Condorcet paradox
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. 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 (the fewest candidates preferred by more voters than all others). 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. 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.
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
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) 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.
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.
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.
See also
- Independence of irrelevant alternatives
- Arrow's impossibility theorem
- Gibbard-Satterthwaite theorem
- Smith set