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

Toby Pereira tdp201b at yahoo.co.uk
Sun Sep 28 16:35:48 PDT 2014


Thank you for your responses Kristofer and Andy.

The problem I have with the Monroe metric is that because it ignores how much you like or dislike the candidates that aren't the one it assigns to you, it can end up with lopsided (I would argue unproportional) results. When several candidates are elected in a proportional election, I don't think saying that each voter has exactly one representative is the best way to look at it. If I vote for several candidates who are elected, then I would feel that I have representation from all of them and would be in a better position than someone who has voted for just one elected candidate, but the Monroe metric would just see us both as catered for and leave it at that.

In Andy's original example (see bottom), Monroe would consider ABC equally proportional to CDE, but clearly under ABC some voters are getting a better deal, whereas CDE is perfectly proportional (although with less support overall). That's why I came up with the metric I did for measuring proportionality, which looks at how you rate every elected 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. 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 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.

Toby


 From: Andy Jennings <elections at jenningsstory.com>
>To: "election-methods at electorama.com" <election-methods at electorama.com> 
>Sent: Sunday, 28 September 2014, 17:48
>Subject: Re: [EM] General PR question (from Andy Jennings in 2011)
>  
>
>
>Yes, there is a tradeoff between proportionality and support.  Kristofer's work speaks to that much better than I can.  But personally, I think proportionality is paramount.  If you're choosing a "representative body", then mirroring the electorate is the ultimate goal, isn't it?  I like Monroe's metric.  If the voters can be divided up equally and assigned to the winners in a way that each voter is perfectly happy with his representative, to me that's a perfect representative body.
>
>But any such method must be non-sequential, and the main problem with a non-sequential method is the losers might be able to complain, "I would've been elected if the council only had 4 seats, but since it has 5 seats, I lost."  Is the answer, "Yes, the optimal 4-person council included you but the optimal 5-person council didn't," good enough?
>
>
>
>If I remember correctly, one of my goals in
 sending that email was to start exploring what multi-winner outcomes felt intuitive to people.  A purpose that you continued later on.  I wonder if you discovered the same thing I did.  That not many people respond.  And that most of us don't have strong intuitions about tricky situations in multi-winner outcomes.
>
>If we could come up with a large set of multi-winner scenarios which had answers that felt intuitive to most people, we could use it to evaluate all existing systems and to quickly get a good handle on any new systems that are proposed.
>
>That's why I made a point to try to respond to your post, indicating which answers felt best to me and how strongly I felt about them.
>
>
>~ Andy
>
>
>
>On Sat, Sep 27, 2014 at 4:28 PM, Toby Pereira <tdp201b at yahoo.co.uk> wrote:
>
>I was thinking recently again about Andy Jennings's PR question (below) and available here http://lists.electorama.com/pipermail/election-methods-electorama.com/2011-July/093278.html, which is about the trade of between proportionality and having candidates with strong support. Warren Smith (http://lists.electorama.com/pipermail/election-methods-electorama.com/2011-July/126111.html) gave the extreme example of a 500-member parliament where two candidates each get 50% approval, and the others each get 0.2% approval. Perfect proportionality could be achieved by electing 500 candidates with 0.2% approval, but in many ways this would seem a perverse result.
>>
>>
>>But the more I think about it, the more I think there isn't a non-arbitrary solution to the problem. What's the exchange rate between proportionality and support? There isn't an obvious answer.
>>
>>
>>I proposed my own proportional approval and score system a few months ago (http://lists.electorama.com/pipermail/election-methods-electorama.com/2014-May/098049.html http://lists.electorama.com/pipermail/election-methods-electorama.com/2014-June/130772.html), and it purely bases result on proportionality, so would elect CDE in Andy's example but would also elect 500 candidates with 0.2% support in Warren's example. However, this also assumes that every possible winning set of candidates would be looked at and the most proportional one found. In practice, the system might be used sequentially. This would force through the most popular candidate, and then each subsequent candidate would be elected to balance it proportionally. This would elect the two most popular candidates in Warren's example, but would fail to elect CDE in Andy's example. But given that there may be no non-arbitrary solution, electing sequentially may be the simplest and least
 arbitrary way around the problems we have. It is also a solution that would likely be forced upon us due to limits on computing power when it comes to comparing all possible sets of candidates. Necessity may force the pragmatic solution upon us.
>>
>>
>>Toby
>>
>>
>>
>>
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>>>Forest and I were discussing PR last week and the following  situation came
>>>up.  Suppose there are five candidates, A, B, C, D, E.  A and B evenly
>>>divide the electorate and, in a completely orthogonal way, C, D, and E
>>>evenly divide the electorate.  That is:
>>
>>>One-sixth of the electorate approves A and C.
>>>One-sixth of the electorate approves A and D.
>>>One-sixth of the electorate approves A and E.
>>>One-sixth of the electorate approves B and C.
>>>One-sixth of the electorate approves B and D.
>>>One-sixth of the electorate approves B and E.
>>
>>>It is obvious that the best two-winner representative body is A and B.  What
>>>is the best three-winner representative body?
>>
>>>CDE seems to be the fairest.  As Forest said, it is
 "envy-free".
>>
>>>Some methods would choose ABC, ABD, or ABE, which seem to give more total
>>>satisfaction.
>>
>>>Is one unequivocally better than the other?
>>
>>>I tend to feel that
 each representative should represent one-third of the
>>>voters, so CDE is a much better outcome.  Certain methods, like STV, Monroe,
>>>and AT-TV (I think) can even output a list of which voters are represented
>>>by each candidate, which I really like.
>>
>>>I also note that if there was another candidate, F, approved by everybody,
>>>it is probably true that ABF would be an even better committee than CDE.  Is
>>>there a method that can choose CDE in the first case and ABF in the second
>>>case?
>>
>>>Andy
>>
>>
>>----
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>>
>>
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