Condorcet criterion

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The Condorcet winner of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. Mainly because of Condorcet's paradox, a Condorcet winner will not always exist in a given set of votes.

The Condorcet criterion for a voting system is that it chooses the Condorcet winner when one exists. Any method conforming to the Condorcet criterion is known as a Condorcet method.

Complying Methods

Black, Smith/IRV, Copeland, Minimax, Smith/Minimax, Ranked Pairs and variations (Maximize Affirmed Majorities, Maximum Majority Voting), Schulze and variations (Schwartz Sequential Dropping, Cloneproof Schwartz Sequential Dropping) comply with the Condorcet criterion.

Approval voting, Cardinal Ratings, Borda count, Plurality voting, and Instant-Runoff Voting do not.

Commentary

Non-ranking methods such as Plurality and Approval cannot comply with the Condorcet Criterion because they do not allow each voter to fully specify their preferences. But Instant-Runoff Voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet Criterion.

Consider, for example, the following vote count of preferences with three candidates {A,B,C}:

499:	A,B,C
498:	C,B,A
3:	B,C,A

In this case, B is preferred to A by 501 votes to 499, and B is preferred to C by 502 to 498, hence B is preferred to both A and C. So according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transfered voted from B; in pluratity voting A wins with the most first choices.