<div dir="ltr"><div>Chris,<br><br>I'm convinced; the default approval should be top set only. How much control the voters have over adjusting that cutoff on their ballots is still up for grabs. I like keeping it simple, but Michael nade some good points about allowing voters to specify individual disapprovals.<br><br></div><div>Good point about presenting half-power-truncation as symmetric completion for truncated candidates. It's easier to motivate it from that perspective.<br><br></div><div>The more I think about it the more I like symmetric completion for all equal rankings below the approval cutoff, not just for truncation.<br><br></div><div>Then not only do we have mono-add-plump, but also this more detailed property:<br><br></div><div>If a new ballot is added that doesn't rank any (previously) disqualified candidate above all of her disqualifiers or equal to any approved disqualifier, then the new winner will be from the candidates that are approved on the new ballot.<br><br></div><div>What more could we ask for by way of participation incentive?<br></div><div><br></div>Forest<br></div><div class="gmail_extra"><br><div class="gmail_quote">On Sun, Nov 20, 2016 at 3:59 PM, C.Benham <span dir="ltr"><<a href="mailto:cbenham@adam.com.au" target="_blank">cbenham@adam.com.au</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<div class="m_8729162978151899294moz-cite-prefix"><span class="">On 11/19/2016 10:26 AM, Forest Simmons
wrote:<br>
<br>
<blockquote type="cite">An interesting fact is that MDDA(pt/2) is
just another formulation of my version of ICA.</blockquote>
<br></span>
Yes, that had dawned on me. It's also like symmetrically
completing ballots only at the bottom, saying<br>
that above-bottom equal-rankings contribute nothing to the
pairwise scores of the candidates with the<br>
same above-bottom ranking versus each other, and then saying that
unless all candidates have a majority-<br>
strength defeat any that do are disqualified.<br>
<br>
But isn't that a little bit different from the normal version of
the "Tied-at-the-Top Rule", because that treats<br>
equal-top equal ranking differently from all below-top
equal-ranking?<br>
<br>
I am a bit concerned about this (sincere) scenario:<br>
<br>
40: A>B<br>
10: A=B<br>
35: B<br>
15: C<br>
<br>
With all the voters' approval cutoffs left in the default position
B wins but A is the CW. Of course if the 40 A>B<br>
preferrers vote A>>B there is no problem, but might there
be a case for the default placement being just below<br>
the top-voted candidate/s?<br>
<br>
For a method with this cute game-theoretic defence of the
'sincere CW who is the smallest faction's favourite' and also<br>
a way of addressing the Chicken Dilemma scenario , this looks very
good.<br>
<br>
It meets FBC, Mono-add-Plump and Irrelevant Ballots Independence,
Plurality and Mono-raise.<br>
<br>
Chris Benham<div><div class="h5"><br>
<br>
<br>
<br>
On 11/19/2016 10:26 AM, Forest Simmons wrote:<br>
</div></div></div><div><div class="h5">
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<div>Does
optional
approval
cutoff wreck
burial
protection?<br>
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Suppose we
have a sincere
scenario<br>
<br>
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40 C>B<br>
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35 A>B<br>
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25 B>C<br>
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and the C
faction
decides to
bury the CWs
B. The B
faction
anticipates
this and
responds by
truncating C.
It is in the
interest of
the A faction
to leave the
default
implicit
approval
cutoff in
place. The C
faction
doesn't want
to give A too
much support
so they use
the explicit
cutoff option:<br>
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40 C>>A<br>
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35 A>B<br>
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25 B<br>
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The approval
winner is B
the CWs.<br>
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<div>If they
left the
implicit
cutoff in
place it would
be worse for
them; their
last choice
would be
elected.<br>
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So I think
MDDA with
optional
explicit
cutoff is fine
with respect
to truncation
and burial.<br>
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How about the
CD?<br>
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In this case
the sincere
profile is<br>
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</div>
40 C<br>
</div>
35 A>B<br>
</div>
25 B>A<br>
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</div>
The B>A faction
threatens to
defect from the AB
coalition.<br>
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The A faction
responds by using
the explicit cutoff:<br>
<br>
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40 C<br>
</div>
35 A>>B<br>
</div>
25 B<br>
<br>
</div>
The approval winner is C, so
the threatened defection
back-fires.<br>
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It seems to me like that is
plenty of chicken defection
insurance.<br>
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The obvious equilibrium position
(for the chicken scenario) is<br>
<br>
40 C<br>
35 A>>B<br>
25 B>>A<br>
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Under MDDA(pt/2) the only
uneliminated candidate is A.<br>
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But if the B faction defects, all
candidates are eliminated, and the
approval winner C is elected.<br>
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This is why I like MDDA(pt/2).<br>
<br>
</div>
An interesting fact is that MDDA(pt/2)
is just another formulation of my
version of ICA. They are precisely
equivalent. Here's why:<br>
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In my version of ICA, X beats Y iff <br>
<br>
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[X>Y] > [Y>X] + [X=Y=T] +
[X=Y=between] , in other words,<br>
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[X>Y] > [Y:>=X] - [X=Y=Bottom],<br>
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which in turn equals<br>
<br>
</div>
100% - [X>Y] - [X=Y=Bottom], since 100%=
[X>Y] + [Y>=X].<br>
<br>
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So X beats Y iff<br>
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[X>Y] > 100% - [X>Y] - [X=Y=Bottom].<br>
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If you add [X.Y] to both sides and divide by 2, you get<br>
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[X>Y] +[X=Y=Bottom]/2 > 50%, <br>
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precisely the "majority-with- half-power-truncation" rule.<br>
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So (my version of) ICA is precisely equivalent to MDDA(pt/2).<br>
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</div>
I believe it to be completely adequate for defending against
burial, truncation, and Chicken Defection.<br>
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<div class="gmail_extra"><br>
<div class="gmail_quote">Now suppose that p<q<r, and
p+q+r=100%, and we have three factions of respective sizes
p, q, and r:, with r + q > 50%.<br>
<br>
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<div class="gmail_quote">p: C<br>
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<div class="gmail_quote">q: A>>B<br>
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<div class="gmail_quote">r: B>>A<br>
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<div class="gmail_extra">Then under the pt/2 rule both C and B
are eliminated, but not A, so A is elected.<br>
<br>
</div>
<div class="gmail_extra">Suppose that the B factions defects.<br>
<br>
</div>
<div class="gmail_extra">Then A is also eliminated, and the
approval winner C is elected.<br>
<br>
</div>
<div class="gmail_extra">Etc.<br>
<br>
</div>
<div class="gmail_extra">So which of the two equivalent
formulations is easier to sell? ICA or MDDA(pt/2) ?<br>
<br>
</div>
<div class="gmail_extra">Forest<br>
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</blockquote></div><br></div>