APR (14): Steve's 14th dialogue with Toby (Steve)
Date: Sat, 3 Jan 2015 12:36:57 +0000
From: tdp201b <at> yahoo.co.uk
To: stevebosworth <at> hotmail.com; election-methods <at> lists.electorama.com
Subject: Re: Subject: APR (13): Steve's 13th dialogue with Toby (Steve)
To Toby and
comments are tagged by “XXS:” Also, to
ease cross referencing, I have numbered all the paragraphs.
1T: 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.
2XXS: Exactly how does your “score system’s” elimination
process work? This same question is
asked or discussed in different ways below in paragraphs 32, 34, 38, 39, 48,
50, 51, 56, 57, 58, 60—63, 66, 67, 69, 70, 73, and 75.
3T: 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. …
4XS: 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?
5T: Yes. It is possible that none of the candidates you give a positive score
to will be elected. However, see below.
6T: … They
would also have the option of simply voting for one candidate and using that
candidate's entire score set of the other candidates.
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?
8T: What I mean is that each candidate would
have a pre-declared set of scores for all the other candidates. They would rate
the other candidates in advance of the election. Then if I as a voter don't
want to individually rate the candidates, then I cast a vote for my favourite
candidate and indicate that I want to use their ratings of the other candidates
rather than rate them myself.
9XXS: Do you
indicate your “favourite candidate” simply by giving the top score only to one
above answer might imply that you now favour score over approval voting. Is
If so, are you
also now adding the following feature to your preferred system, i.e. a score
If you give
your top score only to one candidate and none of the candidates you have given
a positive score to is elected, then that top but eliminated candidate must
continue sequentially to give your top score to each candidate on his rank
ordered pre-declared list until one is elected?
If so, this could
be almost the same as APR’s default voting.
However, what if none of the candidates to which your top but eliminated
candidate gave your top score could be elected? Would this still be your preferred system?
10T: … 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.
11XS: 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?
12T: It's possible that none of the
candidates you give a positive rating to will get elected. Or, if you are using
your favourite candidate's ratings, it's possible that none of the candidates
they have given a positive rating to will be elected. However, under APR,
unless you (or your favourite candidate) has given a complete ranking list of
all candidates then this can happen in APR as well. …
13XXS: Endnote 4 to the article explains an
alternative APR procedure that would also guarantee that your vote will be
added to the weighted vote of the MP most favoured by your first choice but
that since APR has the advantage that its MPs can
have weighted votes (not just one vote each) this allows them more easily to
receive and keep any extra votes that might be offered to them from any first
choice but eliminated candidates; and
that APR’s primary enables each citizen both to
ensure that more attractive candidates (i.e. for his preferred association)
will be available to be ranked in the general election, and that a
pre-established number of these candidates must be elected.
the “ratings” you mention above “scores” or “rankings”?
14T: … Also if
none of my ranked candidates are elected, but my favourite candidate's 78th
ranked candidate is elected as my representative, I might well be justified in
feeling that I'm not really represented in any way. The fact that someone gives
a complete ranking does mean that someone will become their official
representative, but it offers no guarantees about how well someone will feel
represented by them. …
15XXS: I largely accept the point you are making
here. However, I believe it would still
be correct to say that APR maximizes the chances that each citizen will be
represented as well as possible, given currently unchangeable circumstances.
16XS: 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.]
17T: Yes, I did provide the information
below. I also described it (albeit more briefly) in previous e-mails and linked
to a mathematical description of the approval system. It's probably best to
read the whole e-mail before saying things like this. And also, if you weren't
happy with the previous descriptions, you should have said that you weren't
happy with them rather than ignoring them and just stating that I hadn't
provided an explanation.
18XXS: I apologise for misunderstanding these
earlier brief references as attempting fully to provide this information. Thank you for providing the more compete mathematical
information below. Unfortunately,
however, I still need the additional details asked for in paragraphs 2.
19XS: 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.
20T: When the results of the election are published, the number of approvals or
total score given to a candidate would be part of what is published, so you can
know from that.
21XXS: Given this information, I see how each
citizen could eventually calculate her total share, if it were the case that
the 500 most popular candidates are elected by approval or score voting, i.e.
as your next paragraph in this dialogue also seems to imply. However, perplexingly in your 31T: paragraph below, you say that “this is not
how it works”. At that point, I ask you
fully to explain how it does work.
22T: 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.
23XXS: This is an equal share for each approval (and
a proportionate share for each score) only with regard to the one vote of the
relevant elected MP. It is not provide
the much more important equal proportionality guaranteed by APR, i.e. with
regard to all the votes in the Commons. This second kind of “proportionality” which is
guaranteed by APR is what I would term “a perfect level of proportionality”. Do you mean the same thing by this phrase in
your 26T: paragraph below?
24/2T: It is correct that those represented by MP-A
would in this situation have better representation than those represented by
MP-B, and this disproportionality would be measured by the system. ,,,
25XXS: Will this “measurement” be fully explained by
the details of the caIculation I request later in paragraph 32XXS: below?
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. …
“exact proportionality” only in relation to the one vote of each of their elected
MPs (not in relation to all the votes in the Commons)?
28T: …. 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
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:
30XS: 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.
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
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
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 approval or 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”).
