Condorcet paradox

A majority of the dots are closer to B than A, C than B, and A than C.Note that a cycle or circular figure can be drawn pointing from B to C, C to A, and A to B.

A Condorcet cycle example with ice cream flavors, with reference to the Smith set.

The voting paradox, Condorcet paradox, or Condorcet cycle is when within a set of candidates, no one candidate is preferred by at least as many voters as all the other candidates in the set when looking at their pairwise matchups. It essentially means that within that set of candidates, no matter which candidate you pick, more voters always prefer some other candidate in the set. If there is a Condorcet cycle for 1st place (the winner), then all candidates in the cycle will be in the Smith set. It is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive), even if the preferences of individual voters are not i.e. between three candidates, the first can be preferred by a majority over the second, and the second by a majority over the third, yet the first candidate isn't preferred by a majority over the third, or even, the third candidate can be preferred by a majority over the first candidate. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

Another way of thinking about the Condorcet paradox in the context of Condorcet methods is that just because, say, candidate A is better than candidate B by majority rule when only they are running, doesn't mean that candidate B isn't better than candidate A when more candidates are running. Because of this illogicality, all Condorcet methods fail Independence of irrelevant alternatives.

For example, suppose we have three candidates, A, B and C, and that there are three voters with preferences as follows (candidates being listed in decreasing order of preference):

Voter 1: A > B > C

Voter 2: B > C > A

Voter 3: C > A > B

The pairwise preferences can be visualized as:

	A	B	C
A	---	2 (+1 Win)	1 (-1 Loss)
B	1 (-1 Loss)	---	2 (+1 Win)
C	2 (+1 Win)	1 (-1 Loss)	---

If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2 i.e. the first and second) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion.

Notes

Often the number of candidates in the cycle is mentioned as (number of candidates)-cycle i.e. a cycle between 3 candidates will be called a 3-cycle. When a Condorcet method is used to determine an election, a voting paradox among the ballots can mean that the election has no beats-all winner. The several variants of the Condorcet method differ chiefly on how they resolve such ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution based solely off of the ranked preferences to this trivial example because each candidate is in an exactly symmetrical situation.

It is believed to be uncommon for Condorcet cycles to occur, happening in about 9% of elections, depending on the scenario and makeup of the electorate. See w:Condorcet_paradox#Likelihood_of_the_paradox

Condorcet cycles can arise either from honest votes, or from strategic votes. Some cycle resolution methods were invented primarily to elect the "best" candidate in the cycle when the cycle is created by honest voters, whereas others were invented on the assumption that most cycles would be artificially induced so that a faction could change the winner to someone they preferred over the original winner by strategically exploiting the cycle resolution method, and therefore attempt to make such strategic attempts fail or backfire, though this can sometimes mean that these cycle resolution methods elect "worse" candidates if the cycle was induced by honest votes.

Note that the above example demonstrates Condorcet methods' failure of the Favorite Betrayal criterion: if any voter switches their 1st choice and 2nd choice around in their rankings, then their 2nd choice will become the Condorcet winner (for example, if Voter 1 had voted B>A>C, then B would be majority-preferred over A and C and thus win). This may be strategically difficult to exploit when the Condorcet cycle is based on honest preferences, however, because there are often multiple types of voters in the cycle who have an incentive to Favorite Betray, meaning that some voters can actually benefit by not Favorite Betraying; for example, if Voter 1 Favorite Betrays as mentioned above, then Voter 2 need not do anything in order to elect their 1st choice; but if Voter 2, unaware of Voter 1's action, tries to Favorite Betray to make their 2nd choice, C, win, then they will have inadvertently lost the chance to elect their 1st choice. If every voter in this example Favorite Betrays in favor of their 2nd choice, then the ballots will stay exactly the same (i.e. there will still be one voter voting A>B>C, one voter voting B>C>A, and one voter voting C>A>B, though which voter votes which way will change).

Condorcet cycles can never appear in cardinal methods when deciding the winner, because if some candidate (Candidate A) has a higher summed or average score than another candidate (Candidate B), then A will always have a higher summed or average score than every candidate that B has a higher summed or average score over. However, there will still be (if there is no change in voter preferences after the election, and those voters' preferences would create a cycle for 1st place i.e. the winner if ran through a Condorcet method) more voters who prefer someone else over the Utilitarian winner. If "intensity of preference" information is included, the cycle can be resolved by electing the candidate with the highest summed or average score in the cycle, as in Smith//Score and Smith//Approval.

3-candidate Condorcet cycles are one of the easiest situations to check if a voting method that passes the majority criterion fails Favorite Betrayal, because no matter who the voting method elects between the 3, if enough voters Favorite Betray, they can make their 2nd choice become a majority's 1st choice, making it guaranteed that they must win. In order to pass Favorite Betrayal, the method must provide some non-betrayal manner of producing the same result.

See also

• Independence of irrelevant alternatives
• Arrow's impossibility theorem
• Gibbard-Satterthwaite theorem
• Smith set

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