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== Cardinal methods ==
Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates, rather than only the number of voters who prefer one candidate over the other. See the [[rated pairwise preference ballot]] article.▼
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]
 
Note that pairwise counting can be done either by looking at the margins expressed on a voter's ballot, or the "winning votes"-relevant information (see [[Defeat strength]]). For example, a voter who scores one candidate a 5 and the other a 3 on a rated ballot can either be thought of as giving those scores to both candidates in the matchup (winning votes-relevant information) or as giving 2 points to the first candidate and 0 to the second (only the margins). For ranked and choose-one ballots, both margins and winning votes approaches yield the same numbers, since a voter can only give maximal support to at most one candidate in the matchup.
▲Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates.
 
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).
 
The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a [[transitivity]] of strength of preference, just as there is a transitivity of preference for rankings.<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/|title=r/EndFPTP - Score but for every pairwise matchup|website=reddit|language=en-US|access-date=2020-04-05}}</ref>
 
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/> and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1
 
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates.
 
==Notes==
[[File:Pairwise counting table with links between matchups.png|thumb|444x444px|Green arrows point from the loser of the matchup to the winner. Yellow arrows indicate a tie. Red arrows (not shown here) indicate the opposite of green arrows (i.e. who lost the matchup).For example, the B>A matchup points to A>B with a green arrow because A pairwise beats B (head-to-head).]]
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]
 
Line 348 ⟶ 345:
|-
|A
|'''50 points'''
|A>B
|A>C
Line 354 ⟶ 351:
|B
|B>A
|'''3550 points'''
|B>C
|-
Line 360 ⟶ 357:
|C>A
|C>B
|'''2050 points'''
|}
This reduces the amount of space required to store and demonstrate all of the relevant information for calculating the result of the voting method.
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A generalized solution is to, for each candidate the voter ranks or rated, give them one vote, and then for each candidate ranked or rated above that candidate on the voter's ballot, indicate a negative vote in that pairwise matchup. That is, the above B>A voter would be considered as voting for B and A, with -1 vote recorded for A>B. Then, for each vote that a candidate has, that is one vote they get in every head-to-head matchup against all other candidates. So the B>A voter gives 1 vote to B>write-in and A>write-in, and 1 vote to both B>A and A>B, but because of the -1 vote for A>B, this just becomes 1 vote for B>A.
 
For equally ranked candidates, such as A=B>C>D, you put a negative vote for both A>B and B>A. Bonus: The votes for each candidate can be placed in the blank cell comparing themselves to themselves in the pairwise matrix i.e. A>A would contain A's votes.<ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/|title=Possible solution to the Condorcet write-in problem|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref></blockquote>Note that pairwise counting can be done either by looking at the margins expressed on a voter's ballot, or the "winning votes"-relevant information (see [[Defeat strength]]). For example, a voter who scores one candidate a 5 and the other a 3 on a rated ballot can either be thought of as giving those scores to both candidates in the matchup (winning votes-relevant information) or as giving 2 points to the first candidate and 0 to the second (only the margins). For ranked and choose-one ballots, both margins and winning votes approaches yield the same numbers, since a voter can only give maximal support to at most one candidate in the matchup.{{reflist|group=nb}}
 
== References ==
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