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, 3 years ago→Negative pairwise counting approach
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I've made a number of images and GIFs on voting theory. See [[Special:ListFiles/BetterVotingAdvocacy]].
How Score voting and Condorcet can both be thought of in terms of pairwise preferences (as well as [[rated pairwise preference ballot]]): https://imgur.com/a/ssXCQIE
== Description of some common Condorcet methods ==
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One way to understand why a voter should give maximal support to their lesser evil if their favorite isn't viable: if, whenever a candidate is very unlikely to win, the voter pretends they aren't in the election i.e. got eliminated, then eventually the voter will have a candidate who is their "favorite of the remaining candidates". If they are normalizing, then they ought to give this candidate the max score, and likewise give their least favorite of the remaining candidates (i.e. the greater evils) a 0.
It might help when thinking about using negative scores in [[Score voting]] to imagine a Score scale of -1 to 0. This would be equivalent to Approval voting, essentially. In addition, keep in mind that a vote on a scale of - 1 to 1 can be converted to a scale of 0 to 2 by adding 1 point to every candidate in each vote.
One way to understand a voter's absolute score for a candidate is that they are expressing their degree of support for that candidate pairwise against a candidate they don't support at all.
The expressiveness of a rated ballot, as measured by "number of possible votes (permutations) a voter can submit", is greatly reduced if removing non-normalized votes from contention (i.e. on the grounds that they reduce voter power). For example, with a scale of 0 to 3 and 3 candidates, there are 4^3=64 possible votes<ref>This is derived as follows: pretend there are infinite candidates in the election. When there is only 1 candidate in the election, we can pretend everyone except that 1 candidate is scored the same no matter what (let's say they are all scored at the min score, which is usually 0, for this example), and that that 1 candidate can be scored [number of scores in the scale] ways. So with 1 candidate and a scale of 0 to 4, there are 4 possibilities. With 2 candidates, the 2nd candidate is scored a 0 for every possible score the 1st candidate is scored at. But the 2nd candidate could also be scored a 1 for every possible score the 1st candidate is scored at, etc. So the result is 4*4=16 possibilities. For 3 candidates, there are 4 possible scores the 3rd candidate could be at when the first 2 candidates are moving through all of their possible scores, so that is 4*4*4 or 4^3=64 possibilities.</ref>, but when removing non-normalized votes, there are only 18 possible votes.<ref>https://forum.electionscience.org/t/different-ways-of-measuring-expressiveness-for-rated-type-ballots/712 There are 20 votes ignored in that analysis, so that's 20 + 26 = 46 non-normalized votes total to ignore.</ref><ref>(This is because a normalized vote requires one candidate to be at the max score and one candidate to be at the min score, so there are only 4 possibilities for how the 3rd candidate can be scored.)
A:3 B:2 C:0 (=2*3=6, since you could swap B and C here, and you can also swap B or C for A)
A:3 B:1 C:0 (=2*3=6)
A:3 B:0 C:0 (=3, since you can't swap B and C here)
A:3 B:3 C:0 (=3)</ref>
== Miscellaneous ==
One criterion that might be good for PR methods is the "Duplicated Quotas" criterion: if a PR method elects some candidate in the single-winner case, and the ballots are "duplicated" N times, then if N+1 seats are to be filled, the duplicated winner should win. Example for Condorcet PR:
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Here is an example of a situation where, if voters are assumed to normalize their scores, it is possible to justify a non-majoritarian winner even with only ranked preferences: suppose there are very many voters, with there being a majority faction only one voter larger than an opposing minority faction. The majority's preference is A>B>C>etc. while the minority bullet votes B. In this case, B would almost guaranteeably win in Score under the above assumptions, even if decimal scores were allowed, so long as the majority's preference for B was non-infinitesimal, since this would cut into their ability to express their A>B preference.
Regarding normalization, if a voter would end up having to choose between rounding down or rounding up their score for a candidate, then there ought to be a name to describe the voter rounding their score for that candidate closer to their original score i.e. if they originally scored A:2, and after normalization, ought to be scoring A:3.4, then they ought to round to 3 if forced to choose an integer score, because that's closer to 2, the original score.
One way to argue that voters should be able to use only some of their voting power is to consider this thought experiment: suppose every voter had 9 clones of themselves<ref>This can be thought of as similar to the [[KP transform]] and how, in some implementations, it treats each voter as casting multiple ballots.</ref>, and the voter could tell each of the clones whether or not to vote. In such a scenario, should it be required that the voter make all of their clones vote?
Approval voting can be thought of as multiple voters casting FPTP ballots (i.e. approving A and B is equivalent to one voter voting A and another voting B), Score can be thought of as fractional Approval, and FPTP can be thought of as a ranked method. Rated pairwise doesn't seem to easily fall within this framework.
