I think I've made the analysis more rigorous: Suppose that we elect the first choice of the majority, if one exists. Otherwise, we elect a candidate by some other method M. M need not treat all voters and candidates equally, it need not rely solely on rankings (it could also look at approval cutoffs, for instance), in fact, we really don't need to worry about the details of M.
Suppose that in a 3-way race candidate B is one vote away from a majority and M would pick a candidate C whom B defeats pairwise. Voters with the preference A>B>C have an incentive to betray their favorite (there's a loophole that I'll remove later). So, supplement our method to elect any candidate who's a single vote away from a majority if he pairwise defeats the choice of M. Now suppose B is 2 away from a majority, and M selects C. If B defeats C pairwise then there are at least two voters with the preference A>B>C. If one of those voters betrays his favorite, and votes B>A>C, B is a single vote away, and the previous clause applies. So, add another supplement to our method, for the case when a candidate is 2 votes away from a majority and defeats the choice of M. By induction we see that we must also add supplementary rules to cover the case where B defeats C (the choice of M) pairwise but is 3 votes away from a majority, 4 votes,...n votes away. So, as long as there is a candidate who pairwise defeats the choice of M, that other candidate should be elected and M should be circumvented. Before I declare strong FBC to be impossible, there are three things to consider: 1) Suppose that favorite betrayal by the A>B>C faction causes M to select A or B. The majority rule analysis doesn't apply, but M itself fails strong FBC, so by extension the composite method of M plus supplementary conditions for majority rule must also satisfy strong FBC. 2) Suppose that the A>B>C faction need not vote insincerely to defend its interests, i.e. suppose that changing their vote to A>C>B would cause M to elect either A or B instead of C. In that case M is non-monotonic, since the voter has caused C to lose by ranking C higher without changing the relative rankings of A and B. I'm working right now on the case where we allow non-monotonicity. So, if we impose monotonicity we are requiring that (in the case of 3 candidates) our method never elect a candidate who is pairwise defeated. Hence, in general strong FBC, majority rule, and monotonicity are incompatible. Note that imposing monotonicity and strong FBC implies that the A>B>C faction should never be able to obtain a better result from insincere voting if C is the winner. This does not violate the Gibbard-Satterthwaite Theorem, because we've only said that there are particular circumstances where a particular group of voters can't obtain a better result from insincere strategy. We haven't required that NO voter ever be able to obtain a better result from some form of insincere voting. The "vote for top 2" ranked method that I described a few days ago satisfies strong FBC and monotonicity and is still non-dictatorial. 3) Does this analysis apply to more than 3 candidates? I'm thinking about that right now. It's not a priori obvious to me. So, now I'm thinking about removing the monotonicity requirement and extending to the case of 4+ candidates. If I could just do that I'd have a pretty cool result. Alex ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
