General discussion about Congressional Apportionment

[jimrtex]

Since I had nothing else to do, I spent the past weeks of my life (when I wasn't on the forum ) to redo myself the entire 2003 congressional apportionment based on the 2000 Census Bureau's population results. That means I calculated the priority value for each single seat using the formula I found on wikipedia.

Here's what it says :

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To put it clearly, Priority Value=Population/(n*n+1)^0.5

I realized a list of every seat using this system, but it's only at the very end that I realized there was a problem. It obviously came to the very controversial NC-13 and UT-4 seats... And here is how it ends up :

North Carolina : 8,049,313 inh.
8049313/(12*13)^0.5=644461

Utah : 2,233,169 inh.
2233169/(3*4)^0.5=644660

On this count, UT-4's priority value is 199 higher than NC-13's. So, if these population counts are correct, Utah should have been granted the 435th seat.

Instead, if we look at the official census Priority values, we get :

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How can we explain this difference ?

The "apportionment population" is the sum of the resident population and the overseas population.

The overseas population consists of federal government employees, military and civilian and dependents who live with them.  The courts have deferred to the judgment of Congress and the federal government about how the census and apportionment is conducted.  These are recent court decisions on the matter.

Franklin v Massachusetts

Following the 1990 census, Massachusetts had challenged the inclusion of the overseas population in the apportionment population.  This cost Massachusetts its 11th representative.  Because much of the overseas population included in the apportionment population is in the military or their dependents, this tends to have a bias towards states with military bases.  The military has a number of different states that can be attributed to military personnel.  If possible, they use state of residence when they entered the military, but for some people this information isn't available.

Most of the above case deal with legalities of how the Commerce Department and Census Bureau decided to include the overseas population, rather than whether the inclusion was lawful.

Department of Commerce v Montana

Following the 1990 census, Montana lost its 2nd representative, and sued, arguing that the wrong divisors were used in the determining the priority order.

Possible divisors for determining the priority value for the n+1 representative are:

((n)*(n+1))^0.5  what is currently used.

((n)+(n+1))/2  this is more favorable to large states.  It is equivalent to using St.Lague
divisors.  If this were used in 2010, Rhode Island might lose its 2nd seat.

2*(n)*(n+1) / (n)+(n+1) which is more favorable to smaller states.  If you were to repeat your exercise with these divisors, you would find that Montana would get a 2nd seat.  It might be that this would let Utah pass North Carolina, but not gain the 435th seat because Montana would steal it.

(n+1)  This is equivalent to D'Hondt and is very unfavorable to smaller states.

(n)      This is very favorable to small states.

Incidentally, the lawyers making the oral arguments were Marc Racicot, who was Attorney General of Montana at the time, later governor, and then chairman of the Republican National Committee in the early 2000's; and Ken Starr, special prosecutor in the Clinton corruption cases and recently appointed President of Baylor University.

Wisconsin v City of New York

This was about the use of sampling.

Utah v Evans

This was another case involving census methodology.  This was triggered by the determination that North Carolina had narrowly edged out Utah for the 435th seat. 

Utah had considered suing based on the census bureau not counting Mormon missionaries in the overseas, but eventually decided they didn't have a good case.  They did get the Census Bureau to consider other counting methods for US citizens who reside overseas, but are not affiliated with the federal government.  I suspect that if the federal government took a more direct role in the registration of US citizens resident overseas they could get a reasonably accurate count.  After all, the whole purpose of including them is for purposes of apportioning representatives and the federal government requires that they vote in congressional elections.

Anyhow, Utah discovered that if the census had used another method of conducting the resident census, Utah would have got the 435th seat.

The closeness of Utah to getting the 435th seat, is also why it was given the 437th seat under proposals that would have granted the District of Columbia congressional representation.

Muon's projections of apportionment for 2010 do include an adjustment for the overseas population.  The Census Bureau only makes estimates of the the resident population.  The overseas population is only tabulated for apportionment purposes - the Census Bureau does not publish any details other than statewide counts.

