[EM] General PR question (from Andy Jennings in 2011)

[Kathy Dopp]

On Mon, Sep 29, 2014 at 11:53 AM, Toby Pereira <tdp201b at yahoo.co.uk> wrote:

>
> it's quite easy to demonstrate non-monotonicity with it.
>
> 10 voters: A, B
> 10 voters: C
>
> In this case AC and BC are equally proportional. But as soon as any C voters approve A, BC becomes the more proportional result.

If C's 10 voters vote for A, then A has 20 votes total, whereas B and
C only have 10 votes total, so A is the winner with the most votes,
increasing A's chances of winning.  Approval voting is strictly
monotonic.

Your unique and undesirable definition of proportional voting may
induce nonmonotonic results, but normal approval voting does not
because it simply tallies *all* the votes for any candidate.

>
>>>With an approval ballot, if someone has voted for a particular elected candidate, then their representation from that candidate is 1/n where n people in total have voted for the candidate, and 0 if they haven't voted for them.

OK.  I agree with this above statement of yours only with the
restriction that this voter's representation out of all votes for a
particular candidate is their vote (1 or 0) divided by the number of
others who have voted for that candidate.


>> A voter's total level of representation is the sum of their representation from each candidate.

You mean *elected* candidate and you'll need n1, n2, n3,... (different
n values) for each candidate.  How does it even make logical sense to
add up the different proportions of different candidates each voter
contributed to electing? I.e. I contributed 0/n1 to electing candidate
one, I tried to add 1/n2 for candidate 2 but he did not get elected so
I contributed nothing to my representation even if I voted for
candidate 2, but I contributed 1/n3 to electing candidate 3, etc.  How
does it make sense to add these numbers for one voter?


>>For v voters and c elected candidates in total, the mean representation for each voter is c/v (assuming that each elected candidate has at least one vote). Full proportionality is achieved if every voter has representation of c/v.

Nope. It is not. For the reasons I cited before and I'm sure it can
also be shown given your newest formulation.

Also, proportionality is usually defined in terms of groups of voters,
not in terms of individual voters. I.e. a voting system is
proportional if a group comprising a certain proportion of the voters
has (to rounding) the same proportion of seats in a legislature.  Some
voter groups will have too small a proportion of the population to
achieve even one seat.  A single voter cannot have proportional
representation since a single voter does not comprise a large enough
share of the voters to achieve any seats.

>>---
>>
>> The total representation for any voter is the number of candidates
>>the *voter* votes for who are elected to office.  The mean cannot be
>>determined as above.  The mean would be calculated by summing the
>>number of candidates each voter voted for that were elected and
>>dividing by the number of voters.  The terms of this sum would be zero
>>for many voters and less than c for all voters who did not vote for
>>all the elected candidates. The mean would be less than c/v, not as
>>described above.
>>
>>---
>
> "Representation" was just the word I used to describe the thing I was describing. I originally talked about voter "possession" of a candidate when I first described the method. Either way, it's just a semantic point but apologies for any confusion.

How does a voter "possess" a candidate?  You mean if I cast one vote
out of 20,000 for a candidate who won election, and all my other
candidates lost, I possess 0.0005 of candidate 1 and that is my
"representation"?  FYI, calculating the averages of those will still
not be equal to c/v.

When fractions are added, you cannot simply add the numerators and add
up the denominators. Addition of fractions just doesn't work that way.

Try a nontrivial example.

>
>>> The proportionality measure of a set of candidates is the average squared deviation from c/v for the voters' total level of representation (lower deviation being better). There's also a score voting version.
>>>
>>>
>>>If we look at the following approval election with two to elect:
>>>
>>>
>>>10 voters: A, B, C
>>>10 voters: A, B, D
>>>
>>>
>>>Monroe would be indifferent between any set of two candidates, even if it favours one faction over the other. My metric would rate AB and CD as the most proportional.
>>---
>>
>>Since the example above shows 20 voters voted for both A and B, and
>>only 10 voters voted for both C and D, such a metric for
>>"proportional" representational calculation fails to elect candidates
>>that the most number of voters would be happy about and the least
>>dissatisfied with.
>>
>> I have never seen the word "proportional" voting used in this fashion
>>before, so have always supported the notion of "proportional" voting,
>>until now, when I see that using this metric, although strictly
>>proportional, would be undesirable IMO.
>>
>>--
>
> Arguably it would be undesirable but while this case might be fairly clear, we still have the general problem of proportionality v overall support. Instead of the above, you could have:
>
> 10 voters: A, B, C
> 10 voters: A, B, D
> 1 voter: C
> 1 voter: D
>
> What's the best result now? AB or CD?

Since A and B each have 20 votes, they win.  Only 2 voters will be
unhappy. Adding one vote each to C and D increases their chances of
winning from 10/20 to 11/22 but not to equal A and B's chances, which
are now 20/22.

However, I see your point that there would be 2 fewer unhappy people
in this case, using your unique definition of proportional winners.
However,  2 people out of 22 voters is less than 10% of total voters,
not enough to merit proportional representation of even one candidate.
The other 10 voters who voted for either C or D will still be
satisfied with the result.

I claim that if as many as 1/s voters rounded to the nearest integer,
where s is the total number of seats are dissatisfied, then, and only
then, may you claim that the voting system lacks proportionately fair
representation, which your unique definition of "proportionality" is
much more likely to do than is approval voting.


> And we could increase the C and D bullet voters one at a time. When does CD become better than AB? Is there a non-arbitrary answer?

To repeat my answer, if  1/s voters rounded to the nearest integer,
where s is the total number of seats, are dissatisfied, then, and only
then, IMO may you claim that the voting system lacks proportionately
fair representation, at least in the traditional meaning of
proportional voting systems.  Obviously, increasing the number of
seats, s, decreases the size of the group that merits representation
in this meaning.




-- 

Kathy Dopp
Town of Colonie, NY 12304
 "A little patience, and we shall see ... the people, recovering their
true sight, restore their government to its true principles." Thomas
Jefferson

Fundamentals of Verifiable Elections
http://kathydopp.com/wordpress/?p=174

View my working papers on my SSRN:
http://ssrn.com/author=1451051