Condorcet's paradox: when the majority goes round in circles
Perfectly consistent voters can produce a majority that is not. This two-century-old result still sheds light on the way collective decisions are made.
Three voters, three options, no winner
Imagine three people facing three options. The first prefers A to B and B to C, the second B to C and C to A, the third C to A and A to B. Each has a clear ranking, without the slightest internal contradiction.
Now compare the options in pairs. A majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. The collective preference forms a circle: whichever option is chosen, a majority prefers another one.
An eighteenth-century discovery
Condorcet describes this phenomenon in his Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, published in 1785. He was looking for a faithful decision rule, and found that majority voting may designate no winner at all.
In the same period, Jean-Charles de Borda proposed a rival method, which awards points according to the rank of each option. It always designates a winner, but that winner may lose a head-to-head contest against another option, which raises a symmetrical problem.
What Arrow's theorem generalised
In 1951, Kenneth Arrow showed in Social Choice and Individual Values that the difficulty is not peculiar to majority voting. As soon as there are at least three options, no rule satisfies at once a small number of conditions that all seem reasonable.
The result does not say that every collective decision is arbitrary. It says that each rule sacrifices one of these requirements, and that the choice of rule is itself a trade-off between desirable properties that cannot all be combined.
Two readings of the paradox
A first reading, developed notably by William Riker in Liberalism against Populism in 1982, concludes that the majority will has no stable content. Whoever sets the order of the votes can then steer the outcome, and voting serves above all as a way of removing those who govern.
A second reading stresses the conditions of the paradox. Duncan Black showed in 1948 that if preferences can be arranged along a single axis, each with a single peak, the cycle disappears. On this reading, cycles are possible in theory but rare in real choices.
What this changes in reading a result
The result of a vote depends on the preferences expressed, but also on the rule and on the order in which the options are pitted against each other. Before saying what a majority wants, one therefore needs to know which procedure made it speak.
The paradox also sheds light on a common situation: an assembly that rejects each version of a text in turn, although no alternative attracts more support. Such deadlock may reflect a cycle rather than ill will on the part of the participants.
For anyone trying to place themselves, the lesson is a modest one. Having consistent preferences does not guarantee that the collective outcome will be consistent, and a voting procedure deserves to be judged on its properties as much as on the side it favours.
Key points
Consistent individual preferences can produce a circular majority preference.
Arrow's theorem shows that no decision rule combines all the desirable properties.
The debate concerns how often cycles actually occur, which some judge rare and others decisive.
This article describes documented mechanisms and disagreements. It recommends no position, no party and no vote, and commits to no reading other than that of the work it cites.
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