Proportional representation (PR) is listed first in the title of this article, because several progressive parties offer PR in their platforms. But, in this article, apportionment will be discussed first, because apportionment rules are defined more briefly, and the procedures are more direct. Though the allocation rules used in PR are the same ones that have historically been used to apportion the U.S. House of Representatives, the PR allocation procedure rules are usually expressed as a different (though equivalent) procedure. The apportionment allocation procedure rules are more direct and briefly defined, and that’s why this article discusses apportionment before PR.

1. Apportionment:

The U.S. Constitution specifies that House of Representative seats are to be apportioned to the various states in proportion to the states’ populations. Throughout U.S. history, the states have fiercely battled about the apportionment method, because some allocation rules favor small states, and some favor large states—as regards representation per person. Some are nearly un-biased with regard to state-size. Of courses those would be preferable. But the choice of apportionment method has often, usually or always been a matter of choosing the method that best favors the states with the most power in Congress. For example, it has been reported that that was the motivation for the choice of our current apportionment method.

a1) Equal representation among the states isn’t the important thing.

The important thing is for people to be able to freely vote their preferences among policies, parties and the parties’ candidates–Then the 99%  can freely elect the government that it really wants, regardless of the existence of some inequality among the states. Optimal power of choice for the 99% is more important than strict representation-per-person equality among the states. After all, the public in the various states share basically the same policy-interests.

a2) In any case, for one thing, even the particularly biased and un-proportional apportionment method currently in use for apportioning the House seats among the states isn’t too bad.

Sometimes its bias doesn’t even affect the apportionment result, though sometimes it results in various states getting a seat too many or too few (as judged by various objective measures). In the overall scheme of national things, that isn’t a big problem.

a3) And, for another thing, the House’s small representational bias, due to the current apportionment method, is miniscule in comparison to the _huge_ inequality of representation-per-person, among the states, in the Senate.

That’s because each state, regardless of its population, gets 2 seats in the Senate. That’s due to the “Great [bad] Compromise”. So, unless we get rid of the grossly biased Senate, it would be quite pointless to debate how to fine-tune the House’s equality of representation-per-person among the states. Therefore, there’s really no need to even mention apportionment in a political party’s platform.

a4) But suppose that we _did_ achieve a Constitutional amendment to eliminate the grossly biased Senate, and have a unicameral Congress. And suppose we wanted to pursue the (not really necessary) goal of satisfying the states’ rivalry for representational equality. Then we’d want to improve on the current apportionment method.

But the most elegant solution would then be to throw-out apportionment. Instead of apportioning Congressional seat to the states, and then having each state district its seats, why not just district nationally, without regard to state boundaries? After all, the Congress is a national body, and the districts are local. There is no good reason to involve the states as we now do. Sure, the Constitution requires that. But the Constitution can be improved. For the four above reasons, improving apportionment isn’t important.

b) The Current Apportionment method, and some better ones:

But suppose that we enacted a Constitutional amendment to make Congress unicameral. No Senate. Then Congressional representation per person would immediately become much more equal among the states. Then it might be desired to further fine-tune that representational equality among the states. And suppose that discarding apportionment, and replacing it with national districting without regard to state lines were regarded as too controversial, and that, for that reason, a less biased apportionment method were sought.

First, what is the apportionment method currently in use?

Most sources will say that it’s “The Method of [not really] Equal Proportions”. And most sources will echo the attempted justifications for it that were given by its proponents. Some sources refer to that currently-used allocation method as “Hill’s method”, or “Hill”, after its introducer and first proponent. Sometimes we hear it called “Hill-Huntington”, because Huntington was an advocate of Hill’s method. In this article, I’ll refer to the currently-used apportionment method as “Hill”.

How un-biased is Hill? Not very. Though Hill is not as size-biased as the worst apportionment methods that have been used, it nevertheless is more biased than the best method that has been used.

(Henceforth in this article, “biased” means “biased with respect to state-population”, meaning that the states’ representation per person varies consistently with the size of the state).

The best method that has been used was Webster’s method, introduced and initially proposed by Daniel Webster, in the 19th century. Webster’s method was used for a while, but it was replaced by Hill in 1941.

