What level of PR do different systems get?

Disproportionality can be divided into two components:

Random fluctuations from exact proportionality
Bias (by size).

STV & Largest-Remainder have unnecessarily high #1.

Other than that, the divisor (highest average) methods differ unimportantly in #1.

#1 is relatively unimportant because it’s just a tiny (for divisor methods) fluctuation bringing no unfairness.

#2, bias, is what matters.

Sainte-Lague is the allocation rule with virtually zero bias.

It’s bias is very, very nearly zero. …a very, very tiny bias toward the large.

Is you want absolutely zero bias, then use Biad-Free.

Where Sainte-Lague rounds off to the nearest integer, with (a+b)/2, = a + .5 as its rounding point, Bias-Free’s rounding point is determined as follows:

Divide b^b by a^a. Then divide the result by e.

…where e is the base of natural logarithms, = about 2.718…

That rounding point is called an “identical-mean

Bias-Free’s rounding point is called an “Identric-mean. Autocorrect changed “ identric” to “identical”.

That mean has been much discussed in mathematics, but, according to a journal-paper, my EM post was the first proposal to use it for the unbiased divisor-method rounding point.

Toby Pereira

@michaelossipoff I want to return to this "bias-free" method actually.

We briefly had a discussion on EM. Here is my post and here is your reply.

But in summary: I argued that Sainte-Laguë is actually unbiased, whereas you argued that it is slightly biased towards large parties. However, I'm not sure you fully got the point I was making, and I think we are perhaps using different definitions of bias.

As I understand it, your "bias-free" method works to ensure that neither small nor large parties are over-represented as a whole.

Whereas my argument is that Sainte-Laguë still gives objectively the most proportional result. However, due to the statistical make-up of party votes under certain assumptions, large parties as a whole may still be over-represented.

My argument is also that large parties do not form a coalition and neither do small parties, so a bias towards one or the other because of statistical voting patterns should not be countered. I'll use my previous examples:

4 to elect:

A: 38
B: 38
C : 12
D : 12

Under Sainte-Laguë, A and B get two seats each. One might see this as a large-party bias and say that a more balanced result is to take a seat from one of the large parties and give it to a small party. And the other example:

A: 37
B: 37
C : 13
D : 13

In this example under Sainte-Laguë, all four parties get one seat each. One might see this as a small-party bias and argue that a more balanced result would be to take a seat from one of the small parties and give it to a large party.

However, the problem with the reasoning in both these cases is that the parties are not in coalition. In the first example, the votes were just over the threshold so that the large parties won all the seats. They are both over-represented, yes, but giving a seat to a small party would also be over-representation. The two large parties have nothing to do with each other, so talking of a large-party bias is not the right way to look at it. However it is done, two parties will be over-represented. Whether they are both large, both small or one of each is beside the point.

@toby-pereira said

[quote]

But in summary: I argued that Sainte-Laguë is actually unbiased, whereas you argued that it is slightly biased towards large parties. However, I'm not sure you fully got the point I was making, and I think we are perhaps using different definitions of bias.

[/quote]

Here’s what I mean by bias:

The divisor methods divide each party’s votes by the same number, also called the quota.

The 1st quota used is the Hare quota.

So, divide to determine how many quotas each party has.

Then round off each party’s quotas to the nearest integer, for that party’s seat-allocation.

By trial & error or systematic-procedure, find the quota that results in the desired house-size.

Of course if a party has between a quotas & b quota, then its allocation by that quota rounds to a or b.

So, a & b are any two consecutive integers representing whole numbers of quotas.

The region between a & b, I’ll refer to as an “ interval”. Of course each interval has a width, (a-b), of 1.

All this in order to define “interval.

So here’s what I mean by unbias:

Suppose two parties’ quota-amounts are in two very distantly-separated intervals.

e.g. Say one party’s quotas is in the 0 to 1 interval & the other party’s quotas is in the 100 to 101 interval.

The allocation-rule is unbiased if, in every interval, the average seats per quota (including the fractional part of a quota), averaged over all possible non-integer numbers of quotas in that interval, is the same for all intervals.

e.g. including the 0 to 1 interval & the 100 to 101 interval.

Sainte-Lague, roundly to to the nearest integer, has a + .5, = (a+b)/2, as it’s rounding point.. the arithmetical mean.

Bias Fred’s rounding-point is an “identric mean”, determining as follows:

Divide a^a by b^b. Then divide the result by e.

…where e is the base of the natural logarithms, = about 2.718…

Bias-free is unbiased as I defined that term. With Sainte-Lague, the average seats per vote in a higher interval is slightly greater than that average in a lower interval.

That’s bias vs unbias.

There’s always some random fluctuation from exact proportionality. That isn’t bias. The slight random fluctuation from exact proportionality, universally present in all methods, isn’t problematic or objectionable like the CONSISTSNT & SYSTEMATIC unequal treatment by a biased allocation-rule.