Presumably, it will help me to understand this future explanation if you
could also note the following questions and points with regard to the seemingly
relevant part of your contributions to dialogue 13:
Extract from Steve’s 13th
dialogue with Topy (with 14th dialogue (XXS:) additions):
33T: 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. …
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”?
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.
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
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?
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?
At the same time, I believe that your calculations
(starting several paragraphs below) of these squared differences is not mainly
focused on determining how much a given election falls short of “perfect
proportionality”. Instead, I see it as
focused on discovering the fraction of each citizen’s vote which might be
wasted (i.e. might deviate from the average share of each citizen’s vote in the
votes in the Commons), and thus also enable us to discover the total number of
votes wasted by all citizens in a given election.
Instead, for me, a “perfectly
proportional” election in practice would be,
1) if the electoral system used gives us every reason to believe that each
of the various scales of values held by a percentage of the voting citizens in
the country is supported by the same percentage of votes in the Commons, and
2) if we currently cannot think of any other system that would align these
two percentages more completely.
Of course, admittedly, the more a
system wastes votes, the more reasons we have to doubt that it is fully
proportional. 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?
39T: To give a
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). …
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.
41T: …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. …
43T: …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. …
45T: … 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. …
I think you would want to omit “or a 10” here.
47T: …So only one tenth of a voter’s
[vote] approves the candidate with a score of 1 out of 10.
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.
If you think this
conclusion is not correct, please explain.
End of Extract from Dialogue 13
49XS: In any
case, while the resulting voting power of each approving citizen for different
MPs 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 this same total. However,
of these three voting systems, only APR also guarantees that this total will
also equal the total number of voting citizens in the country.
50T: It only makes such a guarantee by
defining some of the information out of existence when it comes to defining
51XXS: Yes, APR assumes that only the highest
available preference of each citizen for a candidate should determine how each
citizen’s one vote should equally count in the election of each MP and the
whole Commons. Of course, lower
preferences could be fruitfully studied by later analysts of any such election
but supporters of APR do not believe any of these lower preferences should
elect any MP. To do so, would be to
violate citizens’ intentions, i.e. to be represented by the MP whose scale of
values seems to be closest to his own. (Please
also see paragraphs 66, 70 and 71.)
agree that with the addition of “asset voting”, both to APR and score voting
(i.e. “default” or “delegated” voting in the hands of the first choice or top scored
but eliminated candidate), score voting could also guarantee that the score
given to the eliminated candidate would be given to the total score of a
pre-declared candidate (and elected) most favoured by that eliminated candidate.
However, the share of that MP’s one vote could easily be more or less than “one”,
i.e. the weight of “one” which each APR citizen’s vote would have in the
contrast, approval voting prevents citizens from distinguishing between their
highest preference candidate, one that is only tolerable, and any in between,
i.e. it excludes information that most people would think is very important in
any election. At least, score voting has
the advantage over approval voting of allowing citizens to express
preferences. However, unlike APR,
neither approval nor score voting can guarantee that the whole of each
citizen’s vote (and only “one” vote) both will be counted, and will only have
the same mathematical weight within the Commons.
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
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?
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”.
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.
your measure ever been used anywhere?
59T: … As I
have said previously, some voters might get chance extra representation by
being better represented by MPs that are not their official representative. If
I polled the voters in an APR election to find their scores (out of e.g. 10)
for each candidate, and measured the APR result as if it was the result in a
proportional score election, I would find disproportionality under that
in order to understand this, I look forward to receiving your full explanation,
requested above, of how this could be calculated for an election of 500 MPs, from thousands of candidates, by
millions of citizens.
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
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.
specify and explain the “flaws” in APR that you still see in the light of all
the above points.
65T: Well, the bottom line is that APR
doesn't use all the information about what voters think of the candidates. …
66XXS: Please see paragraphs
51, 70, and 71 which asks about the relative importance of this extra
If I was coming
at this afresh, without having a system already in mind, I would have thought
that the more voter preference information that is used, the better the result
can potentially be, without worrying exactly how to define "better"
or "more proportional". …
67XXS: If you want to convince other people about
the value of your suggestions, you really need to offer objective definitions
of “better” and “more proportional”.
Otherwise, such words only label your own subjective feelings.
68T: … This is
why I have suggested score voting. Whether the particular method I described
does the job or not is beside the point, because this is really a discussion
about APR. …
69XXS: It is
not “beside the point” if we are attempting to discover the best practical
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?
Secondly, what do you have in mind when you imply that such extra
representation is “officially recognised” in score voting but not in APR?
Is the “recognition” you have in mind any more than the fact that any
researchers could analyse all the scores given by all citizens and, as result,
summarize the information that might not be obvious if one only studied the
scores that elected the 500MPs? I accept
that such summaries could be interesting but you also seem to believe that they
should be essential determinants of which candidates are to be elected. If so, how and why?
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?
72T: …It would be interesting to compare APR with a proportional score system
in the following way: have an election held under APR, but also poll the voters
on what scores they would have given to the candidates if it had been a score
voting election to find what would have been the winning result under these
circumstances. Then measure the proportionality of each result using the score system
to see which does better.
73XXS: Again, I need your definition of “better” and
the details of how you propose to measure it.
74T: … You
might say I'm stacking the odds in favour of the score system by using the
score system to measure proportionality, and it's right to an extent, but it
would be interesting to see how much the weighted power feature of APR worked
in its favour.
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?