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<div class="moz-cite-prefix">I was inspired to compare IA-MPO with
my 3-slot Strong Minimal Defense, Top Ratings method. That has a
been shown<br>
(by Kevin Venzke) to fail the Plurality criterion. This is his
example:<br>
<br>
21 A>C<br>
08 B>A<br>
23 B<br>
11 C<br>
<br>
B>A 31-21, A>C 29-11, C>B 32-31.<br>
<br>
My method took account of 3 types of information: Top Ratings,
Approval, Maximum Approval Opposition. Any candidate<br>
with a MAO score higher their Approval score is disqualified, and
the undisqualified candidate with the highest TR score is<br>
elected. <br>
<br>
Top Ratings scores: B31 A21 C11<br>
(Implicit) Approval: B31 A29 C32<br>
MAO scores: B32 A23 C31<br>
(MPO scores: B32 A31 C31)<br>
<br>
SMD,TR elects A (after disqualifying B) and IA-MPO elects C
(the only candidate with a positive IA-MPO score).<br>
<br>
<blockquote type="cite">"It turns out that IA-MPO does satisfy a
modified version of Plurality: If A is ranked top above C on
more ballots than C is ranked, then C cannot be the IA-MPO
winner."</blockquote>
<br>
Forest, I'm not sure that this isn't the same as the normal
Plurality criterion. The reference to "first preference" in the
Plurality criterion definition I think refers to exclusive first
preference.<br>
<br>
(I gather that Woodall's criteria are only about strict rankings
from the top, which may or may not be truncated,) I suppose it
could and should be extended to applying to ballots<br>
that are symmetrically "completed" only at the top. Doing that to
your example gives:<br>
<br>
41 A<br>
18 C<br>
41 B<br>
<br>
Electing C on these ballots is insane and I don't see how electing
C on the original ballots (where some of the votes are given half
to one candidate and half to another) is<br>
really any more justified.<br>
<br>
Yes, this convinces me that the Plurality criterion should
definitely be applied to to the ballots symmetrically completed at
the top and that we can without regret<br>
kiss IA-MPO goodbye.<br>
<br>
Another version of the criterion is "Pairwise Plurality"
(suggested a while ago by Kevin or me): If candidate X's lowest
pairwise score is higher than candidate Y's highest<br>
pairwise score, then Y must not be elected".<br>
<br>
I like this. Both IA-MPO and SMD,TR fail it, as in the two
examples. <br>
<br>
In yours the pairwise results are A=B 49-49, A>C 33-18,
B>C 33-18.<br>
<br>
Getting back to Approval seeded by MinGS(etrw), that is the least
appealing of 3 different method ideas (all attempting to meet the
FBC) I've had recently.<br>
Given its FBC failure, I withdraw my support for it.<br>
<br>
I'll post the other two soonish. <br>
<br>
Chris Benham<br>
<br>
<br>
On 6/3/2015 8:10 AM, Forest Simmons wrote:<br>
</div>
<blockquote
cite="mid:CAP29ondxnM3S_87Bb7YMZXU=sxrXrfbn4TAEqZi1kjf8o0+3kQ@mail.gmail.com"
type="cite">
<div dir="ltr">
<div>
<div>
<div>
<div>
<div>
<div>
<div>
<div>
<div>Now I remember the interesting example that
shows that IA-MPO can fail Plurality when
equal ranking at top is allowed:<br>
<br>
</div>
33 A<br>
</div>
</div>
16 C=A<br>
</div>
<div>02 C<br>
</div>
16 C=B<br>
</div>
33 B<br>
</div>
<div><br>
</div>
The IA-MPO score for both A and B is 49-49=0, while the
score for C is 34-33=1, so C wins.<br>
<br>
</div>
This is a failure of Plurality because A (for example) is
top ranked on 49 ballots, while C is ranked on only34
ballots.<br>
<br>
</div>
However, any configuration in issue space that could give rise
to this ballot set would be more faithfully reflected in a
ballot like<br>
<br>
33 A<br>
</div>
<div>16 C>A<br>
<div>02 C<br>
</div>
16 C>B<br>
</div>
<div>33 B<br>
<br>
</div>
<div>Why would C voters raise A and B to top if they didn't
really like them as well as the Condorcet Winner C?<br>
<br>
</div>
<div>It could be that (through typical disinformation) voters
thought that C didn't have a chance compared to the two main
party candidates A and B. They raised their lesser evil
compromise candidates to hedge their bets.<br>
