Warren Smith wrote:
Kristofer Munsterhjelm asked me what "proportional representation" (PR) means.
At this time it is probably unwise to make a too-precise definition
since every PR voting method seems to obey a different proportionality
theorem. I say you should just assess each theorem on a case by case
basis to see if you like it.
But a somewhat imprecise definition is:
I would say that any voting method which elects W winers from N
candidates (arbitrary 0<W<N) with the property that
"under an assumption of 'standard racist' voter behavior, it always
elects the same
proportions of different-'color' candidates as the voters (provided
enough candidates of
each color run) up to some reasonable error bound" is PR.
However
* what is the 'standard racist' voter behavior?
* what are the 'error bounds'? (Once they get poor enough, they
would no longer be acceptable, but I propose no precise threshhold)
These differ from theorem to theorem. And for Asset Voting "standard
racism" assumptions also are needed about the candidate-behavior.
I think that the measure of proportionality should be on sets, not just
candidates, because I'd like the method to be better than open party
list. Because of that, I like the Droop proportionality criterion, and
hope something analogous to it can be constructed for Webster, because
it seems that when people *do* vote party list style, Webster beats
Droop in the proportionality department (as you yourself have shown on
your apportionment pages).
A proportionality based on sets would also permit voters to vote for
some semi-popular candidate first and a less-known independent second,
and have the vote support both. Even that does have limits, though,
because it would not guarantee that a vote of "independent first, then
semi-popular" would support semi-popular if independent didn't make it,
a property which I'd also like.
If I can have a pony, metaphorically speaking, the method should capture
people's preferences in orders of sets as well. E.g. if there are n%
libertarian socialists, then the method should pick n% that are both,
not just ensure >n% libertarian, >n% socialist. This might not be
possible, and might reduce to a set covering problem even if technically
possible.
In any case, this is all informal.
HERE'S MY LIST OF KNOWN PR VOTING METHODS:
Webster, and certainly all "divisor methods" for party-list (it is one)
already are known to obey such criteria. (The very definition of
"divisor method"
is a PR theorem.) This should include my new notion of
"generalized divisor methods" where both multiplicative and/or
additive parameters
are involved. Hamilton-Vinton is one. See
http://rangevoting.org/Apportion.html
http://www.RangeVoting.org/NewAppo.html
http://www.RangeVoting.org/BishopSim.html
M.L. Balinski & H. Peyton Young: Fair Representation: Meeting the
Ideal of One
Person, One Vote (2nd edition), Brookings Institution Press 2001
Asset voting also obeys a PR theorem.
http://rangevoting.org/Asset.html
paper #77 at http://www.math.temple.edu/~wds/homepage/works.html
RRV also (RRV is kind of based on "stealing" the
divisor-method idea, inside).
paper #78 at http://www.math.temple.edu/~wds/homepage/works.html
http://rangevoting.org/RRV.html
Hare/Droop STV also.
Nicolaus Tideman: The Single transferable Vote,
J. Economic Perspectives 9,1 (1995) 27-38.
And LPV(kappa) ("logarithmic penalty voting") also.
Invented by F.Simmons. Described in paper #91 at
http://www.math.temple.edu/~wds/homepage/works.html
Also certain PR methods which are "precinct countable"
invented by Forest Simmons, see puzzle#15 at
http://rangevoting.org/PuzzlePage.html .
Finally, there was also a simple one invented by a student at University of
Michigan named Tim Hull. See
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2007-April/020194.html
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2007-April/020195.html
That's my list. Is anybody aware of any other PR methods?
Non-divisor party list PR (closed and open, though neither is
proportional within each list). Party list PR in general might be a good
place to consider how much PR is PR. For instance, is Jefferson party
list PR? Imperiali? Using party list gets rid of the question of
"proportionality of *what*", because there can only be proportionality
of two things, and these two don't interfere: of party lists, and of
candidates within those lists (if open list PR).
I have also constructed some proportional and semiproportional methods.
