Condorcet paradox

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In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory. The result states that it is logically impossible for any voting system to guarantee the winner has a majority of the vote, because it is possible no such winner exists: in some situations, a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.

In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, which a majority of voters support instead. Thus, any attempt to ground social decision-making in majoritarianism must accept such self-contradictions (commonly called spoiler effects). Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called Condorcet methods.

Condorcet's paradox is a special case of Arrow's paradox, which shows that any kind of social decision-making process is either self-contradictory, a dictatorship, or incorporates information about the strength of different voters' preferences (e.g. cardinal utility or rated voting).