Optimal cardinal proportional representation
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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).
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One other post I made on Reddit is basically the topic of this thread. Again, some of the words are links in the original, so I recommend going to that for more info.
Optimal cardinal proportional representation and the "Holy Grail"
By optimal cardinal PR, I mean you remove the restriction of having to elect a fixed number of candidates with equal weight, but can elect any number with any weight. So this is a theoretical thing rather than about coming up with a practical method for use.
But by "Holy Grail", I mean a cardinal method that does elect a fixed number of candidates with equal weight (the usual requirement) and passes certain criteria. So this could be potentially used.
Although this is about cardinal PR, I will make it simpler by talking about approval methods, since I've previously argued for the KP-transformation as the best way to convert scores into approvals.
First of all optimal cardinal PR. It would need a strong form of monotonicity not present in Phragmén-based methods, which would be indifferent between the infinite number of results giving Perfect Representation. To cut a long story short, there are two candidate methods that are proportional, strongly monotonic and pass Independence of Irrelevant Ballots (IIB). They are the optimal version of Thiele's Proportional Approval Voting (Optimal PAV), and COWPEA.
To work out an Optimal PAV result (or an approximation to it), you increase the number of seats to some large number and, allowing unlimited clones, see what proportion of the seats each candidate takes. That proportion would be each candidate's weight in the elected committee. This method would be beyond calculation but exists as a theoretically nice method. If you elect using PAV sequentially it doesn't always give a good approximation, as I think it's possible to end up giving weight to candidates that would actually receive no weight under Optimal PAV, since I think it's possible for Optimal PAV to give zero weight to the most approved candidate. E.g.
150: AC
100: AD
140: BC
110: BD
1: A
1: B
If I've worked it out right, Optimal PAV would give A and B half the weight each, and C and D no weight. This is despite the fact that C has the most votes at 290 (A and B each have 251; D has 210).
COWPEA elects candidates proportionally according to the probability they would be elected in the following lottery:
Start with a list of all candidates. Pick a ballot at random and remove from the list all candidates not approved on this ballot. Pick another ballot at random, and continue with this process until one candidate is left. Elect this candidate. If the number of candidates ever goes from >1 to 0 in one go, ignore that ballot and continue. If any tie cannot be broken, then elect the tied candidates with equal probability.
Because each voter would be the first ballot picked in the same proportion (1/v for v voters), each voter is guaranteed 1/v of the elected body. But where a voter approves multiple candidates, these candidates are then elected proportionally in the same manner according to the rest of the electorate. COWPEA is also beyond calculation for real elections, but can be approximated with repeated iterations of the algorithm.
Both Optimal PAV and COWPEA have the properties that makes them contenders for the optimal approval method, and ultimately it's likely a matter of preference rather than one having objectively the best properties. I compare them both in my non-peer-reviewed COWPEA paper here if you're interested. The current version is not set in stone, and I might tighten certain things up further at some point. But just to give an example of where they differ:
100: AC
100: AD
100: BC
100: BD
1: A
1: B
COWPEA would elect the candidates in roughly equal proportions (with A and B getting slightly more). Optimal PAV would only elect A and B and with half the weight each. This example can be seen as a 2-dimensional voting space with A and B at opposite ends of one axis and C and D at opposite ends of the other. No voter has approved both A and B or both C and D. COWPEA makes more use of the voting space in this sense, whereas Optimal PAV only looks at voter satisfaction as measured by number of elected candidates, and every voter is either indifferent between AB and CD or prefers AB. This is also why the most approved candidates in the previous example gets no weight under Optimal PAV.
Without the extra two voters that approve just A and B respectively, COWPEA would elect all four equally. Optimal PAV would be indifferent between any AB to CD ratio as long as A and B are equal to each other and so are C and D.
Finally, onto the Holy Grail where a fixed number of candidates with equal weight are required. Where unlimited clones are allowed, PAV passes all the criteria, but is not fully proportional where there aren't such clones as I discussed here.
So we need the method to be proportional, strongly monotonic, pass IIB and ideally also Independence of Universally Approved Candidates (IUAC). As far as I'm aware, no known deterministic method passes all of these, but if it doesn't have to be deterministic, then two methods do. And they are versions of the methods above. Optimal PAV Lottery and COWPEA Lottery.
Under Optimal PAV Lottery, the Optimal PAV weights are used as probabilities, but these would need to be recalculated every time a candidate is elected and removed from the pool. This method is clearly not possible to calculate in practice.
COWPEA Lottery is just the lottery used in the COWPEA algorithm. This is easily runnable. And while this may be unrealistic for elections to public office, it can certainly have more informal uses. E.g. friends can use it to determine activities so that choices proportionally reflect the views of the group over time without anyone having to keep count or worrying what to do if not exactly the same people are present each time.
In conclusion, the main contenders for optimal cardinal proportional representation are Optimal PAV Lottery and COWPEA. For the Holy Grail, we have PAV where unlimited clones are allowed, but otherwise Optimal PAV Lottery or COWPEA Lottery, of which only COWPEA Lottery can be reasonably computed.
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Also I've been doing some work on my COWPEA paper to see if I can get it to publication standard. So hopefully that will happen at some point.