EXAMPLE 7 Agenda Manipulation of Sequential Pairwise Voting

Suppose that we have four candidates and three voters who we know will be submitting the following preference list ballots:

Now suppose that we have agenda-setting power in the sense that we get to choose the order in which the one-on-one contests will take place. Remarkably, we can arrange for the winner to be whichever of the four candidates we want.

The intuition behind finding an agenda that will yield a certain candidate as the winner arises from the observation that candidates who appear later in the agenda are favored over candidates who appear early in the agenda. For example, if we want to win, we place last and look for which candidates would, in fact, defeat one on one. Here, only defeats , and so we want to arrange for to be eliminated along the way. But defeats one on one, so if we choose the agenda , , , , we have that is eliminated by in the first round, then is eliminated by in the second round, and finally is eliminated by in the third round, leaving as the winner. Exercise 16 (page 457) asks you to find the three other agendas that will, in turn, yield , , and as the winner.