=== Work-in-progress ===
This section is for things that I'm still working on; they may not be totally correct, but hopefully they have various insights in them.
(Work in progress, calculations and reasoning are off) Expressiveness of rated pairwise vs. other ballot types:
* One way to evaluate the expressiveness of various ballot types is to look at how many possible votes a voter can cast with a certain number of candidates. <ref>https://www.bridgealliance.us/ten_critiques_and_defenses_on_approval_voting See Critique #3.</ref> Rated pairwise, assuming a limited ballot design where voters can indicate their ranking of each candidate, and how they'd vote in a matchup between their 1st choice vs 2nd choice, 2nd choice vs 3rd choice, etc. (with matchups between candidates more than one rank apart calculated by adding up the margins of each intervening matchup i.e. the marginal preference a voter has for 2nd choice over 4th choice is calculated as the margin for 2nd>3rd + 3rd>4th), and if considering the "votes" a voter can cast as the possible rated pairwise preferences they can indicate on such a ballot, allows for at least 72 possible votes with 3 candidates, if allowing a scale of 0 to 3 in each matchup. This compares to a rated ballot with a scale of 0 to 3 itself, which allows for at most 62 possible votes.
** 72 comes from 60 + 12. There are 6 possible rankings of the candidates if disallowing equal-rankings (A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, C>B>A), with each of these having 10 possible preferences (for A>B>C, this can be demonstrated as (with "|" used to separate each possible vote): (A>B 0, B>C 0, A>C 0 | A>B 1, B>C 0, A>C 1 | A>B 0, B>C 1, A>C 1 | A>B 1, B>C 1, A>C 2 | A>B 1, B>C 2, A>C 3 | A>B 2, B>C 1, A>C 3 | A>B 2, B>C 2, A>C 3 | A>B 3, B>C 2, A>C 3 | A>B 2, B>C 3, A>C 3 | A>B 3, B>C 3, A>C 3). In addition, there are a number of rankings that involve equal-ranking (A=B>C, A=C>B, B=C>A, A=B=C, A>B=C, B>A=C, C>A=B), with each of these also allowing for an additional number of possible votes.
* There are two variations of rated pairwise to consider here: one where the voter can only indicate their score for their more-preferred candidate in the matchup (i.e. they can only indicate marginal preference), and one where they can indicate their scores for both the candidates. The figures computed above are for the former variation, but the latter variation greatly increases the level of expressiveness, since, for example, a voter wishing to indicate they prefer A 1 point more than B can cast that as A:4 B:3, A:3 B:2, A:2 B:1, or A:1 B:0, while there is only one way to cast that preference using the margins-based variation.
I'm trying to evaluate whether is a way to essentially "track" a voter if voters are allowed to weaken their votes in Condorcet. By track I mean that you could figure out some voter's preferences by looking at the election result data; here is one example<ref>http://www.votingmatters.org.uk/ISSUE30/I30P2.pdf</ref>. My guess for how you could create an example where such a thing is possible is to have an election with few voters, where only 1 of the voters weakens their vote at all. Keep in mind that this may be somewhat realistic when considering that each precinct releases its own vote totals, such that a very small precinct may be vulnerable to this type of thing, if it exists.
I made some edits (https://electowiki.org/w/index.php?title=Rated_pairwise_preference_ballot&type=revision&diff=12325&oldid=11201) where I discussed some ways of figuring out what scores a voter would give to both candidates in a matchup (Ctrl+F "actual scores in the 1st vs 3rd matchup").
* But I see a problem with the second way I described: it doesn't work properly with regular Score voting. Suppose a voter scores A:4 B:2 C:1. Their 1st vs 2nd preference is A:4 B:2, and 2nd vs 3rd is B:2 C:1. If adding these up as prescribed to obtain the A vs C preference, you get 6 points for A and 3 for C, which reduces to 5 and 2 respectively if fit within a 0 to 5 scale. But the actual A vs C preference was A:4 B:1, which has the same margin, but different absolute scores.
** One way of solving this seems to be to instead do "take the lowest absolute score given to any candidate in any of the relevant matchups, and add the cumulative margin to that candidate's score to find the more-preferred candidate's score, moving this down as necessary to be capped at the max score." This always gives the appropriate result in regular Score voting. I wonder if it's equivalent or not to" take the highest absolute score, subtract cumulative margin, and move this up as necessary to be at least the min score."
== Condorcet ==
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* Any [[:Category:Runoff-based voting methods|Category:Runoff-based voting methods]] will elect a CW who is in the runoff (supposing voters could express their preference for the CW)
* [[Approval voting]] and [[Score voting]] (and most [[cardinal method]]<nowiki/>s) have some kind of strategic equilibrium for the CW, especially if they are a [[Majority Condorcet winner]].
