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<div>Hi Chris,</div><div><br></div><div>I've been short on time so I don't actually have much thought on any of the methods, even my own.</div><div><br></div><div>I suppose Idea 2 is the same as Schwartz-limited MinMax(WV) if nobody submits disapproved rankings. I'm not sure if it makes sense to reject the method over that. Specifically should "positional dominance" have the same meaning whether or not the method has approval in it? As a comparison, I will go easy on these methods over failing MD, because it happens when some of the majority don't approve their common candidate.</div><div><br></div><div>I would have liked to simplify Idea 2, but actually Forest's eventual proposal wasn't all that simple either. As I wrote, if you add "elect a CW if there is one" it can become much simpler, so that it isn't really distinct from Idea 1. I actually tried pretty hard to present three "Ideas" in that post, but kept having that problem.</div><div><br></div><div>I posted those ideas because I thought Forest posed an interesting challenge, and I thought I perceived that he was trying to fix a problem with CD. That said, I am not a fan of Smith//Approval(explicit). If all these methods are basically the same then I probably won't end up liking any of them. I don't think it's ideal if burying X under Y (both disapproved) can only backfire when Y is made the CW.</div><div><br></div><div>Kevin</div><div><br></div><div><br></div>
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Le mercredi 5 juin 2019 à 21:26:23 UTC−5, C.Benham <cbenham@adam.com.au> a écrit :
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<div>Kevin,<br></div><div><div id="ydp50d5cd69yiv9085021920"><div>
<p>I didn't comment earlier on your "idea 2". <br clear="none">
<br clear="none">
If there no "disapproved rankings" (i.e. if the voters all approve
the candidates they rank above bottom),<br clear="none">
then your suggested method is simply normal Winning Votes, which
I don't like because the winner can<br clear="none">
be uncovered and positionally dominant or pairwise-beaten and
positionally dominated by a single other<br clear="none">
candidate.<br clear="none">
<br clear="none">
On top of that I don't think it really fills the bill as
"simple". Approval Margins (using Sort or Smith//MinMax<br clear="none">
or equivalent or almost equivalent algorithm) would be no more
complex and in my opinion would be better.<br clear="none">
<br clear="none">
I would also prefer the still more simple Smith//Approval.<br clear="none">
<br clear="none">
What did you think of my suggestion for a way to implement your
idea 1? </p><div class="ydp50d5cd69yiv9085021920yqt3873327189" id="ydp50d5cd69yiv9085021920yqtfd25173"><br clear="none">
Chris <br clear="none">
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<br clear="none">
<br clear="none">
<blockquote type="cite">
<p><b>Kevin Venzke</b> <a shape="rect" 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" rel="nofollow" target="_blank">stepjak at yahoo.fr </a><br clear="none">
Sat Jun 1 12:48:27 PDT 2019 </p>
<p><br clear="none">
Hi Forest,<br clear="none">
<br clear="none">
I had two ideas.<br clear="none">
<br clear="none">
Idea 1:<br clear="none">
1. If there is a CW using all rankings, elect the CW.<br clear="none">
2. Otherwise flatten/discard all disapproved rankings.<br clear="none">
3. Use any method that would elect C in scenario 2. (Approval,
Bucklin, MinMax(WV).)<br clear="none">
<br clear="none">
So scenario 1 has no CW. The disapproved C>A rankings are
dropped. A wins any method.<br clear="none">
In scenario 2 there is no CW but nothing is dropped, so use a
method that picks C.<br clear="none">
In both versions of scenario 3 there is a CW, B.<br clear="none">
<br clear="none">
If step 3 is Approval then of course step 2 is unnecessary.<br clear="none">
<br clear="none">
In place of step 1 you could find and apply the
majority-strength solid coalitions (using all rankings)<br clear="none">
to disqualify A, instead of acting based on B being a CW. I'm
not sure if there's another elegant way<br clear="none">
to identify the majority coalition.<br clear="none">
<br clear="none">
Idea 2:<br clear="none">
1. Using all rankings, find the strength of everyone's worst
WV defeat. (A CW scores 0.)<br clear="none">
2. Say that candidate X has a "double beatpath" to Y if X has
a standard beatpath to Y regardless<br clear="none">
of whether the disapproved rankings are counted. (I don't know
if it needs to be the *same* beatpath,<br clear="none">
but it shouldn't come into play with these scenarios.)<br clear="none">
3. Disqualify from winning any candidate who is not in the
Schwartz set calculated using double<br clear="none">
beatpaths. In other words, for every candidate Y where there
