Pairwise Elections and the Condorcet Winner

In the late 1700's, the Marquis de Condorcet (pronounced con-dor-SAY) proposed an alternative to the voting method put forth by Jean-Charles de Borda. Since there can be no confusion as to the winner of a two-way election, Condorcet's idea is to consider all possible pairwise elections among the candidates.

Consider the "milk, soda, juice" example. In a pairwise milk versus soda election, the 6 voters that have milk top-ranked vote for milk and the 5 voters that have soda top-ranked vote for soda. The 4 remaining voters have juice top-ranked, but they can't vote for juice. Instead they have to vote for their second choice, soda. Thus soda wins the milk versus soda election by a score of 9 to 6. The other pairwise elections are shown below.

Milk versus Soda Soda wins 9-6	Milk versus Juice Juice wins 9-6	Soda versus Juice Soda wins 11-4

Notice that, using the representation triangle, it is easy to add up a candidate's score in a pairwise election. Simply divide the triangle in half between the two candidates and add up the numbers on each side.

When we look at pairwise elections, if one candidate wins all of the elections they are involved in, we call that candidate the Condorcet winner. If a candidate loses all of the elections they are involved in, that candidate is the Condorcet loser. In this case, soda wins both of the pairwise elections it is involved in, so soda is the Condorcet winner. Since milk loses both of its pairwise elections, milk is the Condorcet loser.

Unfortunately, sometimes pairwise elections do not give us a decisive winner. Consider the following profile:

If we look at pairwise elections in this profile, we find that X beats Y, Y beats Z, and Z beats X. So in this example there is no Condorcet winner or Condorcet loser. We say that instead there is a Condorcet cycle.

In the following interactive diagram, explore pairwise elections and the Condorcet winner. Can you find more examples where there is not a Condorcet winner? Can you create a profile where the Condorcet winner is different from the plurality winner?

Elections with more than two candidates can have disputed results depending on which method is used, and so it seems natural to look at pairwise elections, whose results cannot be disputed. However, pairwise elections can yield no result, forcing us to consider some alternative way to determine a winner.

In the next section, we will summarize the methods we have seen so far, and briefly discuss some other methods.

The Positional Method

Table of Contents

Conclusion