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.

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 or outranking matrix 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

Tennessee's four cities are spread throughout the state

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)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

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

The following square-grid table displays the candidates in the same order in which they appear above.

Square grid

	... 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.

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%

Here is an example of a pairwise victory table for the Burlington 2009 election:

				style="background-color: Template:Green Party (United States)/meta/color;" |	style="background-color: Template:Independent politician/meta/color;" |	style="background-color: Template:Vermont Republican Party/meta/color;" |	style="background-color: Template:Vermont Progressive Party/meta/color;" |	style="background-color: Template:Vermont Democratic Party/meta/color;" |
			wi	JS	DS	KW	BK	AM
style="background-color: Template:Vermont Democratic Party/meta/color; width: 2px;" |	AM	Andy Montroll (5–0)						5 Wins ↓
style="background-color: Template:Vermont Progressive Party/meta/color; width: 2px;" |	BK	Bob Kiss (4–1)					1 Loss → ↓ 4 Wins	4067 (AM) – 3477 (BK)
style="background-color: Template:Vermont Republican Party/meta/color; width: 2px;" |	KW	Kurt Wright (3–2)				2 Losses → 3 Wins ↓	4314 (BK) – 4064 (KW)	4597 (AM) – 3668 (KW)
style="background-color: Template:Independent politician/meta/color; width: 2px;" |	DS	Dan Smith (2–3)			3 Losses → 2 Wins ↓	3975 (KW) – 3793 (DS)	3946 (BK) – 3577 (DS)	4573 (AM) – 2998 (DS)
style="background-color: Template:Green Party (United States)/meta/color; width: 2px;" |	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	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)

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 against another candidate, they "pairwise beat" that candidate, with the other candidate in the matchup pairwise losing.

Pairwise winner and pairwise loser: the candidate who pairwise wins is the pairwise winner, with the other candidate being the pairwise loser.

Pairwise tie: When two candidates receive the same number of votes in their pairwise matchup.

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. One does not always exist due to the Condorcet paradox, though there is always a Smith ranking which more generally holds instead for groups of candidates.