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

[Toby Pereira]

It does seem to be a difficult question of how to balance proportionality v overall support. If we use some function of proportionality and overall support, we have to be careful not to favour larger factions of course. If we have:
Approval voting, 2 to elect
3 voters: A, B1 voter: C
there would be a tie between AB and AC. But if we look at total approval score, then AB beats AC. If we use this in our measure, we would end up simply favouring larger factions. One possibility would be to pick the most proportional result that isn't Pareto dominated, but generally methods that have two stages like this end up a bit clunky and can produce weird jumpy results. (e.g. pick the Condorcet winner but if there isn't one pick the Borda winner). At the moment, I'd still go with sequential electing if proportionality alone isn't acceptable.
But moving onto election methods generally, I do like Jameson's PAL system http://wiki.electorama.com/wiki/PAL_representation. It allows a national proportional vote (as opposed to splitting the country into regions) while still keeping ballot papers simple and keeping local representatives. There are other alternatives such as whether candidates should give cardinal scores to their factions rather than an ordering, and whether individual voters should have the same voting power as the candidates if they choose not to delegate their vote, but it's a very good starting point for a discussion about how countries' electoral systems should be run. I've discussed it here as well http://tobypereira.wordpress.com/2014/07/30/electoral-reform-3-a-reasonable-way-of-electing-mps-proportionally/
Toby
 
      From: Jameson Quinn <jameson.quinn at gmail.com>
 To: Toby Pereira <tdp201b at yahoo.co.uk> 
Cc: Andy Jennings <elections at jenningsstory.com>; "election-methods at electorama.com" <election-methods at electorama.com> 
 Sent: Wednesday, 1 October 2014, 22:09
 Subject: Re: [EM] General PR question (from Andy Jennings in 2011)
   
Several points: 
I think this is a productive discussion.
I generally like Toby's definition as a proportionality measure. However, I think that there should be a majoritarian measure too (something like summed utility for the whole candidate set), and some balance between these. In other words, a method should come from somewhere on the Pareto front.
I'm not sure that we'll be able to find a non-arbitrary way to pick a point on the pareto front.
However, I think that delegated methods are promising in that regard. In particular, if candidates are required to pre-declare some preferences over the other candidates, and if this ordering must satisfy some criteria, then I think that the chances of "pathological" examples can be minimized or eliminated.
That should be balanced with concerns over complexity and transparency. That's why, in designing PAL, I decided that rather than pre-declaring a strict ordering, candidates should only pre-declare a "my faction" subset of their party, and then an ordering over other parties. 
This would leave the possibility of a proportionality/majoritarian tradeoff within a given party. I think such tradeoffs should probably be resolved in the majoritarian direction; since the "disadvantaged" voters from a proportionality standpoint would still be getting fair representation at the party level, and only have grounds for complaint at the faction level.
Obviously, this doesn't resolve cases like the pizza/fry-up example, because you can't delegate to food items. For those cases, I think that the sequential answer is good enough, though I understand why it's unsatisfying.


2014-10-01 16:47 GMT-04:00 Toby Pereira <tdp201b at yahoo.co.uk>:



There are arguably situations where proportionality is desirable but not at the cost of overall support. I gave this example:
10 voters: A, B10 voters: A, C
With two to elect, I would argue that BC is the most proportional. However, imagine a group of people are deciding what to have for dinner on various days. They only have enough to have each particular meal once. For simplicity, Let's say there's 20 people and they have to decide for two days, and they vote approval style on the meals they like.
10 voters: pizza, curry10 voters: pizza, fry-up
This is effectively exactly the same vote as the other example. Curry/fry-up might be more "proportional" but it seems absurd not to have pizza on one of the days. Nothing is gained by preventing the other group from getting more enjoyment at no cost to yourself.
So the question is - what is the best election method in cases such as this? I've struggled with this for a while because it requires a non-arbitrary trade-off between proportionality and positive support. I think my system of proportionality used sequentially would generally give good results, but it's a bit of a cop out and basically hides from the problem. There should be a reasonable non-sequential solution.
Forrest Simmons's PAV would work reasonably well here (although I would argue with Sainte-Laguë rather than D'Hondt divisors), but it still fails independence of commonly rated candidates, and I don't know how to fix it.


      From: Toby Pereira <tdp201b at yahoo.co.uk>
 To: Andy Jennings <elections at jenningsstory.com>; "election-methods at electorama.com" <election-methods at electorama.com> 
 Sent: Monday, 29 September 2014, 13:42
 Subject: Re: [EM] General PR question (from Andy Jennings in 2011)
   
One problem with saying that candidates must be elected non-sequentially is that it can (depending on how you measure proportionality) lead to monotonicity and Pareto violations. If there are two to elect with approval voting:
10 voters: A, B10 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.
But in any case, sequential electing doesn't need to be billed as a system that gives better results (regardless of whether one thinks it does). It could simply be stated (probably truthfully in many cases) that it's unfeasible to check the proportionality of every possible set of candidates, and that close-to-optimal results would still be achievable with sequential electing. I don't have any data for that last statement, but most elections wouldn't be like the contrived examples we've come up with, and I'd be surprised if results ended up being massively different.
Toby

      From: Toby Pereira <tdp201b at yahoo.co.uk>
 To: Andy Jennings <elections at jenningsstory.com>; "election-methods at electorama.com" <election-methods at electorama.com> 
 Sent: Monday, 29 September 2014, 0:35
 Subject: Re: [EM] General PR question (from Andy Jennings in 2011)
   
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, C10 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


>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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