EXAMPLE 1 Manipulating the Borda Count with Four Candidates and Two Voters

Suppose there are two voters and four candidates, and suppose the true preferences of the voters are reflected in the following ballots:

Using the Borda count with point values 3, 2, 1, 0 (or by counting the number of occurrences of other candidates below the one in question, as described in Section 9.3), we see that the Borda scores of the four candidates are as follows:

Thus, Candidate wins this election. Voter 1, however, would have preferred to see Candidate —her top choice, according to her true preferences—win this election rather than Candidate , her second choice.

Assume that Voter 1 had known that Voter 2 planned to submit the ballot that he cast above. Could Voter 1 have secured a victory for Candidate by submitting a disingenuous ballot?

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The answer here, as we'll show, turns out to be "yes." The intuition is fairly transparent: Voter 1 wants to pretend that is not her second choice, but her last choice. Let's see if this is enough to bring about the desired switch in winner from to . The new ballots and Borda scores are as follows:

Close, but not quite what we wanted: Candidates and now tie for the win, and we wanted the winner to be just Candidate . But a moment's inspection reveals that Voter 1 can achieve this if, in addition to plunging Candidate to the bottom of her ballot, she also flip-flops and . That is, the desired ballots (and Borda scores) that yield Candidate as the sole winner are as follows:

Thus, Voter 1 can change her ballot and—with Voter 2 making no change at all— cause the election outcome to go from to . Moreover—and this is very important— Voter 1 prefers to ! The reason we know that Voter 1 prefers to is that we are assuming the original ballots represented the voters′ true preferences, and Voter 1 ranked over on her original ballot.