[EM] Steph: Your criterion
Elisabeth Varin/Stephane Rouillon
stephane.rouillon at sympatico.ca
Sun Dec 8 20:39:47 PST 2002
MIKE OSSIPOFF a écrit :
> Steph--
>
> You wrote:
>
> Statement of Criterion
>
> If an Ideal Democratic Winner (IDW) exists, and if a two-third majority
> prefers the IDW to another candidate, then the other candidate should
> not win if that majority votes sincerely and no other voter falsifies
> any preferences.
>
> I reply:
>
> wv passes that criterion. Does relative margins pass it? Does margins
> pass it?
They all pass that.
> If so, can you demonstrate they they do?
I think so. But maybe Fermat's genes are among mines... It is more an
engineering
proof than a mathematical one. I would prefer improving it first.
Whatever, it would not be the first time I make mistakes and it will go
faster this way. This is what I have:
Suppose a stronger CW (that has a two-third approbation of the total number
of voters against each other candidate). Suppose we know every vote and
allow
only voters to unsincerely truncate their ballot. Suppose a runner-up (R) is
able to be elected with some unsincere truncations. It means that one of the
CW pairwise comparison becomes a defeat for the CW because of these
truncations: namely, more than one third of all the voters, more than half
of those supporting the CW, have to truncate their preferences to give (R)
victory. Note that it does not have to be the CW > R pairwise comparison
that falls, it could be one or more others. However the truncators cannot be
part of the two-third majority who prefers CW to (R) (in a perfect
information world, they would never harm their own preferences). So the
truncators are less than a third of the voters. They cannot be less than a
third and more than a third... (R) does not exist.
Supposing some sincere truncations from start on pairwise comparison not
implicating the CW, only increases the higher limit on the number of
unsincere truncators... (R) still does not exist.
> In any case, that's a weaker version of SFC, which wv meets and
> margins fails, and relative margins fails.
I agree.
I define Safe IDW as having 66.66...% of the total number of votes against
any other candidate
It is just a way to divide the possibilities:
1) a Safe IDW or Stronger CW (call it the way you want) exists:
it is protected from unsincere truncation whatever criteria is used...
2) a Strong CW but not Stronger (50%+1 of total vote against any other
candidate)
exists. (wv) can garantee is election, margin and (rm) no.
3) a weak CW exists. No criteria can garantee its election.
4) no CW exists.
I have not yet produced an example of case 3) that shows that a weak CW
can be stolen with margins or (rm) but I think it should be easy.
Now we could try to evaluate the probability of each case and the mean gain
or loss
in each case to get an expected gain or loss average from unsincere
truncation.
I think it depends highly on the number of candidates (serious ones I
suppose).
Without poll, unsincere truncation should produce a loss in general.
How can we evaluate polls impact on voters behavior?
Steph.
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