It’s been said that, at that time, small states tended to be Democrat, and large states tended to be Republican. When Congress voted on whether or not to replace Webster with Hill, the vote was strictly party-line: All Democrats voted for replacing Webster with Hill, and all Republicans voted in opposition to replacing Webster with Hill.

It was brought out in the pre-vote discussions that Hill is favors small states more than Webster does. The debate was about which method is un-biased. …the matter of whether Hill unfairly favors small states, or whether Webster unfairly favors large states. By the party-line nature of the voting, you can see that congress-members were just voting to favor their own party.

Webster’s method is nearly un-biased. Hill’s method favors small states. Webster very slightly favors large states, but only barely. Hill is considerably more biased than Webster. Webster is the least biased of any divisor method that has been used or proposed in Congress. Webster is the most proportionally-accurate of any apportionment method that has been used or proposed in Congress.

Definitions of some apportionment methods:
The divisor methods:

This class of methods includes Jefferson, Webster, Hill, and Bias-Free, all of which will be defined here.

Here’s how the divisor methods work:

1. An initial divisor is chosen. A good way to choose it is by dividing the states’ total combined population by the total number of House seats (435).
2. For each state, divide its population by the divisor. That gives that state’s “quotient”, or its “q”.
3. Each particular divisor method will have its own way of determining that state’s seat allocation, based on its q. Often, after each state has had its seats allocated to it thereby, the total seats won’t add up to 435. If the total seats are _more_ than 435, then raise the divisor a bit, and then repeat 2. And 3., in the above-listed procedure. The total seats will be less than before. If the total seats are more than 435, then lower the divisor a bit, and then repeat 2. And 3., in the above-listed procedure. The total seats will be more than before. Repeat till the total number of seats is 435.

Actually, the procedure is complicated a bit by the fact that the Constitution requires that each state have at least one House Representative, even if the allocation rule said to not give it any. Incidentally, it wasn’t always done that way. During part of apportionment history, there was no requirement for any particular House size. The total number of seats didn’t have to sum to any particular number. You could divide by that initial value of the divisor, and then just give the resulting allocation. That avoided the need for the trial-and-error procedure described above.

In fact, that could make it possible to avoid the un-proportional results of giving each state at least one seat, regardless of whether the allocation rule said to give it any. That’s because we could adjust the divisor so that (by whichever divisor method is in use), the divisor is just low enough to give at least 1 seat to every state.  It might result in a larger House, but it would make the House more population-proportional.

Now, some particular divisor method rules for determining a state’s seat-allocation based on its q:

Some definitions of terms:

Say a state’s q falls between two consecutive integers (whole numbers). We’ll call the lower of those two integers  “a”, and we’ll call the larger of those two integers “b”.  So, for some two consecutive integers a and b, q is between a and b.

R = the round-up point between a and b.

In other words, if a state’s q is equal to or greater than R, then the state receives b seats. If the state’s q is less than R, then the state receives a seats.

Here are the R for some of the main apportionment allocation methods:

Jefferson’s method rounds q down to the lower integer. In other words, R for Jefferson’s method is b”. So just change “a” to “b”

Webster rounds q to the nearest integer. In other words, for Webster, R = a + .5

(Jefferson very heavily favors large states over smaller ones. Webster, as stated above, is nearly un-biased)

Hill determines R by a formula:

R = the square root of (a times b).

In other words, R = the geometric mean of a and b.

Though Hill is much less biased than Jefferson, Hill is much more biased than Webster. Hill favors small states.

Bias-Free is an allocation rule that has never been used. It was introduced by me, posted to the election-methods mailing list in December, 2006. Bias-Free (BF) uses a formula to determine R. Because BF’s formula is a little more complicated-looking than that of Hill, and because BF’s formula isn’t obvious in its motivation or justification, and because Hill and Webster have been used, and BF has never been used—Therefore of course BF wouldn’t be a very enactable proposal for apportionment.

But because BF is entirely unbiased, in a way that even Webster is not, I mention BF for completeness. BF, like Jefferson, Webster, and Hill, is one of the simple divisor methods.

Here is how R  is determined in BF:

Raise a to the a power. That’s called a^a

Raise b to the b power. That’s called b^b

Divide a^a by b^b.

