Condorcet Paradox

Pairwise majority rule can cycle, so no Condorcet winner exists.

The idea

The Condorcet paradox is the observation that majority voting, extended to three or more alternatives, can fail to produce a consistent group ranking.

Proposition (Condorcet paradox).

Say that $x$ beats $y$ when more voters rank $x$ above $y$. There are profiles of individual rankings, each of them transitive, for which the pairwise majorities are not transitive: the tallies return $x$ beats $y$, $y$ beats $z$, and $z$ beats $x$, so that every alternative loses one of its contests and no alternative beats all the others.

With two options, majority rule elects whichever more voters prefer. With three or more, compare the alternatives in pairs: $x$ beats $y$ when more voters rank $x$ above $y$. An alternative that beats every other alternative head to head is a Condorcet winner, and when one exists it is the natural collective choice. Every individual ranking is transitive: no voter prefers $x$ to $y$, $y$ to $z$, and $z$ to $x$. The group's pairwise verdicts need not inherit that property, and when they form a cycle no Condorcet winner exists.

No votes were miscounted. The majority that puts $x$ over $y$ and the majority that puts $y$ over $z$ consist of different voters, and nothing forces their verdicts to fit into a single ranking.

Ways to work on it

• Walkthrough. Tally the pairs of the classic profile and find the majority cycle.
• Practice. Count a single pairwise majority in a cyclic profile.
• Hardest. Build the full majority relation and decide whether a Condorcet winner exists.

Not sure where to start? Take the ten-question placement test.