Now, just speaking of minimization of that random departure from proportionality, yes Sainte-Lague (Webster) & “Equal-Proportions” (Huntington-Hill) both have claim of minimizing that (harmless & slight) fluctuation.

SL minimizes it I’m terms of difference between 2 parties’ seats/votes numbers…while Huntington-Hill minimizes it in terms of ratio
of those numbers.

To me, of those two, ratio seems more appropriate & meaningful, because s/v is itself a ratio.

Additionally, Huntington gave other good reasons why ratio seems more meaningful.

…& all that is irrelevant, because that slight random fluctuation is harmless & unimportant in comparison to bias, a consistent & systematic disfavoring of small or large parties.

As I understand it, your "bias-free" method works to ensure that neither small nor large parties are over-represented as a whole.

Whereas my argument is that Sainte-Laguë still gives objectively the most proportional result. However, due to the statistical make-up of party votes under certain assumptions, large parties as a whole may still be over-represented.

My argument is also that large parties do not form a coalition and neither do small parties, so a bias towards one or the other because of statistical voting patterns should not be countered. I'll use my previous examples:

4 to elect:

A: 38
B: 38
C : 12
D : 12

Under Sainte-Laguë, A and B get two seats each. One might see this as a large-party bias and say that a more balanced result is to take a seat from one of the large parties and give it to a small party. And the other example:

A: 37
B: 37
C : 13
D : 13

In this example under Sainte-Laguë, all four parties get one seat each. One might see this as a small-party bias and argue that a more balanced result would be to take a seat from one of the small parties and give it to a large party.

However, the problem with the reasoning in both these cases is that the parties are not in coalition. In the first example, the votes were just over the threshold so that the large parties won all the seats. They are both over-represented, yes, but giving a seat to a small party would also be over-representation. The two large parties have nothing to do with each other, so talking of a large-party bias is not the right way to look at it. However it is done, two parties will be over-represented. Whether they are both large, both small or one of each is beside the point.

@Toby-Pereira

I emphasize that SL is so nearly unbiased, & its rounding to the nearest integer is so natural & obvious, that SL is an excellent choice, & the best proposal..,unless absolute unbias is desired.

Huntington-Hill is how we apportion the House of Representatives here. …chosen of course because it favors small states.

Huntington-Hill is twice as biased as SL, in the opposite direction.

The Greens. I’m told, use Huntington-Hill to elect their deliberative-body. SL would be only half as biased.

@Toby-Pereira

I meant divide b^b by a^a.

…& earlier I meant that b-a = 1.

Toby Pereira

@michaelossipoff OK, I'll take it on trust for now that your method has the properties you say. However, there are of course different potential measures of bias.

For example, we can see the amount of representation a voter gets as the number of candidates elected from the party they vote for divided by the number of voters of that party. Sainte-Laguë uniquely minimises the variation of this representation, so can be seen as the most accurate proportional party-list method, or indeed, unbiased. Moving the divisors in the D'Hondt direction creates a large-party bias, whereas moving in the other direction creates a small-party bias.

@toby-pereira said in What level of PR do different systems get?:

@michaelossipoff OK, I'll take it on trust for now that your method has the properties you say. However, there are of course different potential measures of bias.
[\quote]

The alternative definitions aren’t bias.

e.g. see below:

[quote]

For example, we can see the amount of representation a voter gets as the number of candidates elected from the party they vote for divided by the number of voters of that party.
[\quote]

Of course seats per vote (s/v) is what should be equal, & that’s what this is all about.

With Sainte-Lague the average s/v, averaged over a higher interval (like 100 to 101) is greater than the average over a lower interval ( like 0 to 1).

[quote]
Sainte-Laguë uniquely minimises the variation of this representation
[/quote]

No. That random departure from proportionality, that RANDOM variation of s/v, isn’t bias.

Bias is defined as SYSTEMATIC CONSISTENT favoring of one thing or set (party or set of parties) over another. The random s/v fluctuation itself isn’t bias, by any definition.

SL favors large parties over small ones. That’s bias, by any definition.

As I said, fluctuation of the parties’ s/v is inevitable. If it’s random, if it doesn’t consistently systematically favor on set over another (e.g. large parties over small parties, then it’s not bias.

As I said, that fluctuation is inevitable, & it’s completely unimportant compared to a SYSTEMATIC CONSISTENT disfavoring is small parties.

As I said, both SL & Huntington-Hill (HH) can claim to minimize the fluctuation (while consistently giving larger or smaller parties lower s/v).

SL minimizes the fluctuation in terms of difference in s/v. HH minimizes the fluctuation in terms of ratio of s/v.

But both are biased. SL gives higher s/v to large parties. HH give higher s/v to small parties.

Bias-Free (BF) is completely, absolutely unbiased. It doesn’t favor either larger or smaller
parties.

But, as I said:

SL is only half as biased ad HH.

SL’s bias is so tiny, so negligibly tiny, that it’s effectively unbiased. …unbiased for all practical purposes.

e.g. :

150 seats.

17 small parties, each with 3% of the vote.

…totalling 51% of the vote.