So Muon's projects are based on the resident population as estimated by the Census Bureau and then adjusts the population for each state based on an assumption that the ratio of overseas to resident population is constant for each state.

[jimrtex]

One last thing : Do overseas residents vote for House elections in the State they are assigned to ? If not, counting them makes no sense.

No.

The overseas population included in the apportionment count are only federal employees (military and civilian).  They are not enumerated in the manner of US residents, who are sent a census form that they fill out and return.  Instead they are counted based on administrative records of the agency they are affiliated with.  In 1990, 98% of the overseas population was associated with the Department of Defense (DOD), both military and civilian, plus dependents who are living with them.

1990 Overseas Population see pg 62

In the case of the DOD military personnel, the state is generally the "Home of Record", which is where the person lived when they entered the service (or in some cases, re-enlisted).  If this isn't available, then the state indicated on their records for tax withholding purposes.  Some states in the US don't have state income taxes, so many military personnel claim that state as their residence for tax purposes.  Other possibilities include US location for last assignment of 6 months or longer, or last US assignment.

Overseas resident voters may vote in their place of last residence in the United States.  There really isn't any government records of all these persons.  Registering with an embassy or consulate is voluntary.  US citizens are generally free to exit and enter the US.  There might be as many 6 million.

Issues of Counting Americans Overseas in Future Censuses

Whether some are actually resident overseas is vague.  Someone who retires may keep their house in the US, and visit it for a few months a year.  It may be easier to keep their voting residence in the US, and then vote absentee where they live most of the year.

There is a mail forwarding service in Texas that maintains permanent mailing addresses for RV'ers.  So that it doesn't look like a PO box, they use "street names".  Some people have registered to vote based on that address, and it has been an ongoing controversy whether they were resulting in Republicans taking control of the county.

And BTW, I think the arithmetic mean would be fairer as divisor than the geometric one. The current system clearly advantages small States (a State whose population is equivalent to 1.41 seats has as much chances to get its second seat as one of 52.5 to get its 53th). This is particularly senseless, considering that anyways small States get overrepresented thanks to the Senate.

The arithmetic mean minimizes the deviation in the number of representatives per person.  If there was one representative per 1 million persons (for the country).  Then a state that was right on the threshold between having one and two representatives, of having 2/3 as many representatives per person as the ideal, to having 4/3 as many representatives per person.

The harmonic mean minimized the deviation in the number of persons per representative (district size).  A state that was right on the threshold between one and two representatives would go from having a single district that was 4/3 the ideal size to one that was 2/3 the ideal size.  This is what Montana was arguing for.

The geometric mean is somewhere in between.

arithmetic_mean/geometric_mean = geometric_mean/harmonic_mean

[jimrtex]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

Harmonic mean is:  2*(n)*(n+1) / (n)+(n+1)

4/3
12/5
24/7
40/9

or alternatively:

1¹/₃
2²/₅
3³/₇
4⁴/₉

The arithmetic, geometric, and harmonic mean all converge on n¹/₂

If you use the harmonic mean, you minimize the variation in district size.

Let's say that a state has a population that is somewhere between 1 and 2 times the ideal district population.

If the harmonic mean is used, when its population is just below 1.33 the ideal, it will have a single district with a population 33% larger than the ideal.  Increase the population a bit, and it will have two districts whose population is 33% smaller than the ideal.

A similar thing happens around 2.4 times the ideal, when it goes from 2 districts that are 20% too large, to 3 districts that are 20% too small.

If you use the arithmetic mean, a single district would be 50% too large, and then two districts 25% too small.   Or two districts that are 25% too large, and then three districts that are 17% too small.

[jimrtex]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

The harmonic mean was popular with the ancient Greeks, since it had nice geometric properties. In modern use it might come up in a problem like this:

Antonio walks from his house to the post office at a speed of 2 km/hour and returns at a speed of 3 km/hr. What is his average speed during the round trip?

