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Juho wrote (March7, 2007):<br>
<blockquote type="cite">
<pre wrap="">The definition of plurality criterion is a bit confusing. (I don't
claim that the name and content and intention are very natural
either :-).)
- <a class="moz-txt-link-freetext"
href="http://wiki.electorama.com/wiki/Plurality_criterion">http://wiki.electorama.com/wiki/Plurality_criterion</a> talks about
candidates "given any preference"
- Chris refers to "above-bottom preference votes" below
</pre>
</blockquote>
<blockquote type="cite">
<p><em>If the number of ballots ranking <i>A</i> as the first
preference is greater than the number<br>
of ballots on which another candidate <i>B</i> is given any
preference, then <i>B</i> must not be elected.</em></p>
</blockquote>
<blockquote type="cite">
<pre wrap="">Electowiki definition could read: "If the number of voters ranking A
as the first preference is greater than the number of voters ranking
another candidate B higher than last preference, then B must not be
elected".</pre>
</blockquote>
Yes it could and to me it in effect does (provided "last" means "last
or equal-last") The criterion come <br>
from Douglas Woodall who economises on axioms so doesn't use one that
says that
with three candidates <br>
A,B,C a ballot marked A>B>C must always be regarded as exactly
the
same thing as A>B truncates. He <br>
assumes that truncation is allowed but above bottom equal-ranking isn't.<br>
<br>
A similar criterion of mine is the "Possible Approval Winner" criterion:<br>
<br>
"Assuming that voters make some approval distinction among the
candidates but none among those<br>
they equal-rank (and that approval is consistent with ranking) the
winner must come from the set of <br>
possible approval winners".<br>
<br>
This assumes that a voter makes some preference distinction among the
candidates, and that truncated<br>
candidates are equal-ranked bottom and so never approved. <br>
<br>
Looking at a profile it is very easy to test for: considering each
candidate X in turn, pretend that the<br>
voters have (subject to how the criterion specifies) placed their
approval cutoffs/thresholds in the way<br>
most favourable for X, i.e. just below X on ballots that rank X above
bottom and on the other ballots<br>
just below the top ranked candidate/s, and if that makes X the
(pretend) approval winner then X is<br>
in the PAW set and so permitted to win by the PAW criterion.<br>
<br>
<pre wrap="">11: A>B
07: B
12: C</pre>
So in this example A is out of the PAW set because in applying the test
A cannot be more approved<br>
than C.<br>
<br>
IMO, methods that use ranked ballots with no option to specify an
approval cutoff and rank among<br>
unapproved candidates should elect from the intersection of the PAW set
and the Uncovered set<br>
<br>
One of Woodall's "impossibility theorems" states that is impossible
to have all three of Condorcet,<br>
Plurality and Mono-add-Top. MinMax(Margins) meets Condorcet and
Mono-add-Top.<br>
<br>
Winning Votes also fails the Possible Approval Winner (PAW) criterion,
as shown by this interesting<br>
example from Kevin Venzke:<br>
<br>
<pre wrap="">35 A
10 A=B
30 B>C
25 C
A>B 35-30, B>C 40-25, C>A 55-45
Both Winning Votes and Margins elect B, but B is outside the PAW set{A,C}.
Applying the test to B, we get possible approval scores of A45, B40, C25.
ASM(Ranking) and DMC(Ranking) and Smith//Approval(Ranking) all meet the Definite
Majority(Ranking) criterion which implies compliance with PAW. The DM(R) set is
{C}, because interpreting ranking (above bottom or equal-bottom) as approval, both
A and B are pairwise beaten by more approved candidates.
Chris Benham
</pre>
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