Round-robin, pairedcomparison, or tournamentvoting 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.
Most, but not all, election methods meeting the Condorcet criterion are based on pairwise counting.
Summary
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 counting
Pairwise counts are often displayed in a 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]
Opponent
Runner
A
B
C
D
A
—
0
0
1
B
1
—
1
1
C
1
0
—
1
D
0
0
0
—
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]
Pairwise counts
A
B
C
A
A > B
A > C
B
B > A
B > C
C
C > A
C > B
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:
Opponent
Runner
A
B
C
D
A
—
2
2
2
B
1
—
1
2
C
1
2
—
2
D
1
1
1
—
In the sum matrix above, A is the Condorcet winner, because they beats 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 (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.
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.
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.
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.[10]
Tally table
All possible pairs
of candidates
Number of votes with indicated preference
Margin
Prefer X to Y
Prefer Y to X
X − Y
X = Memphis
Y = Nashville
42%
58%
-16%
X = Memphis
Y = Chattanooga
42%
58%
-16%
X = Memphis
Y = Knoxville
42%
58%
-16%
X = Nashville
Y = Chattanooga
68%
32%
+36%
X = Nashville
Y = Knoxville
68%
32%
+36%
X = Chattanooga
Y = Knoxville
83%
17%
+66%
Related Research Articles
In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory. The result implies that it is logically impossible for any voting system to guarantee a winner will have support from a majority of voters: in some situations, a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.
A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner or Pairwise Majority Rule Winner (PMRW). The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
The Copeland or Llull method is a ranked-choice voting system based on counting each candidate's pairwise wins and losses.
The Smithset, sometimes called the top-cycle, generalizes the idea of a Condorcet winner to cases where no such winner exists. It does so by allowing cycles of candidates to be treated jointly, as if they were a single Condorcet winner. Voting systems that always elect a candidate from the Smith set pass the Smith criterion. The Smith set and Smith criterion are both named for mathematician John H Smith.
Ranked Pairs (RP), also known as the Tideman method, is a tournament-style system of ranked voting first proposed by Nicolaus Tideman in 1987.
The Schulze method, also known as the beatpath method, is a single winner ranked-choice voting rule developed by Markus Schulze. The Schulze method is a Condorcet completion method, which means it will elect a majority-preferred candidate if one exists. In other words, if most people rank A above B, A will defeat B. Schulze's method breaks cyclic ties by using indirect victories. The idea is that if Alice beats Bob, and Bob beats Charlie, then Alice (indirectly) beats Charlie; this kind of indirect win is called a beatpath.
A Condorcet winner is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.
A voting system satisfies join-consistency if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion. Systems that fail the consistency criterion are susceptible to the multiple-district paradox, which allows for a particularly egregious kind of gerrymander: it is possible to draw boundaries in such a way that a candidate who wins the overall election fails to carry even a single electoral district.
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
In voting systems, the Minimax Condorcet method is a single-winner ranked-choice voting method that always elects the majority (Condorcet) winner. Minimax compares all candidates against each other in a round-robin tournament, then ranks candidates by their worst election result. The candidate with the largest (maximum) number of votes in their worst (minimum) matchup is declared the winner.
Positional voting is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will or it may form a mathematical sequence such as an arithmetic progression, a geometric one or a harmonic one. The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes.
CPO-STV, or the Comparison of Pairs of Outcomes by the Single Transferable Vote, is a ranked voting system designed to achieve proportional representation. It is a more sophisticated variant of the Single Transferable Vote (STV) system, designed to overcome some of that system's perceived shortcomings. It does this by incorporating some of the features of the Condorcet method, a voting system designed for single-winner elections, into STV. As in other forms of STV, in a CPO-STV election, more than one candidate is elected, and voters must rank candidates in order of preference. As of February 2021, it has not been used for a public election.
The Kemeny–Young method is an electoral system that uses ranked ballots and pairwise comparison counts to identify the most popular choices in an election. It is a Condorcet method because if there is a Condorcet winner, it will always be ranked as the most popular choice.
In social choice theory, the independence of (irrelevant) clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a weak form of the independence of irrelevant alternatives (IIA) criterion that nevertheless is failed by a number of voting rules. A method that passes the criterion is said to be clone independent.
The Borda method or order of merit is a positional voting rule which gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on. Once all votes have been counted, the option or candidate with the most points is the winner.
Dodgson's method is an electoral system based on a proposal by mathematician Charles Dodgson, better known as Lewis Carroll. The method searches for a majority-preferred winner; if no such winner is found, the method proceeds by finding the candidate who could be transformed into a Condorcet winner with the smallest number of ballot edits possible, where a ballot edit switches two neighboring candidates on a voter's ballot.
Ranked voting is any voting system that uses voters' rankings of candidates to choose a single winner or multiple winners. More formally, a ranked system is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.
There are a number of different criteria which can be used for voting systems in an election, including the following
The later-no-help criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.
This article discusses the methods and results of comparing different electoral systems. There are two broad ways to compare voting systems:
Metrics of voter satisfaction, either through simulation or survey.
↑ Darlington, Richard B. (2018). "Are Condorcet and minimax voting systems the best?". arXiv:1807.01366 [physics.soc-ph]. CC [Condorcet] systems typically allow tied ranks. If a voter fails to rank a candidate, they are typically presumed to rank them below anyone whom they did rank explicitly.
↑ Hazewinkel, Michiel (2007-11-23). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN978-0-306-48373-8. Briefly, one can say candidate Adefeats candidate B if a majority of the voters prefer A to B. With only two candidates [...] barring ties [...] one of the two candidates will defeat the other.
↑ Nurmi, Hannu (2012), "On the Relevance of Theoretical Results to Voting System Choice", in Felsenthal, Dan S.; Machover, Moshé (eds.), Electoral Systems, Studies in Choice and Welfare, Springer Berlin Heidelberg, pp.255–274, doi:10.1007/978-3-642-20441-8_10, ISBN9783642204401, S2CID12562825
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