[EM] APR (12): Steve's 12th dialogue with Toby (Steve)

[Toby Pereira]

Steve and everyone.
My new comments are untagged.
SSS: True, but please explain how your highest scored but eliminated candidate is going to pass on your one vote to an elected rep. Are you in favour of this addition to a score system?
I suppose because a score system doesn't work on the same elimination basis as APR, it wouldn't be able to work in exactly the same way. But what you could do is allow the following three options for voters. One would be for the voter to give scores to as many candidates as they like (anyone candidate they ignore gets a default 0) and not transfer any power to their favourite candidate or anyone else. They would also have the option of simply voting for one candidate and using that candidate's entire score set of the other candidates. The other option would be for a voter to give scores (including zeros) to as many candidates as they want (these would be the ones they have specific views about), and then leaving the ratings of every other candidate to their indicated favourite candidate. As for whether I'm in favour of it, yes, I think it should work well.
 T: I would say that a proportional approval/score system could well mean that it is less likely that some people would get extra representation by mere chance because it takes into account your rating of every candidate, not just the one that's deemed to be yours. Therefore it wouldn't have the problem I highlighted in the example (quoted from a previous e-mail below).
SSS: Please explain how you might objectively predict that more chance extra votes would be given to APR reps than to approval/score reps. I do not yet see that this is possible.
The chance nature in APR comes from the fact that it ignores the information about candidates below the transfer line. Approval/score systems use all the information provided by voters and they calculate the most proportional result from this. So any disproportionality comes from the unavoidable fact that perfect proportionality would be impossible rather than the pot luck of what happens to be in the unused information in APR.
T: Approval/score can give better levels of proportionality by using more information, so it doesn't make it more probable that any given individual will have their views better represented in parliament, but I would argue that it reduces the chances of people being over or under-represented making it fairer overall.
 SSS: As it stands, this seems only to be your own vague and subjective opinion. I keep asking you to define what you mean by "proportionality" mathematically in an objective way but you have not yet done so. Why?
I have! In my last post, and previously as well. But I'll give it again with some background. Lots of sources define proportionality in terms of parties or groups of voters, but I think it's best defined in terms of individual voters. If we look at each MP's representation as split among their voters, the basic definition of proportional representation is that every voter has, as close as possible, equal representation. This definition tallies up with other definitions perfectly well, except that it doesn't have to talk about groups of voters, which often aren't clear-cut anyway. For example, under APR if two people vote for candidate A and one votes for candidate B, A has twice the parliamentary power of B. But because A's power is split between two voters, every voter ends up with equal representation.
Under an approval/score system (or certainly one I would advocate) each voter doesn't have just one representative but is represented to some extent by any MP that they have approved or given a non-zero score to. The basic definition of proportionality is unchanged (each voter has equal representation), but how we calculate it is different from APR or STV methods generally. With approval voting, it is fairly simple. Each MP's representation is equally split among all the voters that have voted for them. With score voting, it is split proportionally to the score each voter gives. For example, candidate A is elected and has 1/500 of the parliamentary power. Two voters approved candidate A, so they each effectively have 1/1000 of the total parliamentary representation each plus whatever they might get from other candidates.
Voters' levels of represresentation for all candidates are added up. Exact proportionality is when every voter has the same amount of representation. That is what I mean by proportionality. The total of the voters' levels of representation will always be the same whatever the result (because it always equals the total parliamentary power), and so the average will always be the same. The only exception to this is if a candidate is elected with no support, because there are no voters to split the support between. This wouldn't happen in practice though.
We can therefore measure disproportionality by adding up the squared deviations of representation levels of the individual voters from the mean level of representation.
SS: There is no removable “chance nature” in APR. APR ignores the “information below the transfer line" because the APR citizen has given greater importance to the information above the transfer line.
T: Indeed. There is no removable chance nature. But there is this unremovable chance nature intrinsic to APR that is less present in systems of proportional score/approval.  
SSS: As I see it, it is impossible for you to show that it is less present in score/approval systems than in APR without you first mathematically defining "proportional" (or "overall proportionality", i.e. the goal that seems to be most important to you with regard to electoral systems).
SS: “Overall proportionality” is still too vague to be helpful. Can you not give it a mathematical definition?
 T: As I said in the previous post an MP's representation (and yes this is the amount of weight they have) is split among voters who support that candidate to some degree. It's split equally in approval voting or proportional to the score received from each voter in score voting. From this, each voter then has a numerical score for the amount of representation they have. The total amount of representation is always the same (provided the elected candidate has had non-zero support), and proportionality is measured by the voters' total squared difference from the mean amount of representation.  
SSS: Correct me if I have misunderstood you: Assuming that each MP has one vote in the Commons in your voting system, this means each MP's "total amount of representation [voting power] is always the same". At the same time, each approving elector for an MP would have an equal share of this power. Alternatively, each scoring elector would have a proportion of this power equal to the score he gave to this MP. 
Yes.
However, I do not yet understand your phrases:  "provided the elected candidate has had non-zero support";  "proportionality is measured by the voters' total squared difference from the mean amount of representation". I need these phrases to be explained.  Please explain this calculation using examples both for approval/score and APR.
OK. The phrase "provided the elected candidate has had non-zero support" simply refers to the fact that electing an MP that has no support would mean that the total representation from the MPs is less in this case and you'd have a different mean to calculate deviation from. The system wouldn't work properly in this case. The squared difference isn't really part of the definition of proportionality, but just how you'd measure disproportionality in an approval/score case. To give a simple example:
2 to elect (with equal power), approval voting
3 voters: A, B
1 voters: C
We can say that the representation that a voter gets from a candidate is 1/number ofvoters for that candidate. In this case, because there are two elected candidates and four voters, the average representation level would be 2/4 or 1/2 (the total representation being 2 candidates).
If we elect AB, three voters each have a representation level of 2 * (1/3) or 2/3 and the other has 0. If you add up the squared differences from the average (1/2), you get 3*(2/3 - 1/2)^2 + 1*(1/2 - 0)^2 = 1/3.
If we elect AC, three voters have a representation level of 1/3 and the other has 1/1 or 1. The total of the squared differences from 1/2 is 3*(1/2 - 1/3)^2 + 1*(1 - 1/2)^2 = 1/3.
So this is a tied result.
There is actually a complication I haven't mentioned with score voting. If a voter gives a candidate 1/10 and it's the only score the candidate gets, then according to what I've said, this voter would be considered to have the full representation from this candidate and it would count towards their total level of representation even though they don't like the candidate very much. So with score voting, I would suggest "splitting" each voter into 10 parts (or whatever the score is out of). The "top tenth" only approves candidates given a score of 10, the next tenth approves candidates with a 9 or a 10 and so on. So only one tenth of a voter approves the candidate with a score of 1 out of 10.  
SSS:  Yes, your score voting system (not your approval voting system), like APR, might require your highest scored but eliminated candidate to pass on your score to an MP on his “list of favourite candidates”.  If so, the system would seem to do all that is possible to guarantee the voting satisfaction of each citizen.  However, do you see this is also like APR in guaranteeing that no part of any citizen’s full vote will be wasted?  If so, explain how your score voting could achieve this.  Would this now be an essential part of your preferred system?
I think with the suggestion I made at the top of this post, it does as good a job as APR at ensuring your vote isn't wasted. Also, As I said before, I wouldn't necessarily talk about my preferred system as such. These things are affected by how people would vote in practice. I'm not completely against ranked voting for proportional representation. I think it has flaws, and my current thinking is that score voting in particular would have less but it's always open to evidence and argument.
Toby
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