Difference between revisions of "Pairwise counting"
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'''Pairwise counting''' is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results. 
'''Pairwise counting''' is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results. 

−  Most, but not all, election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting.<ref group=nb>[[Nanson's methodNanson]] meets the [[Condorcet criterion]] and [[Instantrunoff voting]] meets the [[Condorcet loser criterion]].</ref> 
+  Most, but not all, election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting.<ref group=nb>[[Nanson's methodNanson]] meets the [[Condorcet criterion]] and [[Instantrunoff voting]] meets the [[Condorcet loser criterion]].</ref> See the [[Pairwise counting#CondorcetCondorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]]. 
== Example without numbers == 
== Example without numbers == 

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In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells. 
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells. 

+  
+  Note that since a candidate can't be pairwise compared to themselves (i.e. candidate B can't be compared to candidate B, since there's only one candidate in the comparison), the cell that does so is always empty. 

== Example with numbers == 
== Example with numbers == 

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'''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. 

−  '''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 rankingSmith ranking]] that applies to groups of candidates. 
+  '''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 rankingSmith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists. 
+  
+  == Condorcet == 

+  In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties). 

+  
+  In the context of [[Condorcet methods]]: 

+  
+   A [[Condorcet winner]] is a candidate for whom all their cells are shaded green. 

+  
+   The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other. 

+  
+   The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties. 

+  
+   A [[Condorcet loser criterionCondorcet loser]] is a candidate for whom all their cells are shaded red. 

+  
+   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. 

==Notes== 
==Notes== 
Revision as of 18:39, 24 February 2020
Pairwise counting is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results.
Most, but not all, election methods that meet the Condorcet criterion or the Condorcet loser criterion use pairwise counting.^{[nb 1]} See the Condorcet section for more information on the use of pairwise counting in Condorcet methods.
Example without numbers
As an example, if pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:
 Number of voters who prefer A over B
 Number of voters who prefer B over A
 Number of voters who have no preference for A versus B
 Number of voters who prefer A over C
 Number of voters who prefer C over A
 Number of voters who have no preference for A versus C
 Number of voters who prefer B over C
 Number of voters who prefer C over B
 Number of voters who have no preference for B versus C
Often these counts are arranged in a pairwise comparison matrix^{[1]} or outranking matrix^{[2]} table such as below.
A  B  C  

A  A > B  A > C  
B  B > A  B > C  
C  C > A  C > B 
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.
Note that since a candidate can't be pairwise compared to themselves (i.e. candidate B can't be compared to candidate B, since there's only one candidate in the comparison), the cell that does so is always empty.
Example with numbers
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
 Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
 Nashville, with 26% of the voters, near the center of Tennessee
 Knoxville, with 17% of the voters
 Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





As these ballot preferences are converted into pairwise counts they can be entered into a table.
The following squaregrid table displays the candidates in the same order in which they appear above.
... over Memphis  ... over Nashville  ... over Chattanooga  ... over Knoxville  
Prefer Memphis ...    42%  42%  42% 
Prefer Nashville ...  58%    68%  68% 
Prefer Chattanooga ...  58%  32%    83% 
Prefer Knoxville ...  58%  32%  17%   
The following tally table shows another table arrangement with the same numbers.
All possible pairs of choice names 
Number of votes with indicated preference  

Prefer X over Y  Equal preference  Prefer Y over X  
X = Memphis Y = Nashville 
42%  0  58% 
X = Memphis Y = Chattanooga 
42%  0  58% 
X = Memphis Y = Knoxville 
42%  0  58% 
X = Nashville Y = Chattanooga 
68%  0  32% 
X = Nashville Y = Knoxville 
68%  0  32% 
X = Chattanooga Y = Knoxville 
83%  0  17% 
Election examples
Here is an example of a pairwise victory table for the Burlington 2009 election:
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  Writein (0–5)  5 Losses →  3338 (JS) –
165 (wi) 
6057 (DS) –
117 (wi) 
6063 (KW) –
163 (wi) 
6149 (BK) –
116 (wi) 
6658 (AM) –
104 (wi) 
Terminology
The following terms are often used when discussing pairwise counting:
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 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 tie: Occurs when two candidates receive the same number of votes in their pairwise matchup.
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 ranking that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.
Condorcet
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).
In the context of Condorcet methods:
 A Condorcet winner is a candidate for whom all their cells are shaded green.
 The Smith set is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.
 The Schwartz set is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.
 A Condorcet loser is a candidate for whom all their cells are shaded red.
 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.
Notes
 ↑ Nanson meets the Condorcet criterion and Instantrunoff voting meets the Condorcet loser criterion.
References
 ↑ Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
 ↑ Nurmi, Hannu (2012). Felsenthal, Dan S.; Machover, Moshé (eds.). "On the Relevance of Theoretical Results to Voting System Choice". Electoral Systems. Springer Berlin Heidelberg: 255–274. doi:10.1007/9783642204418_10. ISBN 9783642204401. Retrieved 20200116.