I already said that Negative Voting (a person's first and second choices are each given a single vote) passes strong FBC (at least when voters have sufficient information). Granted, it passes in a very dubious way, because the method makes no distinctions between the first and second choices. Nonetheless, if we use ranked ballots it technically does pass strong FBC.
As far as I can tell, negative voting is the only possible system that satisfies the following requirements when there are 3 candidates: (1) strong FBC (2) swapping first and second choices doesn't change outcome (3) symmetry: If every voter swaps his rankings of A and B, and A was the winner, then B now wins, and if C won then C still wins. I haven't proven this, but let's take it as a conjecture and examine its consequences. Now, as always, we have 6 voter types. Constraining the number of voters in each category to add up to some constant gives us 5 essential a 5-dimensional space of possible electorates. Normally, a tie corresponds to a 4-D subset of the 5-D space of possible electorates. Let's resolve at least some 2-way ties with a pairwise comparison of the tied candidates. Our augmented method now makes a meaningful distinction between first and second rankings in at least some circumstances (a 4D subset of the space of possible electorates, to be precise). As far as I can tell, the method satisfies strong FBC. The only time the method will distinguish between your first and second choices is when they tie, and in those cases you never have an incentive to list #2 ahead of #1. You might, of course, have an incentive to demote #2 to third place if your favorite would lose the pairwise contest. But that's consistent with strong FBC. Now, suppose we increase the margin needed to trigger a pairwise contest between the top two. How large the margin is really doesn't matter, as long as it's greater than zero. When the margin is small enough to trigger a pairwise comparison, you're essentially voting on who the 2 contestants will be. We now have a 5 dimensional section of "voter space" where the method makes a meaningful distinction between first and second choices. It's easy to show that although a single person cannot change the outcome to something that he prefers, two voters acting in concert can. Consider this example, where the critical margin for a top-2 runoff is greater than zero: Your preference: A>B>C Society's pairwise preferences: C>A, B>C, A vs. B irrelevant here. Votes for each candidate: A: n-1 votes B: n-2 votes C: n votes There's a runoff between A and C, and C (your least favorite) wins. If you were to insincerely give the ranking B>C>A, A and B would be tied and the result would be indeterminate (who goes into the runoff?). However, if you and a like-minded friend both give the insincere ranking B>C>A, the runoff is between B and C. Now, technically, since a single voter acting alone cannot obtain a better result by favorite betrayal, one could say that the method passes strong FBC. However, that is torturing the concept of strategic voting. If one person notices from polls that the race is really close, the odds are good that a second person will as well, and both will realize "Hey, I have an incentive to do such-and-such." THE POINT OF THIS LONG MESSAGE: We took a method (perhaps the only method?) that (sort of) passes strong FBC but doesn't make a meaningful distinction between first and second choices. We modified the method to make a meaningful distinction between the top two choices on a 4D subset of the 5D "voter space" and saw that it still satisfied strong FBC. When we modified the method further, to make that meaningful distinction in a small 5D subset of voter space we saw that strong FBC is violated, provided that at least 2 people have the same incentives. Although we didn't consider every possible modification of the method, we considered the only obvious modification of the method and saw that, no matter how small of a modification we make, strong FBC is violated. If only I could generalize, and show that ANY modification of negative voting causes a violation of strong FBC (assuming that the modification doesn't violate the symmetry condition)..... ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
