[EM] Subject: APR (13): Steve's 13th dialogue with Toby (Steve)

Fri Jan 2 14:38:25 PST 2015

Subject:
Re: APR (13): Steve's 13th dialogue with Toby (Steve)

Date: Thu, 1 Jan 2015
23:52:41 +0000

From: tdp201b at yahoo.co.uk

To: stevebosworth at hotmail.com; election-methods at lists.electorama.com

Subject: Re: APR (12): Steve's 12th dialogue with Toby (Steve)

To Toby and everyone.

My new comments are tagged by”XS:”

Steve:

SSS: True, but please explain how your highest
scored but eliminated candidate is going to pass on your one vote to an elected
rep. Are you in favour of this addition to a score system?

T:  I suppose because a score system
doesn't work on the same elimination basis as APR, it wouldn't be able to work
in exactly the same way. But what you could do is allow the following three
options for voters. One would be for the voter to give scores to as many
candidates as they like (anyone candidate they ignore gets a default 0) and not
transfer any power to their favourite candidate or anyone else. …

XS:  Do
you agree that this option would not guarantee that your vote would not be
entirely wasted, i.e. not even positively counting in the Commons through the
vote of the MP could have otherwise been given your score by any eliminated candidate
you had scored?

T:  … They
would also have the option of simply voting for one candidate and using that
candidate's entire score set of the other candidates.

XS: Please clarify.  Do you mean this one candidate if eliminated
would be required (some how) to give the score you gave to him to an MP?  If so, how exactly?

T: … The other option would be for a voter to
give scores (including zeros) to as many candidates as they want (these would
be the ones they have specific views about), and then leaving the ratings of
every other candidate to their indicated favourite candidate. As for whether
I'm in favour of it, yes, I think it should work well.

XS:  Again, do you agree that this option
would not guarantee that your vote would not be entirely wasted, not even
positively counting in the Commons through the vote of the MP given your score
by this eliminated candidate whom you had scored?

T: I would say that a proportional
approval/score system could well mean that it is less likely that some people
would get extra representation by mere chance because it takes into account
your rating of every candidate, not just the one that's deemed to be yours.
Therefore it wouldn't have the problem I highlighted in the example (quoted
from a previous e-mail below).

SSS: Please explain how you might objectively predict that more chance extra
votes would be given to APR reps than to approval/score reps. I do not yet see
that this is possible.

T:  The chance nature in APR comes from
the fact that it ignores the information about candidates below the transfer
line. Approval/score systems use all the information provided by voters and
they calculate the most proportional result from this. So any
disproportionality comes from the unavoidable fact that perfect proportionality
would be impossible rather than the pot luck of what happens to be in the
unused information in APR.

XS:  Still
you fail to give a mathematical definition of “proportional”.  Without this, you cannot explain why the
information below APR’s transfer line is important to you.  Similarly, you have not provided me with the
formula by which all the information that would be provided by your
approval/score voting would allow you to calculate what you would see as the
optimal, “overall proportionality” in the Commons as a result of any such
election.  Can you not provide this
information? [I see you have attempted to provide this information below.]

T: Approval/score can give better levels of proportionality by using more
information, so it doesn't make it more probable that any given individual will
have their views better represented in parliament, but I would argue that it
reduces the chances of people being over or under-represented making it fairer
overall.

SSS: As it stands, this seems only to be your own vague and subjective opinion.
I keep asking you to define what you mean by "proportionality"
mathematically in an objective way but you have not yet done so. Why?

T:  I have! In my last post, and
previously as well. But I'll give it again with some background. Lots of
sources define proportionality in terms of parties or groups of voters, but I
think it's best defined in terms of individual voters. If we look at each MP's
representation as split among their voters, the basic definition of
proportional representation is that every voter has, as close as possible,
equal representation. This definition tallies up with other definitions
perfectly well, except that it doesn't have to talk about groups of voters,
which often aren't clear-cut anyway. For example, under APR if two people vote
for candidate A and one votes for candidate B, A has twice the parliamentary
power of B. But because A's power is split between two voters, every voter ends
up with equal representation. …

XS:  This
correctly understands APR but I think it would be clearer to say that “A’s
power is provided by two voters” rather than “split between two voters”.  Also, do you again agree that this option
would not guarantee that your vote would not be entirely wasted, i.e. it will
be wasted if none of the candidates you have approved or scored is sufficiently
popular to be elected?

