, 2 February 2024 (UTC)

GA Review

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

This review is transcluded from Talk:Highest averages method/GA1. The edit link for this section can be used to add comments to the review.

Nominator: Closed Limelike Curves (talk · contribs) 02:34, 10 May 2024 (UTC)

Reviewer: Chiswick Chap (talk · contribs) 11:09, 30 June 2024 (UTC)

Comments

This mature article is very carefully written and fully cited. Copyvios look extremely unlikely. The examples are well chosen to illustrate the differences between the methods. The technicalities are presented simply and clearly.

• The lead is fully-cited. This is not required at GAN, and is disliked by many reviewers. The main reason for citing the lead is where an article is controversial and the claims in the lead have been challenged repeatedly. If that's not the case here (it doesn't seem so) and there isn't any special reason for having them up here, then I'd suggest moving the refs out of the lead.

• The presence of Thomas Jefferson, John Quincy Adams, and [Daniel] Webster (for some reason his forename is never mentioned: it should be) indicates both that there is a lengthy history to this subject, and that it is apparently mainly American. It would be helpful, and arguably essential, to provide a 'Context' or 'History' (or 'Historical context') which summarizes (in a few sentences) the political background to the creation of the method. The bare bones to be described and cited are that the US was created as a democracy; Jefferson was a founding father; it was felt necessary to have a proportional system to allocate seats in each state separately.

• It would be nice if the History could be somewhat internationalized, if (and this is a question) there are instances of methods or suggestions for them from countries other than the US.

• The History just described could well be illustrated with an image of Jefferson, if not of all three men mentioned; and perhaps with a political map of the congressional constituencies in Jefferson's time (showing where the seats were, i.e. the things to be allocated by the method). The mention courts have ruled that the choice between the two constitutes a political question and matter of opinion. could be slightly expanded to mention the state courts involved, and the US Supreme Court's ruling that PR is constitutional.

• A minor point is that the word "method" is used both for the whole article's subject and for each subsidiary procedure. Perhaps one might distinguish 'family of methods' and 'method', or something of that sort. We might need to rename the article (after the GAN).

Images

None. This seems a slight pity as it ought to be possible to find examples where a method has had a dramatic effect on real parties in a real country, which could be illustrated both with (real) data and with a photograph, but this is outwith the GA criteria. See also the comment about History above, which might be the best place for such illustrations.

Sources

• I note with slight concern that 9 of the 28 sources cited are by the same author, Friedrich Pukelsheim: that the first of these is referenced 21 times; and that all of these are from the same edited book. It does seem however that the scholar is well-known and widely-cited; and that the topic would still be unquestionably notable without him.

• [21] is correctly marked as a dead link. It has been archived (e.g. on 28 August 2016), so the ref should be formatted using the 'cite web' template and its three archive parameters.

• [3] is a book, and needs page number[s] for two of its instances.

• [6] is not properly formatted or completely cited.

• [9] is a book chapter; it is incorrectly and incompletely cited.

• Dates are shown as 'April 2008', 'July 10, 2015', and '1979-10-01', i.e. three different systems. Please choose one. Personally I'd avoid the last of these, as the format is ambiguous.

• Spot-checks are all fine.

Summary

There is very little wrong with this article and I hope to see it as a GA very soon. Chiswick Chap (talk) 12:40, 30 June 2024 (UTC)

I believe the sources are fixed (let me know if I've messed any up). I think it'd be very interesting to see a "history of apportionment methods" article, but I feel like this one is getting too long for a substantial digression into the topic. Closed Limelike Curves (talk) 06:14, 3 July 2024 (UTC)

Many thanks for fixing the sources. I'm not asking for a history article or "substantial digression", nor is the article especially long (under 53kbytes) as Wikipedia articles go. All that's needed here is a few sentences of historical context to meet WP:GACR, specifically

3a. "it addresses the main aspects of the topic".

Closed Limelike Curves: It is not the case, as it feels as if you are implying, that mathematical logic is more important or more encyclopedic than the history of mathematics. The history from 1792 onwards is certainly one of the "main aspects" of the highest averages methods topic, along with the connection of the mathematics to the real-world matter of proportional representation in a legislature. The history need only be brief, as its function is to set the mathematics in its historical, political, and real-world context. Such context is not a "digression" but an essential component of the article. Chiswick Chap (talk) 09:56, 3 July 2024 (UTC)

Alright, I added a brief section.

It's not that I feel like the history of this topic isn't important; I just think it's a big enough topic to deserve its own article. Or, alternatively, it could get a long discussion in Apportionment (politics). Otherwise, there will be a lot of duplication between all the apportionment articles. Closed Limelike Curves (talk) 16:20, 5 July 2024 (UTC)

Many thanks. When that article is written we can have a 'main' link here. Good work, it's a GA. Chiswick Chap (talk) 17:04, 5 July 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: rejected by reviewer, closed by SL93 talk 23:21, 19 August 2024 (UTC)

(

• Comment or view
• Article history

)

• ... that in 1840, Congress attempted to legislate 40.5 rounds up to 42?

