Difference between revisions of "Pairwise counting"
(→Example with numbers: Copied source code to fix table error) 
(→Example with numbers: More refinements) 

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{{Tenn_voting_example}} 
{{Tenn_voting_example}} 

As these ballot preferences are converted into pairwise counts they can be entered into a table. 
As these ballot preferences are converted into pairwise counts they can be entered into a table. 

+  
The following squaregrid table uses the popularity sequence calculated by the [[KemenyYoung Maximum Likelihood MethodCondorcetKemeny method]], which does calculations that ensure that the sum of the pairwise counts in the upperright triangular area cannot be increased by changing the sequence of the candidates. 
The following squaregrid table uses the popularity sequence calculated by the [[KemenyYoung Maximum Likelihood MethodCondorcetKemeny method]], which does calculations that ensure that the sum of the pairwise counts in the upperright triangular area cannot be increased by changing the sequence of the candidates. 

{ class="wikitable" 
{ class="wikitable" 

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} 
} 

+  
+  
The following tally table shows another table arrangement. 
The following tally table shows another table arrangement. 

{ class="wikitable" 
{ class="wikitable" 

+  +Tally table 

 
 

−  ! rowspan=2"  All possible pairs<br/>of choice names 
+  ! rowspan="2""  All possible pairs<br />of choice names 
−  ! colspan=3  Number of votes with indicated preference 
+  ! colspan="3"  Number of votes with indicated preference 
 
 

! style="textalign:left"  Prefer X over Y 
! style="textalign:left"  Prefer X over Y 

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! style="textalign:left"  Prefer Y over X 
! style="textalign:left"  Prefer Y over X 

 
 

−   align="left"  X = Memphis<br/>Y = Nashville 
+   align="left"  X = Memphis<br />Y = Nashville 
 align="left"  42% 
 align="left"  42% 

 align="left"  0 
 align="left"  0 

 align="left"  58% 
 align="left"  58% 

 
 

−   align="left"  X = Memphis<br/>Y = Chattanooga 
+   align="left"  X = Memphis<br />Y = Chattanooga 
 align="left"  42% 
 align="left"  42% 

 align="left"  0 
 align="left"  0 

 align="left"  58% 
 align="left"  58% 

 
 

−   align="left"  X = Memphis<br/>Y = Knoxville 
+   align="left"  X = Memphis<br />Y = Knoxville 
 align="left"  42% 
 align="left"  42% 

 align="left"  0 
 align="left"  0 

 align="left"  58% 
 align="left"  58% 

 
 

−   align="left"  X = Nashville<br/>Y = Chattanooga 
+   align="left"  X = Nashville<br />Y = Chattanooga 
 align="left"  68% 
 align="left"  68% 

 align="left"  0 
 align="left"  0 

 align="left"  32% 
 align="left"  32% 

 
 

−   align="left"  X = Nashville<br/>Y = Knoxville 
+   align="left"  X = Nashville<br />Y = Knoxville 
 align="left"  68% 
 align="left"  68% 

 align="left"  0 
 align="left"  0 

 align="left"  32% 
 align="left"  32% 

 
 

−   align="left"  X = Chattanooga<br/>Y = Knoxville 
+   align="left"  X = Chattanooga<br />Y = Knoxville 
 align="left"  83% 
 align="left"  83% 

 align="left"  0 
 align="left"  0 
Revision as of 18:48, 21 January 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 uses the popularity sequence calculated by the CondorcetKemeny method, which does calculations that ensure that the sum of the pairwise counts in the upperright triangular area cannot be increased by changing the sequence of the candidates.
... 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.
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% 
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.