Variable house sizes

Lime

How much can we get out of throwing out the fixed house size assumption?

The Wyoming rule comes up every once in a while as the "obvious" way to handle state apportionment. Just take each state's population, divide by Wyoming's, and round. It's stupidly simple, satisfies a stronger version of quota (everyone is rounded to the nearest integer), you get quota monotonicity, and the smallest state has exact representation. As an apportionment method, it's effectively perfect.

How much can we get out of dropping the constant-size committee constraint if we're working with methods other than party list?

Lime

Mentioning @Toby-Pereira who hopefully knows more on this topic of multi-winner systems than I do. Ideally, dropping the fixed house-size constraint lets you construct a system that satisfies several PR philosophies, by finding a house size where multiple methods give the same apportionment. For instance, in party-list apportionment, having a house size that's not fixed means you can use the generally-superior divisor methods, while still guaranteeing you won't break quota.

As an example of how this can be important, biproportional representation is a harder sell if you don't have a quota, because candidates with fewer votes can defeat candidates with more. A consistent quota lets you transparently explain to every losing candidate why they weren't elected: "You needed a Hare quota's worth of votes to win, and you didn't get that."

Toby Pereira

@lime Well, I'm not sure I do know more than you about this. But yes, you could have a variable house size, though I don't know how variable you would need to make it to give "nice" results every time.

Biproportional representation isn't something I've spent a lot of time on. But if we're talking about party votes, it goes against my general philosophy (and where I've spent most of my time thinking) of voting for candidates. And when you e.g. rank or approve or score candidates it muddies the waters about what exactly counts as a quota of votes. So my current knowledge probably can't help you that much!

Lime

@toby-pereira said in Variable house sizes:

But yes, you could have a variable house size, though I don't know how variable you would need to make it to give "nice" results every time.

Definitely an interesting question. In the party-list case, you only need a very small . The worst case is half the number of parties; in practice, the standard error is sqrt(n/12) IIRC.

@toby-pereira said in Variable house sizes:

Biproportional representation isn't something I've spent a lot of time on. But if we're talking about party votes, it goes against my general philosophy (and where I've spent most of my time thinking) of voting for candidates.

I agree on voting for candidates, which is why biproportional representation uses open lists (and can be done with vote-transfer/panachage). It does include information about partisan affiliation, but only to make sure parties are represented proportionally as a first step, and then candidates are also elected proportionally within parties. In real elections, I think any solidly-proportional system is going to have to involve either biproportional representation, or voters being forced to rate candidates from long lists of politicians they've never even heard of (meaning they'll be going off of nothing but party affiliation).

@toby-pereira said in Variable house sizes:

And when you e.g. rank or approve or score candidates it muddies the waters about what exactly counts as a quota of votes. So my current knowledge probably can't help you that much!

I think that with approval/score voting, the equivalent of a quota would be a priceable election. That way, any candidate who asks why they lost can get a response along the lines of "your supporters didn't have enough votes left."

Lime

Having learned more about Cardinal PR, I'd like to revise my opinion here. From what I can tell, the ideal voting system—the holy-moly grail of proportional representation, if you will—would satisfy, in decreasing order of importance:

Core (stable winner set) approximation better than 2. Ideally down to 1 (see here). This ensures that the maximum number of voters left unrepresented is a single quota. PAV provides a 2-approximation of the core (it's impossible for more than two Hare quotas to be left unrepresented).
Welfarism (or at least Pareto efficiency): some utility function is being maximized.
Some way to stratify along a marginal distribution (like political party or geography).
Budget balance (called "priceability" in the earlier paper, but I don't really like that term).

Conjecture: These conditions are incompatible. Satisfying the first 2 with fixed house and equal-weighted representatives is impossible according to Peters and Skowron (2022); I suspect it's possible if we allow either one of those. Satisfying the 3rd as well feels impossibly greedy, but might be possible if we allow voting weights and variable house sizes.

Toby Pereira

@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.

Lime

@toby-pereira said in Variable house sizes:

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.

All of these impossibility results are correct, but only hold for fixed house sizes, thus the question in this thread. Webster's method occasionally violates upper and lower quota, true. This happens when the method is forced to assign a "leftover seat" that really shouldn't be going to anyone. A simple example is apportioning 7 seats to the vote vector (399, 101, 100).

If we were allowed to apportion only 6 seats, the seat vector (4, 1, 1) would be basically perfect. But if we're forced to apportion that last seat, we're stuck between a rock and a hard place. We can give it to the second-largest party, which would satisfy quota but overrepresent them by a factor of 2. Or instead, we could give it to the largest party, in which case they're only overrepresented by about 20% (much milder).

But in the real world I'd argue we're not actually limited to these options. We can just delete that last seat. Congress used this apportionment method from 1850 to 1910 (they made an effort to select house sizes where Hamilton and Webster agreed). Germany has a variable-size Bundestag to deal with overhang seats (because the CDU/CSU often sweeps too many single-member districts).

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?)

Lime

By the way—I think it's plausible that Pareto is undesirable with approval ballots+fixed house size, but I'd take that as more of an indication that approval ballots are problematic (not enough choice for voters) or fixed house sizes are problematic (they make the tradeoff between efficiency and equity far too "sharp").

Toby Pereira

@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

@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.

Lime

@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:

In some cases, there's more flexibility (if nothing else, small committees like city councils).
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).

Toby Pereira

@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.

Toby Pereira

@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:

In some cases, there's more flexibility (if nothing else, small committees like city councils).

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

@toby-pereira said in Variable house sizes:

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.

I somewhat agree, in that I think those ballots are irrelevant for splitting the A+B between A and B: the relative apportionments of parties A or B shouldn't depend on how many votes another party gets.

On the other hand, I think such a ballot (one that supports C, but not A or B) is actually so irrelevant that, not only should this ballot not affect an "idealized" apportionment (i.e. the number of seats each party would get, if we didn't have to deal with rounding); it actually shouldn't even affect the actual relative apportionment.

Say we declare the relative apportionment of A and B shouldn't depend at all on a ballot that supports C (as the vote ratio of A relative to B has not changed). Sometimes, though, adding a ballot that supports C will cause C to gain a seat. The only way to uphold our restriction, that the seat ratio of A to B should not depend on the addition of an irrelevant ballot (one disapproving both A and B) is to make sure that extra seat for C doesn't come from either A or B.

I actually find this strong monotonicity property (impossible to satisfy in a fixed-size committee) much more pleasing/important than quota. If forced to deal with a fixed committee size, I'll accept violations of quota. I much prefer Webster to Hamilton, but Hamilton+Webster (choosing the house size so both methods agree) results in a stronger population monotonicity property than either one alone.

Lime

@toby-pereira said in Variable house sizes:

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.

BTW, we should probably distinguish different-sized cores—the possibility of an empty Hare core is unknown, but Droop cores can definitely be empty (as the Condorcet paradox proves). What I'm interested in is satisfying the Hare core with high probability and satisfying the anti-Droop core guarantee with certainty; i.e. the share of voters who would prefer some other committee is less than 1 / (seats - 1).