Discussion:
Median-based Proportional Representation
Toby Pereira
2011-07-08 23:19:30 UTC
Permalink
While discussing median-based range voting -
http://rangevoting.org/MedianVrange.html, Warren Smith says "Average-based range
voting generalizes to a multiwinner proportional representation voting system
called reweighted range voting. (See papers 78 and 91 here.) But there currently
is no known way to generalize median-based range voting to do that."

So I was thinking about how you might get a median-based PR system, using range
voting, or some other score system, such as Borda Count. I don't think there is
necessarily a "perfect" method but I did come up with something (possibly
ridiculous). You find a way to convert the scores of the candidates so that a
candidate's median score becomes their mean score. For example, if a candidate's
mean was 5 (out of 10) and their median was 7, their scores would undergo some
sort of transformation so that their mean score became 7. Likewise if someone
had a mean of 7 and a median of 5, their scores would undergo a transformation
to reduce the mean to 5.

One way to do this is as follows: Convert the range so that it becomes 0 to 1
(so in a 0-10 case, just divide all scores by 10). Then for each candidate you
convert their score s to s^n where n is the number for that particular candidate
that will make the original median score the mean of the transformed scores. For
n over 1 the score will be reduced and for n under 1, the score will increase.
So each candidate has their own value of n.

Once all the scores have been converted, you can just do whatever you would have
done in your non-median-based PR system to find the winning candidates.

Obviously, this is a bit of a fudge because although we are fixing the mean for
each candidate to what we want, the rest of the scores just end up how they end
up. There would be different conversion systems that convert median to mean but
give different values for the other scores.

Just looking at the median and mean here could be seen as a bit arbitrary. As
well as converting median to mean, we would ideally also want to convert other
percentiles accordingly. We'd want to convert the 25th percentile score to the
25th "permeantile"*, or whatever the term is. (Is there a term?) But it would
actually be impossible to do this properly. With repeated scores (which would
always happen where there are more voters than possible scores), different
percentile values will have the same score. For example, if someone's median
score is 5, it's likely to also be 5 at the 51st percentile. But, as far as I
understand it, the "permeantile" would not be able to have a flat gradient at
any point, unless it's flat all the way across. So we couldn't have a "perfect"
system that worked on this basis. So for simplicity we can just use the system
as described.

Of course, with range voting, people might vote approval style, so many
candidates might simply have a median of 0 or 10. In that case the only
"reasonable" conversion would be to convert all their scores to 0 or 10
respectively. This problem wouldn't occur to the same degree under Borda Count,
however.

*I was thinking about how you would calculate permeantiles. In a uniform
distribution between 0 and 1, the 25th permeantile would be 0.25. If you weight
the averages of each side 3 to 1 in favour of the smaller side of the
permeantile (0 to 0.25), and average these, then you get 0.25. (3*0.125 +
1*0.625) / 4 = 0.25. So for the 10th permeantile, you have (9*0.05 + 1*0.55) /
10 = 0.1 and so on. I imagine this would work for non-uniform distributions too.
(Sorry for going off topic.)
Jameson Quinn
2011-07-08 23:27:12 UTC
Permalink
Post by Toby Pereira
While discussing median-based range voting -
http://rangevoting.org/MedianVrange.html, Warren Smith says "Average-based
range voting generalizes to a *multiwinner* proportional representation<http://rangevoting.org/PropRep.html>voting system called reweighted
range voting <http://rangevoting.org/RRV.html>. (See papers 78 and 91 here<http://www.math.temple.edu/~wds/homepage/works.html>.)
But there currently is no known way to generalize median-based range voting
to do that."
I've told Warren to change that, and he hasn't given me a clear criterion
for what I have to do so he will. I've created a system called AT-TV which
is PR and reduces to a median-based system in the single-winner case. It's
Bucklin-like, in that there is a falling approval threshold, and when a
candidate gets enough approvals to be elected (a Droop quota) they are,
which "uses up" those votes (except for the excess). So in a one-winner
case, it's based on 50th percentile (median), but in, for instance, a
3-winner case, it would be (pseudo-)maximizing the elected candidates'
75th-percentile score, not their 50th-percentile. I think this is the
appropriate thing to do in the multi-winner "median" case.

