Beats matrix

Pairwise counts are often displayed in a beats, tournament, pairwise comparison,^[4] or outranking matrix^[5] such as those below. In these matrices, each row represents each candidate as a 'runner', while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank.^[6][7]

Imagine there is an election between four candidates: A, B, C and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are B > C > A > D; that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the opponent, while a '0' indicates that the opponent is preferred over the runner.^[6][4]

In this matrix the number in each cell indicates either the number of votes for runner over opponent (runner,opponent) or the number of votes for opponent over runner (opponent, runner).

If pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:

• A vs. B
• A vs. C
• B vs. C

If the number of voters who have no preference between two candidates is not supplied, it can be calculated using the supplied numbers. Specifically, start with the total number of voters in the election, then subtract the number of voters who prefer the first over the second, and then subtract the number of voters who prefer the second over the first.

The pairwise comparison matrix for these comparisons is shown below.^[8]

A candidate cannot be pairwise compared to itself (for example candidate A can't be compared to candidate A), so the cell that indicates this comparison is either empty or contains a 0.

Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using matrix addition. The resulting sum of all ballots in an election is called the sum matrix, and it summarizes all the voter preferences.

An election counting method can use the sum matrix to identify the winner of the election.

Suppose that this imaginary election has two additional voters, and their preferences are D > A > C > B and A > C > B > D. Added to the first voter, these ballots yield the following sum matrix:

In the sum matrix above, A is the Condorcet winner, because they beat every other candidate one-on-one. When there is no Condorcet winner, ranked-robin methods such as ranked pairs use the information contained in the sum matrix to choose a winner.

The first matrix above, which represents a single ballot, is inversely symmetric: (runner,opponent) is ¬(opponent,runner). Or ${\displaystyle (runner,opponent)+(opponent,runner)=1}$. The sum matrix has the property: ${\displaystyle (runner,opponent)+(opponent,runner)=}$ N for N voters, if all runners are fully ranked by each voter.

Number of pairwise comparisons

For N candidates, there are N · (N − 1) pairwise matchups, assuming it is necessary to keep track of tied ranks. When working with margins, only half of these are necessary because storing both candidates' percentages becomes redundant.^[9] For example, for 3 candidates there are 6 pairwise comparisons (and 3 pairwise margins), for 4 candidates there are 12 pairwise comparisons, and for 5 candidates there are 20 pairwise comparisons.