10.3 10.2 Majority Rule and Condorcet's Method
Throughout this section, we assume that the number of voters is odd. In Section 9.2 (page 407), we pointed out that with two candidates, majority rule has three very desirable properties: It treats all voters equally, it treats both candidates equally, and it is monotone, meaning that a single voter's change in ballot from a vote for the loser to a vote for the winner has no effect on the election outcome. More strikingly, May's theorem told us that among all voting systems in the two- candidate case that never result in a tie, majority rule is the only one satisfying these three properties.
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But let's consider for a moment what monotonicity is saying in this two- candidate case for voting systems that never yield ties. It says that if you rank over on your ballot, and the election winner is , then the election winner will remain if you switch to a ballot with over . But there are only two possible choices for a ballot in this two-candidate case: over and over . Monotonicity is thus saying that if you rank over , then no unilateral change in your ballot can make the outcome . This is simply the assertion that you can't manipulate the voting system!
Thus, in the two-candidate case, nonmanipulability and monotonicity are exactly the same thing. This allows us to restate May's theorem from Section 9.2, with the word monotonicity replaced by nonmanipulability.
May's Theorem for Manipulability THEOREM
Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is nonmanipulable.
There are examples of two-candidate voting systems that are manipulable, even though they treat all voters equally and both candidates equally. For example, the voting system that declares the winner to be the alternative with the fewest first-place votes is manipulable, as is the one that declares the winner to be whichever alternative has an odd number of first-place votes (even if that's fewer than half). Exercises 1 and 2 ask you to provide an example of voter manipulation for each of these systems.
Turning to the case of three or more candidates, we begin with Condorcet's method, as we did in Chapter 9. Condorcet's method is based on majority rule, and as we've just seen, majority rule is nonmanipulable. So the following result, as pleasing as it is, comes as no surprise.
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The Nonmanipulability of Condorcet's Method THEOREM
Condorcet's method is nonmanipulable in the sense that a voter can never unilaterally change an election result from one candidate to another candidate that he or she prefers.
Let's see why Condorcet's method is nonmanipulable, regardless of the number of voters. Suppose that we have an election in which you, as one of the voters, prefer Candidate to Candidate , but wins using Condorcet's method. We'll show that any attempt that you might make to manipulate the election so that becomes the winner is doomed to failure, even if there are more than these two candidates in the election.
Because Candidate is the winner, using Condorcet's method, we know that defeats every other candidate in a one-on-one contest based on the ballots cast. In particular, defeats in a one-on-one contest, even with your original ballot that has over . This means that more than half of the other voters ranked over , so, regardless of how you change your ballot, will still defeat in a one-on-one contest. While this need not ensure that remains a winner with Condorcet's method, it certainly guarantees that isn't.
Hence, you cannot unilaterally cause to be a winner using Condorcet's method, and so your attempt at manipulation will have failed.
EXAMPLE 3 Exploiting the Condorcet Voting Paradox
We had to be careful in stating the theorem that asserts Concorcet's method is non-manipulable because, as we've seen, elections occur in which there is no winner using Condorcet's method. With three voters and three candidates, it is possible for a voter (the one on the left in this example) to unilaterally change an election from one that yields his or her second choice as the sole winner (Candidate in the example), to one in which there is no winner at all, as this example shows:
| 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 | |||
A voter's ability to bring about this kind of change in an election unilaterally, however, is not something that falls within the scope of our formal definition of manipulation. nevertheless, one could argue that there are situations in which you might well prefer having an election with no outcome at all to having an election in which a candidate other I than your top choice emerges as the sole winner.
We now move on to voting systems with three or more candidates, systems that, unlike Condorcet's method, always produce at least one winner. As you might expect from the results in Chapter 9, in terms of nonmanipulability, these voting systems are not as perfect as one might hope for.