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

[Toby Pereira]

From:Kathy Dopp <kathy.dopp at gmail.com>
 

On Mon, Sep 29, 2014 at 11:53 AM, Toby Pereira  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.

I understand what you're saying here, but this isn't about normal approval voting, which has nothing to do with proportional representation. It's about achieving proportional representation from approval ballots. It's also not just about the method that I described but about the discussion stemming from Andy Jennings's original example where it was discussed that people's intuitive ideas of proportionality can lead to non-monotonic results and candidates that don't have the most support. My method is not unique in that respect.


>
>>>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?

Well, I suppose I did mean elected candidate, but it doesn't matter either way. They get no representation from unelected candidates, so the sum of their representation from elected candidates is the same as the sum of their representation from all candidates. But anyway I'm using this as the specific definition of representation in this method, so it makes sense that the total representation for each voter is the sum of the representation they get from each elected candidate. But as I said earlier, it might not meet with your everyday definition of "representation". But whatever you call it, adding up these numbers and measuring squared deviation tends to lead to results that seem proportional.


>>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.

Can you give an example where it does not give a proportional result?

>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.

I think it's actually quite tricky to give a definitive definition of proportionality. Voters don't always form clean factions, so it's difficult to even define groups properly. But it can also be looked at in terms of individual voters, such as with single transferable vote and the Monroe method, which both assign candidates to voters rather than looking at groups. Obviously these methods give proportional results at faction level if people vote in clean factions, but they don't actually function at a group level. The same applies for my method.

>>---
>>
>> 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.

Possession, representation - it doesn't matter what you call it. But yes, if you are one of 20,000 people who vote for an elected candidate, then you possess 0.00005 of that candidate or have that level of representation from them.

I'm not sure what you mean about adding fractions. The total level of representation available from each candidate is 1. Where 20,000 people vote for an elected candidate, each have a representation score of 0.00005. These add up to 1. The total level of representation is c (the number of candidates) - as long as each elected candidate has received at least one vote. This representation is split among v voters, so the average (arithmetic mean) representation is c/v.


>
>> 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.

This isn't about my "unique definition of proportionality" as you put it, but this stems from previous discussion about the same thing. In any case, I would argue that proportionality isn't simply an all-or-nothing thing. We don't simply say that because only a group smaller than 1/s has no elected candidate then anything goes. I prefer to look at measurable degrees of proportionality.


>> 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.

In that case take the following example:

4 to elect, proportional representation, approval voting

10 voters: A
10 voters: B
10 voters: C
9 voters: D
1 voter: D, E

I declare the winning set of candidates to be ABCE.
 

>Kathy Dopp

Toby
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