Optimal cardinal proportional representation

@lime In the example I gave, electing AB would mean that 300 of the 302 voters would have approved exactly one elected candidate, whereas electing CD would mean that all 302 would have approved exactly one elected candidate. So by that measure, CD would be better.
But  under AB, 150 people have approved A and 150 have approved B. Under CD, 199 have approved C and 103 have approved D. So CD is a disproportional result in that the 103 D voters wield a disproportionate amount of power in parliament. Or perhaps more relevantly, the D party has only about 1/3 of the support but half the power. AB would be more balanced in that respect. Methods that use a measure of proportionality rather than satisfaction (e.g. Phragmen) would tend to elect AB.
This doesn't matter in the purely optimal case, because PAV would elect CD but in the respective proportions. COWPEA would elect all four in varying proportions. If these were reallife votes, it would be likely that AB and CD focus on different issues. A and B are opposed on the issues that they focus on. C and D are opposed on the issues they focus on. By electing all four, COWPEA would be making sure that the issue space is better covered.
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.

@tobypereira said in Optimal cardinal proportional representation:
But  under AB, 150 people have approved A and 150 have approved B. Under CD, 199 have approved C and 103 have approved D. So CD is a disproportional result in that the 103 D voters wield a disproportionate amount of power in parliament. Or perhaps more relevantly, the D party has only about 1/3 of the support but half the power. AB would be more balanced in that respect. Methods that use a measure of proportionality rather than satisfaction (e.g. Phragmen) would tend to elect AB.
Right. I suppose that's what I meant by disliking the idea of making an underrepresented group worseoff just to make the overrepresented one even worseoff.
@tobypereira said in Optimal cardinal proportional representation:
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.
That's surprising. I know there are local councils and similar that use weighted votes, but I can't imagine any legislature or council (especially a small one) using a random method.

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@lime said in Optimal cardinal proportional representation:
@tobypereira said in Optimal cardinal proportional representation:
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.
That's surprising. I know there are local councils and similar that use weighted votes, but I can't imagine any legislature or council (especially a small one) using a random method.
Which bit is surprising? I'm only saying that COWPEA Lottery could be used (i.e. there would be no computational problems)  not that it's likely to be. In any case, from my point of view, I don't have a problem with nondeterministic methods in some situations.

I think that a simple model of perfectly proportional representation is to make a network flow problem and just simply set each voter's flow to be equal.
In this network flow model, a cardinal ballot would be represented as capacity for flow between a voter and each candidate. I'll put a link here to wikipedia just for general reference: https://en.wikipedia.org/wiki/Network_flow_problem
How is it possible to have each voter's flow equal? I am thinking there would have to be a change to the rules for the elected body. We might have to allow winners to have different weights. We might have to allow any amount of winners. The benefit is we get perfectly proportional representation.

@paretoman said in Optimal cardinal proportional representation:
I think that a simple model of perfectly proportional representation is to make a network flow problem and just simply set each voter's flow to be equal.
Possibly.
Personally, I'm wondering whether we could figure out optimal representation from a model that pins down what, precisely, the utility of a given committee for each voter is, under a particular definition of what the scores are supposed to mean.
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.

@lime said in Optimal cardinal proportional representation:
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.
(Oh, quick note @TobyPereira : I think this is a good example of why perfect location or scale invariance may not actually be ideal. If you rescale every voter's ballot to fall between 50% and 100% instead of 0100%, that might indicate a meaningfully different situation.)

@lime said in Optimal cardinal proportional representation:
@lime said in Optimal cardinal proportional representation:
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.
(Oh, quick note @TobyPereira : I think this is a good example of why perfect location or scale invariance may not actually be ideal. If you rescale every voter's ballot to fall between 50% and 100% instead of 0100%, that might indicate a meaningfully different situation.)
Yes, that is a way that someone could vote. However, I wouldn't see it as some sort of objective standard, so would still see scale invariance as desirable. I think you mentioned this point previously in a discussion about normalising ballots, and I didn't get round to responding at the time.
The method being discussed renormalises using ratios when candidates are eliminated, so if someone gives scores of 1 and 0 (out of e.g. 5) to two candidates, this could later become 5 and 0 after some elimination, whereas scores of 5 and 1 would be locked in as that. However, even if it is assumed that people vote in the manner you described, I still don't think this is good voting method behaviour. If I give scores of 5, 1 and 0, I would still prefer the 5 to the 1 by more than I prefer the 1 to the 0, and I think voters generally would not be happy that they lose some of the normalisation power in the 51 situation but not in the 10 situation.

@Lime As mentioned upthread, there is the probabilistic transformation as well, which you might prefer from the point of view being discussed. I'm not a big fan of it though.
@tobypereira said in Optimal cardinal proportional representation:
There is also the probabilistic transformation, which I see as inferior to KP as well. Someone might give scores of 9 and 1 (out of 10) to 2 candidates A and B respectively. KP would split the voter as follows:
0.1: 
0.8: A
0.1: ABThe probabilistic transformation would give:
0.09: 
0.81: A
0.09: AB
0.01: B 
@tobypereira said in Optimal cardinal proportional representation:
By the way, since PAV with infinite clones passes core (which it doesn't with a limited number of candidates), I presume the optimal version probably is properly proportional (passes perfect representation). I might update my paper with this in at some point.
I have updated the paper to mention the proportionality of Optimal PAV (with variable candidate weight allowed), which allows for a proper comparison with COWPEA  these two methods being the main candidates for a truly optimal cardinal PR method (practicalities aside).