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<div>In a previous reply on this subject, I said I'd prefer the definition to be something like:</div><div><br></div><div>"<span style="font-family: Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;">If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is ranked anything other than last or joint last (either explicitly or through implication on a truncated ballot), then A's probability of winning must be no less than B's.</span>"</div><div><br></div><div>In Woodall's definition you quote, there's no mention of an implicit approval cutoff, but maybe that's because it's implicit rather than explicit! But the point is that the definition refers to votes - "If some candidate <i>a</i> has strictly fewer votes in total..." and by vote it seems to means an explicit ranking of a candidate. And that wording suggests some form of approval because you are in some sense voting for that candidate. Whereas if I decide to rank every candidate, then I wouldn't consider this to be a vote for my second-to-bottom, any more than it would be if I just truncated and left the bottom two or more.</div><div><br></div><div>On the face of it, it does suggest that it might encourage truncation. I might hate my second-to-bottom candidate, so I wouldn't want to cast a vote for them, and if the plurality criterion says that's what I'm doing, then a method passing this criterion might punish me for it. So it seems that I should not rank any of the candidates I hate and just truncate.</div><div><br></div><div>Of course, it might be that in practice a method passing plurality won't punish me for a full ranking (it's not something I've greatly studied so I don't know), but the wording of the criterion itself suggests that it might.</div><div><br></div><div>Toby</div><div><br></div><div><br></div>
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On Sunday, 30 June 2019, 17:01:02 BST, C.Benham <cbenham@adam.com.au> wrote:
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<p>Juho (and interested others),<br>
<br>
The Plurality criterion was coined in 1994 by Douglas Woodall.
Quoting him exactly from then:<br>
</p><blockquote type="cite">The following rather weak property was
formulated with single-seat elections in mind, but it makes
sense also for multi-seat elections and, again, it clearly holds
for STV .<br>
<br>
Plurality. If some candidate <i>a</i> has strictly fewer votes
in total than some other candidate <i>b</i> has
first-preference votes, then <i>a</i> should not have greater
probability than <i>b</i> of being elected.<br>
<b><br>
</b></blockquote>
No mention of any "implicit approval cutoff". I know that at the
time Woodall was only thinking about strict rankings from the top
with truncation allowed. <br>
If equal-first ranking is allowed, then for the purpose of this
criterion we should be using the fractional (summing to 1)
interpretation of the number of <br>
"first-preference votes".<br>
<p></p>
<p>Juho seems to think that the Plurality criterion is a "feature"
or strategy device that somehow encourages truncation. It isn't
and doesn't. <br>
<br>
If the method uses one of the traditional Condorcet algorithms
that are almost the same as each other (Smith//MinMax, Schulze,
River, Ranked Pairs)<br>
and uses Winning Votes as the measure of pairwise defeat strength,
then the method meets Plurality and also has, at least in the
zero-info case, a weak<br>
<b>random-fill incentive.<br>
</b></p>
<p>IRV, and IRV modified to meet Smith by before each elimination
checking to see if there is pairwise-beats-all candidate among
those remaining, both meet<br>
the Plurality criterion. In those methods do the voters have any
have any incentive "not to rank the candidates of the competing
groupings" ? No they <br>
don't.<br>
<br>
So what is the point of the Plurality criterion? To my mind it is
simply about not offending obvious fairness and common-sense.<br>
<br>
Juho, try to imagine that you have no interest in or knowledge
about voting algorithms, you've never thought about the split-vote
problem. You are accustomed<br>
to voting in plurality elections (or even perhaps Approval
elections) and you've never been interested in doing anything
other than voting for your sincere<br>
favourite, who regularly wins. You are content with the current
voting method and can't see any point in changing it.<br>
<br>
Now imagine some voting-reform movement succeeds and the new
method is, say, MinMax(Margins). You hear that voters can now
rank more<br>
than one candidate and you simply seek assurance that you will be
allowed to go on voting as before and you assume that the
government must<br>
more-or-less know what it's doing and assume the method won't in
any way be less fair than before.<br>
</p>
<p>In this election your favourite is A.<br>
46: A<br>
44: B>C<br>
10: C<br>
<br>
It is announced that the winner is B. At first you think "A got
more first-preference votes than B, it must have something to do
with some voters'<br>
second preference votes", but then you notice that B got the same
number of second-preference votes as A (zero), and then you ask
"How on earth<br>
did this crazy new method elect B over my favourite A, who very
clearly got more "votes" (marks next to his name on the paper
ballots) all of which<br>
were first-preference votes!"<br>
<br>
On hearing the reply "Oh, that's because B was the fewest votes
shy of being the Condorcet winner" do you (a) say "Oh how silly of
me, obviously<br>
that's fair!" or (b) say .. something much less understanding and
accepting ?<br>
</p>
<p>This scenario also works if the old method was IRV. You might
also notice that this first MMM election scenario is also a
massive egregious failure<br>
of the Later-no-Help criterion (because if the B voters had
truncated then B wouldn't have won). Do you like that criterion?<br>
<br>
If the old method had been Approval, you would then presumably be
understanding and resigned if it is announced that C won. <br>
In fact electing A is a failure of the Minimal Defense criterion.
