Condorcet Systems

A Condorcet system is simply a system in which the Condorcet criterion is obeyed above all other rules. In particular, elections are implemented in the following way.

• Ranked Ballots. A Condorcet ballot asks the voter to rank the candidates from top to bottom. There is some variation in different methods; some allow the voter to rank candidates “equally,” and often the voter need not rank all candidates, but only their top choices.
• Pairwise Comparisons. Election results are interpreted as being a set of runoffs between any pair of candidates. Thus the act of ranking a candidate above another (or ranking one, but not the other) is interpreted as one vote for the higher candidate in a runoff between the two.
• Condorcet Victory. If there is a Condorcet winner, that is, a candidate that wins in its runoff with every other candidate, that candidate is immediately declared the winner.

The typical way of representing the votes of a Condorcet election are in a matrix. For example, the following might be the result of a Condorcet election between three candidates. Each number corresponds to the number of voters who chose the candidate for that row over the candidate for that column. In this case, Bob Jones is the Condorcet winner, John Hennessy second place.

Ambiguity Resolution

All Condorcet methods agree on the basic procedure for choosing a Condorcet winner. It is a simple extrapolation of majority rule, and when it works, it satisfies all of the criteria we set forth. However, as discussed under The Paradox, an ambiguity can arise, and what distinguishes various Condorcet methods is the way in which they resolve this ambiguity, if it arises. In all these methods, the candidate pool is first reduced to what is called the Smith Set, which is the smallest set of candidates, each of which defeats every candidate outside of the set. Intuitively, this eliminates all candidates who cannot possibly be considered winners. Next the ambiguities must be resolved within the Smith Set. Three prominent methods are these. See Example to see a demonstration of each of these methods for a particular electoral result.

• Minimax. In this system, the winner is the candidate whose worst defeat is best. Intuitively, the idea is to pick the candidate who was most nearly a Condorcet winner in the sense that even their worst defeat was closest to being a victory.
• Ranked Pairs. In this system, all pairwise defeats are ranked in terms of strength. The defeats are then examined from strongest to weakest. The strongest defeat is “locked in” but ranking the winner above the loser. Each defeat is then examined, and if it does not violate any previous decisions, it is also locked in. This is a bit subtle. The intuition is that we should regard each defeat as sufficient cause to rank the winner above the loser, but since this cannot be done for defeats, we at least ensure that it is done for the defeats of strongest margin.
• Deferral to Alternative Method. In this case, an alternative method is used on the same election result, such as Instant Runoff, Borda, Approval, or Plurality. These alternate methods are discussed elsewhere on this site.

For a discussion of how one might go about choosing the best resolution procedure, see Method Selection.