Wikipedia:Reference desk/Archives/Mathematics/2020 August 20

= August 20 =

Geometry question
I read on a website that someone had had a geometry class where they had to find a polygonal shape with four edges of which two were parallel and only one right angle. They couldn't do it, and it turned out the whole question was wrong.

I am quite sure such a shape is impossible. The two parallel edges can't have an angle between them, so they must both have an angle with a third edge. However, because the edges are parallel, either both or neither of these angles are right.

Is this reasoning correct? J I P &#124; Talk 20:37, 20 August 2020 (UTC)


 * Yep. See Transversal (geometry) for a little supplementary (nyuk nyuk) info. –Deacon Vorbis (carbon &bull; videos) 20:51, 20 August 2020 (UTC)


 * Hmm. This depends on your rules for excluding degenerate cases.  I see no exclusion of a right triangle with a fourth vertex midway along one side from being a quadrilateral (tetragon), or of an angle being neither 0° not 180°.  This allows two adjacent edges to be parallel and only one angle to be a right angle.  —Quondum 23:38, 22 August 2020 (UTC)

Single transferable vote eliminations
I'm trying to understand how eliminations work under Single transferable vote. Sometimes it's possible to eliminate a whole group of candidates in a single count if it's clear that their elimination separately could not affect the outcome of the vote. What I don't understand is, how do you find this group? Is there somewhere that explains the algorithm? Thanks, 78.19.190.3 (talk) 22:21, 20 August 2020 (UTC)


 * Let us take a concrete example, where there is one position to be filled and four candidates: Amy, Brian, Crystal and David. After the first round, the vote tally is:
 * 120: Crystal
 * 60: Brian
 * 30: Amy
 * 20: David
 * So David will be eliminated. The 20 votes that went to David in this round will now be transferred in the next round to other candidates. The best thing that can happen for Amy is that all of these 20 votes go to her, so in the best case she will get 50 votes. But both Crystal and Brian are already assured that they will receive a higher number than 50 (numbers can only go up, not down). So without looking at the preference ranking lists of all the voters we know in advance that Amy will finish last in the next round – so she too can be eliminated. Then for Brian the best scenario is that in fact all the 20 original votes for David are transferred to him, and that, moreover, all of Amy's 30 original votes now also go to him. In that best-case scenario for Brian he now gets their combined 20+30 = 50 votes and hits 110 – still at least 10 shy of Crystal's guaranteed 120 votes. And so there goes Brian too. Congratulations, Crystal. The general idea is that for each not-yet-eliminated candidate you can compute a best-case-scenario upper bound on how many votes they might theoretically accumulate. If that is lower than the last-round numbers of all other candidates, they will be the sure loser of the next round. --Lambiam 00:08, 21 August 2020 (UTC)
 * Crystal will have been declared the winner after the first count as she had received more than 50% of the vote, so no further transfers would be necessary.  — Preceding unsigned comment added by 82.38.235.128 (talk) 11:03, 22 August 2020 (UTC)
 * True, this guarantees that she will eventually emerge as the winner – although it depends on how the procedure has been specified whether she will already be declared the winner in this stage; there is no such shortcut clause in the formulation of our article Single transferable vote. That a candidate who obtains in any round the absolute majority will eventually win the election is a consequence of the type of reasoning I sketched: no other candidate's best-scenario can beat this. But what if Crystal receives only 100 votes in the first round? We can still see that David and Amy can be eliminated already now, but Crystal may still lose to Brian. --Lambiam 21:05, 22 August 2020 (UTC)

If there is only one position to be filled then STV would not be used. See third sentence of paragraph 7 of the Wikipedia article. — Preceding unsigned comment added by 82.38.235.128 (talk) 09:03, 23 August 2020 (UTC)
 * Which paragraph is that? The one that starts with "STV also provides approximately proportional representation"? I do not see how the third sentence of that paragraph – or any sentence of any paragraph of the article – applies specifically to the situation in which there is only one position to be filled. --Lambiam 17:26, 23 August 2020 (UTC)


 * It’s where it refers to multiple winners. See also the first few words in the first sentence in the section under By-elections. — Preceding unsigned comment added by 82.38.235.128 (talk) 11:00, 24 August 2020 (UTC)
 * These sentences say STV can be used if there are many seats to be filled, which is true. That does not mean it cannot be used in cases where there is a single vacancy. This is in fact not unusual. See the book Approval Voting by Brams & Fishburne. There may be arguments to prefer other methods in case of one seat, such as the Condorcet method or variants, but no voting system can be perfect, so such arguments are at best suggestive, not compelling. --Lambiam 14:31, 24 August 2020 (UTC)

How do the edge cases work? Given this concrete example on count one with a quota of 101:

Which candidates get excluded on the first count and why? Obviously you'd start with the candidates with one vote, but can you do all six as a group or would you have to do them one at a time? 93.107.224.16 (talk) 14:15, 24 August 2020 (UTC)


 * Using the best-case scenario criterium I gave above, none of these six could hope to receive more than six votes, so you can clearly eliminate all. If all six votes now go to candidate C7, they will be the winner of the election. Otherwise, C7 can be eliminated after that round (assuming at least one voter's preference list is longer than six), but more rounds may be needed. In other cases it may be necessary to break a tie, such as when all six got 16 votes, in which case one of them, if not eliminated, might end up with six times 16 = 96 votes, enough to eliminate C7 and perhaps to go on and win the election. There are several ways of doing this, and the one to be used should be chosen and made known in advance.