# Talk:Pairwise counting

## Alternative pairwise counting table

I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election

Pairwise preference combinations:[21][26]

wi JS DS KW BK AM

AM Andy

Montroll (5–0)

5 Wins ↓

BK Bob

Kiss (4–1)

1 Loss → ↓ 4 Wins

4067 (AM) – 3477 (BK)

KW Kurt

Wright (3–2)

2 Losses → 3 Wins ↓

4314 (BK) – 4064 (KW)

4597 (AM) – 3668 (KW)

DS Dan

Smith (2–3)

3 Losses → 2 Wins ↓

3975 (KW) – 3793 (DS)

3946 (BK) – 3577 (DS)

4573 (AM) – 2998 (DS)

JS James

Simpson (1–4)

4 Losses → 1 Win ↓

5573 (DS) – 721 (JS)

5274 (KW) – 1309 (JS)

5517 (BK) – 845 (JS)

6267 (AM) – 591 (JS)

wi Write-in (0–5) 5 Losses → 3338 (JS) –

165 (wi)

6057 (DS) – 117 (wi)

6063 (KW) – 163 (wi)

6149 (BK) – 116 (wi)

6658 (AM) – 104 (wi)

BetterVotingAdvocacy (talk) 08:50, 17 January 2020 (UTC)

- When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). VoteFair (talk) 18:45, 17 January 2020 (UTC)

- Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." BetterVotingAdvocacy (talk) 21:40, 17 January 2020 (UTC)

- If you want to refer to Condorcet methods feel free to add a section on that topic.
- I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."
- Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. VoteFair (talk) 04:57, 19 January 2020 (UTC)

- To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. VoteFair (talk) 18:51, 21 January 2020 (UTC)

## Placement of image on page

User:VoteFair, with regards to this edit (https://electowiki.org/w/index.php?title=Pairwise_counting&oldid=8660) which moved the large image to a lower section, I think that image should be in the section relating to how to do pairwise counting on various ballot types (what you titled as "Example using rated (score) ballots"), since that's what the image described. I'd like to ask you what you think before making any edits, though. Edit: I decided to just move that image even further down the article, and to add a few details to the section on doing pairwise counting with various ballot types instead. BetterVotingAdvocacy (talk) 17:19, 15 March 2020 (UTC)

- That image needs to be converted into paragraphs of text with inserted graphics/images where it isn't just text. Currently it is much, much too tall! In addition it might need to be put into a new article -- or several existing articles -- because it appears to be about a few specific vote-counting methods. This article (about pairwise counting) should JUST be about pairwise counting, and not about specific ways of using the pairwise counts. VoteFair (talk) 22:52, 16 March 2020 (UTC)

- I will see if there are ways to do what you asked regarding the image. But I don't see how the image says anything about what to do with the pairwise counts (i.e. finding the Condorcet winner, or something like that) or how it pertains only to certain vote-counting methods (e.g. Approval voting, Score voting, etc.); rather, it only speaks about how to extract pairwise counts from ballots, which is very important information to document in this article (otherwise, where else would it go?). Also, I can understand putting information "JUST" about pairwise counting higher in the article, but I don't see the issue if information about "specific ways of using the pairwise counts" is put lower in the article. I'm willing to compromise on that, but at the very least, I'd like to have small sections explaining various ways of using the pairwise counts with links to larger articles. BetterVotingAdvocacy (talk) 23:14, 16 March 2020 (UTC)

- I see your point. The text in the image is so poorly formatted that I had just skimmed it and thought it was progressing to a single winner.
- The lower of the two images definitely needs to be converted into text and images. The upper image looks shorter than I remember, so that's good.
- I think there should be separate sections for how to do pairwise counting using: ranked ballots, score/cardinal ballots, approval ballots, and single-mark ballots. Their headings will clarify context, which is difficult to figure out from the image versions.
- Thank you for your help with this article! It keeps getting better! Hopefully this long-overdue article will find its way to Wikipedia someday. VoteFair (talk) 23:39, 16 March 2020 (UTC)

## Ways of speeding up pairwise counting

User:Kristomun, I wanted to discuss your edit https://electowiki.org/w/index.php?title=Pairwise_counting&diff=next&oldid=9855. I later generalized the point that I was making, which you removed, so let me explain my generalization: if there are, say, 5 candidates A through E, and a voter bullet votes A, there is no need to record A's victory in all 4 matchups, because you can just say "A gets a vote in every match-up" and move on from that. This information can be stored in the cell comparing A to themselves. Likewise, if someone ranked A>B, you only need to record in addition "B gets a vote against everyone except A" which can be shown with a negative vote in the B>A column. In other words, instead of recording 4+3 matchups (A beats B through E and B beats C through E), the work can be shortened to recording 2+1=3 (2 votes for A and B in every matchup, and 1 negative vote for B>A) things. If there are a lot of candidates, this can create quite a lot of time savings. I should note that someone who votes A>B=C would need a negative vote for both B>C and C>B with this approach to preserve the accurate winning votes total in the B vs C matchup (though the margin will be accurate either way), so that's the only time it might require more markings than the regular approach. BetterVotingAdvocacy (talk) 03:24, 12 April 2020 (UTC)

- It doesn't seem to gain you much. Suppose you're using a method like minmax. You need to determine the strength of each candidate's weakest pairwise showing (or the closest to a landslide in favor of the other candidate). If you've used a shortcut/speedup when noting the pairwise matrix, you still have to unpack that speedup, i.e. "decompress" the Condorcet matrix in order to determine the relevant pairwise contest for each candidate. And if you're doing that, you could just as well decompress the matrix as part of the counting procedure itself.