T: … Under an approval/score system (or certainly one I would advocate) each voter doesn't have
just one representative but is represented to some extent by any MP that they
have approved or given a non-zero score to. …

XS:  Again,
do you accept that according to your system it is possible that none of the
candidates you have approved will be elected? 
However, if any of these are elected, will each voter be able to know
exactly to what “extent” he is represented by each of these MPs?  Will this also work for score voting?  If so, please explain how.

T:  The
basic definition of proportionality is unchanged (each voter has equal
representation), but how we calculate it is different from APR or STV methods
generally. With approval voting, it is fairly simple. Each MP's representation
is equally split [????shared????] among all the voters that have voted for
them. With score voting, it is split proportionally to the score each voter
gives. For example, candidate A is elected and has 1/500 of the parliamentary
power. Two voters approved candidate A, so they each effectively have 1/1000 of
the total parliamentary representation each plus whatever they might get from
other candidates.

XS: 
Correct me if my following understanding is mistaken:  Let us assume that your candidate A received
only a number of approvals equal to one 500th of all the citizens voting in the
country.  In this case, each citizen who
approved MP-A has 1/500 of one vote in the Commons.  Again, assume that  this one 500th of all the voting power in the
Commons was produced by 3,000 approvals. 
However, a different group of citizens might elect MP-B with 4,000
approvals.  If so, each voter in this
second group of citizens who gave their approvals to candidate B will have a
smaller share in the one vote in the Commons held by MP-B.  This means that the share of the voting power
in the Commons held by each of the voters for MP-B is less than the share held
by the voters for MP-A.  Is this
correct?  

If so, each citizen’s vote is not equal, “each
voter does not have equal representation”. 
For simplicity, I have assumed that each of the citizens voting for these
two MPs only approved of one candidate. 
However, the possibility of this inequality of representation would
remain even if each had approved of more than one candidate.  Please explain why you agree or disagree.

T: …Voters' levels of representation for all candidates are added up. Exact
proportionality is when every voter has the same amount of representation. That
is what I mean by proportionality. The total of the voters' levels of
representation will always be the same whatever the result (because it always
equals the total parliamentary power), and so the average will always be the
same. …

XS:  The
meaning you give to the above words must be different from the one I see in
them because I take them as just another way of expressing what APR
offers.  However, I would like you to
comment on my following attempt to rewrite your above words to show you how
they could be used correctly to characterize approval, score, and APR systems from
the point of view of each voter:

In your countrywide approval election of a 500
member Commons, the 500 candidates who receive the most approvals are elected
as MPs. The share that each approving citizen will have in the total power of
the Commons is discovered by adding together each of the shares he has of each
of the MPs he approved.  The more such
MPs he has approved, the larger will be his share of total voting power in the
Commons.  Because citizens may approve
different numbers of candidates, and thus different numbers of MPs, approval
voting (unlike APR voting) by no means guarantees that each voting citizen will
have “the same amount of representation”, or any representation at all.  

In any case, while the resulting voting power of
each approving citizen may be different, the total voting power either of all
approval, score, or APR MPs “will always be the same whatever the result
because it always equals the total parliamentary power”.  At the same time, “the average” voting power
of each approval, score, or APR MP will be one 500th of the total.  Of these three voting systems, only APR also
guarantees that this total will also equal the total number of voting citizens
in the country.

T: … The only exception to this is if a
candidate is elected with no support, because there are no voters to split the
support between.

XS: An approval, score, or APR candidate “with
no support” could not be elected unless the total number of candidates is equal
or less than the number of MPs to be elected. 

T: …We can therefore measure disproportionality by adding up the squared deviations
of representation levels of the individual voters from the mean level of
representation.

XS: Here you are admitting that “individual
voters” will have “deviations of representation levels from the mean”, i.e.
each citizen’s vote may not count equally in an approval or score election,
i.e. in the Commons. 