<br>
</div>
<div>It turns out that IA-MPO does satisfy a modified version of
Plurality: If A is ranked top above C on more ballots than C
is ranked, then C cannot be the IA-MPO winner.<br>
<br>
</div>
<div>In any case where this Plurality' would allow C to win
while an ordinary Plurality requirement would preclude C's
right to win, the C voters would (under perfect information
conditions) rightly have an incentive to change each instance
of C=A to C>A .<br>
<br>
</div>
<div>Proof that IA-MPO satisfies this modified Plurality':<br>
<br>
</div>
<div>First note that the IA winner cannot have a negative IA-MPO
score, because it is ranked on as many (or more) ballots than
any other candidate, including the one that gives it max
opposition.<br>
<br>
</div>
<div>Next note that if A is ranked top above C on more ballots
than C is ranked, then A's pairwise opposition against C is
greater than C's IA score, therefore C's IA-MPO score is
negative, and therefore smaller than the IA-MPO score of the
IA, winner, and therefore not maximal.<br>
<br>
</div>
<div>In a way IA-MPO automatically compensates for voters'
hypercautious raising of compromises to equal top status.
This should attract voters that don't like Approval because
they know that (under Approval) approving their compromise can
take the win away from a Condorcet Winner.<br>
<br>
</div>
<div>For this reason, I suggest that even on two slot approval
style ballots, we use the Approval-MPO score to determine the
winner instead of Approval alone.<br>
<br>
</div>
<div>Forest<br>
</div>
<div><br>
</div>
</div>
<div class="gmail_extra"><br>
<div class="gmail_quote">On Mon, Jun 1, 2015 at 7:10 PM, Forest
Simmons <span dir="ltr"><<a moz-do-not-send="true"
href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span>
wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr">
<div>Chris,<br>
<br>
</div>
it is interesting to me that IA-MMPO (implicit approval
minus max pairwise opposition) gives the same results as
Approval Seeded by MinGS (etrw) in the four examples that
you offered:<br>
<div class="gmail_extra"><br>
<div class="gmail_quote"><span class="">
<blockquote class="gmail_quote" style="margin:0px
0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex"> <br>
46 A<br>
44 B>C<br>
10 C<br>
</blockquote>
</span>
<div> The respective IA-MPO scores for A, B, and C
are 46-54, 44-46, and 54-46, the only positive one.<br>
</div>
<span class="">
<div><br>
</div>
<blockquote class="gmail_quote" style="margin:0px
0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex"> 46 A<br>
44 B>C (sincere might be B or B>A)<br>
05 C>A<br>
05 C>B</blockquote>
</span>
<div>The respective IA-MPO scores are 51-54, 49-51,
and 54-46, again the only positive one.<br>
<br>
</div>
<span class="">
<blockquote class="gmail_quote" style="margin:0px
0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex"> 46 A>C<br>
10 B>A<br>
10 B>C<br>
34 B=C<br>
</blockquote>
</span>
<div>The respective IA-MPO scores are 56-54, 54-46,
and 90-56. C wins again.<br>
<br>
<br>
</div>
<span class="">
<blockquote class="gmail_quote" style="margin:0px
0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex"> 40 A>C<br>
15 B>A<br>
20 B<br>
15 C>A<br>
10 C<br>
</blockquote>
</span>
<div>The respective scores are 70-35, 35-65, and
65-55. This time A wins with an IA-MPO score of 35
compared to C's 10.<br>
<br>
</div>
<div>This IA-MPO method does satisfy the FBC, but is
not chicken proof. However, small defensive moves
can thwart chicken threats.<br>
<br>
</div>
<div>It seems like I might have suggested IA-MPO
before, but we were trying for something fancier at
the time.<span class="HOEnZb"><font color="#888888"><br>
</font></span></div>
</div>
<span class="HOEnZb"><font color="#888888"><br>
</font></span></div>
<span class="HOEnZb"><font color="#888888">
<div class="gmail_extra">Forest<br>
</div>
<div class="gmail_extra"><br>
</div>
</font></span></div>
</blockquote>
</div>
<br>
</div>
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