- In other words, suppose you happen upon a voter who bullet-votes for A. You can either indirectly mark that he bullet-votes for A by using negative counts, or you can just increment the entire A vs everybody row without looking at the ballot more than once. It seems to me that the latter is easier to do and less messy: that the benefit you get by speedup isn't as great as it seems because you have to translate into a canonical representation at some point anyway.

- Perhaps it could be used as a sort of shorthand if lots of voters bullet-vote or only rank a few candidates each, out of very many candidates. But even so, the point remains that when you're done counting, you have to e.g. add 2917 to every A>X pairwise matrix cell if you recorded 2917 bullet votes for A. So this would make sense if adding 2917 to every one is significantly less expensive than adding 1 to every one, 2917 times, as part of the count. Kristomun (talk) 23:57, 14 April 2020 (UTC)

- In the example in your second paragraph, we don't need negative counts to indicate a bullet voter; we can just say "A is marked on 1 ballot", and then we are done processing that ballot. The key thing I think you missed is that the unpacking happens at the central counting place using this approach, rather than in the precincts. So, for example, if there are 5 candidates, instead of the vote-counter marking A>B, A>C, A>D, A>E, the central counting place gets the information "A is marked on 1 ballot" and then they can unpack this by saying "OK so A must have gotten 1 vote in A>B, A>C, A>D, and A>E." Thus, the unpacking doesn't actually take any significant amount of work to do. Another thing that may have been misinterpreted is the negative count approach; you only need negative counts when a voter ranks one candidate above or equal to another candidate. So, for example, someone voting A>B only needs a negative vote recorded in B>A in order for us to figure out which matchups they don't prefer B in, because in all other matchups we know they prefer B, therefore we can just record that "B is marked on 1 ballot" and this one negative vote, which allows us to collectively say "B is preferred in every matchup except against A". Thus, it still only requires looking at the ballot once per candidate. Regarding your point in your third paragraph, it seems to me that it would always be significantly easier to record a bullet vote with only 1 marking rather than several? I made an example of this at https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fn75b3g/ if it helps. A broader point I should mention is that, ignoring equal-rankings, this approach will always require at most a few more markings than the regular approach (at most it's the number of markings in the regular approach plus the number of candidates), and often will require far fewer. I'll show this for the 4-candidate case: if someone votes A>B>C>D, then in the usual approach, we do 3 markings for A's matchups, 2 for B's, and 1 for C's. With this approach, we do 4 markings, one for each candidate to indicate that they were ranked by the voter, and then we do 3 negative votes for D, 2 negatives for C, and 1 for B. Now, if this voter had only ranked A>B, then in the usual approach that's 3+2=5, whereas with this approach, it's 2+1=3. As the number of on-ballot candidates increase, the time-savings starts to possibly become worth it. Anyways, I think one thing we can probably agree on is that even if you're using the regular pairwise counting approach, it's smart to, for every voter who has only one 1st choice candidate, report the bullet votes for that candidate and skip counting that candidate's matchups, while still manually counting the matchups of all lower-ranked candidates. BetterVotingAdvocacy (talk) 01:07, 15 April 2020 (UTC)

- You added "However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods." to a part of the page discussing how negative counting required less markings than the regular approach. I think you properly understand how negative counting works, but I'd just like to reiterate that this part of the procedure doesn't add any work for the vote-counters, and thus it doesn't work against the claim that negative counting requires less marks, or the general idea of it being less work. BetterVotingAdvocacy (talk) 21:55, 22 May 2020 (UTC)

- Yes, that's true. I simply meant to show that you don't get quite as much for free as it might seem like you're getting, particularly if the counts are computerized, because you have to add some numbers to correct the non-marked candidate counts at some point, however you do the precinct counts. Kristomun (talk) 12:11, 25 May 2020 (UTC)
- I'm not sure I understand what you mean by "correcting the non-marked candidate counts". But if you understand that this post-processing stage requires maybe a couple of minutes of work at most for regular elections, then that's good; I just wanted to clarify that the math can be done in 2 seconds by a computer (Excel spreadsheet with the value for number of voters ranking a candidate added to all other values in that row, which will be the number of voters ranking that candidate below another candidate in a head-to-head matchup) whereas the tallying in the precincts could take days, so it's not as big a caveat as the wording of the sentence might suggest. BetterVotingAdvocacy (talk) 21:55, 27 May 2020 (UTC)

- Yes, that's true. I simply meant to show that you don't get quite as much for free as it might seem like you're getting, particularly if the counts are computerized, because you have to add some numbers to correct the non-marked candidate counts at some point, however you do the precinct counts. Kristomun (talk) 12:11, 25 May 2020 (UTC)

I ought to note that I just realized that simply counting 1st choices separately from all other ranks actually has the potential to rival the speedup produced by negative counting in many election scenarios. For Burlington 2009, for example, doing regular pairwise counting with the 1st choice trick is actually faster than negative counting. https://electowiki.org/wiki/Negative_vote-counting_approach_for_pairwise_counting#Burlington_2009_mayoral_election BetterVotingAdvocacy (talk) 21:14, 3 June 2020 (UTC)