One big party with 49% of the vote.

The small parties together get a majority of the seats, & can form a majority coalition & government.

…& of course SL, with its rounding point of (a+b)/2, = a + .5, is the natural obvious intuitive
divisor-method.

Toby Pereira

@michaelossipoff I'm not talking about random fluctuation. I'm talking about Sainte-Laguë being the unique method that minimises the representation variance. For any given election the result from "Bias Free" will have an equal to or greater variation than Sainte-Laguë. So before we get to bias, I would say that Sainte-Laguë is objectively more proportional. So for "Bias Free" to indeed be less biased, we'd be saying that proportionality and bias can be varied independently, to some extent at least.

But as for bias, take the following example with 2 to elect:

Party A: 75
Party B: 25

Sainte-Laguë gives a tie between 2-0 (both to party A) and 1-1. "Bias Free" systematically favours smaller parties in such a tie case, awarding the parties 1 seat each. This is, as I would see it, bias.

The point is that in any given election, Sainte-Laguë minimises the variation and gives the most proportional result. It might still be that if you look at a large number of elections, large parties fare better on average, but this does rely on certain assumptions about the voting distribution, as I've said previously, rather than being intrinsic to the method. Plus one large party being favoured doesn't mean this should be balanced elsewhere as the large parties are separate entities and not in coalition with each other.

@toby-pereira said in What level of PR do different systems get?:

@michaelossipoff

[quote]
I'm not talking about random fluctuation.
[\quote]

Talk about it or not, but, with BF, the fluctuation is random only. No bias.

[quote]
I'm talking about Sainte-Laguë being the unique method that minimises the representation variance.
[\quote]

Fine, then you aren’t talking about bias.

I explained the difference.

…&, as I also explained 2 or 3 times, SL & HH both have their claim to minimizing that variation.

I pointed out that HH’s measure based on ratio makes more sense because s/v is, itself, a ratio.

Huntington pointed out that the ratio measure of the distance between two s/v values is consistent with a lot of measures, & is therefore less arbitrary. For his argument, I’ll have to refer you to Huntington’s article.

Google “Huntington, Huntington-Hill vs Webster.”

[quote]
For any given election the result from "Bias Free" will have an equal to or greater variation than Sainte-Laguë.
[/quote]

…at least by SL’s more questionable (compared to HH) measure of distance between s/v values.

In any case, you’re still confusing bias with variation, something that I’ve explained several times.

[quote]
So before we get to bias, I would say that Sainte-Laguë is objectively more proportional.
[/quote]

Less s/v variation by SL’s more questionable difference measure of distance between two s/v values. But as you agreed above, that isn’t bias, & isn’t relevant to the matter of bias.

So for "Bias Free" to indeed be less biased, we'd be saying that proportionality and bias can be varied independently, to some extent at least.

I’ve been trying to explain to you that they’re different topics.

[quote]

But as for bias, take the following example with 2 to elect:

Party A: 75
Party B: 25

Sainte-Laguë gives a tie between 2-0 (both to party A) and 1-1.
[\quote]

SL doesn’t give an answer, in your special & atypical example.

BF gives each party one seat.

[quote]
"Bias Free" systematically favours smaller parties in such a tie case…
[\quote]

Incorrect. One example isn’t a basis for saying “systematically”. “Systematically” refers to something that happens consistently in many
instances.

So no, BF doesn’t allocate biasedly in that example.

What is it about that result in that example that makes you think it’s biased?

By the definition of bias, it’s meaningless to say that a single result in a single example is “biased”. I’ve been trying to explain to you what bias means, but evidently I haven’t been getting through.

Bias means what I said. BF is entirely unbiased for the reason that I said. SL is biased in favor of large parties for the reason that I said.

[quote]
…awarding the parties 1 seat each. This is, as I would see it, bias.
[/quote]

How so? See above.

[quote]
The point is that in any given election, Sainte-Laguë minimises the variation…
[/quote]

Only by its questionable measure of distance between two s/v values.

[quote]*
It might still be that if you look at a large number of elections, large parties fare better on average, but this does rely on certain assumptions about the voting distribution, as I've said previously, rather than being intrinsic to the method.
[\quote]

What assumptions? I averaged over all the values that a party’s number of quotas could have in a particular interval.

[quote]
Plus one large party being favoured doesn't mean this should be balanced elsewhere as the large parties are separate entities and not in coalition with each other.
[\quote]

Not sure what you mean by that. I told you what bias means.

In your special & atypical example, as I said, SL doesn’t have a result. BF & HH give each party one seat.

In actual SL elections, the 1st rounding-point, in most implementations without a higher threshold, raise that 1st rounding-point from.5 to .7, in order to discourage or prevent splitting-strategy.

BF & HH, in PR, should do the same, for the same reason.

Then, in your example, SL, BF, & HH give one party 0, & give the other party 2.

With single-winner methods, disagreements are often a matter of opinion: “Which problem is more undesirable?”

That isn’t the case in this instance. …& usually isn’t, with PR.

