# Condorcet paradox

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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. This illogicality means that 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.

## Known Condorcet cycles

While rare, Condorcet cycles have been documented. They are more common in small elections or where the outcome is sufficiently close to a tie that they can be produced by noise. Some examples are:

## Modeling Condorcet cycles

Simple mathematical models may under- or overestimate the probability of Condorcet cycles. For instance, the impartial culture model maximizes the chance of a Condorcet cycle among a very broad range of voting models. Failing to take voter indifference into account can also lead to models that overstate the chance of a cycle.[3]

## 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 (which contrasts to how there is always a utilitarian 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.

However, most Condorcet methods still do narrow down their selection of a final winner to a best set of candidates. Most Condorcet advocates unambiguously agree that the Smith set is a minimum standard for this best set (i.e. a Condorcet method must be Smith-efficient, meaning that it always picks someone in the Smith set.) However, some advocates further prefer subsets of the Smith set i.e. they prefer some members of the Smith set over others, though they still agree that anyone in the Smith set would be better than anyone not in the Smith set. One of the most prominent of these subsets is the Schwartz set.

Condorcet rankings can be modified to work when there are Condorcet cycles by using Smith set rankings instead.

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.

Strategic Condorcet cycles arise when voters use burial strategy to help their preferred candidate pairwise beat someone who beats the buried candidate. In essence, it involves making the buried candidate no longer be a Condorcet winner, and be in a cycle with the candidate preferred by those attempting burial, such that the Condorcet cycle resolution method elects the candidate preferred by the buriers. This is one major point of comparison between different Condorcet methods; some methods allow voters to resist burial without Favorite Betrayal, such as Condorcet-cardinal hybrid methods.

Note that the above example demonstrates some potential for Favorite Betrayal in Condorcet methods: 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). Or if they equally rank their 1st and 2nd choice as 1st choices. 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.

## References

1. McCune, David (2023-03-10). "Ranked Choice Bedlam in a 2022 Oakland School Director Election". arXiv:2303.05985 [econ.GN].
2. The parliamentary investigation committee for Gardermoen. "Dokument nr. 18 (in Norwegian)" (PDF). The Norwegian Parliament web site. It's likely that the choice of voting order determined the outcome. In all likelihood there was a cyclical majority ('roterende flertall'). In such a case, the term 'the will of the majority' is meaningless, and one cannot assert that the actual outcome respects the will of the majority...
3. Fishburn, Peter C.; Gehrlein, William V. (1980). "The paradox of voting. Effects of individual indifference and intransitivity". Journal of Public Economics. Elsevier BV. 14 (1): 83–94. doi:10.1016/0047-2727(80)90006-7. ISSN 0047-2727.
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