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

• 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:

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

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 ${\displaystyle h}$, that number of votes is equal to ${\displaystyle 2+\left[{\frac {h\cdot p_{i}}{\sum p_{i}}}\right]}$, 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)