Toby, if you just keep re-asserting your assumptions, instead of even considering what someone is telling you, then you thereby cheat yourself out of the opportunity to find out about the subject.

BTW, a 2-member district is unusual, though there are sometimes such small districts, even in party-list PR.

Toby Pereira

@michaelossipoff OK, I think we can still get to the bottom of this.

Firstly, I see Huntington-Hill as a red herring. Its measure is somewhat arbitrary compared to Sainte-Laguë, despite looking superficially sensible. Huntington-Hill looks at number of seats the parties will get and then looks at the geometric mean, whereas Sainte-Laguë looks at the arithmetic mean. Superficially both might make sense. But with Sainte-Laguë you can look at the representation per voter (rather than seats per party), which is what makes more sense (as it is about giving voters, not parties, representation), and it minimises the variance of that. It is also more generally accepted I believe that Sainte-Laguë gives the closest to exact proportionality possible (I can find sources if need be).

The example I gave was just one example, but it was indicative of what the "Bias Free" method (or indeed any non-Sainte-Laguë method) can do. In any exact tie case under Sainte-Laguë (not just the example I gave), "Bias Free" will award in favour of the smaller party. I see that as systematic bias.

As for what I was talking about with assumptions about distributions, according to Warren Smith's page here:

As our starting point, we shall assume the state populations are independent identically distributed exponential random variables

Is this not the assumption you are making about the distribution?

But the point is that even if under real life conditions, Sainte-Laguë does award larger parties more seats than they are entitled to on average, it is still doing so by using the objectively most proportional method. The bias is not in the method, but what you get from the distribution.

I don't think it makes sense to use a less proportional method to balance this out. If party A and party B are big parties, with C and D small parties, is it better to have one of each over-represented rather than both small or both large represented? I would argue not, because the parties are all separate from each other. It's not small v large, but A v B v C v D.

Lime

@toby-pereira said in What level of PR do different systems get?:

It is also more generally accepted I believe that Sainte-Laguë gives the closest to exact proportionality possible (I can find sources if need be).

There's a lot of literature on this, but there's just no single "best" PR system. They all optimize different measures of misrepresentation. Huntington-Hill does better on some metrics and worse on others.

Any sensible rule (Dean, HH, Webster, identric) will give almost identical results. The only real discussions are:

Jefferson (less strategy) vs. everything else (more proportional), and
Trying "something weird" like varying house sizes (I haven't seen enough research on this, TBH) or fractional votes. Variable house sizes seem like the "least weird" thing you could do here.

I'd suggest picking a house size where the HH+Webster apportionments agree, or using the Webster technique and ignoring the house size constraint (pick a divisor and then round, without updating the divisor). This "does the impossible" (quota and strong monotonicity: every party's number of seats depends only on their vote count, not on the vote counts of other parties; Balinsky-Young theorem assumes a fixed house size).

Toby Pereira

@lime It's not that I'm saying there is a "best" method, but that Sainte-Laguë/Webster is most accurate in terms of pure PR. Other methods might still have certain advantages.

Another thing about Huntington-Hill is that it breaks if a party with any votes has zero seats. It's clearly not the "right" measure.

@toby-pereira said in What level of PR do different systems get?:

[quote]

@michaelossipoff OK, I think we can still get to the bottom of this.
[\quote]

No, it’s now evident that that was a hopeless cause.

Firstly, I see Huntington-Hill as a red herring. Its measure is somewhat arbitrary compared to Sainte-Laguë, despite looking superficially sensible.

It’s too bad that you weren’t there to set PhD mathematician Huntington straight, & explain to him where he was wrong when he thought he was minimizing variation in s/v !

[quote]
Huntington-Hill looks at number of seats the parties will get…
[\quote]

Hello? All allocation-rules are about how many seats a party (or state) will get.

Most of the divisor-methods use a mean as their rounding-point. Arithmetical, geometric, identric, harmonic.

[quote]
and then looks at the geometric mean, whereas Sainte-Laguë looks at the arithmetic mean.

Superficially both might make sense. But with Sainte-Laguë you can look at the representation per voter (rather than seats per party)
[\quote]

If you’d actually looked at Huntington’s article, as I suggested, you’d find that he was talking about minimizing variation in s/v. He wasn’t trying to make seats per state equal

which is what makes more sense (as it is about giving voters, not parties, representation), and it minimises the variance of that. It is also more generally accepted I believe that Sainte-Laguë gives the closest to exact proportionality possible (I can find sources if need be).

For at least the 4th time, HH minimizes variation measured in ratio of s/v values, while SL minimizes variation measured in difference of s/v values.

I don’t know where you get your latest notion, but both minimizations are about s/v.

[quote]

The example I gave was just one example, but it was indicative of what the "Bias Free" method (or indeed any non-Sainte-Laguë method) can do. In any exact tie case under Sainte-Laguë (not just the example I gave), "Bias Free" will award in favour of the smaller party. I see that as systematic bias.
[\quote]

Then you see it wrong.