exists a candidate X such that X has a<br clear="none">
double beatpath to Y and Y does not have a double beatpath to
X, then Y is disqualified.<br clear="none">
4. Elect the remaining candidate with the mildest WV defeat
calculated earlier.<br clear="none">
<br clear="none">
So in scenario 1, A always has a beatpath to the other
candidates, no matter whether disapproved<br clear="none">
rankings are counted. The other candidates only have a
beatpath to A when the C>A win exists. So<br clear="none">
A has a double beatpath to B and C, and they have no path
back. This leaves A as the only candidate<br clear="none">
not disqualified.<br clear="none">
<br clear="none">
In scenario 2, the defeat scores from weakest to strongest are
B>C, A>B, C>A. Every candidate has<br clear="none">
a beatpath to every other candidate no matter whether the
(nonexistent) disapproved rankings are<br clear="none">
counted. So no candidate is disqualified. C has the best
defeat score and wins.<br clear="none">
<br clear="none">
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 clear="none">
beats C pairwise no matter what, so B has a double beatpath to
C. However C has no such beatpath<br clear="none">
to A, nor has A one to B, nor has B one to A. The resulting
Schwartz set disqualifies only C. (C needs<br clear="none">
to return B's double beatpath but can't, and neither A nor B
has a double beatpath to the other.)<br clear="none">
Between A and B, B's score (as CW) is 0, so he wins. <br clear="none">
<br clear="none">
Scenario 3, second version: B again has no losses, and also
has double beatpaths to both of A and<br clear="none">
C, neither of whom have double beatpaths back. So A and C are
disqualified and B wins.<br clear="none">
<br clear="none">
I must note that this is actually a Condorcet method, since a
CW could never get disqualified and<br clear="none">
would always have the best worst defeat. That observation
would simplify the explanation of<br clear="none">
scenario 3.<br clear="none">
<br clear="none">
I needed the defeat strength rule because I had no way to give
the win to B over A in scenario 3<br clear="none">
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 clear="none">
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 clear="none">
the same place with this question.<br clear="none">
<br clear="none">
Oh well, I think the ideas are interesting enough to post.<br clear="none">
<br clear="none">
Kevin<br clear="none">
<br clear="none">
>Le jeudi 30 mai 2019 à 17:32:42 UTC−5, Forest Simmons
<fsimmons at pcc.edu> a écrit : <br clear="none">
><br clear="none">
>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 clear="none">
><br clear="none">
>I am interested in simple methods that always ...<br clear="none">
><br clear="none">
>(1) elect candidate A given the following profile:<br clear="none">
><br clear="none">
>P: A<br clear="none">
>Q: B>>C<br clear="none">
>R: C,<br clear="none">
>and <br clear="none">
>(2) elect candidate C given<br clear="none">
>P: A<br clear="none">
>Q: B>C>><br clear="none">
>R: C,<br clear="none">
>and <br clear="none">
>(3) elect candidate B given<br clear="none">
<br clear="none">
><br clear="none">
>P: A<br clear="none">
>Q: B>>C (or B>C)<br clear="none">
>R: C>>B. (or C>B)<br clear="none">
><br clear="none">
>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 clear="none">
><br clear="none">
>Here's the rationale for the requirements:<br clear="none">
><br clear="none">
>Condition (1) is needed so that when the sincere
preferences are<br clear="none">
<br clear="none">
><br clear="none">
>P: A<br clear="none">
>Q: B>C<br clear="none">
>R: C>B,<br clear="none">
>the B faction (by merely disapproving C without
truncation) can defend itself against a "chicken" attack
(truncation of B) from the C faction.<br clear="none">
><br clear="none">
>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 clear="none">
><br clear="none">
>Condition (2) is needed so that when sincere preferences
are<br clear="none">
<br clear="none">
><br clear="none">
>P: A>C<br clear="none">
>Q: B>C<br clear="none">
>R: C>A,<br clear="none">
>then the C faction (by proactively truncating A) can
defend the CW against the A faction's potential truncation
attack.<br clear="none">
><br clear="none">
>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 clear="none">
>control over their own approval cutoffs (as opposed to
implicit approval) with default approval as top rank only. The
other <br clear="none">
>method I have in mind is not quite as <br clear="none">
>simple, but it has the added advantage of satisfying the
FBC, while almost always electing from Smith.<br clear="none">
<br clear="none">
<br clear="none">
<br clear="none">
<br clear="none">
<br clear="none">
<br clear="none">
</p>
</blockquote>
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