APR (15):
Steve's 15th dialogue with Toby (Steve)

Date: Mon, 9 Feb 2015
22:51:00 +0000

From: tdp201b <at> yahoo.co.uk

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

Subject: Re: APR (14): Steve's 14th dialogue with Toby (Steve)

To Topy and everyone

Here's my latest replies tagges with”S:”

Steve

…………………..

T: I do overall prefer score voting to approval
voting. It's slightly more complex (obviously), but it is still my preference.
It wouldn't be quite as you suggest. ……….. ………………………..

T: By measuring the amount of representation each voter gets from each
candidate (in the way described), we attempt to select the set that minimises
any difference in representation between voters. …

S: By what calculations do you make this “attempt”?

T: … It doesn't guarantee that they will be exactly
equal. …

S: What is your formula or mathematical definition of
“exactly equal”?

T: … APR does so but only by its own measure. For
example, if one voter is represented by their favourite candidate and another
by their third favourite, then APR doesn't measure this difference. It just
considers them both to be fully represented.

S: Yes, as
“fully represented” as possible given the preferences expressed by all the
other citizens. Also, in what sense does
APR not “measure this difference”? Each
voter knows when his first choice MP was his favourite rather than his third
favourite candidate. This difference is
also fully recorded on his ballot and in the whole counts of the ballots. These records could be available for later
researchers to analyse and report. At
the same time, APR allows each citizen to guarantee that her vote will be added
to the weighted vote of her favourite MP.
Score voting offers no such guarantee.
Again, please explain the special benefit that you see as guaranteed by
score voting but not by APR.

26T: …The overall most proportional set of candidates
would be the set elected, but I am not claiming that you would be able to find
a set of candidates that gives a perfect level of proportionality. Voters'
levels of representation for all candidates are added up. Exact proportionality
is when every voter has the same amount of representation. …

…………………………………..

T: Exact proportionality would occur when everyone has
the same level of representation in the commons by adding up the representation
they get from everyone in the commons, so it refers to the whole result.

S: You admit that score voting may not produce this
“same level” so why not welcome the fact that APR does guarantee it?

……………………………

31T: That's not how it would works. It's not simply the 500 candidates with the
most approvals that are elected. It would be the 500 MPs that would minimise
the disproportionality as I have described it. For example, if 51 people
approve A and B and 49 approve C and D, and two are elected, then it would be
one of A/B and one of C/D, even though A and B have the most approvals. We can
therefore measure disproportionality by adding up the squared deviations of
representation levels of the individual voters from the mean level of
representation.

32XXS: I understand this, but would not APR’s solution
for this example better? Assuming 51 citizens ranked A and B 1st and 2nd, and
49 citizens ranked C and D 1st and 2nd, A and C would be elected and have 51
and 49 weighted votes respectively, each citizen being entirely satisfied and
represented with exact mathematical proportionality. Your solutions gives each
vote of the 49 a slightly higher and disproportionate weight in the Commons. …

T: A weighted system, such as APR, would
give a more proportional result in that case, yes. We have debated the pros and
cons of weighted systems in previous e-mails, …

S: Please list
any of the “cons” that you believe have survived those debates.

T: … but I would agree that a weighted system allows
for better mathematical proportionality (under virtually any definition you
might think of for proportionality). A score system can also be used with
weighted representation, so it's not a difference between score and APR per se.

S: Please give me your formula for determining the
weighted vote of each MP using score voting.
Also, later, you refer to “the score measure of proportionality”. Please
explain it and how it might be related to determining weighted votes.