Divide the result by e.  e is a mathematical constant. e is the base of the natural logarithms. To get e on a scientific calculator, enter 1 in the display, and then press ex, or inverse ln, or exp, depending on how that particular scientific calculator indicates the exponential function.

To write BF’s formula for R as a concise formula:

R = (b^b/a^a) divided by e

As stated above, Bias-Free (BF) is entirely un-biased, in a sense in which none of the other divisor methods are. In BF, for each interval between two integers (whole numbes), the average of the seats/q values that go with the various possible values of q in the interval are the same for every interval. That can’t be said for any other divisor method. Here’s another possibile method for avoiding bias in apportionment: The Pearson’s Correlation-Coefficient Method (PCCM).

I mention it because, unlike BF, Pearson’s correlation coefficient is a very familiar and much used thing, maybe giving PCCM a better chance of enactment someday, for apportionment or PR. I emphasize that Jefferson and Webster are widely used for PR, under different names, and that Bias-Free and the Pearson Correlation Coefficient Method would both be good PR methods, to minimize bias. There will be more about PR in the PR section of this article, after this apportionment section.

First a definition: The Hare quota is the combined population of all the states, divided by the total number of seats (435). A state’s “Hare quotas” is the number of Hare quotas of population it possesses.

Pearson’s correlation coefficient is familiar in statistics, and is much used, to find a relation between two sets of variables. For instance, we’re looking for a correlation between the states’ populations, and the states’ seats per Hare quotas. (Each state’s seats per Hare quotas is its number of seats divided by its number of Hare quotas of population). A state’s seats per Hare quotas is a measure of its representation per person. We don’t want that representation per person to be correlated with the states’ population. If it is, then that’s bias.

So we want the Pearson correlation-coefficient for populations and seats per Hare quotas to be zero, or at least as close to zero as possible. So: By trial and error, find the seat allocation among the states whose Pearson correlation coefficient is as close to zero as possible, for the correlation that is of interest to us—the correlation between population and seats per Hare quotas.

That would be computation-intensive. But the computation can be much eased and shortened by first doing a Webster allocation, or, better yet, a BF allocation. Then try various adjustments to it, to find one that brings that Pearson correlation coefficient closer to zero. Then continue trying adjustments to find out if you can find one that brings the Pearson correlation coefficient even closer to zero. Repeat till it’s no longer possible to find an adjustment of the seat allocation that will bring a Pearson correlation coefficient closer to zero.

A word about bias and proportional accuracy:

Different people have different definitions of bias, different considerations by which to evaluate, measure or compare bias. Likewise for proportional accuracy. In a fully meaningful sense, Bias-Free is entirely un-biased, and Webster is by far the least biased divisor method that’s ever been used or proposed in Congress, and the most proportionally accurate allocation rule that’s ever been used or proposed in Congress.

The other divisor methods are so biased that, it would be difficult or impossible to convincingly claim that any of them is as un-biased as Webster or Bias-Free.   …or that any allocation rule that’s been used or proposed in Congress is as proportionally accurate as Webster or Bias-Free. Though Webster’s bias is miniscule, it would be difficult to argue that it’s as low as that of Bias-Free.

2. Proportional Representation:

This proportional representation section will be brief, because I described the allocation topic in the apportionment section.

a) The Proportional Representation Methods Are the Same As Our Apportionment Methods:

The proportional representation allocation rules that are currently in use were all used for apportionment of the U.S. House of Representatives, at various times in U.S. history. In proportional representation (PR), Jefferson’s method is called d’Hondt. Just as Jefferson strongly favors large states, d’Hondt strongly favors large parties.

In PR, Webster’s method is called Sainte-Lague. Just as Webster is by far the most unbiased divisor method that has been used or proposed in Congress for apportionment, and by far the most proportionally accurate method that has been used or proposed in Congress for apportionment—Those same things can be said for Sainte-Lague, in proportional representation in Parliaments and Congresses.

Hill’s method hasn’t been used in PR. But, if it were, it would favor small parties just as strongly as it now favors small states in House apportionment. Of course Bias-Free hasn’t been used at all. But, if it were, it would be the un-biased divisor method in a fully meaningful sense, just as it is in apportionment.

