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<div class="moz-cite-prefix">If my old memory serves me tolerably
well, isn't this paper something like an article entitled The Best
of Both Worlds, where the authors did a survey of a tendency for
European electoral systems, over the decades, to have decreased
their average magnitude. I forget the details, just about
everything actually. But it may have gone something like: the
constiuencies shrank and the thresholds got higher.<br>
It was an informative statistical survey. <br>
But I think it went awry on what academics are fond of calling
"normative" considerations. Or on the stricture of David Hume,
that what is, is not necessarily right. <br>
I would have put to the authors, as a critic. That was this trend,
they so diligently exposed, but the moving to a "sweet spot" for
political incumbents, with precious little to do with democracy
and effective elections for the voters?<br>
<br>
from<br>
Richard Lung.<br>
<br>
On 07/03/2018 19:19, Jack Santucci wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CAESNRmcO9ga2d-W3aWfAxYxPGtHjkDzvAtDSSZpUG4NoSCPwuw@mail.gmail.com">
<div dir="ltr">Consensus in academia? Maybe that cigarettes cause
cancer. Maybe.
<div><br>
</div>
<div>I jest.</div>
<div><br>
</div>
<div>This paper may be helpful: <a
href="http://personal.lse.ac.uk/hix/Working_Papers/Carey-Hix-AJPS2011.pdf"
moz-do-not-send="true">http://personal.lse.ac.uk/hix/Working_Papers/Carey-Hix-AJPS2011.pdf</a></div>
</div>
<div class="gmail_extra"><br>
<div class="gmail_quote">On Wed, Mar 7, 2018 at 2:14 PM, Richard
Lung <span dir="ltr"><<a
href="mailto:voting@ukscientists.com" target="_blank"
moz-do-not-send="true">voting@ukscientists.com</a>></span>
wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">So, the
academic world has no consensus or standard model of
election method?
<div class="HOEnZb">
<div class="h5"><br>
<br>
On 03/03/2018 19:57, Kristofer Munsterhjelm wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
Say we have a consensus method M that works by
choosing the council C that minimizes the maximum
penalty p(C, v) for the voter that maximizes this
penalty. That is, the method finds C according to<br>
<br>
C = arg min max p(c, v)<br>
c v<br>
<br>
where ties are broken in a leximax fashion (i.e.
considering next to max, then next to next to max and
so on). Furthermore let the penalty "nonnegative" in
the sense that any voter with a real preference has at
least as great a penalty as a voter with no preference
(the zero voter, as it were).<br>
<br>
Now let the modified consensus method M' be one that
has the same optimization objective, but the method is
permitted to remove a Droop quota of votes to help
minimize the penalty.<br>
<br>
So M says "what council displeases the most displeased
voter the least?", while M' says "what council
displeases the most displeased voter the least, if we
can discard a Droop quota of voters from
consideration?"<br>
<br>
Then, are there any properties for p that makes M'
satisfy Droop proportionality? Can we in general turn
consensus methods of this form into PR methods by
adding a "you can discard a Droop quota" rule?<br>
<br>
If we can, then we easily get a multiwinner version of
Bucklin/MJ by doing the following:<br>
<br>
Let g(c, v) be the grade voter v gives to the least
preferred candidate in c.<br>
<br>
Let the consensus method M be<br>
<br>
C = arg max min g(c, v)<br>
c v<br>
<br>
Let M' permit the method to remove a Droop quota, i.e.
if |V| is the number of voters, and V is the set of
voters itself:<br>
<br>
C' = arg max c:<br>
max x subset of V so that |x| = |V|/(seats+1):<br>
min v in V \ x:<br>
g(c, v)<br>
<br>
For a single-winner election, M' is (up to tiebreaker)
just MJ, because for each potential winner c, the
removal step will remove the voters who grade c the
worst, and the Droop quota for single-winner is a
majority. Then the voter grading the c the worst after
half of the voters have been removed is just the
median voter.<br>
<br>
<br>
<br>
Some thoughts about two-winner remove-voter minimax
Approval follow. Reasoning about what voter removal
actually does can get kinda hairy, thus I may very
well be wrong:<br>
<br>
In minimax Approval, p(c, v) is the Hamming distance
between c and voter v's ballot, i.e. the number of
candidates in c but not approved by v plus the number
of candidates approved by v not in c.<br>
<br>
Say we have an analogous Droop criterion for Approval:
if more than k Droop quotas approve of a set of j
candidates (and nobody else), then at least min(k, j)
of these must be elected.<br>
<br>
For two winners, there are these possibilities:<br>
1. no Droop constraints<br>
2. k = 2, j >= 2<br>
3. k = 2, j = 1<br>
4. k = 1, j >= 1<br>
5. k = 1, j = 1<br>
<br>
1. is no problem, because we can elect anyone we want
without running afoul of the Approval DPC.<br>
<br>
2. Since there can only be three Droop quotas in
total, when we're considering A = {C_1, C_2} with C_1
and C_2 in the set of j candidates (call it J), we can
eliminate all but the J-voters and the maximum penalty
is j-2.<br>
In contrast, for some B = {C_x, C_y} not a subset of
j, the best it can do is eliminate a Droop quota of
the J-voters. In the best case (for B), everybody but
the J-voters approve of B alone. But there still
remains a Droop quota (plus one voter) of the
J-voters, and each of them gives penalty j. So A is
preferred to B.<br>
If B = {C_1, C_x}, then even if everybody but the
J-voters approve of B alone, the J-voters give penalty
j-1. So A is still preferred to B.<br>
<br>
3. Same as in 2, but let A = {C_1, C_x}, J = {C_1}.
With A, we eliminate so that only the J-voters are
left, and then max penalty is 1 (for C_x).