32XXS: … In any case, it seems to me that to use an
example of an election with only 1-3 winners is not very useful when we >are
talking about a system that would in practice elect 500 MPs, from thousands of
candidates, by millions of >citizens. Therefore, please fully explain the
formula and how in practice these 500 MPs would be elected by … score voting by
these millions of citizens (i.e. 500 “minimally disproportional” MPs,(or
better, the >500 MPs that would have the smallest total of “squared
deviations”). …

T: Well, I think a small case is useful
to give an example of how things would work. …

S: But you have said above that APR gives “a more
proportional result” in that “small case”.
If so, why would you still prefer score voting? I would ask the same question even if we
assumed for the sack of the argument that your following suggestion is
valid: “In any case, the explanation of
the system would be the same for any number of candidates/voters, except to say
that candidates would have to be elected sequentially if there are a lot of
them.” ….

T: You can sequentially elect candidates rather than
sequentially eliminate them. If you don't have the computing resources to check
every possible slate of candidates, you pick the candidate with the highest
total score, and then the most proportional two-candidate set that includes
that first candidate and so on.

S: I do not
recall any earlier discussion of the above “sequential” election process for
score voting. How do you mathematically
define “most proportional two-candidate set”?
Given unlimited time, how would you calculate it on paper?

……………………………

T: As I say, the measurements would be
the same for any number of candidates/voters. But for it do be done by computer
in a reasonable time-frame, candidates would be elected sequentially, as I said
above. The highest scoring candidate would be elected, and then we'd find the
most proportional two-candidate set that included the first candidate and so
on. I think I've given enough information in the previous e-mail, but as I say
I'm unwilling to re-describe it all again given that I'm not committed to one
particular method at present. I hope to be in the near future and if so will
give a more detailed description in its own message rather than as part of a
reply to this.

……………………………………

T: … I think
I've discussed previously how it would work and to be honest, I don't want to
give a full description again, partly because what I have described was only
ever a working model anyway. There are a couple of similar systems with their
own pros and cons, rather than one definitive system that I would fully
advocate. …I'm hoping that this will
change, however. …

S: Fine, but I
do not recall you ever completing your explanations in response to my questions.
Perhaps I missed somehow. Sorry. If you can find the time, I would very much
appreciate it if you could cut and paste to me.
I would not need an account of the “similar systems” you mention above
but jus I that you think would do the job.

S: On the other hand, you may prefer to postpone our
next dialogue until you have identified a “system that you would fully
advocate”.

T: … My point of bringing up score systems was not to
take over the discussion of APR with them, but to point out that they can look
at aspects of voter preference that APR ignores.

S: Please specify each of the important “aspects of
voter preference that APR ignores”.

……………………………..

34XXS: Correct me I am misunderstanding this by my
following report: As I see it, “electing an MP that has no support” is
impossible. Please explain how “the total representation from the MPs is less
in this case and you'd have a different mean to calculate deviation from.”

35T: … The system wouldn't work properly in this case.
…

36XXS: What “system wouldn't work properly in this
case”?

T: I wouldn't worry too much about that. It would never elect a candidate with
no votes so it wouldn't come up. But basically, if a candidate is elected, then
their power is divided among the voters who have voted for them. If the
candidate has one voter, each voter has 1/1 of the representation, if they have
two voters, then they each have 1/2 each and so on. If it's 0 voters, then it's
0/0 each. Er... Also, if there are 500 representatives, then that power is
split in some manner among the voters. The same amount of power is always there
regardless of who is elected (500 candidates-worth), but the distribution of
representation among the voters can change. That is unless someone is elected
with no votes. That would mean that the power of only 499 representatives is
split among the voters. It could be that this is split equally, so fully
proportional in that sense, but it's not directly measurable against a result
with 500 voted-for candidates. But none of this matters. The system would not
elect a candidate with no votes.

37T: …The squared difference isn't really part of the
definition of proportionality, but just how you'd measure disproportionality in
an approval/score case.

38XXS: I agree that they are not the same thing.
Still, if so, how do you propose to define “perfect proportionality” other than
to say it is when the sum is zero when you add

1) the “squared difference” between each citizen’s actual share and the average
share of each citizen in the total voting power in the Commons

2) to the similar squared difference in each of all the other citizen’s actual
and average votes.

