The voting paradox (also known as Condorcet's paradox or the paradox of voting) 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): Marie Jean Antoine Nicolas Caritat, marquis de Condorcet (September 17, 1743 - March 28, 1794) was a French philosopher, mathematician, and early political scientist who devised the concept of a Condorcet method. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... In mathematics, the word transitive admits at least two distinct meanings: A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. ... Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist. ...

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.

Also, if an election were held with the above three voters as the only participants, nobody would win, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop their vote for C, then Voter 3 can choose between either A or B - and become the agenda-setter.

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. Any election method conforming to the Condorcet criterion is known as a Condorcet method. ... The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. ... Any election method conforming to the Condorcet criterion is known as a Condorcet method. ...

See also

• Kenneth Arrow, Section 1 with an example of a distributional difficulty of intransitivity + majority rule
• Arrow's impossibility theorem
• Gibbard-Satterthwaite theorem
• Independence of irrelevant alternatives
• Smith set
• Social Choice and Individual Values
• voting system
• Paradox of Majority Rule, a Madisonian rendering