JQ
Post by Toby Pereira
So I was thinking about how you might get a median-based PR system, using
range voting, or some other score system, such as Borda Count. I don't think
there is necessarily a "perfect" method but I did come up with something
(possibly ridiculous). You find a way to convert the scores of the
candidates so that a candidate's median score becomes their mean score. For
example, if a candidate's mean was 5 (out of 10) and their median was 7,
their scores would undergo some sort of transformation so that their mean
score became 7. Likewise if someone had a mean of 7 and a median of 5, their
scores would undergo a transformation to reduce the mean to 5.
One way to do this is as follows: Convert the range so that it becomes 0 to
1 (so in a 0-10 case, just divide all scores by 10). Then for each candidate
you convert their score s to s^n where n is the number for that particular
candidate that will make the original median score the mean of the
transformed scores. For n over 1 the score will be reduced and for n under
1, the score will increase. So each candidate has their own value of n.
Once all the scores have been converted, you can just do whatever you would
have done in your non-median-based PR system to find the winning candidates.
Obviously, this is a bit of a fudge because although we are fixing the mean
for each candidate to what we want, the rest of the scores just end up how
they end up. There would be different conversion systems that convert median
to mean but give different values for the other scores.
Just looking at the median and mean here could be seen as a bit arbitrary.
As well as converting median to mean, we would ideally also want to convert
other percentiles accordingly. We'd want to convert the 25th percentile
score to the 25th "permeantile"*, or whatever the term is. (Is there a
term?) But it would actually be impossible to do this properly. With
repeated scores (which would always happen where there are more voters than
possible scores), different percentile values will have the same score. For
example, if someone's median score is 5, it's likely to also be 5 at the
51st percentile. But, as far as I understand it, the "permeantile" would not
be able to have a flat gradient at any point, unless it's flat all the way
across. So we couldn't have a "perfect" system that worked on this basis. So
for simplicity we can just use the system as described.
Of course, with range voting, people might vote approval style, so many
candidates might simply have a median of 0 or 10. In that case the only
"reasonable" conversion would be to convert all their scores to 0 or 10
respectively. This problem wouldn't occur to the same degree under Borda
Count, however.
*I was thinking about how you would calculate permeantiles. In a uniform
distribution between 0 and 1, the 25th permeantile would be 0.25. If you
weight the averages of each side 3 to 1 in favour of the smaller side of the
permeantile (0 to 0.25), and average these, then you get 0.25. (3*0.125 +
1*0.625) / 4 = 0.25. So for the 10th permeantile, you have (9*0.05 + 1*0.55)
/ 10 = 0.1 and so on. I imagine this would work for non-uniform
distributions too. (Sorry for going off topic.)
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Toby Pereira
2011-07-09 10:52:34 UTC
Permalink
I had a look at your system -
http://www.mail-archive.com/election-***@lists.electorama.com/msg07066.html -
I think I might have to look at it again to get it! But one thing about
percentiles. As I understand it, people often disagree about how to calculate
percentiles. The one I agree with on here -
http://en.wikipedia.org/wiki/Percentile - is the one recommended by NIST. Where
P is the percentile and N is the population, the ranked posistion would be P/100
* (N+1). Does your system have an inbuilt assumption about this?




________________________________
From: Jameson Quinn <***@gmail.com>
To: Toby Pereira <***@yahoo.co.uk>
Cc: electorama list <election-***@lists.electorama.com>
Sent: Sat, 9 July, 2011 0:27:12
Subject: Re: [EM] Median-based Proportional Representation




I've told Warren to change that, and he hasn't given me a clear criterion for
what I have to do so he will. I've created a system called AT-TV which is PR and
reduces to a median-based system in the single-winner case. It's Bucklin-like,
in that there is a falling approval threshold, and when a candidate gets enough
approvals to be elected (a Droop quota) they are, which "uses up" those votes
(except for the excess). So in a one-winner case, it's based on 50th percentile
(median), but in, for instance, a 3-winner case, it would be (pseudo-)maximizing
the elected candidates' 75th-percentile score, not their 50th-percentile. I
think this is the appropriate thing to do in the multi-winner "median" case.


JQ
Warren Smith
2011-07-09 02:40:28 UTC
Permalink
Sorry, as Jameson pointed out, he has invented a voting method he calls AT-TV
which (he claims)
1. obeys a proportional representation theorem
2. in the single-winner case reduces to median-based range voting.