>Do you agree that this sum would be zero for APR?

T: It would be zero for APR, but as I say, only because the measure is based
around APR. APR has no way of measuring disproportionality when one voter has
their favourite candidate and another has their third favourite.

38XXS: … Are you willing to compare APR, approval
voting, and score voting systems according to the same test, e.g. the one
mentioned in the previous paragraph (and which might also be the one you
already want to use for testing the latter two systems), or any other test of
“proportionality you may be willing to define and explain?

T: I agree that PR systems should be put
to the same tests as each other to see how they fare. For example, you could
ask voters for ranks and scores, and see how the APR result would fare when
proportionality is measured using the scores. For example, when one voter is
represented by their third favourite candidate (who they might give 8 out of
10), and also when some voters also happen to also like several of the elected
candidates who aren't their official representative and other voters are less
lucky.

S: Your formula for testing both is still not clear to
me. Is it different from the one you had
in mind with regard to the above “small case”?
If not, APR would seem to prove more proportional as a result of most if not all of the above
tests. Please explain.

38XXS: …At least with respect to wasting votes, do you
agree that APR has the advantage of allowing each citizen to guarantee that her
vote will not be wasted, that it will at least be added to the weighted vote of
the MP pre-declared to be most preferred by her first choice but eliminated
candidate?

T: As long as all the candidates give a
full ranking of the other candidates, then no vote will be wasted in that
sense.

39T: To give a simple example:

>2 to elect (with equal power), approval voting

> 3 voters [approvals]: A, B

1 voter [approval]: 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). …

40XXS: I added “desired” average because the actual
denominator of the fraction producing the average would be smaller to the
extent that some of the voter’s votes might be wasted.

T: The average would always have to be
the same, so it's not just the desired average. The total representation is
always two candidates-worth, and this is split among the four voters, so the
average representation has to be 2/4 (1/2). As I explained above, the only way
this could ever not happen is if a candidate was elected with no votes, but the
system wouldn't do that.

48XXS: As I see it, your above complicated attempt at
an explanation does not remove the validity of the following conclusion I
offered for your consideration in dialogue 13:

You have admitted 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 [wasted votes] disproportionality” would
usually show that there is some [wasted votes] “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.

T: There would be disproportionality in the system I have described, yes. And
there would be no disproportionality in APR, but only under APR's own measure.
If we surveyed everyone for their true scores out of 10 in an APR election, and
then fed the APR result into the score system that I have described, it would
have disproportionality by that measure.

S: Sorry. Perhaps
you think you have already given it but, I still do not understand your formula
for this measure, and thus, nor how you would feed “the APR result into the
score system”. Please explain.

52XS: 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.

53T: Yes. Proportionality would never be perfect in a system that uses this
amount of information.

54XXS: I’m a bit confused here. You seem to believe that APR is not perfect
because it ignores important information. Now you are also saying that your
system cannot be perfect because it offers too much information. I need you to
explain both of these claims more fully. Please start by defining what you mean
by “perfect” because it seems to contain the prime value by which you are
judging both systems. Would the “perfect” guarantee no squared difference
between the average and the actual mathematically expressed share each voter
has in the voting power in the Commons?

T: We need to distinguish between our measure and the result. I would argue
that APR is problematic because its measure of proportionality is too crude
(ignores lower preferences). …

S: Why do you
want “lower preferences” to help elect MPs?
The full value of an APR citizen’s vote is fully used up in the MP’s
weighted vote that she has helped to elect.

T: …Its own result is proportional by its own measure,
but would not be if measured using a score system (if we surveyed people for
their scores for each candidate). I would argue that a measure of
proportionality that uses voters' scores for all candidates is better, as all
preferences contain relevant information. But because of this more complex
measure, it would be much harder for a result to achieve full proportionality.
What I am saying is that any system would fail to achieve full proportionality
by this measure, but that this is a strength of the measure in that it looks
more deeply for flaws in a result by using more information.

