In group decision-making it is possible for a voting paradox to form.

Social choice theory dates from Condorcet's formulation of the voting paradox.

In this respect it is comparable to analysis of the voting paradox from use of majority rule as a value.

This is illustrated by the following example of Condorcet's voting paradox:

The indifference maps there generate the voting paradox.

Where this condition (or slightly weaker versions - see Kramer, 1973) fails to hold, the celebrated voting paradox may arise.

A Condorcet winner will not always exist in a given set of votes, which is known as Condorcet's voting paradox.

The result generalizes the voting paradox, which shows that majority voting may fail to yield a stable outcome.

The result generalizes and deepens the voting paradox to any voting rule satisfying the conditions, however complex or comprehensive.

Such a set can be constructed with Unrestricted Domain and an adaptation of the voting paradox to imply a still smaller set.

According to Brennan and Lomasky, the voting paradox can be resolved by differentiating between expressive and instrumental interests.

This famous voting paradox, noted as early as the eighteenth century by Borda and Condorcet, has given rise to a voluminous literature.

Strong empirical evidence that people engage in strategic voting, and important differences between voting and applause, make this resolution of the voting paradox inadequate.

When a Condorcet method is used to determine an election, a voting paradox among the ballots can mean that the election has no Condorcet winner.

Pressman has also written on economic methodology, arguing that the voting paradox cannot be resolved by claiming that voting is like clapping or applauding for a candidate.

Majority voting satisfies conditions P, I and D. Where it breaks down is that it does not satisfy U, as we have seen with the example of the voting paradox given earlier.

(The minimum number of alternatives that can form such a cycle (voting paradox) is 3 if the number of voters is different from 4, because the Nakamura number of the majority rule is 3.

Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet voting method in which ranking several candidates can produce a loop of preference when the weights are compared (see voting paradox).

The Marquis de Condorcet publishes Essai sur l'application de l'analyse á la probabilité des décisions rendues á la pluralité des voix including his voting paradox, the Condorcet method of voting and his jury theorem.

The single-crossing property as reformulated by Milgrom and Shannon was subsequently shown by Joshua Gans and Michael Smart not only to resolve Condorcet's Voting paradox in majority voting and social choice theory but also to give rise to a complete characterization of social preferences.

The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive), even if the preferences of individual voters are not.