I should update http://rangevoting.org/MedianVrange.html
to reflect that. Why haven't I? Partly laziness/busyness, and partly because I
do not really understand AT-TV and the theorem it satisfies (which is
related). Sorry for my faults. I've been busy working on a different
project.

Jameson ran AT-TV on a real-world 9-winner election and claimed in that election
it gave the same results as STV.

Toby Pereira's suggestion for turning median-based-range voting into a PR system
is a pretty ridiculous "kludge" but yes, technically, it works. It
is kind of a matter of
opinion what is a "natural generalization" of median-based range to
multiwinner PR,
and what is an "unnatural kludge." In my subjective view TP's
suggestion is clearly the latter and I suppose that kind of ugly
approach could be used to (technically) turn virtually any
single-winner voting method into a PR multiwinner method. It is
really ugly though
and his transformation can distort a voter's preference A>B to B>A,
which I would
hope a lot of people would find very disturbing as step 1 in any election.
I offer no opinion on whether AT-TV is "natural generalization" or
"unnatural kludge." I also have little clue whether AT-TV or RRV is
better as a voting method.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html
----
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Jameson Quinn
2011-07-09 16:11:01 UTC
Permalink
Post by Warren Smith
Sorry, as Jameson pointed out, he has invented a voting method he calls AT-TV
which (he claims)
1. obeys a proportional representation theorem
Yes. It's instructive to see what PR criterion AT-TV satisfies, and what RRV
satisfies.

The standard Droop criterion, if I'm not mistaken, is: If a group of N droop
quotas of voters votes a set of >N candidates above all other candidates,
then at least N candidates from that set must win.

The AT-TV version would be: If a group of N droop quotas of voters votes a
set of >N candidates at or above a given rating, and all other candidates
below that rating, then at least N candidates from that set will win.

The RRV version... well, I'm not sure, but my guess is that it would be
something like: If a group of N droop quotas of voters votes a set of >N
candidates each with at least N times the rating of any candidate outside
that set, then at least N candidates from that set must win.

Note that these are successively weaker criteria on the systems; that is,
the coordination of a given party must be successively stronger to ensure PR
for that party. Purely on a subjective level, I think that AT-TV criterion
is about right, and that the RRV one is too weak.

JQ
Warren Smith
2011-07-09 18:19:27 UTC
Permalink
[Minor bug in Jameson;s claims: I think he meant >=N not >N.]
[Major ambiguity in Jameson's post: when you say "given rating" did
you mean this rating
is allowed to depend upon the party, or must it be party-independent?]

The RRV PR-theorem was if the voters all are "totally racist" that is vote
max-score for their color, and min-score for all others, then the
winners will have the same
color fraction as the voters (if enough candidates run from each
color, and up to
integer roundoff effects). Further if any particular party has
at least N droop-quotas worth of total-racist supporters, they'll win
at least N seats.

If RRV voters merely vote at least X times higher score for their
color than any other color,
that by itself will not assure PR. For example if the red voters
voted 0.001 for each red and
0.000001 for each non-red, that'd have little effect on helping reds win seats.

The STV PR guarantee is pretty much the same as the RRV one, that is,
the "racist"
voters need to vote their color absolute top, all rival colors non-top.

Incidentally in RRV and STV, the "colors" do not need to have anything
to do with parties.
They can be an arbitrary coloring.

In AT-TV, suppose
30% of the voters vote Red=9, Blue=Green=0;
30% of the voters vote Blue=9, red=green=0;
40% of the voters vote Green=5, red=green=4.
Will it then elect 30% reds, 30% blues, and 40% greens?
And do you consider that the right thing to do?
Post by Jameson Quinn
Post by Warren Smith
Sorry, as Jameson pointed out, he has invented a voting method he calls AT-TV
which (he claims)
1. obeys a proportional representation theorem
Yes. It's instructive to see what PR criterion AT-TV satisfies, and what RRV
satisfies.
The standard Droop criterion, if I'm not mistaken, is: If a group of N droop
quotas of voters votes a set of >N candidates above all other candidates,
then at least N candidates from that set must win.
The AT-TV version would be: If a group of N droop quotas of voters votes a
set of >N candidates at or above a given rating, and all other candidates
below that rating, then at least N candidates from that set will win.
The RRV version... well, I'm not sure, but my guess is that it would be
something like: If a group of N droop quotas of voters votes a set of >N
candidates each with at least N times the rating of any candidate outside
that set, then at least N candidates from that set must win.
Note that these are successively weaker criteria on the systems; that is,
the coordination of a given party must be successively stronger to ensure PR
for that party. Purely on a subjective level, I think that AT-TV criterion
is about right, and that the RRV one is too weak.
JQ
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html
----
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Warren Smith
2011-07-09 18:38:13 UTC
Permalink
Post by Warren Smith
In AT-TV, suppose
30% of the voters vote Red=9, Blue=Green=0;
30% of the voters vote Blue=9, red=green=0;
40% of the voters vote Green=5, red=blue=4.
Will it then elect 30% reds, 30% blues, and 40% greens?
And do you consider that the right thing to do?
--(in this situation, by the way, I think
RRV would elect <=20% Greens and the rest would be evenly split between
reds and blues.)

