Voting paradox

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 appears paradoxical, because it means that majority wishes can be in conflict with each other. This paradox can be explained away by the fact that in that case the majorities are made up of different groups of individuals. The paradox is highlighted by the Condorcet method of voting, which will fail to determine a winner in such a situation — an alternate technique must then be used.

This is best illustrated by an 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 majority of voters (two thirds in each case) prefer A to B, B to C, and C to A.

See Condorcet method#Resolving Disputes for the ways in which these situations can be resolved. Note that the trivial example given above is unresolvable fairly because each candidate is in an exactly symmetrical situation.

See also: Arrow's Impossibility Theorem, Gibbard-Satterthwaite theorem, Smith set, voting system.