10.4 10.3 The Manipulability of Other Voting Systems for Three or More Candidates
Manipulability and the Borda Count
Example 1 showed how a single voter can manipulate an election in which the Borda count is being used. But Example 1 involved four candidates. Is there a simpler example involving only three candidates?
The answer turns out to be "no," provided that we continue to interpret the notion of a "more preferred election outcome" to be a switch from a single winner to another single winner (as opposed to a switch creating or breaking a tie). This negative answer is formalized in the following theorem.
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The Nonmanipulability of the Borda Count with Exactly Three Candidates THEOREM
With exactly three candidates, the Borda count cannot be manipulated in the sense of a voter unilaterally changing an election outcome from one single winner to another single winner that he or she prefers according to that voter's ballot in the first election, which we take to be sincere preferences.
Let's see why this is true. Suppose the candidates are , , and , and that you prefer to , but is the election winner using the Borda count. We'll show that any attempt you make to manipulate the election by changing your ballot so that emerges as the winner (using the Borda count) is doomed to failure.
Because you prefer to , your sincere ballot can be one of only three possibilities, corresponding to whether is ranked first, second, or third. We'll consider each case in turn.
- Case 1. Your sincere ballot is over over . No ballot change on your part can increase 's Borda score, and you can decrease 's Borda score by no more than 1. Thus, at best, you can make a unilateral change that results in and having the same Borda score, whereas successful manipulation on your part requires that have a strictly higher Borda score than after your ballot change.
- Case 2. Your sincere ballot is over over . No ballot change on your part can decrease 's Borda score, and you can increase 's Borda score by no more than 1. Thus, at best, you can make a unilateral change that results in and having the same Borda score, whereas successful manipulation on your part requires that have a strictly higher Borda score than after your ballot change.
- Case 3. Your sincere ballot is over over . No ballot change on your part can increase 's Borda score or decrease 's Borda score. Thus, after your ballot change, will still have a higher Borda score than , so your attempt at manipulation has failed in this case as well.
So with three candidates, the Borda count is nonmanipulable. With more than three candidates, the Borda count does not fare as well, regardless of how many voters there are.
The Manipulability of the Borda Count with Four or More Candidates THEOREM
With four or more candidates (and two or more voters), the Borda count can be manipulated in the sense that there exists an election in which a voter can change the election outcome unilaterally from one single winner to another single winner that he or she prefers according to that voter's ballot in the first election, which we take to be sincere preferences.
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As we've seen in Example 1, the Borda count can be manipulated in the case of four candidates and two voters. This is really half the battle, as we can modify that example to serve in any case in which the number of voters is even, as follows:
- Any candidates in addition to , , , and can be placed below those four on every ballot.
- The rest of the voters can be paired off with the members of each pair holding ballots that rank the candidates in exactly opposite orders (thus "canceling each other out" in terms of the Borda scores).
The following example illustrates this method of generalizing our earlier instance of manipulation of the Borda count to the case of five candidates and six voters.
EXAMPLE 4 Manipulating the Borda Count with Five Candidates and Six Voters
Consider the following two elections:
| Election 1 | |||||
|---|---|---|---|---|---|
| Election 2 | |||||
|---|---|---|---|---|---|
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.
To handle the case where the number of voters is odd, we need to start with a four-candidate, three-voter example of manipulation of the Borda count. Exercise 8 (page 457) provides this. We can then modify this example to work for any odd number of voters by again adding pairs of ballots that cancel each other out exactly as we did before. Exercises 9 and 10 (page 457) fill in some of the details needed for this part of the argument and ask you to provide the necessary explanations and calculations.
Manipulability of Runoff Systems
EXAMPLE 5 Manipulability of Runoff Systems
Both the plurality runoff rule and the Hare system are manipulable. But rather than give the whole story away, we'll just present the sequences of sincere ballots in each case. Exercises 13 and 14 (page 457) ask you to figure out how the leftmost voter in each case can secure a more preferred outcome by a unilateral change of ballot.
