A method that elects the "most stable" candidate set

@andydienes said in A method that elects the "most stable" candidate set:
But my point is that "goodness" isn't one of them.
Various notions of "fairness" and "proportionality" are though. There are dozens of proposed formalizations. Inasmuch as those things are "good" we can measure them.
Sure, but then the goodness of Websterreducing approval methods wasn't refuted where you told me to look.
it would be desirable to have notions of proportionality that are agnostic to the underlying apportionment method.
So I don't think your claims are correct.
That's exactly my point, there are very few notions of proportionality that extend Webster. If you can propose one I will listen. These are not "my" opinions or "my" claims, I'm just telling you what the current state of research looks like. Pretty much all the active conversations and open questions across the various academic groups that study Approvalbased PR or participatory budgeting revolve around extensions of lower quota; very few extensions of Webster have been proposed or studied.
But this wasn't your original point that started this. Your original point was that the likes of Monroe and ChamberlainCourant (and presumably varPhragmen but we didn't explicitly mention that) were not proportional because they didn't pass PJR (and lower quota). Your point wasn't that methods that don't pass PJR and lower quota could be proportional but haven't been extensively studied.
I'm unaware of any proof that e.g. multiwinner Pareto is incompatible with perfect representation in the limit. Because passing perfect representation in the limit is not the same as passing the criterion.
Just look at proposition A.9 in the Lackner&Skowron book. There is a proof right there. If you mean something else by "in the limit" then you will have to be more precise.
Proposition A.9 gives a ballot profile where there is one possible candidate set that gives perfect representation, but where that set is Pareto dominated by another set.
My "in the limit" thing was that as the number of candidates is increased towards infinity, or even just equal to the number of voters actually. E.g. if there are 100 voters and 100 candidates to be elected, I would say that perfect representation should hold, as long as the ballots allow it, for a deterministic method to be properly proportional. Proposition A.9 has 8 voters and 2 to elect, so I'd want to see a proof with 8 to elect.
In any case I'm not saying that Pareto is compatible with this, just that I don't think it's been shown that it isn't. Also, there are cases where I don't necessarily think this form of Pareto is desirable, but I might start a separate thread on that, and how it also relates to consistency.

@tobypereira Among the metrics/axioms to measure approvalbased proportionality that I have ever seen been studied, Monroe and ChamberlinCourant pass very few (and the latter almost none). Nearly all of these axioms reduce to lowerquota on partylist profiles.
If you can provide a compelling and intuitive axiom which implies Webster on partylists and relates to other notions in fairness and proportionality as comprehensively as the PJR family do then I will be open to that discussion. Until then, I will consider such rules not proportional.

@tobypereira said in A method that elects the "most stable" candidate set:
Also, any election where k > n (i.e. more seats to elect than voters) cannot have Perfect Representation, so the notion of Perfect Representation "in the limit" is kind of nonsensical except when k exactly equals n.
If you want to take that as your guiding axiom I can't stop you, but it seems rather contrived to me. Especially because when k == n exactly, then (I think) any outcome providing Stable Priceability as defined in http://www.cs.utoronto.ca/~nisarg/papers/priceability.pdf will also provide Perfect Representation, suggesting that even in this restricted case where k == n that stability is still the superior metric.

@andydienes said in A method that elects the "most stable" candidate set:
@tobypereira said in A method that elects the "most stable" candidate set:
Also, any election where k > n (i.e. more seats to elect than voters) cannot have Perfect Representation, so the notion of Perfect Representation "in the limit" is kind of nonsensical except when k exactly equals n.
If you want to take that as your guiding axiom I can't stop you, but it seems rather contrived to me. Especially because when k == n exactly, then (I think) any outcome providing Stable Priceability as defined in http://www.cs.utoronto.ca/~nisarg/papers/priceability.pdf will also provide Perfect Representation, suggesting that even in this restricted case where k == n that stability is still the superior metric.
If there cannot be perfect representation when k>n, then that's really just because of the narrow way perfect representation is defined, as they probably didn't consider this case when defining it. It doesn't really change the general principle, and I would just extend it in the way it naturally should be.
Essentially if there are n voters, then if a result allows each voter to be uniquely assigned to 1/n of the representation (whether that's a fraction of a candidate or more than one candidate), then that's "perfect representation" in the way I would extend the definition. So I would still use this in the limit as my defining feature of a proportional approval method.
I think it's simpler than what the acedemics in the field have been trying to do by coming up with a whole zoo of different axioms trying to capture the essence of proportionality.

each voter to be uniquely assigned to 1/n of the representation (whether that's a fraction of a candidate or more than one candidate), then that's "perfect representation" in the way I would extend the definition
I just can't shake the feeling that this better describes priceability and (stable priceability) than it does perfect representation. I think many academics, and I personally, agree more or less with this intuition, but formalizing it without introducing unforeseen sideeffects of the definition is the hard part. As it happens, I think the attempted formalization in the axiom labeled "perfect representation" does have some unfortunate sideeffects, but you can get the same spirit of proportionality in (to me) a more principled way via priceability.