From an assumption that SL is unbiased, then that must mean that to differ from SL is to be biased

[quote]
As for what I was talking about with assumptions about distributions, according to Warren Smith's page here:
[\quote]

A morass of gibberish. Warren expounded & theorized about the distribution of the populations of the U.S. states, & felt that it was necessary to base an allocation-rule on such a theory.

I’ve never heard of anyone agreeing with him on that.

At least two academic journal-paper authors agree about BF. I posted reference to two such papers in my September’23 EM posts. One of the authors referred to BF as the Ossipoff-Agnew method.

(Agnew independently proposed the use of the identric mean as the rounding-point of an unbiased divisor-method a few years after I did.)

At the time that I proposed BF in 2006, I hadn’t heard of the identric mean. I determined the average s/v over all the possible numbers of quotas that a party could have in an a-to-b interval, with a given rounding-point. …& solved for the rounding-point that would make that average equal in the various intervals.

I was surprised to hear that that mean had a name, & had been discussed a lot in connection with other matters.

As our starting point, we shall assume the state populations are independent identically distributed exponential random variables

Is this not the assumption you are making about the distribution?

No, that’s Warren.

[quote]
But the point is that even if under real life conditions, Sainte-Laguë does award larger parties more seats than they are entitled to on average, it is still doing so by using the objectively most proportional method.
[\quote]

For about the 7th time:

SL & HH differ in regards to how they measure variation in s/v. Huntington gives good reasons why his measure (ratio) is more meaningful. I referred you to his article. His use of ratio seems more meaningful to me, because s/v, itself is a ratio.
Variation in s/v, & bias are two separate matters. Their minimizations are separate goals.
s/v variation is inevitable. It’s small & unproblematic for divisor-methods. What’s problematic, because it’s an unfair SYSTEMATIC & CONSISTENT inequality, is bias.

Yes, HH goes wrong when it gives a seat to a party with one vote, but Huntington’s argument that HH best minimizes variation in s/v is more convincing than the claim that difference is the meaningful measure.

[quote]
The bias is not in the method, but what you get from the distribution.
[/quote]

Nearly all allocation rules have their intrinsic bias. It’s a property of an allocation rule. Perhaps you’ve been listening to Warren. You make far too many careless assertions.

[quote]

I don't think it makes sense to use a less proportional method to balance this out.
[/quote]

Suit yourself, & feel free to try to convince people that small random variation is more important than systematic consistent disfavoring.

[quote]
If party A and party B are big parties, with C and D small parties, is it better to have one of each over-represented rather than both small or both large represented? I would argue not, because the parties are all separate from each other. It's not small v large, but A v B v C v D.
[/quote]

I have no idea what that means, but it has now become particularly evident that this conversation isn’t accomplishing any purpose.

It obviously isn’t helping you. I often reply to erroneous posts so that they won’t deceive others, but this could obviously go on forever.

I’m done with this conversation.

Toby Pereira

@michaelossipoff As I put above, Huntington-Hill breaks if a party (that has at least one vote) gets no seats. So if there are the same number of seats as parties, then the only possible result is that every party wins exactly one seat. If there are more parties than seats, then every result is infinitely bad. It's not a sensible measure. It doesn't pass the sniff test. Minimising variance using ratios fails.

This isn't to say it can't have any uses, but as an objective measure of the most proportional result, it is not in the running. Similarly D'Hondt doesn't maximise objective proportionality, but it can be a useful method. By the way, you could have given me a link to Huntington's paper rather than telling me to Google something, which didn't come up with it in the search results. I have, however, found it here. This was why I used the Wikipedia page, which just talked in terms of parties. So apologies for any confusion, but ultimately it makes no difference.

There is in fact a reason why minimising the variance using arithmetic differences is the uniquely non-arbitrary measure. And that is if you add up the representation of each voter under any result, you get the same total. If you multiply them together, you don't always get the same product. If you add up the reciprocals, you don't get the same total etc. So using arithmetic means (as opposed to geometric, harmonic or anything else) is uniquely non-arbitrary. You have to use the variance that mathematically fits the data. Geometric does not. Huntington-Hill therefore does not.

So, given that Sainte-Laguë is the uniquely non-arbitrary way of maximising proportionality (by the sensible measure), any method that systematically favours small parties or large parties relative to it is biased.

Toby Pereira

I know Michael has said he's done with this conversation, but I just want to add a couple of things. Firstly in my last post I should have linked the first and third paragraphs. Huntington-Hill breaks when a party gets zero seats. Also it uses the wrong variance (geometric instead of arithmetic). These two are connected. It breaks precisely because it uses the wrong variance. For geometric to make sense, the product of representation per voter should be the same regardless of result and zero should be impossible. I was probably a bit dense in understanding what Michael was saying about geometric variance earlier, but it's all understood now, and taken into account in any conclusions.

