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Kevin,<br>
<br>
As something fairly simple I think I like this version of your "Idea
1":<br>
<br>
1. If there is a CW using all rankings, elect the CW.<br>
<p>
2. Otherwise flatten/discard all disapproved rankings.</p>
<p>3. If there is a "CW" based on the remaining rankings (i.e. the
rankings among approved plus approved over not approved)<br>
then elect that candidate.</p>
<p>4. Otherwise elect the most approved candidate.<br>
<br>
That strikes me as something not too hard to explain or sell.<br>
<br>
Chris Benham<br>
</p>
<blockquote type="cite">
<p><b>Kevin Venzke</b> <a title="[EM] What are some simple
methods that accomplish the following conditions?"
href="mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20What%20are%20some%20simple%20methods%20that%20accomplish%20the%20following%0A%20conditions%3F&In-Reply-To=%3C1931864740.14928463.1559418507456%40mail.yahoo.com%3E">stepjak
at yahoo.fr </a><br>
Sat Jun 1 12:48:27 PDT 2019 </p>
<p><br>
Hi Forest,<br>
<br>
I had two ideas.<br>
<br>
Idea 1:<br>
1. If there is a CW using all rankings, elect the CW.<br>
2. Otherwise flatten/discard all disapproved rankings.<br>
3. Use any method that would elect C in scenario 2. (Approval,
Bucklin, MinMax(WV).)<br>
<br>
So scenario 1 has no CW. The disapproved C>A rankings are
dropped. A wins any method.<br>
In scenario 2 there is no CW but nothing is dropped, so use a
method that picks C.<br>
In both versions of scenario 3 there is a CW, B.<br>
<br>
If step 3 is Approval then of course step 2 is unnecessary.<br>
<br>
In place of step 1 you could find and apply the
majority-strength solid coalitions (using all rankings)<br>
to disqualify A, instead of acting based on B being a CW. I'm
not sure if there's another elegant way<br>
to identify the majority coalition.<br>
<br>
Idea 2:<br>
1. Using all rankings, find the strength of everyone's worst WV
defeat. (A CW scores 0.)<br>
2. Say that candidate X has a "double beatpath" to Y if X has a
standard beatpath to Y regardless<br>
of whether the disapproved rankings are counted. (I don't know
if it needs to be the *same* beatpath,<br>
but it shouldn't come into play with these scenarios.)<br>
3. Disqualify from winning any candidate who is not in the
Schwartz set calculated using double<br>
beatpaths. In other words, for every candidate Y where there
exists a candidate X such that X has a<br>
double beatpath to Y and Y does not have a double beatpath to X,
then Y is disqualified.<br>
4. Elect the remaining candidate with the mildest WV defeat
calculated earlier.<br>
<br>
So in scenario 1, A always has a beatpath to the other
candidates, no matter whether disapproved<br>
rankings are counted. The other candidates only have a beatpath
to A when the C>A win exists. So<br>
A has a double beatpath to B and C, and they have no path back.
This leaves A as the only candidate<br>
not disqualified.<br>
<br>
In scenario 2, the defeat scores from weakest to strongest are
B>C, A>B, C>A. Every candidate has<br>
a beatpath to every other candidate no matter whether the
(nonexistent) disapproved rankings are<br>
counted. So no candidate is disqualified. C has the best defeat
score and wins.<br>
<br>
In scenario 3, the first version: B has no losses. C's loss to B
is weaker than both of A's losses. B<br>
beats C pairwise no matter what, so B has a double beatpath to
C. However C has no such beatpath<br>
to A, nor has A one to B, nor has B one to A. The resulting
Schwartz set disqualifies only C. (C needs<br>
to return B's double beatpath but can't, and neither A nor B has
a double beatpath to the other.)<br>
Between A and B, B's score (as CW) is 0, so he wins. <br>
<br>
Scenario 3, second version: B again has no losses, and also has
double beatpaths to both of A and<br>
C, neither of whom have double beatpaths back. So A and C are
disqualified and B wins.<br>
<br>
I must note that this is actually a Condorcet method, since a CW
could never get disqualified and<br>
would always have the best worst defeat. That observation would
simplify the explanation of<br>
scenario 3.<br>
<br>
I needed the defeat strength rule because I had no way to give
the win to B over A in scenario 3<br>
version 1. But I guess if it's a Condorcet rule in any case, we
can just add that as a rule, and greatly<br>
simplify it to the point where it's going to look very much like
idea 1. I guess all my ideas lead me to<br>
the same place with this question.<br>
<br>
Oh well, I think the ideas are interesting enough to post.<br>
<br>
Kevin<br>
<br>
>Le jeudi 30 mai 2019 à 17:32:42 UTC−5, Forest Simmons
<fsimmons at pcc.edu> a écrit : <br>
><br>
>In the example profiles below 100 = P+Q+R, and
50>P>Q>R>0. One consequence of these constraints is
that in all three profiles below the cycle >A>B>C>A
will obtain.<br>
><br>
>I am interested in simple methods that always ...<br>
><br>
>(1) elect candidate A given the following profile:<br>
><br>
>P: A<br>
>Q: B>>C<br>
>R: C,<br>
>and <br>
>(2) elect candidate C given<br>
>P: A<br>
>Q: B>C>><br>
>R: C,<br>
>and <br>
>(3) elect candidate B given<br>
<br>
><br>
>P: A<br>
>Q: B>>C (or B>C)<br>
>R: C>>B. (or C>B)<br>
><br>
>I have two such methods in mind, and I'll tell you one of
them below, but I don't want to prejudice your creative efforts
with too many ideas.<br>
><br>
>Here's the rationale for the requirements:<br>
><br>
>Condition (1) is needed so that when the sincere preferences
are<br>
<br>
><br>
>P: A<br>
>Q: B>C<br>
>R: C>B,<br>
>the B faction (by merely disapproving C without truncation)
can defend itself against a "chicken" attack (truncation of B)
from the C faction.<br>
><br>
>Condition (3) is needed so that when the C faction realizes
that the game of Chicken is not going to work for them, the
sincere CW is elected.<br>
><br>
>Condition (2) is needed so that when sincere preferences
are<br>
<br>
><br>
>P: A>C<br>
>Q: B>C<br>
>R: C>A,<br>
>then the C faction (by proactively truncating A) can defend
the CW against the A faction's potential truncation attack.<br>
><br>
>Like I said, I have a couple of fairly simple methods in
mind. The most obvious one is Smith\\Approval where the voters
have <br>
>control over their own approval cutoffs (as opposed to
implicit approval) with default approval as top rank only. The
other <br>
>method I have in mind is not quite as <br>
>simple, but it has the added advantage of satisfying the
FBC, while almost always electing from Smith.<br>
<br>
<br>
<br>
<br>
<br>
<br>
</p>
</blockquote>
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