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 on this voter's ballot over the 'opponent', while a '0' indicates that the runner is defeated. For example, in the following matrix, A beats D:
Using a matrix like the one above, one can find the overall results of an election. Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using matrix addition. The sum of all ballots in an election is called the sum matrix.
Suppose that in the imaginary election there are two other voters. Their preferences are (D, A, C, B) and (A, C, B, D). Added to the first voter, these ballots would give the following sum matrix:
When the sum matrix is found, the contest between each pair of candidates is considered. The number of votes for runner over opponent (runner,opponent) is compared with the number of votes for opponent over runner (opponent,runner) to find the Condorcet winner. In the sum matrix above, A is the Condorcet winner because A beats every other candidate. When there is no Condorcet winner, many Condorcet completion methods, such as Ranked Pairs and the Schulze method, use the information contained in the sum matrix to choose a winner.
Cells marked '—' in the matrices above have a numerical value of '0', but a dash is used since candidates are never preferred to themselves. The first matrix, that 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 were fully ranked by each voter.
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.
The candidates for the capital are:
The preferences of the voters would be divided like this:
To find the Condorcet winner every candidate must be matched against every other candidate in a series of one-on-one contests. In each pairing the winner is the candidate preferred by a majority of voters. The results for every possible pairing are as follows:
The results can also be shown in the form of a matrix:
As can be seen from both of the tables above, Nashville beats every other candidate. This means that Nashville is the Condorcet winner. So Nashville would win an election held under any Condorcet method.
While any Condorcet method will elect Nashville as the winner, first-past-the-post and instant-runoff voting would select Memphis and Knoxville respectively, using the same ballots. This would occur despite the fact that most people would have preferred Nashville to either of those options. Condorcet methods make these preferences obvious rather than ignoring or discarding them.
On the other hand, note that in this example Chattanooga also defeats Knoxville and Memphis when paired against those cities. If Memphis voters had preferred Chattanooga as a second choice rather than as a third choice, then Chattanooga would be the Condorcet winner even though finishing in last place in a first-past-the-post election.
An alternative way of thinking about this example if a Smith-efficient Condorcet method that passes ISDA is used to determine the winner is that 58% of the voters, a mutual majority, ranked Memphis last (making Memphis the majority loser and Nashville, Chattanooga, and Knoxville above Memphis, ruling Memphis out. At that point, the voters who preferred Memphis as their 1st choice could only help to choose a winner among Nashville, Chattanooga, and Knoxville, and because they all preferred Nashville as their 1st choice among those three, Nashville would've had a 68% majority of 1st choices among the remaining candidates and won as the majority's 1st choice.
Some Condorcet methods first do a series of pairwise comparisons and then, if there is no Condorcet winner, use an entirely different, non-Condorcet method to determine a winner. The simplest such fall-back methods completely ignore the results of the pairwise comparisons. For example, the Black method chooses the Condorcet winner if it exists, but uses the Borda count instead if there is a cycle (the method is named for Duncan Black).
A more complex two-stage process is, when there is a cycle, to first eliminate all candidates not in some group which is found by looking at the pairwise comparisons, and then use a separate voting system to find the winner. Sets used for this purpose are defined so that they will always contain only the Condorcet winner if there is one, and will always, in any case, contain at least one candidate. Such sets include the
Some Condorcet methods use a single procedure that meets the Condorcet criteria and, without any extra procedure, also pick winners when there are cycles. In other words, these methods do not involve separate procedures for different situations. Typically these methods base their calculations on pairwise counts. These methods include:
Some Condorcet methods produce not just a single winner, but a ranking of all candidates from first to last place. A Condorcet ranking is a list of candidates with the property that the Condorcet winner (if one exists) comes first and the Condorcet loser (if one exists) comes last, and the same holds for the candidates ranked between them.
Methods that satisfy this property include:
Though there won't always be a Condorcet winner or Condorcet loser, there is always a Smith set and "Smith loser set" (smallest group of candidates who lose to all candidates not in the set in head-to-head elections). Some voting methods produce rankings that sort all candidates in the Smith set above all others, and all candidates in the Smith loser set below all others, with this also applying to all candidates ranked between them. Because the Smith set and Smith loser set are equivalent to the Condorcet winner and Condorcet loser when they exist, methods that always produce Smith set rankings also always produce Condorcet rankings.
Many proponents of instant-runoff voting (IRV) are attracted by the belief that if their first choice does not win, their vote will be given to their second choice; if their second choice does not win, their vote will be given to their third choice, etc. This sounds perfect, but it is not true for every voter with IRV. If someone voted for a strong candidate, and their 2nd and 3rd choices are eliminated before their first choice is eliminated, IRV gives their vote to their 4th choice candidate, not their 2nd choice. Condorcet voting takes all rankings into account simultaneously, but this makes it fail the later-no-harm criterion and the later-no-help criterion. With IRV, indicating a second choice will never hurt your first choice. With Condorcet voting, it is possible that indicating a second choice will cause your first choice to lose.