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 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: 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.