Pairwise counting: Difference between revisions
→Terminology
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|}To read this, take for example the cell where BK is compared to AM; "4067 (AM)" means that 4067 voters preferred AM (Andy Montroll) over BK (Bob Kiss), and "3477 (BK)" means that 3477 voters preferred BK over AM. Because AM got more votes than BK in that matchup, AM won that matchup.
== Terminology ==
The following terms are often used when discussing pairwise counting:
'''Pairwise matchup''': Also known as a head-to-head matchup, it is when voters are asked to indicate their preference between two candidates, with the one that voters prefer winning. It is usually done on the basis of majority rule (i.e. if more voters prefer one candidate over the other than the number of voters who have the opposing preference, then the candidate preferred by more voters wins the matchup) using [[Choose-one voting|choose-one voting]], though see the [[Pairwise counting#Cardinal methods|Cardinal methods]] section for alternative ways. Pairwise matchups can be simulated from ranked or rated ballots and then assembled into a table to show all of the matchups simultaneously; see above.
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."▼
▲'''Pairwise win/beat''' and '''pairwise lose/defeated''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses." Often this is represented by writing "Pairwise winner>Pairwise loser"; this can be extended to show a [[beatpath]] by showing, for example, "A>B>C>D", which means A pairwise beats B, B pairwise beats C, and C pairwise beats D (though it may or may not be the case, depending on the context, that, for example, A pairwise beats C).
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.▼
▲'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins a matchup is the pairwise winner of the matchup (not to be confused with the pairwise champion; see the definition two spots below). The other candidate is the pairwise loser of the matchup. (Note that sometimes "pairwise loser" is also used to refer to a [[Condorcet loser]], which is a candidate who is pairwise defeated in all of their matchups).
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.▼
▲'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup. (Note that sometimes it is also called a tie when there is pairwise cycling, though this is different; see the definition two spots below.)
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.▼
'''Pairwise champion''': Also known as a beats-all winner or [[Condorcet winner]], it is a candidate who pairwise beats every other candidate. Due to pairwise ties (see above) and pairwise cycling (see below), there is not always a pairwise champion.
'''Pairwise cycling:''' Also known as a [[Condorcet cycle]], it is when within a set of candidates, each candidate has at least one pairwise defeat.
'''Minimal pairwise dominant set''': Also known as the [[Smith set]], it is the smallest group of candidates who pairwise beat all others. The [[Pairwise champion|pairwise champion]] will always be the only member of this set when they exist.
▲'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], it is
== Condorcet ==
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- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.
== Cardinal methods ==
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]
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