RRV incentivizes exaggerating in your vote to max and min scores; but it also
de-incentivizes that because if you exaggerate then your vote gets
de-weighted more
when your guy wins a seat. So the net result is probably that you are
incentivized
to exaggerate on candidates whose winning chances seem poor,
but to try to "free ride" for candidates that seem sure-thing winners by
dishonestly not giving them high scores.

With STV, a big flaw is that it pays no attention to anybody you
ranked below top.
If you ranked Bush>Gore>>>Hitler, it pays no attention to the "Gore>>>Hitler"
part of your vote, until after Bush is elected. And basically, any
party voter-bloc
which ranks Green-party candidates above all others, will have zero effect on
non-Green winners until after the last Green has won, then some small
fraction of
"leftover" green power will be able to exert some (tiny and varying with
the size of the leftover in a rather random-seeming way) effect.

This ignoring, seems a very serious flaw of STV, that is not present with RRV.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html
----
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Toby Pereira
2011-07-09 18:58:08 UTC
Permalink
Is this just the Instant Run-off version of STV that you're talking about that
pays no attention to who you ranked below top? Also your example here -
http://rangevoting.org/PRcond.html - would that be resolved under any form of
STV in existence?




________________________________
From: Warren Smith <***@gmail.com>
To: Jameson Quinn <***@gmail.com>
Cc: election-methods <election-***@electorama.com>
Sent: Sat, 9 July, 2011 19:38:13
Subject: Re: [EM] Median-based Proportional Representation
Post by Warren Smith
In AT-TV, suppose
30% of the voters vote Red=9, Blue=Green=0;
30% of the voters vote Blue=9, red=green=0;
40% of the voters vote Green=5, red=blue=4.
Will it then elect 30% reds, 30% blues, and 40% greens?
And do you consider that the right thing to do?
--(in this situation, by the way, I think
RRV would elect <=20% Greens and the rest would be evenly split between
reds and blues.)

RRV incentivizes exaggerating in your vote to max and min scores; but it also
de-incentivizes that because if you exaggerate then your vote gets
de-weighted more
when your guy wins a seat.  So the net result is probably that you are
incentivized
to exaggerate on candidates whose winning chances seem poor,
but to try to "free ride" for candidates that seem sure-thing winners by
dishonestly not giving them high scores.

With STV, a big flaw is that it pays no attention to anybody you
ranked below top.
If you ranked    Bush>Gore>>>Hitler, it pays no attention to the "Gore>>>Hitler"
part of your vote, until after Bush is elected.    And basically, any
party voter-bloc
which ranks Green-party candidates above all others, will have zero effect on
non-Green winners until after the last Green has won, then some small
fraction of
"leftover" green power will be able to exert some (tiny and varying with
the size of the leftover in a rather random-seeming way) effect.

This ignoring, seems a very serious flaw of STV, that is not present with RRV.
--
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html
----
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Warren Smith
2011-07-09 20:23:19 UTC
Permalink
Is this just the Instant Run-off version of STV that you're talking about
that
pays no attention to who you ranked below top?
--all forms. They all go thru your ballot in descendign order and pay
no attention to the part they haven't reached yet.
Also your example here -
http://rangevoting.org/PRcond.html - would that be resolved under any form
of
STV in existence?
--I don't think so.
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Kristofer Munsterhjelm
2011-07-10 09:56:02 UTC
Permalink
Is this just the Instant Run-off version of STV that you're talking
about that pays no attention to who you ranked below top? Also your
example here - http://rangevoting.org/PRcond.html - would that be
resolved under any form of STV in existence?
The DPC says that at least one candidate should be elected from each
coalition preferred by more than 16 + 2/3 of the voters. Since the first
place votes are disjoint, that in effect means

Elect B.
Elect C.
Elect D.
Elect E.
Elect F.

