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@lime well, people who oppose reform (aka those in power) will find ways to concoct detailed arguments against the adoption of any reform proposal, and questionable constitutionality is a low bar.

@jack-waugh that will work if everybody does it. However, it’s likely that people will not go through with that unless they have the same kind of discipline it takes not to constantly check stock values.

@gregw I honestly think approval voting is one of the best footholds we have for moving forward with actual voting reform. Especially considering the language you reference in the constitutions, it fits the bill. It may not be perfect but it’s miles ahead of choose-one voting and would have dramatically positive consequences if adopted. While IRV is a flawed incremental change, approval voting would be a real game changer.

Score will certainly be more questionable than approval in terms of the constitutional language. I think these kinds of practical constraints are likely to focus reformer support for approval.

@lime since we would be iterating through the edges that belong to cycles in order of ascending weight, any edge under consideration would automatically be the minimum weight edge of every cycle it is a part of. I think maybe it wasn't clear that equivalently, we can ask for each edge, "Is there some cycle to which this edge belongs?" If yes, it is in the search space of edges, and then we identify the edges in the search space of minimum weight.

Generally, whether an edge (u,v) belongs to a cycle can be checked by removing the edge and doing a (depth-first) search for a path from u to v, since (u,v) belongs to a cycle if and only if such a path exists.

But in any case it's just a concept, similar to Young's method and Dodgson's method, since it tries to perturb the ballot set in a conservative way until a Condorcet winner emerges.

@jack-waugh I think it would be best to have a multifaceted vote on voting systems. Here is the ballot format I suggest, since it can be flexibly transformed into ballots that are compatible with other systems.

Once the candidate voting systems under consideration are chosen, each voter should submit a ballot assigning each candidate an independent integer score ranging from 0 to 100. For scaling purposes, two pseudo-candidates who cannot win will be introduced to the election as well, one automatically receiving a 0 on every ballot, and the other automatically receiving a 100 on every ballot. Each voter will also submit an integer from 0 to 100 as their approval threshold, and scores above that threshold will count as an approval.

This way, we can examine the winner under various different systems using self-consistent ballots. We can also see which voting systems if any end up electing themselves, just for curiosity.

If the election gets organized well, I’ll participate.

@gregw said in STLR - Score Than Leveled Runoff might not be too complex for voters:

@toby-pereira said in STLR - Score Than Leveled Runoff might not be too complex for voters:

Using score ratios is always a bad idea. It only makes sense (as with utility scores) to look at the absolute difference between scores. If someone scores three candidates 0, 1, 2, then they are equidistant. To see the 1 as infinitely more than the 0 is bad voting method behaviour.
It would also be insane if (under a 0 to 5 ballot) if someone scoring two candidates 5 and 1 had less voting power in the run-off than someone scoring them 1 and 0.

Therefore, STLR’s leveled runoff is not a good idea, but it might be less bad using a 1 - 6 scoring range than a 0 - 5 range. Also, normalizing a ballot with scores of 4, 2, 1, and 0 to scores of 5, 3, 2, and 1, keeping the absolute differences, makes more sense. Call it STLR2? It would be a little easier to explain the regular STLR. Thank you for the help!

Well, if I was normalising, I would always stretch scores to fill the whole range. So every voter would have a 0 and a 5 in this case.

@lime said in Variable house sizes:

@toby-pereira said in Variable house sizes:

Also, looking at proportional voting methods rather than apportionment (which is more relevant if we're talking about the Holy Grail of cardinal PR), some countries are split into regions where they might each have 5 or so representatives elected (normally using STV). We would not be able to reduce this number in a particular region if doing so would give a more proportional result, because it would mean the entire region would be be under-represented nationally.

There's two reasons I still think this question would be relevant:

1. In some cases, there's more flexibility (if nothing else, small committees like city councils).
2. Even if we have to accept a fixed-size legislature, proving a method satisfies the core in the variable-size case would probably imply a nearly-satisfied core property in the case of a fixed-size legislature (think of how Webster satisfies lower quota 99.9% of the time). It could also lead to randomized methods that can satisfy the core with a bare minimum of randomness (requiring random selection or weighting only very, very rarely).

