Let's look at an example to illustrate how the Borda count can be manipulated.
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:
| Voter 1 | Voter 2 |
|---|---|
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:
| Voter 1 | Voter 2 |
|---|---|
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:
| Voter 1 | Voter 2 |
|---|---|
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.
Voter 2 could also change the outcome of the original election by moving to the bottom of his ballot (with Voter 1 making no change in her ballot). Explain why this would be a pointless thing for Voter 2 to do.
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In presenting an example of a voting system's susceptibility to manipulation, we will typically present two elections—the original one ("Election 1") in which we assume all ballots are sincere, and the one that contains a disingenuous ballot from a voter ("Election 2"). For example, if we collect the pieces of what we just did, this instance of manipulation of the Borda count could be presented succinctly as follows:
| Election 1 | ||
|---|---|---|
| Rank | Number of Voters (2) | |
| First choice | ||
| Second choice | ||
| Third choice | ||
| Fourth choice | ||
| Election 2 | ||
|---|---|---|
| Rank | Number of Voters (2) | |
| First choice | ||
| Second choice | ||
| Third choice | ||
| Fourth choice | ||
There are three aspects of manipulation taking place in this example that deserve comment.
First, there is only one voter (the voter on the left, in this example) changing his or her ballot; we call this a unilateral change in ballot. An example involving a unilateral change of ballot is sometimes referred to as an instance of "single-voter manipulation" to distinguish it from a situation wherein a group of voters, acting in concert, can change their ballots so that all of them prefer the new winner to the original winner. We'll see examples of group manipulation in Section 10.2.
Second, the original election produced a single winner, as did the new election held after we finished constructing Voter 1's disingenuous ballot. Thus, because we know each voter's sincere preference ranking for the candidates, we also know exactly which of the two election outcomes each voter will prefer. Ties, on the other hand, present a problem. For example, if a voter has sincere preferences that rank over over over , then it's not at all obvious whether this voter will prefer an election outcome that ties and to an election outcome that ties and or vice versa.
Third, the voter who is changing her ballot changes the election outcome to one that she prefers. It really is pointless to submit an insincere ballot if doing so only makes the election outcome worse in your opinion.
A voting system is manipulable if there is at least one scenario in which some voter can achieve a more preferred election outcome by unilaterally changing his or her ballot. The precise definition follows.
Manipulability DEFINITION
A voting system is said to be manipulable if there exist two sequences of preference list ballots and a voter (call the voter Jane) such that
With this definition at hand, we can now turn to the study of the manipulability of some of the particular voting systems introduced in the last chapter.
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EXAMPLE 2 Finding a Successful Manipulation with the Borda Count
Suppose there are two voters and four candidates, and suppose the true preferences of the voters are reflected in the following ballots:
| Voter 1 | Voter 2 |
|---|---|
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, is the winner of this election using the Borda count. Voter 2 can indeed change the outcome of the election, but only in a pointless way. That is, the winning alternative is her top choice, and so it is quite impossible for her to do any better than this!
Voter 1, however, is quite unhappy with being the winner, and she would like to change her ballot so as to make either or the winner (as she prefers both of these to ). Let's first see if Voter 1 can change her ballot so as to make the winner. Voter 1 certainly cannot increase 's Borda score of 3 from the original election. But had a Borda score of 4 and the most Voter 1 can do is to reduce this by 1 (by moving to the bottom of her ballot). Hence, there is no way that Voter 1 can change her ballot to make the unique winner.
Voter 1, however, can make the unique winner. To do this, she first moves to the top of her ballot (an obvious thing to do because she is trying to make the winner). That raises 's Borda score to 4. But still has a Borda score of 4, so she must move to the bottom of her ballot, reducing the Borda score of to 3. Now we must check that neither nor have a Borda score of 4, but this is easily done. Hence, Voter 1 can submit a disingenuous ballot and achieve a result she prefers to that of the original election. Hence, this example also serves to show that the Borda count is manipulable.
Show that Voter 2 can change her original ballots so that (assuming Voter 1 submits her original ballot) the winner becomes . And again, why is this a pointless move on Voter 2's part?