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

[Toby Pereira]

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On Mon, Sep 29, 2014 14:43 BST Kathy Dopp wrote:

> Just a few words of arithmetic:
>
>---
>> From: Toby Pereira <tdp201b at yahoo.co.uk>
>
>>
>> 10 voters: A, B
>> 10 voters: A, C
>>
>> I would argue that the most proportional result is BC even though everyone has voted for A. (Monroe would be indifferent between the three possible results, however.) Sequential electing is likely to lead to less failures of monotonicity, and perhaps less prone to strategic voting as a result.
>---
>
>Monotonicity simply means that increasing the vote share, increases
>the probability of winning the election.  The above example does not
>provide an instance of nonmonotonicity.
>
>---

I wasn't stating that that was a non-monotonic result itself but 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.

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

>> 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? 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?

>
>Kathy Dopp

Toby