One simple method is "multiwinner Bucklin", which goes like this: do
ordinary Bucklin until some candidate has the support of at least a
Droop quota. Elect him and remove from all ballots, reweighting the
ballots who contributed to his victory, then restart.
I also tried to make a Droop proportional summable version of Bucklin,
but it failed because of an ambiguity problem I called "shadowing". See
http://www.mail-archive.com/[email protected]/msg03893.html
As a consequence, I conjectured that no method where you can say "oh,
these winners were elected because more than a Droop quota supported
them" can be summable, because you can pad with irrelevant candidates
and voters to make the Droop quota any number, and thus extract DAC/DSC
type data (which requires more than polynomial space) from it.
You should already know about Setwise Highest Average, another method of
mine. It's house proportional (i.e. produces a proportional ordering)
and reduces to DAC (or DSC). Unfortunately, it's highly nonmonotonic.
My latest attempt at a multiwinner method is the "continuous forced
clustering Kemeny" method. It has horrible runtime, but is meant to be a
proof of concept more than a viable method in itself. The method uses
linear programming to determine an assignment of fractional voters to
"clusters" (each of which hold equally many voters, hence
proportionality) so as to minimize a distance measure (could also
maximize a utility measure). That provides an optimal least distance for
a given council, and we "just" have to try all of them to find out which
gives the minimum optimal least distance (i.e. it's easiest to fit the
voters into clusters if the clusters represent these candidates).
See
http://www.mail-archive.com/[email protected]/msg04312.html
for that. It could be applied to any weighted positional system, or to
Range; applying it to Condorcet methods in general seems to require more
than just linear programming (or else horrible runtimes like for Kemeny)
Enough about me.
There is PSC-CLE ( http://wiki.electorama.com/wiki/PSC-CLE ). It passes
Droop proportionality, but isn't very good past that (caveat simulator,
disclaimer, etc).
Better is Woodall's Quota-Preferential by Quotient (
http://www.votingmatters.org.uk/ISSUE17/I17P1.PDF ), especially if the
quotient is altered from d'Hondt to Sainte-Laguë, although that does
significantly compromise its single-winner performance.
Plain old SNTV (plurality) meets a weaker definition of PR: the method
has a strategic equilibrium where no party has an incentive to field
more (or fewer) candidates, and that does achieve party PR when that is
the case and voters vote in a certain way; but this is not, I think,
true PR. It is summable, but so is party list PR.
CPO-STV and Schulze STV try to "Condorcet-ize" STV by running a
Condorcet election between all possible assemblies, picking the
Condorcet winner assembly. Because of that, they're not polytime. I
think the latter is better than the former, although I haven't tested
that. The Condorcet Internet Voting Service, CIVS, has a similar "gotta
try them all" method:
http://www.cs.cornell.edu/w8/~andru/civs/proportional.html . When the
methods employ a virtual Condorcet election, Forest calls them
"Condorcet flavored PR methods" - see also
http://www.mail-archive.com/[email protected]/msg08378.html
, and for another such method, based on Borda:
http://www.mail-archive.com/[email protected]/msg08507.html .
Anthony O'Neal generalized BTR-IRV to STV-ME: in an n-seat election, do
STV as usual, but when you have to eliminate, run an n+1 candidate
"bottom runoff" (using, in reverse, either Plurality or a Condorcet
method) to determine who to eliminate. (
http://lists.electorama.com/pipermail/election-methods-electorama.com/2006-June/018293.html
)
D'Hondt without lists (
http://www.mail-archive.com/[email protected]/msg08230.html
) uses Condorcet with reweighting. It doesn't seem to do very well, but
it is house monotone.
Since you mention RRV, why not complete the list with PAV? Proportional
approval voting. Rob LeGrand coauthored a paper about a minmax version
of approval voting (find the council which is least unlike the Approval
vote it's most unlike), but that seems to me to be more a
consensus-seeking method than a proportional one.
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