55XS: 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.

56T: There would be no disproportionality in APR, but only under APR's own
measure. That there will always be some disproportionality? Yes, but this would
also happen in APR if measured by the same metric.

57XXS: I am open to being convinced of this upon
receipt of your full explanation of this “same metric”.

T: I would want to survey voters in an APR election for their scores for
candidates and test the APR result under the score measure of proportionality.
You could, of course, do the reverse. Use people's ranks and measure the result
of a score election using APR's measure of proportionality. It's just that we
would differ on which we think is the better measure.

S: Please
specify what you think is “better” and explain why.

58XXS: We should be aiming to use the same “measure”
and the one I would propose starts with the definition of “perfect
proportionality” I define above in

38XXXS: Would
your proposal be different? If so, please explain how and why.

T: I'd want to use people's scores of
candidates to see how well represented they are by the representatives, whether
their official representative or otherwise.

58XXS: … Secondly, has your measure ever been used
anywhere?

T: Not as far as I know.

61XS: Also, because your 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.

62T: It's something to consider, but I don't think citizens all understand how
STV works, and it is used in some places anyway.

63XXS: Yes, but do you believe your full score system
with the above “measure” is also easier than existing STV (with their quotas,
fractional transfers of votes, many iterations, etc.) systems to understand? In
any case, APR’s count with weighted votes is easier to understand than previous
STV systems because it only transfers whole votes from eliminated candidates
without losing any votes.

T: APR is probably easier to understand
than what I have presented in terms of how results are calculated, yes. …

70T: …The main point is that APR doesn't use all the information and another
system (whether the system I described or not) might, so APR could end up lacking
something as a result, including in the examples I gave in previous e-mail
about some voters getting chance extra representation that isn't officially
recognised by APR.

71XXS: I thought we previously agreed that all systems
can produce “chance extra representation” for some voters. Do we not agree
about this now?

T: We agreed that they can do this, but
that APR does not even attempt to measure it and reduce it. A score system,
such as I described, attempts to minimise this.

S: Why would
you want to “reduce it”? Any such
so-called “extra representation” occurs by chance in any system only to the extent that one citizen’s scale
of values happens to agree with those of is fellow citizens. Why would anyone want to remove the electoral
effects of such agreement?

>Secondly, what do you have in mind when you imply
that such extra representation is “officially recognised” in score voting but
not in APR?

T: If a voter's first and second
preferences are both elected then they would have better representation than
someone whose first and fifth are elected (assuming that the first person's
second preference is preferred to the second person's fifth), but APR does not
measure this. A score system can measure this as disproportionality and can look
for a less disproportional result.

S: Do you think
it would be better, for example, if score voting could instead guarantee that
both of these voters would only see that their first and fifth candidates are
elected? In any case, I do not see how
score voting could guarantee this, do you?

71XXS: …Thirdly, if for the sake of the argument, we were to agree that they
are essential, why would you say that these summaries would be more revealing
or more important than the similar summaries that could be made as a result of
equally rigorous analyses of all the citizens’ rankings in an APR election?

T: Rankings tell you less about how much
a voter likes the candidates. A voter might like ranks 1 to 20 similarly or
hate everyone from 2 down. A score ballot allows for a more rigorous analysis
of whether voters are getting what they want.

S: No. Rankings
also tell us “whether voters are getting what they want”. If a voter mentally scores 20 candidates in
the same way, in an APR election she should number this group 1 - 20 in any
order. She would be equally happy if any
one of these 20 is elected. Why is it
more important for you or anyone else also to know that this voter valued each
of these 20 equally?

75XXS: Of course, before we could compare the two systems in this way, you
would need, as requested in paragraph 32, to explain all the essential details
of how these calculations could be done in practice for an election of 500 MPs,
from thousands of candidates, by millions of citizens. Will you do this?

………………………

Toby.