Round-robin,pairedcomparison, ortournamentvoting methods, are a set ofranked voting systems that choose winners by comparing every pair of candidates one-on-one, similar to around-robin tournament.^[1] In each paired matchup, the total number of voters who prefer each candidate is recorded in abeats matrix. Then, amajority-preferred (Condorcet) candidate is elected, if one exists. Otherwise, if there is acyclic tie, the candidate "closest" to being a Condorcet winner is elected, based on the recorded beats matrix. How "closest" is defined varies by method.

Round-robin methods are one of the four major categories ofsingle-winner electoral methods, along withmulti-stage methods (likeRCV-IRV),positional methods (likeplurality andBorda), andgraded methods (likescore andSTAR voting).

Most, but not all, election methods meeting theCondorcet criterion are based on pairwise counting.

In paired voting, each voterranks candidates from first to last (orrates them on a scale).^[2] For each pair of candidates (as in around-robin tournament), we count how many votes rank each candidate over the other.^[3]

Pairwise counts are often displayed in apairwise comparison^[4] oroutranking matrix^[5] such as those below. In thesematrices, 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 andD. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences areB >C >A >D; that is, the voter rankedB first,C second,A third, andD 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 namedA,B, andC, 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.

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

A candidate cannot be pairwise compared to itself (for example candidateA can't be compared to candidateA), 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 usingmatrix 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 areD >A >C >B andA >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 asranked 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 forN voters, if all runners are fully ranked by each voter.

ForN candidates, there areN · (N − 1) pairwise matchups, assuming it is necessary to keep track oftied 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.

SupposeTennessee is holding an election on the location of itscapital. The population is split between four cities, andall the voters want the capital to be as close to them as possible. The options are:

These ranked preferences indicate which candidates the voter prefers. For example, the voters in the first column prefer Memphis as their 1st choice, Nashville as their 2nd choice, etc. 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.

The following tally table shows another table arrangement with the same numbers.^[10]

• Single-winner electoral systems
• Preferential electoral systems

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