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| Election 1 for the Hare System | ||||
|---|---|---|---|---|
| Election 1 for the Plurality Runoff Rule |
||||
|---|---|---|---|---|
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 .
| Election 1 | |||
|---|---|---|---|
| Rank | Number of Voters (3) | ||
| First choice | |||
| Second choice | |||
| Third choice | |||
| Election 2 | |||
|---|---|---|---|
| Rank | Number of Voters (3) | ||
| First choice | |||
| Second choice | |||
| Third choice | |||
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.
Sequential Pairwise Voting and Agenda Manipulability
Thus, sequential pairwise voting can also be manipulated by a single voter, even in the case of three voters and three candidates. But there is another aspect of manipulability that arises with this particular voting system that is of even more interest, and this is something called agenda manipulation.
Agenda Manipulation DEFINITION
Agenda manipulation refers to the ability to control who wins an election with sequential pairwise voting by a choice of the agenda.
William H. Riker, in his book The Art of Political Manipulation, spoke of the possibility that "those in control of procedures can manipulate the agenda by, for example, restricting alternatives [candidates] or by arranging the order in which they are brought up." The following example provides a striking illustration of this with sequential pairwise voting.
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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:
| Rank | Number of Voters (3) | ||
|---|---|---|---|
| First choice | |||
| Second choice | |||
| Third choice | |||
| Fourth choice | |||
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.
Self Check 3
The agenda , , , yielded as the winner in Example 7. There is a trivial change to this agenda that also yields as the winner. What is this trivial change?
- Switch the first two to get , , , . The trivial change of switching the order of the first two in the agenda never has an effect on which one-on-ones take place.
Self Check 4
In Example 7, is there an agenda in which is first in the agenda, is last in the agenda, and is the winner?
- Yes. There are only two agendas with first and last, and wins with both of them.
Self Check 5
In Example 7, why is it the case that if is first in the agenda, then will definitely not win?
- If is first in the agenda, then one of two things must happen. Either loses to some alternative in the agenda between and , or defeats all of these and then loses to .
Plurality Voting and Group Manipulability
In the real world, all other voting systems pale in comparison to plurality voting in terms of the significance of the role played by disingenuous voting. "Throwing away your vote"—as some accuse Nader voters in Florida of doing in the 2000 presidential election—represents a choice, conscious or otherwise, to forgo obtaining a more desired outcome through strategic considerations.
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Ironically, plurality voting, like Condorcet's method, is nonmanipulable according to the formal definition given on page 444. However, a group of voters, acting together, can change an election outcome into something they all prefer. This observation gives rise to the following definition, theorem, and explanation of why the theorem is true.
Group Manipulability DEFINITION
A voting system is group manipulable if there are elections in which a group of voters can change their ballots so that the new winner is preferred to the old winner by everyone in the group, assuming that the original ballots represent the true preferences of each voter in the group.
The Group Manipulability of Plurality Voting THEOREM
Plurality voting cannot be manipulated by a single individual. However, it is group manipulable.
First of all, let's see why no individual can manipulate plurality voting. Suppose that you prefer to , but is the winner with plurality voting. Then has at least one more first-place vote than . Now, because you prefer to , we know that is not on top of your sincere ballot, so no ballot change that you make can subtract from 's number of first-place votes. Moreover, by moving to the top of your ballot, you only increase 's number of first-place votes by 1. Thus, the best you can do with a unilateral change in ballots is to move into a tie with .
To see that plurality voting is group manipulable, we only have to look at any real-world election in which a third-party candidate acted as the "spoiler." As we've said, Ralph Nader was exactly that in the state of Florida in the 2000 presidential election. Another example occurs in Exercise 18 (page 458).
At this point, we've seen that several of our familiar voting systems for three or more candidates—the Borda count, runoff systems, sequential pairwise voting—can be manipulated. Can't we do better than this in attempting to improve on Condorcet's method? We turn to this question next.