So the DPC would exclude A, yes. The Droop proportionality criterion
considers it more important, in this case, to give each
greater-than-Droop block of voters "what they're due" than to give them
a compromise.

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Kristofer Munsterhjelm
2011-07-10 09:44:12 UTC
Permalink
Post by Warren Smith
Sorry, as Jameson pointed out, he has invented a voting method he calls AT-TV
which (he claims)
1. obeys a proportional representation theorem
Yes. It's instructive to see what PR criterion AT-TV satisfies, and what
RRV satisfies.
The standard Droop criterion, if I'm not mistaken, is: If a group of N
droop quotas of voters votes a set of >N candidates above all other
candidates, then at least N candidates from that set must win.
It's actually, if *more* than n droop quotas vote for a set of k
candidates, then min(k,n) must win. So to win exactly the Droop quota is
not enough, as there may be too many candidates, e.g. for 90 voters to 2
seats:

30: A>B>C
30: D>E>F
30: G>H>I

Then if it had been "n Droop quotas", not "more than", it would say
"elect one of {ABC}, and one of {DEF}, and one of {GHI}", which is
impossible.
Post by Warren Smith
The AT-TV version would be: If a group of N droop quotas of voters votes
a set of >N candidates at or above a given rating, and all other
candidates below that rating, then at least N candidates from that set
will win.
The RRV version... well, I'm not sure, but my guess is that it would be
something like: If a group of N droop quotas of voters votes a set of >N
candidates each with at least N times the rating of any candidate
outside that set, then at least N candidates from that set must win.
Note that these are successively weaker criteria on the systems; that
is, the coordination of a given party must be successively stronger to
ensure PR for that party. Purely on a subjective level, I think that
AT-TV criterion is about right, and that the RRV one is too weak.
I agree that the RRV one is too weak.

I further think we could classify proportionality criteria into
categories such as these:

- very weak: It is possible for a party or group of candidates to get a
proportional outcome if they coordinate and can control their voters.
(SNTV, Taiwanese birthday type strategies.)

- weak: It is possible for a group of voters to force a proportional
outcome if they can coordinate among themselves.

- middle: It is possible for a group of voters to force a proportional
outcome with regards to a group of candidates they like (but not within
that group) by employing a common strategy (RRV "totally racist" criterion).

- stronger: If a group of voters reward a group of candidates with some
desirable relative aspect of their votes (top rank, high scores), then
the candidates will receive seats in proportion to the size of the voter
group and the desirable aspect, and this also counts recursively. (Droop
proportionality, divisor analogs.)

Would that let us find a rated version of the DPC? If proportional
representation was to be by rating instead of by seat count, it might
involve that more than a Droop quota can determine the ratio of the
rating of a candidate they rate highly to the rating of a candidate they
rate lower... but it seems difficult to get any closer than that.

Perhaps, for the utilitarian "proportionality of ratings" approach, it
would make more sense to look at party list methods first (since the
concept of having different ratings for winners makes sense there), then
generalize.

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Toby Pereira
2011-07-09 10:45:03 UTC
Permalink
Just to clarify this, for the nth "permeantile", I think you'd weight each point
on the n side (100-n)^2 and on the (100-n) side it would be n^2 .




________________________________
From: Toby Pereira <***@yahoo.co.uk>
To: electorama list <election-***@lists.electorama.com>
Sent: Sat, 9 July, 2011 0:19:30
Subject: [EM] Median-based Proportional Representation


*I was thinking about how you would calculate permeantiles. In a uniform
distribution between 0 and 1, the 25th permeantile would be 0.25. If you weight
the averages of each side 3 to 1 in favour of the smaller side of the
permeantile (0 to 0.25), and average these, then you get 0.25. (3*0.125 +
1*0.625) / 4 = 0.25. So for the 10th permeantile, you have (9*0.05 + 1*0.55) /
10 = 0.1 and so on. I imagine this would work for non-uniform distributions too.
(Sorry for going off topic.)
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