Pairwise preference: Difference between revisions

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'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], it is a 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 order/ranking''': Also known as a [[Condorcet ranking]], it is a 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.
==Election examples==

Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:
{| class="wikitable"
| colspan="3" rowspan="2" | 
| 
| 
| 
| 
| 
| 
|-
!wi
!JS
!DS
!KW
! BK
!AM
|-
! 
!AM
| colspan="6" | Andy
Montroll (5–0)
|5 Wins ↓
|-
! 
!BK
| colspan="5" |Bob
Kiss (4–1)
|1 Loss →
↓ 4 Wins
|4067 (AM) –
3477 (BK)
|-
! 
!KW
| colspan="4" |Kurt
Wright (3–2)
| 2 Losses →
3 Wins ↓
|4314 (BK) –
4064 (KW)
|4597 (AM) –
3668 (KW)
|-
! 
! DS
| colspan="3" |Dan
Smith (2–3)
|3 Losses →
2 Wins ↓
|3975 (KW) –
3793 (DS)
|3946 (BK) –
3577 (DS)
|4573 (AM) –
2998 (DS)
|-
! 
! JS
| colspan="2" |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)
|}To read this, take for example the cell where BK is compared to AM (the cell with BK on the left and AM on the top); "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.
== Condorcet ==
== 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.
* 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.


<br />
== Strength of preference ==
== Strength of preference ==
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 for a way to do this on a per-matchup basis.
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 for a way to do this on a per-matchup basis.
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Another criticism is that it can be harder to do [[pairwise counting]] than it is to count the vote in other methods, such as [[Approval voting]]. The [[Rated pairwise preference ballot#Rated or ranked preference]] implementation can potentially mitigate this criticism, because for every voter who indicates a rated preference, at most only one piece of information need be collected from their ballot for every candidate they marked (their score for the candidate), rather than several pairwise preferences.
Another criticism is that it can be harder to do [[pairwise counting]] than it is to count the vote in other methods, such as [[Approval voting]]. The [[Rated pairwise preference ballot#Rated or ranked preference]] implementation can potentially mitigate this criticism, because for every voter who indicates a rated preference, at most only one piece of information need be collected from their ballot for every candidate they marked (their score for the candidate), rather than several pairwise preferences.

<references />
One of the notable aspects of pairwise preferences is that a Condorcet winner or member of the Smith set can be found in a simple manner without needing to be done with written ballots; see [[:Category:Sequential comparison Condorcet methods]] for more information.

=== Reading the pairwise table ===
When looking at two candidates, a quick way to figure out the number of votes for the first candidate>second candidate and vice versa is to first locate the cell for "first candidate>second candidate", count the minimum number of cells diagonally one must go to be adjacent to the middle dividing line of the matrix (where there is a --- cell), and then going one cell further diagonally (meaning you'll be starting from the closest cell on the opposite side of that dividing line), go that number of cells further diagonally to reach the other cell. For example:
{| class="wikitable"
!
!A
!B
!C
! D
!E
|-
|A
| ---
| 2
|2
|'''2'''
|2
|-
|B
| 0
| ---
|2
|2
| 2
|-
|C
|0
| 0
| ---
| 2
|2
|-
| D
|''<u>0</u>''
|0
|0
| ---
|0
|-
| E
|0
|0
|0
|0
| ---
|}Try locating A>D (the fifth cell in the second row). To find the reverse, D>A, first you check and see that you have to go one cell down and to the left to be adjacent to the middle dividing line. Then, starting from the cell one cell down and to the left of the middle dividing line, go one cell further down and to the left to reach D>A. In doing this, you would start at A>D, go down to B>C, then jumping over the middle dividing line to C>B, go down to D>A.<references />
[[Category:Condorcet-related concepts]]
[[Category:Condorcet-related concepts]]
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