Regarding references (as I mentioned them), in "Fair Representation: meeting the ideal of one man", Balinski and Peyton Young conclude that Webster (equivalently Sainte-Laguë) is "the unique unbiased divisor method". Also, though I can't seem to access the paper now, I believe this was also the conclusion of Kenneth Benoit in "Which Electoral Formula Is the Most Proportional? A New Look with New Evidence".

Finally, regarding "Bias Free":

@michaelossipoff said in What level of PR do different systems get?:

With Sainte-Lague the average s/v, averaged over a higher interval (like 100 to 101) is greater than the average over a lower interval ( like 0 to 1).

I'd like to see an example of this and where it is that Sainte-Laguë supposedly does go wrong. Because everything really hinges on this. It's the only unturned stone.

Toby Pereira

OK, so about what I was talking about regarding assumptions about the distribution:

@michaelossipoff said in What level of PR do different systems get?:

What assumptions? I averaged over all the values that a party’s number of quotas could have in a particular interval.

This does itself assume some sort of uniform distribution. (As said up the thread, Warren Smith made a different assumption. Edit - If I understand correctly a uniform distribution would mean that e.g. for every party with 100 supporters, there would be 100 with 1 supporter. Which is presumably why Warren thought an exponential distribution made more sense. Not that this makes a difference to any conclusions drawn.) So the argument seems to be that under such an assumption, Sainte-Laguë will award larger parties more seats than they should get on average and smaller parties fewer seats.

But one can ask questions about the most realistic distribution and also question whether it is worth getting less proportional results (introducing another form of bias), to eliminate this type of bias. (See my examples that started my discussion on this upthread.)

In fact, if we imagine a scenario where voting behaviour is relatively fixed across time (or indeed populations of states relatively fixed in an apportionment scenario), then every election could potentially get the same result and so the same parties (or specifically voters of those parties) will always end up on the wrong side of a rounding. It is possible to find voting distributions that make all deterministic methods appear biased. It might be more complex than simply large v small, but that doesn't really matter.

And while that might seem unrealistic, we can see the case of very small parties that never get enough votes to win a seat. A particular party might be due about 0.1 seats at every election but never win one under a particular method. Is that bias? Or is it simply that the most proportional method won't award them a seat at each election? (Michael's method involves a 0^0 in the 0 to 1 seat range, so appears to break, so I'm not sure how it is supposed to handle this case.)

But arguably it is not simply the voting method itself that is biased (after all Sainte-Laguë simply returns the most proportional result), but the voting method in combination with specific voting patterns.

The only way to get rid of bias under any assumptions about voting distributions would be to use a non-deterministic method. That way even the party that should be getting 0.1 seats per election will get someone elected once every 10 years or so.

So in conclusion, Michael's "Bias-Free" method is a theoretical method that under certain voting assumptions will balance out small/large party discrepancies. But it does this at the cost of proportionality (introducing another form of bias). And unless the voting distribution happens to exactly match the theory, it will contain the biases he was trying to avoid. A non-deterministic method can avoid both forms of bias, regardless of distribution.

@lime said in What level of PR do different systems get?:

@toby-pereira said in What level of PR do different systems get?:

It is also more generally accepted I believe that Sainte-Laguë gives the closest to exact proportionality possible (I can find sources if need be).

There's a lot of literature on this, but there's just no single "best" PR system. They all optimize different measures of misrepresentation. Huntington-Hill does better on some metrics and worse on others.

Any sensible rule (Dean, HH, Webster, identric) will give almost identical results. The only real discussions are:

Jefferson (less strategy)

Jefferson/d’Hondt (DH) calls for aggregation-strategy, because it’s relatively strongly large-biased.

d’Hondt’s guarantee is that if a party has a majority of the votes, it will have a majority of the seats. But how often does one party get a majority of the votes.

Sainte-Lague is, by far, the least biased divisor method that is, or ever has been, in use.

That, & its uniquely obvious & natural rounding-point, make it the natural choice for unbias.

…even though BF is the unbiased divisor-method.

I like BF & SL because bias is the important & unfair kind of disproportionality.

But yes, the difference between SL & DH isn’t really great enough to be very important, is it.

DH is the most widely-used allocation-rule. I don’t know why, unless there’s been a desire to discourage small parties.

It has been suggested here that a voter should be able to vote for 2 parties. If even one ballot’s 1st choice party doesn’t get a seat, then the count is repeated, using the 2nd-choice party of every ballot whose 1st party didn’t get a seat.

That’s no significant extra count-work, because party list allocation counts are so undemanding.

Of course a lot of people would agree that would be better. I’d prefer it. But it would be rejected as a new-idea, & would probably have no chance of acceptance.

So, one argument in support of DH could be that voting for a small party risks wasting your vote. So one should instead vote for a party that one likes, & is sure to win seats ( if there is one).

Well, since the whole point of unbias is fairness to smaller parties, & if it’s wiser to not vote for the seat-questionably small ones, but to instead add to aggregation of a larger one, then might that make DH’s one-party majority guarantee more important? Maybe, but isn’t it rare for one party to get a majority of the votes?

