Round-robin, paired comparison, Condorcet completion, or tournament voting methods, are a set of ranked voting systems that choose winners by comparing every pair of candidates one-on-one, similar to a round-robin tournament.[1] In each paired matchup, we record the total number of voters who prefer each candidate in a beats matrix. Then, a majority-preferred (Condorcet) candidate is elected, if one exists. Otherwise, if there is a cyclic 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 of single-winner electoral methods, along with multi-stage methods (like RCV-IRV), positional methods (like plurality and Borda), and graded methods (like score and STAR voting).
Most, but not all, election methods meeting the Condorcet criterion are based on pairwise counting.
In paired voting, each voter ranks candidates from first to last (or rates them on a scale).[2] For each pair of candidates (as in a round-robin tournament), we count how many votes rank each candidate over the other.[3]
Pairwise counts are often displayed in a pairwise comparison[4] or outranking matrix 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.[5] [6]
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.
| A | B | C | D | ||
|---|---|---|---|---|---|
| A | — | 0 | 0 | 1 | |
| B | 1 | — | 1 | 1 | |
| C | 1 | 0 | — | 1 | |
| D | 0 | 0 | 0 | — |
If pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:
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.[7]
| A | A > B | A > C | ||
|---|---|---|---|---|
| B | B > A | B > C | ||
| C | C > A | C > B |
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:
| A | B | C | D | ||
|---|---|---|---|---|---|
| A | — | 2 | 2 | 2 | |
| B | 1 | — | 1 | 2 | |
| C | 1 | 2 | — | 2 | |
| D | 1 | 1 | 1 | — |
A
The first matrix above, which represents a single ballot, is inversely symmetric: (runner,opponent) is ¬(opponent,runner). Or (runner,opponent) + (opponent,runner) = 1. The sum matrix has this property: (runner, opponent) + (opponent, runner) = N for N voters, if all runners are fully ranked by each voter.
For
N
N ⋅ (N-1)
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.
| Prefer Memphis ... | - | 42% | 42% | 42% | |
| Prefer Nashville ... | 58% | - | 68% | 68% | |
| Prefer Chattanooga ... | 58% | 32% | - | 83% | |
| Prefer Knoxville ... | 58% | 32% | 17% | - |
| Number of votes with indicated preference | Margin | |||
|---|---|---|---|---|
| Prefer X to Y | Prefer Y to X | X − Y | ||
| X = MemphisY = Nashville | 42% | 58% | -16% | |
| X = MemphisY = Chattanooga | 42% | 58% | -16% | |
| X = MemphisY = Knoxville | 42% | 58% | -16% | |
| X = NashvilleY = Chattanooga | 68% | 32% | +36% | |
| X = NashvilleY = Knoxville | 68% | 32% | +36% | |
| X = ChattanoogaY = Knoxville | 83% | 17% | +66% |