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A voting paradox in which collective majority preferences become cyclical (A beats B, B beats C, C beats A), even when individual voters have consistent rankings.
The Condorcet Paradox, identified by the French mathematician and philosopher Marquis de Condorcet in his 1785 Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, shows that majority rule can produce intransitive group preferences even when every individual voter holds transitive (logically consistent) preferences.
The classic illustration uses three voters and three options:
Pairwise majority voting yields A beating B (voters 1 and 3), B beating C (voters 1 and 2), and C beating A (voters 2 and 3). No option is a Condorcet winner — an alternative that defeats every other in head-to-head contests — and the social ranking cycles endlessly.
The paradox has several major implications for political research:
Practical voting systems respond differently. Condorcet methods (such as Schulze and Ranked Pairs, used by the Wikimedia Foundation and the Debian project for internal elections) explicitly search for a Condorcet winner and apply tie-breaking rules when a cycle exists. Plurality, instant-runoff (IRV), and Borda count do not guarantee selection of a Condorcet winner even when one exists.
The paradox remains central to social choice theory, mechanism design, and debates over electoral reform.
Empirical studies (e.g., Gehrlein, 2006) suggest cycles are mathematically possible in roughly 6–9% of three-option elections under impartial culture assumptions, but documented real-world cycles are rarer because voter preferences tend to cluster ideologically.
The paradox is the simplest concrete example of the problem Arrow generalized: no ranked voting rule can guarantee transitivity, non-dictatorship, unanimity, and independence of irrelevant alternatives simultaneously when there are three or more alternatives.
A Condorcet winner is an option that defeats every other option in pairwise majority comparisons. When the paradox occurs, no Condorcet winner exists, which is why the cycle arises.