EXAMPLE 6 Manipulating Sequential Pairwise Voting

Sequential pairwise voting can also be manipulated by a single voter, even in the case of three voters and three candidates. For example, consider the following two elections with the agenda , , and .

In Election 1, defeats by a score of 2 to 1, so moves on to meet . But defeats by a score of 2 to 1, so is the winner in Election 1. Election 2 is the result of Voter 1 (on the left) submitting a disingenuous ballot in which he or she has elevated (his or her actual second choice) to first place. It is now clear that first defeats by a score of 2 to 1 and then moves on to defeat by this same score. Hence, is the winner in Election 2. This is an instance of manipulation in which Voter 1 has secured a more preferred outcome by submitting an insincere ballot, because Voter 1 actually prefers to (assuming that his or her ballot in Election 1 represents his or her true preferences). This shows that sequential pairwise voting is manipulable.