<html>
<head>
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
</head>
<body text="#000000" bgcolor="#FFFFFF">
<p>Kevin,<br>
</p>
<p>I didn't comment earlier on your "idea 2". <br>
<br>
If there no "disapproved rankings" (i.e. if the voters all approve
the candidates they rank above bottom),<br>
then your suggested method is simply normal Winning Votes, which
I don't like because the winner can<br>
be uncovered and positionally dominant or pairwise-beaten and
positionally dominated by a single other<br>
candidate.<br>
<br>
On top of that I don't think it really fills the bill as
"simple". Approval Margins (using Sort or Smith//MinMax<br>
or equivalent or almost equivalent algorithm) would be no more
complex and in my opinion would be better.<br>
<br>
I would also prefer the still more simple Smith//Approval.<br>
<br>
What did you think of my suggestion for a way to implement your
idea 1? <br>
<br>
Chris <br>
</p>
<div class="moz-forward-container"><br>
<br>
<br>
<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"
moz-do-not-send="true">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>
<i><br>
</i> </div>
<div id="DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2"><br />
<table style="border-top: 1px solid #D3D4DE;">
<tr>
<td style="width: 55px; padding-top: 13px;"><a href="http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient" target="_blank"><img src="https://ipmcdn.avast.com/images/icons/icon-envelope-tick-green-avg-v1.png" alt="" width="46" height="29" style="width: 46px; height: 29px;" /></a></td>
<td style="width: 470px; padding-top: 12px; color: #41424e; font-size: 13px; font-family: Arial, Helvetica, sans-serif; line-height: 18px;">Virus-free. <a href="http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient" target="_blank" style="color: #4453ea;">www.avg.com</a>
</td>
</tr>
</table><a href="#DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2" width="1" height="1"> </a></div></body>
</html>