Round-robin voting

"Paired voting" redirects here. For the strategic practice of trading votes across states, see vote pairing.

Round-robin voting (also called paired/pairwise comparison or tournament voting) refers to a set of ranked voting systems that elect winners by comparing all candidates in a round-robin tournament. Every candidate is matched up against every other candidate, where their point total is equal to the number of votes they receive; the method then selects a winner based on the results of these paired matchups.

Round-robin methods are one of the four major categories of single-winner electoral methods, along with multi-stage methods (including instant-runoff voting and Baldwin's method), positional methods (including plurality and Borda), and graded methods (including score and STAR voting).

While most methods satisfying the Condorcet criterion are pairwise-counting methods, some are not. A handful of sequential-loser methods satisfy the Condorcet criterion, as do many Condorcet-hybrid methods.

Summary

In paired voting, each voter ranks candidates from first to last (or rates them on a scale); candidates not ranked by voters are given the lowest rank or score.

For each pair of candidates (as in a round-robin tournament), we count how many votes rank each candidate over the other candidate. Thus each pair will have two totals, the size of its majority and the size of its minority.

Pairwise counting

In the pairwise-counting procedure, we compare each pair of candidates (as in a round-robin tournament), counting how many voters rank each candidate over the other.

Pairwise counts are often displayed in a pairwise comparison or outranking matrix. In these matrices, each row represents 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.

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 runner is defeated.

Alternatively, the margin matrix can be used for most methods. The margin matrix considers only the difference in the vote shares of the two candidates, making it antisymmetric (i.e. the top half is the negative of the bottom half).

Example

Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:

• Memphis, the largest city, but far from the others (42% of voters)
• Nashville, near the center of the state (26% of voters)
• Chattanooga, somewhat east (15% of voters)
• Knoxville, far to the northeast (17% of voters)

The preferences of each region's voters are:

42% of voters Far-West	26% of voters Center	15% of voters Center-East	17% of voters Far-East
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

A pairwise-comparison matrix can be constructed as:

A B	Memphis	Nashville	Chattanooga	Knoxville
Memphis		[A] 58% [B] 42%	[A] 58% [B] 42%	[A] 58% [B] 42%
Nashville	[A] 42% [B] 58%		[A] 32% [B] 68%	[A] 32% [B] 68%
Chattanooga	[A] 42% [B] 58%	[A] 68% [B] 32%		[A] 17% [B] 83%
Knoxville	[A] 42% [B] 58%	[A] 68% [B] 32%	[A] 83% [B] 17%
Copeland score:	0-3-0	3-0-0	2-1-0	1-2-0
Minimax score:	58%	42%	68%	83%