The voting paradox 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 Condorcet 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 to this trivial example because each candidate is in an exactly symmetrical situation.
A paradox is an apparently true statement or group of statements that seems to lead to a contradiction or to a situation that defies intuition, such as "This statement is false".
Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context (or language) to lose their paradox quality.
Supplee's paradox: the buoyancy of a relativistic object (such as a bullet) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest
Condorcet continued to receive prestigious appointments: in 1777, he was appointed Secretary of the Académie des Sciences, and, in 1782, secretary of the Académie Française.
The paradox states that it is possible for a majority to prefer A over B, another majority to prefer B over C, and another majority to prefer C over A, all from the same electorate and same set of ballots.
Condorcet was interred in The Pantheon in 1989, in honor of the bicentennial of the French Revolution and Condorcet's role as a central figure in the Enlightenment.
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