Also if we're using a candidate-based system (e.g. STV or a cardinal method as opposed to party list), it wouldn't make sense to talk about parties being over-represented since candidates would be elected in a party-agnostic manner. So to talk of quota violations would become meaningless.

Thus why I asked about whether we could guarantee the core instead, which seems like the most attractive generalization of quota for me. (As well as having very appealing strategy-resistance properties).

Methods that guarantee core stability are of interest to me (see this thread, which I linked to earlier) even if it's not my priority. From what I've read, I think it's still unproven that it's guaranteed that the core is non-empty. But if you use a stability measure (as suggested in the thread) rather than an all-or-nothing, it could be workable regardless.

@lime said in Variable house sizes:

@toby-pereira said in Variable house sizes:

In your example, allocating the extra seat wouldn't violate either of these (I don't think divisor methods fail population monotonicity anyway). With 399, 101 and 100, the (almost) exact number of seats each with 7 to elect would be 4.655, 1.178 and 1.167. So the 399 getting 4 or 5 wouldn't violate quota and neither would the others getting 1 or 2.

Ack, I botched the example. Here's an example from Balinski and Young that illustrates the example better. The problem with satisfying quota in this example is it would require having some states be dramatically over (or under) represented.

@toby-pereira said in Variable house sizes:

Obviously, I understand that having just the 6 seats would give better proportionality, but without any specific violations, where do you draw the line of how many seats to remove? If 8 was the highest available available number to give, would you still cut it down to 6 for better proportionality? Or what if the proportions were pretty much exactly 3/7, 2/7, 2/7, and you could give a theoretical maximum of 13 seats. Would you cut it all the way down to 7?

In the party-list apportionment case, the line-in-the-sand gets drawn at exactly half the number of parties, which is the most the initial apportionment can be off by, and the worst-case difference between Webster and Hamilton. In practice, the actual is ~0.

Do you think a method not violating quota is a matter of principle or simply that it looks better if it doesn't? Because interestingly enough, while Hamilton doesn't violate quota, Webster is simply Hamilton but without the IIB failures. For two parties they are identical.

If you take your example above where D is below the lower quota under Webster, you can do a Hamilton comparison between D and any of the others head-to-head and the result won't change.

E.g. compare A and D. They won 27 seats between them under Webster. Of those 27, A should now have 1.508 seats and D should have 25.492. So it is better for A to keep the 2nd seat rather than D get the 26th under both Hamilton and Webster. The same would apply if you compared D against either B or C.

This is why I regard occasional quota violations as a natural proportional result not requiring a change in house size even if it might look unfair at a first glance.

@lime said in Variable house sizes:

I consider any rule that violates either quota or population monotonicity to be unfair; instead I reject the assumption (usually left unstated) that we must have a fixed number of seats with equally-weighted representatives. (Why?)

In your example, allocating the extra seat wouldn't violate either of these (I don't think divisor methods fail population monotonicity anyway). With 399, 101 and 100, the (almost) exact number of seats each with 7 to elect would be 4.655, 1.178 and 1.167. So the 399 getting 4 or 5 wouldn't violate quota and neither would the others getting 1 or 2.

Both Webster and Jefferson would give the 399 the extra seat. Obviously, I understand that having just the 6 seats would give better proportionality, but without any specific violations, where do you draw the line of how many seats to remove? If 8 was the highest available available number to give, would you still cut it down to 6 for better proportionality? Or what if the proportions were pretty much exactly 3/7, 2/7, 2/7, and you could give a theoretical maximum of 13 seats. Would you cut it all the way down to 7?

This could also lead to another sort of paradox - a state's population could increase and they lose a seat, because due to a more proportional result becoming available, every state loses seats.

