Voting Paradoxes

Condorcet's Paradox

Some paradoxes arise because we erroneously attribute to collective bodies the same properties as we assign to individuals. A case in point is Condorcet's voting paradox, also known as the phenomenon of cyclic majorities. Even though each individual in a voting body may have a (transitive) preference ordering over the alternatives or candidates, the collective choices based on pairwise majority comparisons may result in a preference cycle: A is better than B, B is better than C, but C is better than A.

Suppose that a group of three persons, 1, 2, and 3, is to elect a person from a set of three applicants, Brown, Jones, and Smith, for a position. Suppose that Elector 1 considers Brown best, Jones the second best, and Smith third. Elector 2's ranking, in turn, is Jones, Smith, Brown, while Elector 3's preference is Smith, Brown, Jones. As is common in voting bodies, two candidates may be paired in an initial vote and the winner is then paired with the third candidate (and so on if there are more than three candidates), with the candidate surviving the final vote declared the overall winner. Suppose that Brown and Smith are first paired. Elector 1 prefers Brown, Elector 2 Smith, and Elector 3 [Page 696]Smith. Hence, Smith gets the majority of votes and is paired in the second vote with Jones. Now, Jones gets two votes and Smith one. Hence, Jones is the overall winner. The initial paradox consists in the observation that Brown, eliminated on the first vote, would have beaten Jones. The underlying paradox is that majority preference is cyclical: Brown is beaten by Smith, Smith is beaten by Jones, and Jones is beaten by Brown. Thus any of the candidates can be elected, given an appropriate order for pairing them. This is Condorcet's paradox, named after the 18th-century French social philosopher Marquis de Condorcet.

The method of pairwise majority comparisons where two comparisons are conducted for three candidates, three comparisons for four candidates, and, in general, k− 1 comparisons for k candidates, which underlies Condorcet's paradox, is based on the mistaken assumption that if candidate A beats candidate B, A also beats all other candidates that B beats. In the above example, even though the declared overall winner Jones defeats Smith—the winner of the first contest—he does not defeat Brown, the loser of the first contest. In fact, Condorcet's paradox can be expressed as the possibility that no matter which candidate is elected, a majority of voters will be frustrated because they would have preferred another candidate to the elected one.

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