Pairwise counting: Difference between revisions
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+  == 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: 
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election: 

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The following terms are often used when discussing pairwise counting: 
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, 
+  '''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''': 
+  '''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''': 
+  '''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]], 
+  '''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. 
==Notes== 
==Notes== 
Revision as of 17:45, 21 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]}
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.
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.
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.