So I don’t know. I still prefer unbias.

But, the difference between SL & DH is slight enough that I’d propose whichever one is more likely to be accepted.

I like Finland’s exemplary-simple open-list system, & maybe the proposal should also copy their allocation-rule too: DH.

I’ve only heard of 4 countries that use SL. The near-universal use of DH might make it the best allocation rule for a list-PR proposal.

vs. everything else (more proportional), and

Trying "something weird" like varying house sizes (I haven't seen enough research on this, TBH) or fractional votes. Variable house sizes seem like the "least weird" thing you could do here.

I'd suggest picking a house size where the HH+Webster apportionments agree, or using the Webster technique and ignoring the house size constraint (pick a divisor and then round, without updating the divisor). This "does the impossible" (quota and strong monotonicity: every party's number of seats depends only on their vote count, not on the vote counts of other parties; Balinsky-Young theorem assumes a fixed house size).

@toby-pereira said in What level of PR do different systems get?:

[quote]
I know Michael has said he's done with this conversation, but I just want to add a couple of things.
[/quote]

Sorry about the delay in replying. First I wanted to wait till after the poll, & then there were posts at EM that I wanted to answer right away.

[quote]
Firstly in my last post I should have linked the first and third paragraphs. Huntington-Hill breaks when a party gets zero seats.
[/quote]

Not for U.S. apportionment, where the Constitution requires each state to get at least 1 seat.

But yes in PR no one would agree with it giving every party a seat regardless of their votes.

Not that that’s a problem in PR, because SL moves the 1st rounding-point from .5 to.7, to avoid & discourage splitting-strategy. ….the 1st rounding-point that should be used BF & HH, for the same reason.

[quote]

Also it uses the wrong variance (geometric instead of arithmetic).
[\quote]

You say you found Huntington’s article or paper. Maybe you should read it. What, you read it? Then re-read Huntington’s argument for ratio vs difference, because you didn’t understand it.

You know, that must be the most annoying form of Internet-abuse in forums: Continuing to repeat a refuted argument without answering the criticisms of it. Very common. Most forums specifically forbid it.

[quote]
These two are connected. It breaks precisely because it uses the wrong variance. For geometric to make sense, the product [He means ratio or division-result] of representation per voter should be the same regardless of result and zero should be impossible.
[/quote]

Not quite sure what you mean by that, but I’ll assume that you mean that HH’s automatic 1st seat for every party regardless of its votes should be impossible?

Shall we call that “Pereira’s Law”? The fact that HH gives an unacceptable result when it gives a 1st seat to a by party with 1 vote, doesn’t make ratio the wrong measure for other seats. …unless you have an argument instead of just an assertion.

[quote]

Regarding references (as I mentioned them), in "Fair Representation: meeting the ideal of one man", Balinski and Peyton Young conclude that Webster (equivalently Sainte-Laguë) is "the unique unbiased divisor method".
[\quote]

For one thing I doubt that they said that. It was known, at their time of writing, that SL has some (but very little) bias.

Maybe they said that SL is the only divisor-method ever having been used that is nearly unbiased. …or at least that it was the least biased of all divisor-methods that have been used.

[quote]
Also, though I can't seem to access the paper now, I believe this was also the conclusion of Kenneth Benoit in "Which Electoral Formula Is the Most Proportional? A New Look with New Evidence".
[\quote]

Wrong guess. It’s unlikely that he’d say that. See above.

[quote]

Finally, regarding "Bias Free":

@michaelossipoff said in What level of PR do different systems get?:

With Sainte-Lague the average s/v, averaged over a higher interval (like 100 to 101) is greater than the average over a lower interval ( like 0 to 1).

I'd like to see an example of this and where it is that Sainte-Laguë supposedly does go wrong. Because everything really hinges on this. It's the only unturned stone.

I was going to answer that until, in a later message, you deny the accepted meaning of bias, & assert your own personal definition.

…to claim that SL is the completely unbiased divisor method.

Bias means consistent or systematic favoring or disfavoring of a particular thing, person, class of things, or class of persons.

Specifically, for proportional-representation, it means systematically & consistently, overall, giving more or s/v to larger parties.

Period. That’s what bias means. You can make up your own personal definition if you want to.

Anyway, when you tried to impose your own personal definition of “bias”, that meant that it would be a waste of time to explain why SL is biased by the actual meaning of bias. …a waste of time because you don’t even accept the universally-accepted definition. …instead making up a bizarre definition of your own.

@toby-pereira said in What level of PR do different systems get?:

OK, so about what I was talking about regarding assumptions about the distribution:

@michaelossipoff said in What level of PR do different systems get?:

What assumptions? I averaged over all the values that a party’s number of quotas could have in a particular interval.

[quote]
This does itself assume some sort of uniform distribution.
[\quote]

No, it doesn’t. What your statement shows is that you’ve been reading Warren.

I’ve never heard of anyone agreeing with that peculiar notion of Warren’s.

I averaged s/v over an interval (between two consecutive integer numbers of quotas).

