Condorcet paradox

The possibility that pairwise majority preferences form a cycle, even when every individual voter has consistent preferences. With three options, a majority may prefer A to B, B to C, and C to A.

Majority rule: A decision rule that selects the option supported by more than half of voters. The paradox arises when majority rule is applied separately to each pair of options.
Preference relation: A formal relation expressing how one option is ranked relative to another. The paradox concerns how individual and collective preference relations differ.
Condorcet method: A family of voting methods that selects an option defeating every other option in pairwise majority contests, when one exists. A Condorcet winner avoids the cycle; methods differ when no such option exists.
Condorcet winner criterion: The principle that an election outcome should select an option that beats every rival in pairwise majority contests. The paradox matters because some preference profiles have no option satisfying this criterion.
Marquis de Condorcet: An eighteenth-century French mathematician and political thinker who studied voting and social choice. He described the possibility that majority preferences could be cyclic.
Pairwise comparison: A method that compares alternatives two at a time. The cycle is built from the outcomes of three pairwise comparisons.
Transitive relation: A relation in which x related to y and y related to z implies x related to z. Each voter can rank options transitively while the majority relation is not transitive.
Borda count: A positional voting method that assigns points according to each option’s rank on each ballot. It always produces a ranking but can choose differently from pairwise majority comparisons.
Smith set: The smallest nonempty set of candidates that each member defeats every candidate outside the set in pairwise majority contests. When a cycle exists, the Smith set identifies the group of candidates involved at its top.
Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix: Condorcet’s 1785 work applying probability and mathematics to decisions made by voting. It presented his analysis of majority decisions and the cyclic-preference problem.

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