EXAMPLE 4 Manipulating the Borda Count with Five Candidates and Six Voters

Consider the following two elections:

The ballots of the first two voters (in both elections) are the same as in Example 1 (the manipulation of the Borda count with four candidates and two voters), with the new candidate placed at the bottom of both ballots. The last four voters contribute exactly 8 to the Borda score of each candidate, and so, taken together, they have no effect on who is the winner of the election. This is what we mean by "canceling each other out."

In the first election, as in Example 1, Candidate wins. But if we take these ballots to represent true preferences, the voter on the far left prefers to . Moreover, that voter can achieve this better outcome—Candidate A—by submitting the disingenuous ballot that he or she cast in Election 2.