2012/2/23 Kristofer Munsterhjelm <[email protected]> > On 02/20/2012 03:13 AM, robert bristow-johnson wrote: > >> On 2/19/12 8:53 PM, David L Wetzell wrote: >> >>> It seems quite a few election rules get quirky in one way or the other >>> with a 3-way competitive election. >>> >>> That might be a point worth considering in the abstract in a paper or >>> something.... why are 3-way single-winner elections quirky? >>> >>> >> isn't it obvious? >> >> http://en.wikipedia.org/wiki/**Duverger%27s_law<http://en.wikipedia.org/wiki/Duverger%27s_law> >> >> to wit: Duverger suggests two reasons why single-member district >> plurality voting systems favor a two party system. One is the result of >> the "fusion" (or an alliance very like fusion) of the weak parties, and >> the other is the "elimination" of weak parties by the voters, by which >> he means that the voters gradually desert the weak parties on the >> grounds that they have no chance of winning. >> > > I'd call Duverger's law more an effect rather than a cause. The question > in itself is why some methods seem to have a much harder time dealing with > three-way (and n > 3 way) races than 2-candidate or 2.5 candidate races. > > I think the answer is simple enough: > > - When you have two candidates, there's one strategy-proof deterministic > method, and its name is majority rule. > > - When you have two candidates and a bunch of tiny ones, it's usually > pretty easy to know who the tiny ones are and remove them so they don't > upset the outcome. (IRV does this) > > - But when you have three or more candidates, Arrow's impossibility > theorem says that you can't have a ranked method that's independent of > irrelevant alternatives. So no such method can be perfect. The concept of > removing irrelevant candidates to reduce to majority rule no longer works, > because you can't say "these candidates are obviously tiny and so should > never win" when they're all strong contenders. > > As a consequence, among ranked methods, some really bad methods (like > Plurality) gets it wrong when there are two candidates plus no-hopes; some > slightly better methods (like IRV, and perhaps I'd also put DAC/DSC here > since it uses the same logic) can identify and remove the no-hopes but then > gives bad results when the going gets tough; while yet other methods (such > as Condorcet) use more consistent logic and, though not perfect, handle > three-way (and n>3 n-way) races much better. > > Rated method supporters, like Warren, would likely say that the rated > methods are even better since they can pass IIA and so can scale to any > number of candidates. They do pass IIA, but in exchange people have to be > able to say how much they like a candidate rather than just > better/worse-than, and it doesn't get around Gibbard-Satterthwaite. >
Note that SODA avoids most pathologies up to 4 candidates. It does not, as I've previously claimed, meet FBC even for three candidates*. But it is monotonic, consistent, participation, IIA, and cloneproof up to 4 candidates, and it handles the chicken dilemma both honestly and strategically. Of course, in order to accomplish this, you must give up some freedom; in this case, the freedom to vote anything more expressive than approval if you don't happen to agree with your favorite candidate's preferences. *There is an FBC fix for SODA which works for n candidates, is n^2 summable, and does not break the other properties given here; I'll write more on that later, when I've had more time to think about it. Both the problem and the fix are more theoretically than practically interesting; I would never advise FBC strategy in SODA with anything less than perfect polling. > > (Finally, just to preemptively head that off: just because no ranked > method can meet IIA doesn't mean they are all equally bad. Just because > there's no such thing as perpetual motion doesn't mean a modern steam > turbine is just as inefficient as the aelopile. I don't think you think so, > but certain others on the list might, so I'll make that clear.) Agreed. In a similar sense, just because all rated methods ask for degrees of preference and aren't 100% strategy-proof, doesn't mean that they are all equally sensitive to strategies involving preference degrees. It's clear that people on this list seem to have preferences for ranked or rated ballot formats; but regardless of those preferences, I think both sides can agree that a good method even of the "worse" ballot format is better than a bad one with the "better" format. Jameson
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