• Source: https://archive.org/details/fairrepresentati00bali

• ALT0a: ... that in 1840, Congress passed a bill with the effect of rounding 40.5 up to 42?
• ALT1: ... that the 1876 United States presidential election was decided by Congress's failure to use the correct rounding procedure? Source: https://archive.org/details/fairrepresentati00bali
• ALT1a: ... that the 1876 presidential election was decided by rounding errors? Source: https://archive.org/details/fairrepresentati00bali
• ALT1b: "... that the result of the 1876 presidential election was decided in part by the algorithm used to handle rounding?
• Reviewed:

Improved to Good Article status by Closed Limelike Curves (talk). Number of QPQs required: 0. Nominator has less than 5 past nominations.

Closed Limelike Curves (talk) 17:27, 10 July 2024 (UTC).

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

QPQ: None required.

Overall: no concerns; it's a new GA. prefer ALT1 as clearer and hook-ier to a general audience. good work! ... sawyer * he/they * talk 01:24, 18 July 2024 (UTC)

• @Closed Limelike Curves, Sawyer777, AirshipJungleman29, Black Kite, and David Eppstein: per the discussion at DYK's noticeboard and the fact that this is up in a day or two, I've pulled this for now. Discussion should continue here :) theleekycauldron (talk • she/her) 08:35, 31 July 2024 (UTC)
• @Closed Limelike Curves: I have added your amended Alt1 as Alt1a as the original Alt1 has already been discussed. TSventon (talk) 15:21, 31 July 2024 (UTC)
  • Sounds good, thanks! Closed Limelike Curves (talk) 18:14, 1 August 2024 (UTC)
• I think all the hooks are misleading, even ALT0. There are many systems of rounding in apportionment. All must produce rounding errors, in the sense that rounded numbers rather than exact numbers are a necessary outcome of the process. Therefore, rounding errors are not a cause of anything. Different apportionment procedures have different priorities and it is an inappropriate editorialization to call one of them correct and others incorrect. Congress did not legislate the result of rounding a particular number; they legislated a rounding procedure that applies to a system of numbers (rather than a single number at a time) that happens to have this result, because it prioritized something else and did not prioritize getting within one of the unrounded value. As a simple example (different from what actually happened) this could easily happen in a system that prioritized relative error rather than absolute error. The nominator appears to have an agenda for promoting certain electoral methods and for putting down certain other ones, rather than treating them all equally and neutrally; we should not further this agenda. —David Eppstein (talk) 18:04, 31 July 2024 (UTC)
  • Hi David, do you think ALT0A is acceptable? I agree different rounding procedures are inevitable, and all of them will have various quirks and paradoxes. I'm just highlighting this as an interesting example of such a paradox. (Though I'd note that rounding 40.5 up to 42 is a difference of 1.5, making this an unusually severe violation of the quota rule, which is why it's notable/surprising.)

I'm not sure what you mean by "Therefore, rounding errors are not a cause of anything." If the results of an election would have been different with a different rounding algorithm, and also would have been correct if no rounding algorithm was used, I think it's fair to say the election results were caused by round-off errors. Closed Limelike Curves (talk) 18:10, 1 August 2024 (UTC)

You keep saying "correct". Stop. That is the problem. An imputation that some procedures are correct and others are not is what is causing this issue. If what you really mean is that, for one particular election, one rounding method caused the results to agree with direct democracy and a different rounding method caused a different result, then say so, but for any rounding method one can find scenarios where it will differ from direct democracy and others will not. This is not an argument for one being correct and another not.

As for ALT0a: No. It did not have that effect. It had the effect that rounding a system of numbers caused one of the numbers to be rounded from 40.5 to 42, but that is not interesting, cause for alarm, or problematic. To spell out a simple example: suppose we are trying to round numbers to achieve minimum relative error, that the total number of seats is 48, and that the numbers we are trying to round are (1.25,1.25,1.25,1.25,1.25,1.25,40.5). Then the obvious way to round it is (1,1,1,1,1,1,42). Anything else would assign one of the small numbers a number of seats far out of proportion. Your hook describes political grandstanding from the time but by cherry-picking a detail from the rounding is misleading about the actual effect of the bill.