Do you like that one? So methods that meet both MD and Plurality
(such as Winning<br>
Votes and Smith//implicitA) must elect C.<br>
<br>
</p><blockquote type="cite"> ... methods might not elect the best
winner (sincere Condorcet winner).
</blockquote>
If voters decline to (or don't bother to) express some or all of
their very weak (possibly light-minded) pairwise preferences by
truncating, then I don't<br>
classify that as "insincere" voting. Since therefore there could
be several (or even many) alternative "sincere voting" profiles it
follows that there<br>
could be more than one "sincere CW". It seems obvious to me that
the one of of those that is based on only the relatively strong
pairwise preferences<br>
will have a higher "social utility" than one based on all pairwise
preferences which include a lot of very weak ones.<br>
<br>
49: A>>>B>C<br>
03: B>A>>>C<br>
48: C>>>B>A<br>
<br>
Say these are the sincere preferences. If the voters care to
express all their pairwise preferences then the "sincere CW" is B,
but if they choose<br>
what I consider to be an alternative way of sincere voting and
truncate where that will only "conceal" some weak pairwise
preferences then<br>
an alternative "sincere CW" is (the apparently higher Social
Utility candidate) A.<br>
<p></p>
<p>In fact if the method used was the tweaked IRV method with an
explicit approval cutoff that I recently suggested and the cast
votes were<br>
49: A>>B<br>
03: B>A>><br>
48: C>>B<br>
<br>
then only C would be disqualified (because A both pairwise beats C
and is more approved than C) and then B is eliminated and A wins.<br>
I doubt that there would much blood flowing in the streets caused
by the failure to elect the voted CW (B).<br>
</p>
<p>As consolation for not meeting the Condorcet criterion we would
have a method much more resistant to Burial strategy than any
Condorcet<br>
method (and maybe more appealing to people who like IRV).<br>
<br>
<b><br>
Juho Laatu</b> <a 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=%3C06DA7FE6-AAF5-4684-B448-7FD93DCA0E35%40gmail.com%3E" rel="nofollow" target="_blank">juho.laatu
at gmail.com </a><br>
<i>Sat Jun 29 07:43:29 PDT 2019</i> <br>
</p><blockquote type="cite">P.S. I don't like the plurality criterion.
It actually sets an implicit approval cutoff at the end of the
listed candidates. The worst part of that idea is that it
encourages voters not to rank the candidates of the competing
groupings. That (potentially huge amount of missing information)
is not good for ranked methods. If voters learn to use that
feature, methods might not elect the best winner (sincere
Condorcet winner).
</blockquote>
<br>
<br>
<br>
The following rather weak property was formulated with single-seat
elections in mind, but it makes sense also for multi-seat
elections and, again, it clearly holds for STV .<br>
Plurality. If some candidate a has strictly fewer votes in total
than some other candidate b has first-preference votes, then a
should not have greater probability than b of being elected.<br>
<p></p>
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