Note that the vote totals from a Condorcet election can be combined with the vote totals from FPTP, Approval, and Score. This can come in handy if doing a national popular vote for U.S. President. See [[Negative vote-counting approach for pairwise counting#Connection to cardinal methods]] and the image in that section for more details.
=== Condorcet-cardinal hybrid methods ===
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=== Rated pairwise preference ballot ===
(See the rated pairwise preference ballot subsection of the vote-counting section below for more information on how to count the votes with this ballot type)
It is possible to treat every pairwise matchup as being a [[Score voting]] match-up where the voter can score both candidates on a scale. In this sense, the traditional pairwise preference idea of a voter giving one vote to their preferred candidate in a match-up is akin to them giving the max score to that candidate, and a min score to the other candidate. This may make it easier to think about the rated support and pairwise support that are both allowed with the [[rated pairwise preference ballot]].
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A way to measure how much voters are taking advantage of the ability to indicate maximal support in multiple transitive matchups in rated pairwise relative to ranking and scoring is to, for each voter, add up the margin they express in each transitive matchup from 1st choice to last i.e. add up 1st>2nd, 2nd>3rd+...+2nd to last>last. If, for a given voter, this cumulative margin exceeds 1 vote/100% support/(max score - min score) points, then that voter's rated pairwise preference can't be compressed into a scale. If this cumulative margin is added up for all voters, and is also calculated for each voter's ranked preference (this can be found from the data they submit on a rated pairwise ballot), then this also shows how much voters weakened their vote relative to if they had been forced to rank the candidates in Condorcet. One thing that confounds this analysis is that some voters might indicate a weak rated pairwise or rated preference between two candidates, but rank the two equally rather than differentiate the two, to avoid putting too much power in the matchup. There may be a way to do similar analysis simply by looking at the number of votes candidates have overall in rated pairwise versus ranked (compare the bottom 2 tables in <ref>https://forum.electionscience.org/t/how-should-transitivity-be-handled-with-rated-pairwise-preferences/693/34</ref> for an example).
Here is an example where the Score winner, Condorcet winner, and the "Condorcet winner based on rated pairwise preferences" (RPCW) are all different:
26 A:5 B:4
25 B:5 A:4
49 C:5
C is the Score winner. Pairwise, A beats B 26 to 25, and C 51 to 49, so A is the CW. But using rated pairwise preferences:
26 A:0.2 B:0, B:1 C:0 (by transitivity this implies A:1 C:0)
25 B:1 A:0, A:1 C:0
49 C:1 (this just means C:1.A/B:0)
Here, B beats A 25 to 5.2 (125 to 26 on a scale of 0 to 5), and C 51 to 49. So because A-top voters indicated weak preference between A and B, but maximal preference for them over C, they ensured one of the two won while still conceding to B.
Regarding [[Rated pairwise preference ballot#Preference threshold]] implementation, there is probably a way to allow a voter to indicate multiple preference thresholds, such that candidates within each threshold are weakly preferred at most to one another, but maximally preferred over all candidates in lower thresholds. The thresholds could even be allowed to overlap (i.e. by having the voter indicate the upper and lower bound scores for each threshold). See <ref>https://www.reddit.com/r/EndFPTP/comments/h7wszd/comment/fuqkrrp</ref> for context; in the described situation, supposing there are two sides whose preferences are Favorite:5 2nd Choice:4 Compromise:1 Other Side:0, overlapping thresholds could allow each side to maximally boost their own candidates against the other side's candidates, while indicating their cardinal (weak) preference for their side's candidates over the compromise, and the compromise over the other side. This would only allow the compromise to win if they were scored highly enough by voters weakening their votes like this.
==== Transitivity ====
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A voter's rated pairwise preference can be partially expressed as a ranked preference augmented with margins. So, for example, A 30%>B 50%>C 100%>D 100% (which could also be written as A 0.3>B..., or A 40% to 10%>B 80% to 30%, etc.) would indicate the voter gave a 30% margin to A>B, a 50% margin to B>C (implying, by transitivity, that they gave at least a 50% margin to A>C as well), etc. This doesn't tell you how the voter scored matchups involving candidates who are more than one rank apart, however.
One of the reasons it would be difficult to allow a voter to directly express their preference for their 1st choice>3rd choice is because they could theoretically give 1st>2nd and/or 2nd>3rd large margins, but then give 1st>3rd a small margin, which would violate rated pairwise transitivity. One way of solving this would be to allow a voter to express their 1st>3rd preference by indicating how much larger they want the margin in that matchup to be than what would be required by transitivity. So for example, if 1st>2nd: 30% and 2nd>3rd: 40%, and the voter indicates that 1st>3rd should be 20% higher than usual, then that could be interpreted as "add up 1st>2nd and 2nd>3rd to find the minimum margin transitively required for 1st>3rd (i.e. 30%+40%=70%), and then add in the voter's additional 20% preference to yield a margin of 90% for 1st>3rd."