You might just as well say that the current US electoral college rounds 0.9 to 3 (the proportional fraction of electoral college seats that should be held by Wyoming vs the seats it gets). Is that cause for alarm? Is that cause for saying that the highest averages method used in part to allocate these seats is incorrect? —David Eppstein (talk) 18:49, 1 August 2024 (UTC)

Dr. Eppstein: If the problem is the word "correct", I've removed the it from ALT1a. I agree it was sloppy phrasing on my part in the DYK, which I only included because I did not expect that hook to be used, and definitely not that it would be used without first being workshopped a bit. (This is my first DYK, so I'm a bit unfamiliar with the procedure.) However, I don't see the relevance of any of this to the newest version of the hook, given I've removed the word "correct" from it. Otherwise, when I say "correct", I'm only defining this to mean the results with the idealized procedure, using fractional apportionments, which does not introduce any rounding errors.

Direct democracy is wholly unrelated to this topic, and I don't understand why you keep bringing it up. If you mean a direct popular vote, then no, I'm not talking about the popular vote. My point is that A) who won the election depended on the specific rounding rule for the House (which is interesting); and B) all the rounding rules well-regarded by experts for this purpose (Webster, Huntington-Hill, some for Hamilton) produce the same winner, and this winner disagrees with the actual results of the election. Closed Limelike Curves (talk) 15:49, 3 August 2024 (UTC)

I keep bringing it up because it is the only way to make sense of your comparison between rounded and unrounded outcomes. —David Eppstein (talk) 18:12, 3 August 2024 (UTC)

Each state has a certain number of votes. Those votes go to the candidate who wins the most votes in that state (in this election, all states used a winner-take-all rule for choosing electors). In a House of size $h$, that number of votes is equal to $2+\left[{\frac {h\cdot p_{i}}{\sum p_{i}}}\right]$, where the brackets denote rounding by whatever apportionment method. I am saying that if you dropped the brackets, i.e. if every state's electoral college apportionment was equal to two senators plus its exact entitlement in the house, the result would be different. In addition, if the entitlement had been done using any common rounding procedure (Webster, Huntington-Hill, Hamilton), the election results would have been different. Closed Limelike Curves (talk) 22:03, 3 August 2024 (UTC)

And I am saying that dropping the rounding and determining the result of an election by the exact unrounded vote tally is exactly the definition of direct democracy. Whether you divide all vote counts by the same quota (without rounding) or whether you leave them as integer numbers of voters, the result is identical. What about this is difficult to understand? —David Eppstein (talk) 22:15, 3 August 2024 (UTC)

@David Eppstein: What changes would you propose to the current ALT1a? Closed Limelike Curves (talk) 22:07, 3 August 2024 (UTC)

I would not use ALT1a at all, because it is misleading. All (unfixed) elections are decided by the choices of the voters and the voting system used for the election. All representative systems round, and all rounding systems produce rounding errors. ALT1a suggests to the reader, incorrectly, that the result of the 1876 election was somehow the wrong result, and that if only people had known how to perform arithmetic correctly then the outcome would have been different. It was the correct result, for the voting system chosen for that election, and the arithmetic was performed correctly. Get off this hobbyhorse of correctness and error. Leave 1876 politics behind. Find a different and unrelated hook for this article. —David Eppstein (talk) 22:14, 3 August 2024 (UTC)

Also there were many problems with the 1876 United States presidential election apart from rounding methods, so it is not an ideal example of the effect of rounding. TSventon (talk) 15:30, 4 August 2024 (UTC)

I'm sure they were, but round-off error was definitely involved as well, as per the sources, no? —Closed Limelike Curves (talk) 18:20, 4 August 2024 (UTC)

• The nominator brought this up in the WP Discord asking for input. For what it's worth, David Eppstein's commentary is accurate. It is not acceptable in Wikivoice to say that these were "rounding errors" (ALT1A) or that it was not "correct" (ALT1), and ALT0/0A are deeply misleading. A new hook should be offered. This was an unavoidable quirk of the system chosen, but as already stated, there is inherently going to be drift in any system attempting to lodge fractional pegs into integer holes. That's exactly the problem being solved. Any alternative system could have other "haha look this number rounded to this wrong number" issues as well. SnowFire (talk) 18:27, 4 August 2024 (UTC)
  • Quick edit to ward off an objection: there's two sense of "error". Of course there were rounding errors in the mathematical sense of the distance from the real number to the result, but there's also errors in the sense of "being wrong", which is how a standard reader will read hook 1A. But as discussed, no such error in that sense of the word was made. SnowFire (talk) 18:37, 4 August 2024 (UTC)
    • That's reasonable! I have no objections to tidying up the phrasing, and I can see how someone might misunderstand what I meant by "rounding errors". Do you have any suggestions for how to rephrase this more clearly? Do you think ALT1c looks good? Thanks! —Closed Limelike Curves (talk) 18:48, 4 August 2024 (UTC)

  • This was an unavoidable quirk of the system chosen, but as already stated, there is inherently going to be drift in any system attempting to lodge fractional pegs into integer holes.