Also, looking at proportional voting methods rather than apportionment (which is more relevant if we're talking about the Holy Grail of cardinal PR), some countries are split into regions where they might each have 5 or so representatives elected (normally using STV). We would not be able to reduce this number in a particular region if doing so would give a more proportional result, because it would mean the entire region would be be under-represented nationally. Also if we're using a candidate-based system (e.g. STV or a cardinal method as opposed to party list), it wouldn't make sense to talk about parties being over-represented since candidates would be elected in a party-agnostic manner. So to talk of quota violations would become meaningless.

So I think the proportionality criteria you want really only apply to apportionment and nationwide party-list voting. It doesn't apply to candidate-based voting or the Holy Grail of cardinal PR (which you wouldn't use for apportionment). I know this thread started as an apportionment thing, but since cardinal PR came up, I think all this has become relevant.

@lime Cardinal PR and Holy Grail criteria is my main area of interest when it comes to voting methods, so any mention of them and I'm back in the thread.

Your criteria aren't the same that I would pick. There was also some discussion in this thread, but Sainte-Laguë/Webster does not pass core stability or priceability. I (along with many other people) would consider it to be the most accurate party PR method, so if a cardinal method reduces to Sainte-Laguë/Webster where there is party voting, it should not be disqualified.

As I understand it, D'Hondt is the only divisor method that satisfies lower quota, so therefore the only one in the running to pass core stability. As I also understand it, non-divisor methods all fail Independence of Irrelevant Ballots, so I think we would be forced into a method that reduces to D'Hondt under party voting if we also wanted to pass that, which isn't satisfactory.

It is an inconvenient truth actually that most PR criteria that have popped up in recent years for cardinal methods are failed by Sainte-Laguë/Webster, so I would argue that they are not really fit for purpose.

Single-candidate Pareto efficiency is pretty non-controversial. If candidate A is approved on all the ballots that B is and at least one other, then if B is elected so should A.

However, the multiple-candidate criterion, while intuitive, is at least debatable. Basically: for the winning set of candidates, there should not be another set for which every voter has approved at least as many elected candidates as they have in the winning set, and at least one voter has approved more. Take the following example with 500 voters and 2 to elect:

150 voters: AC
100 voters: AD
140 voters: BC
110 voters: BD

AB and CD are the only two electable sets for a PR method. In either case every voter will have one elected candidate. PAV and welfarist methods in general would regard them as the same because they just look at number of candidates elected for a voter.

However, under AB each elected candidate is has been approved by 250 voters. Under CD, it's 290 for C and 210 for D. AB is therefore more proportional than CD, and certainly the result I would prefer. But then you could have:

150 voters: AC
100 voters: AD
140 voters: BC
110 voters: BD
1 voter: C
1 voter: D

So unless AB's win over CD in the previous example was just of a tie-break nature, AB should still win here. But under the Pareto criterion above, CD must win. I regard this as unsatisfactory.

The four main criteria I look for in the Holy Grail are:

Perfect Representation in the Limit (the primary PR criterion I use, which does not disqualify Sainte-Laguë/Webster)
Strong Monotonicity (so an extra approval should count in favour of a candidate rather than merely not against them and not just as a tie-break measure)*
Independence of Irrelevant Ballots
Independence of Universally Approved Candidates

*Phragmén, while monotonic, fails strong monotonicity. E.g. two to elect:

1: AB
1: AC

It regards all results (AB, AC, BC) as equally good other than possibly in a tie-break (despite A's universal support). This makes a difference in the following example:

99: AB
99: AC
1: B
1: C

Phragmén prefers BC in this case, which does not seem right. Electing sequentially avoids this obviously, but examples can easily be found where electing sequentially does not save it from a bad result.

Anyway, my criteria may be incompatible, in a deterministic method at least. COWPEA Lottery passes them all but is non-deterministic. Optimised PAV Lottery might also pass as well, but this is unproven as far as I am aware.

Pretty much everything I have discussed here is discussed in detail in my non-peer-reviewed arXiv paper on COWPEA and COWPEA Lottery. I think it's at least worth a look. Arguably the most exciting part is the section on COWPEA v Optimised PAV as the ultimate in cardinal PR for cases where there are no limits to the number and weight of elected candidates.