# [EM] élection de trois élection de trois

Kristofer Munsterhjelm km_elmet at lavabit.com
Thu Feb 23 10:40:59 PST 2012

```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
>
> 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.

(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.)

```