I'm not sure what you're disagreeing with here, though. Yes, there is unavoidable drift (although I'd note the largest remainder methods don't violate quota rule) and in this case, the method used by Congress had an unusual/unavoidable quirk (which eventually led them to reject it). This quirk is interesting, which is why I think it makes a good DYK. —Closed Limelike Curves (talk) 18:48, 4 August 2024 (UTC)

It's better, but TSventon's concern remains. The 1876 election is just an unusually weird example to pick - sure, the apportionment system mattered, but so did the decision for electoral votes to include Senators (i.e. Nevada having 3 votes rather than 1 vote). So did voter suppression in the South (this & many elections until the 1960s, alas). So did the Compromise of 1877. You've picked an election which was so close that just about everything could be said to have affected the result. Moreover, it's not even clear that this was the fault of the "algorithm used to decide rounding" - it depends on what exactly was going on with the "supplemental apportionment" that the Balinksi & Young source describes.

Are there any non-1876 election related hooks to be had? If you really want to do one there, then I think we need some deep, ironclad sourcing from someone who both knows the politics AND the math behind it. So more than just passing mention or the half-page in Balinkski & Young. SnowFire (talk) 19:40, 4 August 2024 (UTC)

There is a more detailed account of the 1876 allocation on pages 71 and 72 of a US Government report here. But I am still not keen on an 1876 hook. TSventon (talk) 22:50, 4 August 2024 (UTC)

Agree an 1876 source is not ideal. That said, if the supplemental apportionment was a true "fudge", then that wasn't the fault of any rounding method, that was just an exercise in raw power politics, which is a little off-topic from the article and thus not a great hook for a different reason. SnowFire (talk) 23:44, 4 August 2024 (UTC)

The source is dated 1981 on its front cover, but I am still not keen on a hook about the highly contested 1876 election. TSventon (talk) 17:36, 5 August 2024 (UTC)

@Closed Limelike Curves: Please address the above. Z1720 (talk) 14:55, 12 August 2024 (UTC)

at a week since last ping, i believe it's time to mark this one for closure. theleekycauldron (talk • she/her) 18:36, 19 August 2024 (UTC)

One-sided?

Sorry for writing after the GA review has concluded, but after reading the great book Electoral systems by Lebeda and co. I stumbled upon this article and it's assosiates and was quite astonished by how 'pro-divisor' this and all the other articles are.

Though after going through the references, it started to make sense, considering that the only two larger sources cited are pretty much just Pukelsheim, who is very reliable on a susceptibility to paradoxes (which isn't bad, but there are two sides to the story) and Balinski (whom I didn't read, so perhaps he makes great arguments, sorry if the does). I'd say one needs a bit more than that to claim that divisor methods are "generally preferred by social choice theorists" (which changes to mathematicians in the latter part of the article), especially considering that when describing d'Hondt, Gallagher is cited with "However, Jefferson's method performs poorly when judged by most metrics of proportionality".

In my opinion, the parts of the article where the comparison is made, should be edited as to include a more neutral overview of the topic. Using Gallagher's work (for example Gallagher 1992) for a bit more balanced outlook or Lijphart (Lijphart 1994) for a bit more pro-quota outlook to balance things out.

To be clear, I do not admonish divisors or love quotas (my belief is that STV rules anyways and for regular PR Sainte-Laguë might just be the best of both worlds), but the current text is lopsided.

In summary, making the text more neutral on the divisor/quota divide would be of use. Quotas have their problems (population and Alabama paradoxes (though sidenote, I don't think participation can be called as a paradox intrinsic to quotas, as unrounded quotas or Hare-Niemeyer don't produce it)) and Divisors have theirs (breaking the quota rule obviously, generally considered less proportional, because we use Hare as a point of reference most of the time (though again, consult Gallagher for a better explanation)), so that should be included. In my experience and from my reading I have not encountered a broad consensus that divisors are inherently superior. JerenMapper (talk) 10:28, 23 August 2024 (UTC)

Excuse me, but aren’t the divisor-methods the topic of this article?? …& the meaning of the article’s title?

Discussion of STV & Largest-Remainder have no place here.

There are other Wikipedia articles about them, or about the larger general topic of Proportional Representation & the relative-merits of its types.

2600:6C55:7900:2B8:89A7:F9D5:59FC:4E65 (talk) 10:32, 24 December 2024 (UTC)

First Webster (SL) example to the side of the Adams

Looking at the Webster example that is in the single comparison to the Adams method, should the result not be 12, 4,3,1,1

With the 12th seat line being 4,783 1,503 1443 483 483

I.e. the 12th seat for Yellow wins with 4,783 over the 4th seat for Red with 4,743— Preceding unsigned comment added by Blueflatsky (talk • contribs) 00:00, 29 October 2024 (UTC)