[dane-pgp]

What if an election is held to determine a set of two "finalists"? People might disagree about who the top two candidates are, but they might be satisfied as long as one of their top preferences makes it through to the second round.

Obviously once there are two candidates remaining, Gibbard's Theorem doesn't apply, so I'm guessing that any procedure which reduces the set to two outcomes must itself be subject to strategy, but it would be interesting if allowing people to cast a separate vote in a run-off election was enough to make the first round no longer require strategy.

[cdsmith]

If the result of the election is the choice of two finalists, then as long as there are more than two candidates, there are more than two possible outcomes. For example, if the candidates are A, B, and C, then the possible outcomes are: (A and B), (A and C), or (B and C), so 3 in all. In general, for n candidates, there are n * (n - 1) / 2 outcomes. By Gibbard's Theorem, then, either there is a dictator, or there is strategy.

Another way to see this: the entire process of choosing two finalists and then having a runoff to choose an ultimate winner counts as a collective decision-making process. That there are two separate votes doesn't actually matter. By Gibbard's theorem, then, if there isn't a dictator, then the entire process is strategic. Since the chance to use strategy doesn't occur in the simple-majority final runoff, by process of elimination, it must occur in choosing the candidates who qualify for the runoff.