Condorcet Paradox
What is Condorcet Paradox?
The Condorcet paradox is when majority preferences cycle (A beats B, B beats C, C beats A) even though each individual voter has consistent rankings.
Majority rule can produce intransitive group preferences: a majority prefers A to B, another majority prefers B to C, yet another prefers C to A, so no option is a stable winner. This 'voting cycle' means the outcome can depend on the agenda or order of votes rather than true preferences. It is the simplest illustration of why fair aggregation of preferences is hard and is the seed of Arrow's impossibility theorem.
Condorcet Paradox: a worked example
Three trustees rank three uses of a grant: art (A), a bike path (B), a clinic (C). Ana ranks A over B over C, Ben ranks B over C over A, Cara ranks C over A over B. Vote A against B and Ana and Cara pick A, so A wins two to one. Vote B against C and Ana and Ben pick B, so B wins two to one. Vote C against A and Ben and Cara pick C, so C wins two to one. Every option loses to something. Worse, whoever sets the agenda picks the winner: A against B first, then the winner against C, funds the clinic, while starting with B against C funds art.
The mistake students make with condorcet paradox
People assume a cycle proves someone voted irrationally or gamed the ballot. Every individual ranking in the example is perfectly consistent; the intransitivity belongs to the majority rule doing the adding up, not to any voter. The other error is filing the paradox as a rare curiosity. Its likelihood climbs as the number of options grows and as voters' ranking patterns become more varied, which is precisely why the order in which pairs are voted on can decide a real outcome.
Condorcet Paradox questions
What is an example of the Condorcet paradox?
The Condorcet paradox shows up whenever three voters hold the three rotations of one ranking. Say a club picks a trip: the first member ranks the coast above the mountains above the city, the second ranks the mountains above the city above the coast, and the third ranks the city above the coast above the mountains. Then the coast beats the mountains, the mountains beat the city, and the city beats the coast, each by two votes to one.
What is a Condorcet winner?
A Condorcet winner is an option that beats every other option in a head-to-head majority vote. Many elections have one, and voting rules are often judged by whether they manage to pick it. The paradox is the case where no Condorcet winner exists at all, because the pairwise results form a loop, and then the rule you use and the order you vote in determine the outcome.
Why does majority voting produce cycles?
Majority voting produces cycles because each pairwise vote discards everything except the two options being compared, so a different majority can assemble for each pair. No voter is inconsistent; the majorities are simply different groups of people. The information thrown away in each comparison is exactly what would reveal that the three verdicts cannot be stacked into one group ranking.
Related terms
Common comparisons
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