I wouldn’t hesitate to recommend Bias-Free for PR—except that its newness, its lack of precedence in use, its somewhat more complicated-looking formula, and the fact that its justification isn’t obvious at a glance. For all those reasons, I’d recommend Webster/Sainte-Lague instead.

Likewise, the Pearson Correlation Coefficient Method has no precedent in use, and that would make its enactment feasibility less than that of Sainte-Lauge. Otherwise, the Pearson Correlation Coefficient Method would be a good proposal. (I prefer the implementation simplicity of BF, but, as I mentioned before, the Pearson Correlation Coefficient Method isn’t as unfamiliar as BF, because of the great familiarity and wide use of Pearson’s Correlation Coefficient.)

Sainte-Lague is already used for PR by a number of countries. If we were to get a unicameral Senate for the U.S.,  and it were desired to fine-tune House proportionality and un-bias, then I’d recommend Webster (instead of Bias-Free, for the reasons stated two paragraphs before this paragraph), But, if those considerations weren’t a problem, then of course I’d recommend BF—for apportionment &/or for PR.

As I mentioned above, the actual PR procedures are a little different from, but completely equivalent to, the apportionment procedures. A systematic approach is usually used instead of trial and error. If there are a lot of seats, that systematic procedure, as defined, could take longer than the trial and error procedure. As mentioned above, I described apportionment first, and defined the allocation rules as apportionment defines them, because the apportionment procedures are more direct, and the definitions briefer.

b) Considerations For the Choice of a Proportional Representation Method:

I’ve already told of Webster’s advantages over the other allocation rules that have precedent, and BF’s advantage over all of them, including Webster, and BF’s enactment-feasibility disadvantages.

I’d just like to discuss one other consideration: Strategic splitting. Any method that achieves as good proportional accuracy as Webster or BF do, can have strategic splitting. That possible disadvantage comes with the good proportional accuracy. Say the method is Sainte-Lague or BF. Say a large or slightly large party has a pretty good estimate of what its vote-total will be. It could divide into smaller parties, in such a way that each party would have just enough votes to get a seat, by whichever of those good-proportionality methods is in use.

Sure, there’s some risk of not getting any seats, if the party over-estimates its predicted vote-total. But it depends on how sure the party is about its predictive estimate. Where Sainte-Lague is used, it usually or always raises R in the 0 to 1 interval. Typically R is raised from .5 to .7

Sainte-Lague continues to be used, and has been used for a long time. So, at least with the .7 value of R in the 0 to 1 interval, there must not be a significant strategic splitting problem. Obviously, then, there isn’t a problem. That goes for Sainte-Lague, and also for Bias-Free, if its R were raised to .7 in the 0 to 1 interval.

But, just hypothetically, for completeness, because I want to cover everything, what if there were a problem? What if strategic splitting still occurred even with an R of .7 in the 0 to 1 interval? One suggestion has been to raise R in the 0 to 1 interval to 1.  …so it’s like d’Hondt in that interval. That would avoid that problem, at least in that interval. The best possible reward of strategic splitting is considerably less, while the risk remains just as menacing, in the intervals above 0 to 1. But still, for completeness, what if people felt that, in the 0 to 1 interval, an R of 1 is unfair to small parties?

And what if the strategic splitting were occurring when the 0 to 1 R is at .7?

Then I’d suggest the Largest-Remainder allocation rule, under those quite non-existent hypothetical conditions.

Largest-Remainder:

Divide the total number of votes in the election by the total number of seats to be allocated in the Parliament. That number is called the “Hare quota”. A party gets a seat for every whole Hare quota of votes that it has. Some votes will usually be left over. One at a time, allocate each left-over seat to the parties, in order of how large the party’s fractional part of a Hare quota is. For instance, if a party has 7.35 Hare quotas, that party’s fractional part of a Hare quota is .35    That’s also called that party’s “remainder”.

So, the first left-over seat goes to the party with the largest remainder. The 2nd left-over seat goes to the party with the 2nd largest remainder…etc. …until all of the left-over seats have been allocated in that way.

Largest Remainder doesn’t have the proportional accuracy of Sainte-Lague or BF, but it isn’t susceptible to strategic splitting. So I’d recommend Largest Remainder only if strategic splitting remained a problem no matter which of the above-discussed remedies were tried.

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This concludes this discussion of proportional representation and apportionment.