<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type">
<title></title>
</head>
<body bgcolor="#ffffff" text="#000000">
<br>
<br>
Juho wrote:<br>
<blockquote type="cite">
<div>The "Possible Approval Winner" criterion looks actually quite
natural in the sense that it compares the results to what Approval
voting could have achieved.</div>
<div><br class="khtml-block-placeholder">
</div>
</blockquote>
I'm glad you think so.<br>
<br>
<blockquote type="cite">
<div>The definition of the criterion contains a function that can be
used to evaluate the candidates (also for other uses) - the possibility
and strength of an approval win. This function can be modified to
support also cardinal ratings.</div>
<div><br class="khtml-block-placeholder">
</div>
<div>In the first example there is only one entry (11: A>B) that
can
vary when checking the Approval levels. B can be either approved or
not. In the case of cardinal ratings values could be 1.0 for A, 0.0 for
C and anything between 0.0001 and 0.9999 for B. Or without
normalization the values could be any values between 0.0. and 1.0 as
long as value(A) > value(B) > value (C). With the cardinal
ratings version it is possible to check what the "original utility
values" leading to this group of voters voting A>B could have been
(and if the outcome is achievable in some cardinal ratings based
method, e.g. max average rating).</div>
</blockquote>
This concept looks vulnerable to some weak irrelevant candidate being
added to the top of some ballots, displacing a candidate down to <br>
second preference and maybe thereby causing it to fall out of the set
of "possible winners". It probably has other problems regarding
Independence <br>
properties, and I can't see any use for it.<br>
<br>
Chris Benham<br>
<br>
<br>
<br>
<blockquote cite="mid3555C37A-102E-485F-A943-302A84B26723@yahoo.co.uk"
type="cite">
<div>The "Possible Approval Winner" criterion looks actually quite
natural in the sense that it compares the results to what Approval
voting could have achieved.</div>
<div><br class="khtml-block-placeholder">
</div>
<div>The definition of the criterion contains a function that can be
used to evaluate the candidates (also for other uses) - the possibility
and strength of an approval win. This function can be modified to
support also cardinal ratings.</div>
<div><br class="khtml-block-placeholder">
</div>
<div>In the first example there is only one entry (11: A>B) that
can vary when checking the Approval levels. B can be either approved or
not. In the case of cardinal ratings values could be 1.0 for A, 0.0 for
C and anything between 0.0001 and 0.9999 for B. Or without
normalization the values could be any values between 0.0. and 1.0 as
long as value(A) > value(B) > value (C). With the cardinal
ratings version it is possible to check what the "original utility
values" leading to this group of voters voting A>B could have been
(and if the outcome is achievable in some cardinal ratings based
method, e.g. max average rating).</div>
<div><br class="khtml-block-placeholder">
</div>
<div>The max average rating test is actually almost as easy to make
as the PAW test. Note that my description of the cardinal ratings for
candidate B had a slightly different philosophy. It maintained the
ranking order of the candidates, which makes direct mapping from the
cardinal values to ordinal values possible. The results are very
similar to those of the approval variant but the cardinal utility
values help making a more direct comparison with the "original
utilities" of the voters.</div>
<div><br class="khtml-block-placeholder">
</div>
<div>Now, what is the value of these comparisons when evaluating the
different Condorcet methods. These measures could be used quite
straight forward in evaluating the performance of the Condorcet methods
if one thinks that the target of the voting method is to maximise the
approval of the winner or to seek the best average utility. This need
not be the case in all Condorcet elections (but is one option). There
are several utility functions that the Condorcet completion methods
could approximate. The Condorcet criterion itself is majority oriented.
Minmax method minimises the strength of interest to change the selected
winner to one of the other candidates. Approval and cardinal ratings
have somewhat different targets than the majority oriented Condorcet
criterion and some of the common completion methods, but why not if
those targets are what is needed (or if they bring other needed
benefits like strategy resistance).</div>
<div><br class="khtml-block-placeholder">
</div>
<div>I find it often useful to link different methods and criteria to
something more tangible like concrete real life compatible examples or
to some target utility functions (as in the discussion above). One key
reason for this is that human intuition easily fails when dealing with
the cyclic structures (that are very typical cases when studying the
Condorcet methods). In this case it seems that PAW and corresponding
cardinal utility criterion lead to different targets/utility than e.g.
the minmax(margins) "required additional votes to become the Condorcet
winner" philosophy. Maybe the philosophy of PAW is to respect clear
majority decisions (Condorcet criterion) but go closer to the
Approval/cardinal ratings style evaluation when the majority opinion is
not clear. You may have different targets in your mind but for me this
was the easiest interpretation.</div>
<div><br class="khtml-block-placeholder">
</div>
<div>Juho</div>
<div><br class="khtml-block-placeholder">
</div>
<div><br class="khtml-block-placeholder">
</div>
<div>P.S. One example.</div>
<div>1: A>B</div>
<div>1: C</div>
<div>Here B could be an Approval winner (tie) but not a max average
rating winner in the "ranking maintaining style" that was discussed
above (since the rating of B must be marginally smaller than the rating
of A in the first ballot).</div>
<div><br class="khtml-block-placeholder">
</div>
<br>
<div>
<div>On Mar 7, 2007, at 16:28 , Chris Benham wrote:</div>
<br class="Apple-interchange-newline">
<blockquote type="cite"> <br>
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>
<br>
</blockquote>
</div>
<br>
<pre wrap="">
<hr size="4" width="90%">
----
election-methods mailing list - see <a class="moz-txt-link-freetext" href="http://electorama.com/em">http://electorama.com/em</a> for list info
</pre>
</blockquote>
</body>
</html>