Condorcet's method, as we've seen, has a number of very desirable properties, including the following four:
Unfortunately, Condorcet's voting paradox on page 411 shows that there are elections in which Condorcet's method produces no winner at all.
Can we find a voting system that satisfies all four of these properties and that, unlike Condorcet's method, always yields a winner? Several possibilities suggest themselves. For example, to avoid ties, we could modify any of our usual methods by agreeing to use a fixed ordering of the candidates to break any ties that occur. Or we could extend Condorcet's method by making the winner be the candidate with the best "win-loss record" in one-on-one contests (a method called Copeland's rule).
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Alas, any such attempt is doomed. In the early 1970s, Allan Gibbard and Mark Satterthwaite independently proved the following remarkable result.
The Gibbard-Satterthwaite Theorem THEOREM
With three or more candidates and any number of voters, there does not exist—and there never will exist—a voting system that always produces a winner, never has ties, satisfies the Pareto condition, is nonmanipulable, and is not a dictatorship.
The Gibbard-Satterthwaite theorem (often called the GS theorem for short) is a deep result that is related in important ways to Arrow's impossibility theorem. In particular, you shouldn't find it at all obvious, and we won't be saying anything about the proof. But we can state and prove a much weaker result that is of some interest in its own right.
A Weak Version of the GS Theorem THEOREM
Any voting system for three candidates that agrees with Condorcet's method whenever there is a Condorcet winner—and that additionally produces a unique winner when confronted by the ballots in the Condorcet voting paradox—is manipulable.
Let's see why this is true. With the Condorcet voting paradox, the winner is either or or . For the moment, we'll assume it is (and leave the other two cases to you—see Exercise 24 on page 458). Consider the following two elections:
| 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, the winner is (our assumption in this case), and in Election 2, the winner is (because we are assuming that our voting system agrees with Condorcet's method when there is a Condorcet winner, as is here). Notice that the voter on the left, by a unilateral change in ballot, has improved the election outcome from being his or her third choice to being his or her second choice. This is what that voter set out to do and this is the desired instance of manipulation.
But the nonintuitive nature of voting and manipulation does not end here. It also turns out that sometimes "more is less" when it comes to "voting power." We illustrate this with the so-called chair's paradox.