[EM] Re: Markus, criteria

[Alex Small]

>FBC is a great example of a Mike-style criterion that does nothing but 
>complicate the idea it attempts to express. Why did Mike create this 
>"criterion"? Probably because he didn't understand that other election 
>method criteria are based on cast and tally rules votes only.

That's not entirely true.  The famous Gibbard-Satterthwaite (GS) Theorem makes reference to preferences.  The theorem, crudely speaking, says that there's no ranked election method that doesn't, from time to time, give voters an incentive to vote insincerely.  You could think of the method by first imagining that every voter votes sincerely.  Then, based on tallies of sincere votes, ask if any of the voters would have preferred the outcome that would have obtained if he or she individually had voted differently.  The theorem says that there will always be cases where at least one voter would have been happier if he or she had voted insincerely.
 
FWIW, I prefer to think of strategic voting in terms of groups of voters with identical preferences, rather than in terms of individual voters.  In elections for public office it will be exceedingly rare that a single person could change the outcome, unless the margin is 1 vote.  Even the 2004 gubernatorial election in Washington was decided by (depending on the lawsuit) at least 30 votes, so no individual could say that it would have been different if he had just voted differently.  On the other hand, it will more frequently be the case that a faction of voters regret voting sincerely.  e.g. a few percent of the population votes Green in a race where a Republican narrowly wins, or a few percent vote Libertarian in a race where a Democrat narrowly wins.  Or maybe an IRV election where the moderate is eliminated because he came up short by just a few percent, and after the left (or right) candidate wins the people who supported the right (or left) candidate regret that they di
 dn't
 switch to the moderate (whom they'd prefer over their polar opposite candidate).
 
>But wait ... isn't voting another candidate *equal* to your favorite a 
>kind of "betrayal" too? If I told my wife that she has equal standing 
>with some other woman, I'll bet she'd feel "betrayed"! Which just goes 
>to show that Mike's "criteria" can be misleading.
 
Well, the name might not be the best.  Some of us have divided the FBC into 2 criteria, "strong FBC" and "weak FBC."  Weak FBC is the one satisfied by Approval.  Strong FBC would be the one that, in the spirit of your post, your wife would prefer.  In my spare time I've worked on a proof that strong FBC is impossible to satisfy with purely ranked voting methods.  I've been close to completing it, but there are a few points missing.  I've recently taken up these geometric issues again because I think they'll get me closer to a proof.
 
If my intuition is born out by the complete proof, what I'll basically show is that only 1 ranked method satisfies strong FBC, sometimes referred to as "Top 2 Voting" or "Anti-Plurality Voting":  Each voter ranks the candidates and the top 2 candidates on each ballot get one point each.  The only reason this can be called "strong FBC" is that ties can be broken by a pairwise comparison of the 2 leading candidates.  But ties are rare ("a set of measure zero" in technical parlance), so it isn't a very satisfying way to achieve strong FBC.  Which is sort of the point of the proof.  (Well, there is at least one other really bizarre method, but in some sense it's strategically equivalent to Anti-Plurality Voting.) 
 
Where it becomes more interesting is that if my result holds, it's easy to see that by allowing equal rankings (including equal rankings for last place) in some sense the "most strategy-proof" method is equivalent to Approval Voting (well, when there are 3 candidates).  I do hope to complete the proof some day, but I'm facing some obstacles in the geometry, and in these higher dimensions it's harder for me to visualize things.  It's harder to move quickly because I can't just start with intuition and then fill in the formal steps.  Not to mention that I'm trying to write my dissertation.
 
 
 
Alex Small

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