EXAMPLE 1 Condorcet’s Method

The mathematics department is hiring a new faculty member and the five-person hiring committee has interviewed four candidates: Adam, Beth, Carol, and Dan. They have decided to use Condorcet’s method on their five ballots (reproduced in the table below). Let’s see who gets the offer.

To find a winner using Condorcet’s method, we begin by choosing an ordering of the candidates, which we’ll take to be alphabetical for this example. Thus, we first pit Adam against each of the others in a one-on-one contest based on the ballots. In an Adam-versus-Beth election, based on these ballots, Voters 1 and 4 would vote for Adam; and Voters 2, 3, and 5 would vote for Beth. Hence, Beth would win this one- on-one contest against Adam, so we know Adam is not going to be the winner using Condorcet’s method. But Beth still has a chance, so we move on to see how Beth would fare in a one-on-one contest against Carol (knowing already that Beth would defeat Adam). But here it is easy to see that Beth would lose to Carol (with Voters 2, 3, and 4 voting for Carol).

Hence, Beth is not a winner with Condorcet’s method, but Carol has not yet been eliminated (and neither has Dan, but remember that we have chosen to check out the candidates in alphabetical order). First pitting Carol against Adam, we see that Carol wins 3 to 2. And pitting Carol against Dan, we see that Carol again wins by this same 3 to 2 score. This shows that Carol is the winner using Condorcet’s method!

By the way, if it seems like we never gave Dan a chance, notice that our determination that Carol is the Condorcet winner means that we already know that she beats Dan one on one. So Dan can’t be a winner in this election using Condorcet’s method.