One way to understand why, regardless of transitivity, a voter may indicate their strength of preference independently in pairwise matchups between candidates one rank apart (i.e. how you score 1st choice vs 2nd choice is totally separate from 2nd vs 3rd) is because if your 1st choice dropped out of the race, then your 2nd choice would become your new 1st choice, so you'd then want to have accurately expressed the strength of your 2nd>3rd preference.
==== Criterion compliances ====
If using rated pairwise in a Condorcet method:
* The majority criterion and mutual majority criterion are passed if the majority indicates maximal preference for all of their preferred candidates in matchups against all candidates they don't prefer.
* Condorcet and Smith criteria are passed if the voters who prefer a candidate in the Smith set over a candidate not in the Smith set always maximize their preference.
=== Connection between Condorcet, Smith set, and Asset ===
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A basic justification for using some kind of pairwise counting procedure where every candidate a voter ranks 1st can be counted with only one mark each: suppose you use the "rated or ranked preference" implementation of [[Rated pairwise preference ballot]], and a voter does [[min-max voting]] with their scores and casts a rated preference. This voter would only need one mark to count each candidate they gave a max score to, and no marks for the min-scored candidates. And in effect, this voter is giving one set of candidates maximal support against all candidates not in the set, while casting no preference between the candidates in the set, which is equivalent to ranking them 1st and all other candidates last. But, if this voter were to switch to now casting a ranked preference, the vote-counters would have to increase the number of marks they count for the voter's ballot, while not essentially capturing any different information (except that the voter would now be essentially treated as giving 0 votes to both candidates in the matchup between two equally-ranked candidates, rather than potentially giving both of them 1 vote i.e. because they might have max-scored both).
==== Semi-negative counting procedure ====
Technically, the markings required for the negative counting approach can be reduced almost by half in the following manner: when a voter ranks a candidate last, make no marks for them. When a voter ranks a candidate one rank above last, the only mark made is that the voter prefers this candidate over the last-place candidate; this way, rather than marking negative votes in almost all of this candidate's matchups, only one mark has to be made. And so on. For ballots that rank all candidates, the top-ranked half of the candidates would be counted negatively, while the bottom-ranked half would be counted in this way. But this could potentially be more confusing and/or require more data storage (i.e. separately counting the negative and positive pairwise votes for each candidate).
When doing the "semi-negative" counting procedure mentioned in the previous paragraph, some voters will be able to contribute votes to both candidates in a matchup, while other voters won't, purely based on how highly or lowly they ranked them. If this creates legal or procedural issues, it is possible to have each precinct only submit the [[Margins|margin]] they found in every pairwise matchup, rather than the votes on both sides as well. In other words, if, for the A vs B matchup, in Precinct 1 A has 15 votes and B 10, while in P2 A has 7 and B 8, then it is possible for P1 to submit that A has 5 votes more than B, and P2 to submit that B has 1 vote more than A. This can be used to find that A has 4 votes more than B in the combined electorate of the two precincts.
===== Rated pairwise preference ballot =====
For the rated pairwise ballot, semi-negative counting can sometimes be beneficial, but there is some nuance in how to apply it. Take the following two rated pairwise votes for example (on a scale of 0 to 5):
A:5 B:0, B:5 C:0 (so A:5 C:0)
A:5 B:4, B:5 C:3 (transitivity implies something like A:5 C:2)
For the former vote, it'd be easier to assume the voter gave the max score to every candidate in each matchup, and then do negative counting i.e. count it as
{| class="wikitable"
|+
!
!A
!B
!C
|-
|A
|5
|
|
|-
|B
| -5
|5
|
|-
|C
| -5
| -5
|5
|}
Note that with a scale of 0 to 1, this particular example looks exactly like negative pairwise counting as applied to a ranked ballot.
For the latter vote, it'd be easier to assume the voter is somehow constantly scoring the candidates in each matchup, and then only add or subtract points in as necessary i.e.
{| class="wikitable"
|+
!
!A
!B
!C
|-
|A
|5
|
|
|-
|B
|
|4
| +1
|-
|C
| -1
|
|3
|}
So it's as if the voter had filled out a rated ballot, with the only difference being that they gave B 1 more point in the B vs C matchup than would normally be allowed, which then forced the voter to give C 1 less point against A (depending on how you implement the ballot design and transitivity requirements).
==== Examples ====
| |||