Optimal cardinal proportional representation
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Thank you so much for this post! It's great
@toby-pereira said in Optimal cardinal proportional representation:
There are several possible methods of converting an approval method to a score method, but the KP-transformation keeps the Pareto dominance relations between candidates and allows the methods to pass the multiplicative and additive versions of scale invariance, so my current thinking is that this is the optimal score conversion.
I'm not 100% sure about this myselfâwon't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the core-approximation properties that make PAV so appealing (i.e. 2-approximation of the core, and satisfying core with enough similar candidates).
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@lime said
I'm not 100% sure about this myselfâwon't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the core-approximation properties that make PAV so appealing (i.e. 2-approximation of the core, and satisfying core with enough similar candidates).
Can you remind me exactly what these mean?
And I'm glad you liked the post!
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@toby-pereira said in Optimal cardinal proportional representation:
@lime said
I'm not 100% sure about this myselfâwon't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the core-approximation properties that make PAV so appealing (i.e. 2-approximation of the core, and satisfying core with enough similar candidates).
Can you remind me exactly what these mean?
And I'm glad you liked the post!
K-approximation means K Hare quotas would prefer another committee (instead of just 1).
And if every candidate has infinitely many clones, you can guarantee the method will choose from the core (see here).
I'm wondering if some method can keep these properties in the score voting case.
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@lime OK, I'm not sure how the KP-transformation would affect these things. Do you specifically think it's likely to be any worse than any other transformation, or is it general concerns about any transformation that hasn't been demonstrated to pass these things?
In any case, I definitely think PAV + KP is better than RRV or SDV because of its scale invariance, and I don't see any particular advantages of these methods over it.
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: BThis wrecks both Pareto dominance and scale invariance.
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
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.
Also, in the optimal scenario, both Phragmén and Monroe would be unsuitable as contenders. Both would be indifferent between an infinite number of different candidate proportions. They are concerned only with perfect representation, and this is very easy to achieve in the optimal case with any proportions allowed, and they have nothing to say to distinguish between them. Monroe is also essentially the Hamilton version of Phragmén (which can be made D'Hondt or Sainte-Laguë), so essentially the same but with more IIB failures.
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@toby-pereira said in Optimal cardinal proportional representation:
OK, I'm not sure how the KP-transformation would affect these things. Do you specifically think it's likely to be any worse than any other transformation, or is it general concerns about any transformation that hasn't been demonstrated to pass these things?
General concern about transformations. My worry is transforming score ballots to approval ballots discards information about which voter gave which ratings, so I'm not sure it will preserve the stable winner set properties of PAV.
@toby-pereira said in Optimal cardinal proportional representation:
In any case, I definitely think PAV + KP is better than RRV or SDV because of its scale invariance, and I don't see any particular advantages of these methods over it.
Oh, definitely, harmonic voting is great! (Although it needs a less intimidating name.)
@toby-pereira said in Optimal cardinal proportional representation:
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
This is a much bigger hangup for me personally. If everyone agrees a different committee would be better, then leveling-down (making some people worse-off, just to make the outcome more equal/proportional) strikes me as wrong.
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@lime said in Optimal cardinal proportional representation:
@toby-pereira said in Optimal cardinal proportional representation:
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
This is a much bigger hangup for me personally. If everyone agrees a different committee would be better, then leveling-down (making some people worse-off, just to make the outcome more equal/proportional) strikes me as wrong.
Right, but it's debatable whether a voter's utility is purely determined by approved candidates elected. A voter is better represented in parliament if they share their representative with fewer other voters. So in the example up the thread:
250: AC
250: AD
250: BC
250: BD
2: C
2: DA and B are more attractive options to most voters.
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@lime Who would you elect in the following election with 2 to elect:
150: AC
100: AD
140: BC
110: BD
1: C
1: DEdit - I can give a more extreme example:
99: AC
51: AD
99: BC
51: BD
1: C
1: D -
@toby-pereira said in Optimal cardinal proportional representation:
99: AC
51: AD
99: BC
51: BD
1: C
1: DI'm not really seeing what the problem with electing C & D here is supposed to be
It seems like a gain for only 2 voters, so I might be missing something, but I'm not seeing what would make that bad.@toby-pereira said in Optimal cardinal proportional representation:
Right, but it's debatable whether a voter's utility is purely determined by approved candidates elected.
I'll briefly set aside the "approved" part and focus on score voting more broadly (since voters rarely have dichotomous preferences).
I'm not sure why the distribution of like this would particularly matter. The way I'd model is that each candidate is assigned a utility equal to the (importance-weighted) probability that they'll break a tied vote in my favor. I'm not sure why it would be better for me to have a legislator who casts votes that represent my interests less often, or why it would be better for me to have a legislator supported by fewer voters.
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@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 real-life 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.
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@toby-pereira 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 worse-off just to make the overrepresented one even worse-off.
@toby-pereira 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:
@toby-pereira 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 non-deterministic methods in some situations.
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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.
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@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.
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@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 @Toby-Pereira : 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 0-100%, that might indicate a meaningfully different situation.)
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@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 @Toby-Pereira : 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 0-100%, 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 5-1 situation but not in the 1-0 situation.
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@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.
@toby-pereira 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 -
@toby-pereira 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).