 Condorcet’s second insight, often called Condorcet’s paradox, is the observation that majority preferences can be ‘irrational’ (specifically, intransitive) even when individual preferences are ‘rational’ (specifically, transitive). Suppose, for example, that one third of a group prefers alternative \(x\) to \(y\) to \(z\), a second third prefers \(y\) to \(z\) to \(x\), and a final third prefers \(z\) to \(x\) to \(y\). Then there are majorities (of two thirds) for \(x\) against \(y\), for \(y\) against \(z\), and for \(z\) against \(x\): a ‘cycle’, which violates transitivity. Furthermore, no alternative is a Condorcet winner, an alternative that beats, or at least ties with, every other alternative in pairwise majority contests.