# Pairwise counting

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.

Pairwise counts
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)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

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

The following square-grid table uses the popularity sequence calculated by the Condorcet-Kemeny method, which does calculations that ensure that the sum of the pairwise counts in the upper-right 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.

Tally table
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%

## References

1. Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
2. 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:10.1007/978-3-642-20441-8_10. ISBN 978-3-642-20440-1. Retrieved 2020-01-16.