@andydienes said in A method that elects the "most stable" candidate set:
each voter to be uniquely assigned to 1/n of the representation (whether that's a fraction of a candidate or more than one candidate), then that's "perfect representation" in the way I would extend the definition
I just can't shake the feeling that this better describes priceability and (stable priceability) than it does perfect representation. I think many academics, and I personally, agree more or less with this intuition, but formalizing it without introducing unforeseen sideeffects of the definition is the hard part. As it happens, I think the attempted formalization in the axiom labeled "perfect representation" does have some unfortunate sideeffects, but you can get the same spirit of proportionality in (to me) a more principled way via priceability.
OK, but does Webster pass priceability? I wouldn't want to throw that out.

@tobypereira said in A method that elects the "most stable" candidate set:
@andydienes said in A method that elects the "most stable" candidate set:
each voter to be uniquely assigned to 1/n of the representation (whether that's a fraction of a candidate or more than one candidate), then that's "perfect representation" in the way I would extend the definition
I just can't shake the feeling that this better describes priceability and (stable priceability) than it does perfect representation. I think many academics, and I personally, agree more or less with this intuition, but formalizing it without introducing unforeseen sideeffects of the definition is the hard part. As it happens, I think the attempted formalization in the axiom labeled "perfect representation" does have some unfortunate sideeffects, but you can get the same spirit of proportionality in (to me) a more principled way via priceability.
OK, but does Webster pass priceability? I wouldn't want to throw that out.
As Andy said, it does not, since Webster does not pass lower quota.

@spelunker Fair enough. I'm not as familiar with priceability and what it rules out, though I have read the definition. I knew Webster failed PJR.

@A Former User said in A method that elects the "most stable" candidate set:
Also "more stable" I think might not even be resolute whatsoever, forget resolvable in polynomial time. Imagine like a condorcet cycle but of committees.
Just use MES or partylist and be happyJust to revisit the initial topic  If there was a cycle of "most stable" then like with singlewinner Condorcet methods, there could be a system in place to determine the winner. Also if this sort of thing were ever adopted in any sort of election, it would likely have to be used sequentially for computational reasons, which should make it workable.
And on Condorcet methods, it has been said that this core stability is a sort of multiwinner analogue of Condorcet.
Another thing about Condorcet methods is that the Condorcet winner is, as far as I understand, the game theoretically stable winner for approval voting. I don't think core stability in the sense that it's discussed for multiwinner cardinal methods is necessarily the same as game theoretically stable. So perhaps that would be an interesting thing to investigate. I think it would be interesting to see what winner sets were game theoretically stable under, for example, single nontransferable vote.
Has Andy left the forum by the way?

If this was extended to score voting, I think it should elect the Condorcet winner in the singlewinner case, if there is one. Otherwise, obviously it would have to choose between the candidates in some way, like other Condorcet methods do so it's not a big problem.
When there's more than one winner, what happens depends on how you interpret the scores. You could measure a voter's satisfaction by adding up the scores the voters have given to the elected candidates, but I think that might be unsatisfactory in a few ways. There's always debate about how to interpret scores and what they mean, and whether absolute numerical values should really be used in their raw form.
Instead, the scores could be used as layers of approval. This basically means that a voter's satisfaction with a candidate set is determined by the single highest score they've given to a candidate in the set, next best used as a tiebreak, and so on. So for scores out of 5, a single 5 is better than multiple 4s etc.
This should keep it relatively simple. Also if candidates are elected sequentially, it should be simple enough to calculate the results.
I think this should be a decent enough method and I think I'd prefer it to things like Allocated Score and Sequentially Spent Score.
Obviously COWPEA Lottery using scores as layers of approval is Godtier in terms of criterion compliance, and very simple to implement, but it is nondeterministic, which might be too much for some people, so this method could be a good compromise.
Edit  You'd have to work out exactly how to measure the stability of a candidate set though. Let's say the first 2 candidates elected are AB. Then you need to test e.g. ABC, ABD, ABE etc. to find the 3rd candidate. But I think you might be able to test them against each individual candidate not in the set. So test ABC against, D, E, F etc. separately.
Edit 2  You'd probably have to test each potential set against all the other subsets. So ABC would go against ABD, ABE etc., plus AD, AE, BD, BE, as well as D, E etc. Still not that many in the general scheme of things.