Set three voters' preference rankings and run a round-robin vote. Does majority rule produce a winner — or a cycle?
Condorcet's Paradox (majority cycles): Even when every individual voter has rational preferences, aggregating those preferences through majority rule can produce a collective preference that is intransitive — a cycle where A beats B, B beats C, and C beats A. Because every alternative can be beaten by another, no stable majority winner exists (no Condorcet winner). The paradox shows that individual rationality does not guarantee collective rationality.
Condorcet winner: If an alternative beats every other alternative in pairwise majority voting, it is the Condorcet winner and represents the majority-preferred outcome.