[EM] What does "proportional representation" MEAN? And list of known PR methods (know any more?)

[Kristofer Munsterhjelm]

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/election-methods@lists.electorama.com/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/election-methods@lists.electorama.com/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/election-methods-list@eskimo.com/msg08378.html 
, and for another such method, based on Borda: 
http://www.mail-archive.com/election-methods-list@eskimo.com/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/election-methods-list@eskimo.com/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.