Again in the light of the above attempt to
rewrite your words, your above “measure of disproportionality” would usually
show that there is some “disproportionality” in approval or score systems but
none in APR.  This is true of APR because
the total voting power in the Commons would be equal to the total number of
voting citizens, each citizen’s vote being present in the weighted vote of his
MP, i.e. each citizen’s voting power is one – exactly one of the voting
population.

What do you think?

……………………………….

SSS: Correct me if I have misunderstood you:
Assuming that each MP has one vote in the Commons in your voting system, this
means each MP's "total amount of representation [voting power] is always
the same". At the same time, each approving elector for an MP would have
an equal share of this power. Alternatively, each scoring elector would have a
proportion of this power equal to the score he gave to this MP.

T: Yes.

SSS:  However, I do not yet understand
your phrases: "provided the elected candidate has had non-zero
support"; "proportionality is measured by the voters' total squared
difference from the mean amount of representation". I need these phrases
to be explained. Please explain this calculation using examples both for
approval/score and APR.

T:  OK. The phrase "provided the
elected candidate has had non-zero support" simply refers to the fact that
electing an MP that has no support would mean that the total representation
from the MPs is less in this case and you'd have a different mean to calculate
deviation from. The system wouldn't work properly in this case. The squared
difference isn't really part of the definition of proportionality, but just how
you'd measure disproportionality in an approval/score case. To give a simple
example:

2 to elect (with equal power), approval voting

3 voters: A, B

1 voter: C

We can say that the representation that a voter gets from a candidate is
1/number of voters for that candidate. In this case, because there are two
elected candidates and four voters, the [desired] average representation level
would be 2/4 or 1/2 (the total representation being 2 candidates).

If we elect AB, three voters each have a representation level of 2 * (1/3) or
2/3 and the other has 0. If you add up the squared differences from the average
(1/2), you get 3*(2/3 - 1/2)^2 + 1*(1/2 - 0)^2 = 1/3.

If we elect AC, three voters have a representation level of 1/3 and the other
has 1/1 or 1. The total of the squared differences from 1/2 is 3*(1/2 - 1/3)^2
+ 1*(1 - 1/2)^2 = 1/3.

So this is a tied result.

There is actually a complication I haven't mentioned with score voting. If a
voter gives a candidate 1/10 and it's the only score the candidate gets, then
according to what I've said, this voter would be considered to have the full
representation from this candidate and it would count towards their total level
of representation even though they don't like the candidate very much. So with
score voting, I would suggest "splitting" each voter into 10 parts
(or whatever the score is out of). The "top tenth" only approves
candidates given a score of 10, the next tenth approves candidates with a 9 or
a 10 and so on. So only one tenth of a voter approves the candidate with a
score of 1 out of 10.

XS:  As I
see it, your above complicated attempt at an explanation does not remove the
validity of the conclusion I offered for your consideration in
my above “XS:” comment immediately preceding this one.  If you think that
conclusion is not correct, please explain.

Also, because such a mathematical explanation
would be much more difficult for most citizens to comprehend, in contrast to
the relative mathematical simplicity of APR, I would see it, instead, as proving
an argument that APR should be preferred both over approval or score systems.

SSS: Yes, your score voting system (not your approval voting system), like APR,
might require your highest scored but eliminated candidate to pass on your
score to an MP on his “list of favourite candidates”. If so, the system would
seem to do all that is possible to guarantee the voting satisfaction of each
citizen. However, do you see this is also like APR in guaranteeing that no part
of any citizen’s full vote will be wasted? If so, explain how your score voting
could achieve this. Would this now be an essential part of your preferred
system?

T:  I think with the suggestion I made at
the top of this post, it does as good a job as APR at ensuring your vote isn't
wasted. …

XS:  If you
still believe this please explain why.

T:  ….I'm not
completely against ranked voting for proportional representation. I think it
has flaws, and my current thinking is that score voting in particular would
have less but it's always open to evidence and argument.

XS:  Please
specify and explain the “flaws” in APR that you still see in the light of all
the above points.

Toby

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