Condorcet Method - Circular Ambiguities

Circular Ambiguities

As noted above, sometimes an election has no Condorcet winner because there is no candidate who is preferred by voters to all other candidates. When this occurs the situation is known as a 'majority rule cycle', 'circular ambiguity', 'circular tie', 'Condorcet paradox', or simply 'cycle'. This situation emerges when, once all votes have been added up, the preferences of voters with respect to some candidates form a circle in which every candidate is beaten by at least one other candidate. For example, if there are three candidates, Candidate Rock, Candidate Scissors, and Candidate Paper, there will be no Condorcet winner if voters prefer Candidate Rock over Candidate Scissors and Scissors over Paper, but also Candidate Paper over Rock. Depending on the context in which elections are held, circular ambiguities may or may not be a common occurrence. Nonetheless there is always the possibility of an ambiguity, and so every Condorcet method must be capable of determining a winner when this occurs. A mechanism for resolving an ambiguity is known as ambiguity resolution or Condorcet completion method.

Circular ambiguities arise as a result of the voting paradox—the result of an election can be intransitive (forming a cycle) even though all individual voters expressed a transitive preference. In a Condorcet election it is impossible for the preferences of a single voter to be cyclical, because a voter must rank all candidates in order and can only rank each candidate once, but the paradox of voting means that it is still possible for a circular ambiguity to emerge.

The idealized notion of a political spectrum is often used to describe political candidates and policies. This means that each candidate can be defined by her position along a straight line, such as a line that goes from the most right wing candidates to the most left wing, with centrist candidates occupying the middle. Where this kind of spectrum exists and voters prefer candidates who are closest to their own position on the spectrum there is a Condorcet winner (Black's Single-Peakedness Theorem). Real political spectra, however, are at least two dimensional, with some political scientists advocating three dimensional models.

In Condorcet methods, as in most electoral systems, there is also the possibility of an ordinary tie. This occurs when two or more candidates tie with each other but defeat every other candidate. As in other systems this can be resolved by a random method such as the drawing of lots. Ties can also be settled through other methods like seeing which of the tied winners had the most first choice votes, but this and some other non-random methods may re-introduce a degree of tactical voting, especially if voters know the race will be close.

The method used to resolve circular ambiguities is the main difference between Condorcet methods. There are countless ways in which this can be done, but every Condorcet method involves ignoring the majorities expressed by voters in at least some pairwise matchings.

Condorcet methods fit within two categories:

• Two-method systems, which use a separate method to handle cases in which there is no Condorcet winner
• One-method systems, which use a single method that, without any special handling, always identifies the winner to be the Condorcet winner

Many one-method systems and some two-method systems will give the same result as each other if there are fewer than 4 candidates in the circular tie, and all voters separately rank at least two of those candidates. These include Smith-Minimax, Ranked Pairs, and Schulze.