# 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."
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)