Furthermore, every remaining voter gives penalty 1.
Let B = {C_x, C_y}. In the best case for B, a Droop
quota of J-voters are eliminated and we have a Droop
quota plus one left. These all give penalty 2, which
is worse than penalty 1. So A is preferred to B.<br>
<br>
5. Let A = {C_1, C_x} and B = {C_x, C_y}. In the best
case for B here, two Droop quotas minus a voter
approve only of B, and the remaining Droop quota plus
one voter approves of J = {C_1}. Eliminating all but
one of the J-voters gives a max penalty of 3 from that
one J-voter: one point for not having C_1, and two
points for having C_x and C_y. A eliminates one of the
two B-approving Droop quotas and gets a penalty of 1
from every remaining voter, which is better.<br>
Note that I assume that C_x is approved by the
B-voters. If that were not the case, then {C_x, C_y}
would already be beaten by some {C_z, C_y} where C_z
is. Note also that I don't consider the case where the
B-voters also approve of a whole load of other
candidates, with the idea of raising the penalty under
A. The problem is that because only two candidates can
be elected, this would also raise their penalty under
B.<br>
<br>
4. Let A = {C_1, C_x} and B = {C_x, C_y}. The best
case for B has worst penalty j+2, since after a Droop
quota of J-voters have been eliminated, there remains
a single voter who only approves of J. After
eliminating some of the B-voters, A gets penalty j
from the J-voters (j-1 for the members of J not part
of {C_1, C_x} and one more for C_x which is not
approved by them), and one penalty point from the
B-voters.<br>
Here it'd seem that adding loads of candidates to the
B-voters would make things hard. Can it be salvaged?<br>
<br>
Suppose there are J-voters and C-voters. B is a subset
of C.<br>
When considering outcome B, before excluding a Droop
quota, every J-voter gives a penalty of j+2 and every
C-voter gives a penalty of c-2 where c=|C|.<br>
Under outcome A, before excluding, every J-voter gives
j, and every B-voter gives c (-1 for having C_x, +1
for having C_1).<br>
If j+2 > c, then we're in the domain above, and no
problem.<br>
If c > j+2, then the excluded candidates under both
A and B are C-voters.<br>
So under B we have a Droop quota of C-voters with
penalty c-2, and a Droop quota plus one of J-voters at
j+2.<br>
Under A we have a Droop quota of C-voters with penalty
c, and a Droop quota plus one of J-voters at j.<br>
<br>
So unless I made a mistake, Hamming distance is not
good enough. But I might have made a mistake, because
it seems strange that even in ordinary minimax
Approval, a coalition can increase its power by
approving a lot of clones. E.g. suppose in ordinary
minimax Approval that there are two coalitions of
almost equal size:<br>
<br>
n+1: A B<br>
n: C1 C2 C3 ... Cq<br>
<br>
{A, B} gets worst penalty q+2 (there are n of these
and n+1 zeroes)<br>
{A, C1} gets worst penalty q (n voters like C1 but not
A)<br>
{C1, C2} gets worst penalty q-2 (n voters give this
penalty, and then n+1 give penalty 4).<br>
<br>
... does that mean an arbitrarily small minority can
force its own council to win by just approving enough
clones that they set the worst penalty in every
outcome? That feels rather wrong.<br>
----<br>
Election-Methods mailing list - see <a
href="http://electorama.com/em" rel="noreferrer"
target="_blank" moz-do-not-send="true">http://electorama.com/em</a>
for list info<br>
<br>
</blockquote>
<br>
</div>
</div>
<span class="HOEnZb"><font color="#888888">
-- <br>
Richard Lung.<br>
<a href="http://www.voting.ukscientists.com"
rel="noreferrer" target="_blank"
moz-do-not-send="true">http://www.voting.ukscientists<wbr>.com</a><br>
Democracy Science series 3 free e-books in pdf:<br>
<a href="https://plus.google.com/106191200795605365085"
rel="noreferrer" target="_blank"
moz-do-not-send="true">https://plus.google.com/106191<wbr>200795605365085</a><br>
E-books in epub format:<br>
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rel="noreferrer" target="_blank"
moz-do-not-send="true">https://www.smashwords.com/pro<wbr>file/view/democracyscience</a><br>
<br>
<br>
---<br>
This email has been checked for viruses by Avast
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<div class="HOEnZb">
<div class="h5"><br>
<br>
----<br>
Election-Methods mailing list - see <a
href="http://electorama.com/em" rel="noreferrer"
target="_blank" moz-do-not-send="true">http://electorama.com/em</a>
for list info<br>
</div>
</div>
</blockquote>
</div>
<br>
<br clear="all">
<div><br>
</div>
-- <br>
<div class="gmail_signature" data-smartmail="gmail_signature">
<div dir="ltr">
<div>
<div dir="ltr">
<div>
<div><span style="font-size:small">Jack Santucci,
Ph.D.</span>
<div style="font-size:small">Independent scholar</div>
<div style="font-size:small"><a
href="http://www.jacksantucci.com"
target="_blank" moz-do-not-send="true">http://www.jacksantucci.com</a></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
<p><br>
</p>
<pre class="moz-signature" cols="72">--
Richard Lung.
<a class="moz-txt-link-freetext" href="http://www.voting.ukscientists.com">http://www.voting.ukscientists.com</a>
Democracy Science series 3 free e-books in pdf:
<a class="moz-txt-link-freetext" href="https://plus.google.com/106191200795605365085">https://plus.google.com/106191200795605365085</a>
E-books in epub format:
<a class="moz-txt-link-freetext" href="https://www.smashwords.com/profile/view/democracyscience">https://www.smashwords.com/profile/view/democracyscience</a>
</pre>
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