I assume nothing about the distribution. I speak of the average s/v, over all possible numbers of quotas in an interval (as defined above).

The bias that I speak of is differing s/v averages over low & high intervals.

Nothing whatsoever to do with a distribution.

Warren is talking about a completely different issue of his own.

[quote]
. (As said up the thread, Warren Smith made a different assumption.
[/quote]

You got that right.

He, & you when you quote him, seem to be talking about a time-average or something. I referred instead to an average over numbers of quotas.

[quote]

And while that might seem unrealistic, we can see the case of very small parties that never get enough votes to win a seat. A particular party might be due about 0.1 seats at every election but never win one under a particular method. Is that bias?
[/quote]

Sure, but it’s not the kind that has always been meant when speaking of bias. Fractional-quota small parties were traditionally never really wanted in PR countries.

I’m willing to settle for traditional list-PR, with its exclusion of fractional-quota parties. Must we have every reform at once? How about we settle for traditional PR first.

Later we could propose letting voters choose 2 parties to vote for, I. either the closed-list or open-list way.

First allocate by the #1 parties.

If 1 or more parties are in-seated, then start over, doing the allocation again, but, for each ballot for an un-seated party, use, this time, that ballot’s #2 party.

Of course I’d prefer that, & that’s how it would be done if it were up to me.

But you just don’t ask for a “new invention” in a 1st-reform proposal.

Netherlands uses DH, at-large, for a 150-seat parliament.

No added-threshold.

Any party with at least one quota gets a seat by that quota.

(Of course as with any divisor-system, a quota is found (by a systematic-procedure) that results, after the rounding—always down in DH—in the desired house-size of 150 seats.)

That means that typically a party with 6.67% of the votes gets a seat. That isn’t unfair to small parties.

[quote]
Or is it simply that the most proportional method won't award them a seat at each election?
[/quote]

Yes. Even the most best divisor method, by whatever measure, won’t give a seat to a party that doesn’t have a big enough fraction of a quota to qualify by that method’s rules.

(In DH, that fraction is unity. In SL as officially designated, that fraction is .7. …as it should likewise be in BF & HH, to discourage splitting-strategy.)

I don’t want any additional threshold.

I don’t want districts. They’re arbitrary, contentious & often unfair, often intentionally, by gerrymandering.

Any population group with a quota of people can elect someone local if they want to, even in an at-large system, as is their right. …just as it’s a voter’s right to help elect someone not local.

Districts are an unfair mess, & that’s even without mentioning how they impose an effective threshold, depending on how small they are.

[quote]
(Michael's method involves a 0^0 in the 0 to 1 seat range, so appears to break, so I'm not sure how it is supposed to handle this case.)
[\quote]

No, it still works, though the usual formula doesn’t work. There’s a way to do the integral from that 0^0 point. It’s an exception that has to be separately integrated as a separate problem.

The answer to that problem is a rounding point equal to 1/e.

[quote]
But arguably it is not simply the voting method itself that is biased…
[\quote]

It’s the SL allocation-rule that’s biased…ever so slightly.

[quote]
(after all Sainte-Laguë simply returns the most proportional result)…
[/quote]

…according to the difference measure of s/v-variation. …which doesn’t make as much sense as ratio. But of course your preference is entirely your business. …but if you’re going to say that Professor Huntington was wrong, you’ll need more than an assertion. You’ll need to say where you think he was wrong. Help that mathematics professor out by explaining where he made his error.

…& as I‘be explained to you many times, if by “most proportional” you mean “ having least maximum variation in s/v, that’s an entirely different matter from bias, whose meaning I’ve already told you several times.

[quote]

The only way to get rid of bias under any assumptions…
[\quote]

BF has no bias.

[quote]

So in conclusion, Michael's "Bias-Free" method is a theoretical method that under certain voting assumptions
[\quote]

I told you what bias means & how I define its measure. Feel free to have your own original definition.

[quote]
will balance out small/large party discrepancies. But it does this at the cost of proportionality (introducing another form of bias)
[\quote]

Maybe you’re catching on. Unbias & variation m-minimization are two separate goals. Bias is more important than minimizing the small fluctuations, because bias is, by definition, consistent & systematic unfairness to large or small parties.

[quote]
And unless the voting distribution happens to exactly match the theory, it will contain the biases he was trying to avoid.
[\quote]

According to Warren’s unusual definitions, which you’re welcome to embrace.

Alright, I don’t have time for this.

I don’t want to receive more your repetition about matters that have been many times explained to you.

I’m blocking your email, though because this is a website-based forum, it will be necessary to block you at this forum too.

Well, I wanted to check this forum out, & there was talk about doing a lot of polling, which I consider very useful to demonstrate how the methods work.

But the amount of participation in the recent poll wasn’t very promising.

I tried to block Toby, but got a notification that he can’t be blocked, no matter how much bullshit he posts, because he’s an administrator.

So I quit this forum if that’s the only way to block him.

I hereby notify this forum to not send any more message or topic notifications to me.