The answer is not 2.5 km/hr, but instead it is the harmonic mean of 2 and 3 which is 12/5= 2.4 km/hr. If the post office were 3 km away it would take 1.5 hours to get there and 1 hour to return at total of 2.5 hours. The round trip is 6 km, so the average speed is 6 km/2.5 hr = 2.4 km/hr.

Or, Antonio lives in a country which apportions one representative in its Congress per 100,000 people.  His state has 240,000 persons or the equivalent of 2.4 representatives.  So the question is whether his state should have 2 or 3 representatives.  And it so happens that 2.4 is the harmonic mean of 2 and 3.

If Antonio was a congressman, and the state was apportioned 2 seats, he would represent 120,000 persons or 20,000 (20%) more than the ideal.  If the state was apportioned 3 seats, he would represent 80,000 persons or 20,000 (20%) less than the ideal.  It could be considered fair to apportion the state either 2 or 3 representatives, since the deviation in the district size from the ideal would be the same in either case.  If Antonio's state had 1000 more people, then  the deviation for 2 representatives would be +20,500 but for 3 representatives would be -19,667.  So it would be better to apportion the 3rd representative since it reduces the deviation from the ideal.  And if the population were slightly less than 240,000, the opposite would be true.  The deviation would be less if there were two representatives.

On the other hand, perhaps fairness should not be determined based on persons per representative, but rather representatives per person.  Don't we want to let each person have about the same chance of influencing the decision of his country through his representative.

It will be easier to see if instead of using representatives / person, we use representatives per 100,000 persons.  The result is the same, but instead of small fractions (0.00001) we end up with numbers closer to 1.

If Antonio is no longer a representative, but just an ordinary citizen, then if his state of 240,000 persons is apportioned 2 representatives, there will 0.833 representatives per 100,000 persons.  If there are 3 representatives, there will be 1.250 representative per 100,000 persons.  In one case, Antonio will receive 16.7% less than the ideal representation, while in the other, he will receive 25.0% more representation.  We can't give him the ideal amount of representation, but we can give him as close to the ideal as possible by apportioning his state 2 representatives.

It would not be until his state grew to 250,000 persons (2.5 times the ideal, and the arithmetic mean of 2 and 3) that it would be fairer in terms of representation to apportion a 3rd seat.  If there are 3 representatives for the 250,000 persons, there will be 1.20  representatives per 100,000; and if there are 2 representatives there will be 0.80 representatives per 100,000.  So the over-representation on the one hand and under-representation on the other are balanced at 20%.

[jimrtex]

Las Vegas might have happened anyway. Without the state of Nevada, the Clark county area was part of New Mexico territory. It wasn't even part of NV until 2 years after statehood. This is an 1857 map of NM terr:

There would be a village there, but there likely wouldn't be any casinos.  For a long time Nevada was the only State that allowed legal gambling on anything besides horseracing.

In 1910 Las Vegas, NM had 3755 people; Las Vegas, NV had 945.  Clark County, NV was formed between 1900 and 1910, and its largest town was Searchlight.

The original border between Utah and Nevada was one degree further west.  If you extend that border south, then the Colorado River would be east of the line, though Las Vegas would be west.  There would really have been not much reason to shift the Utah border south, unless you made the Colorado River the border between Arizona and Utah.  And it would be unlikely to move the tip of Nevada southward when all the population was up around Virginia City and Reno.

[jimrtex]

Which method was used before 1941 ? I'm extremely interested by these issues.

The ranking methods such as Huntington-Hill are not particularly intuitive.  You can crank through the arithmetic and get an apportionment for each State, but it may not be obvious why it works.

Traditionally, there are a couple of other approaches.  The first decides the number of persons that are to be represented by each representative.   Let's say 30,000 (the Constitution requires each representative to represent at least 30,000 persons; so this is good starting point).

You divide the population of each state by this target population, and apportion it that many representatives.  A state with 210,000 persons would get 210,000/30,000 or 7 representatives.  But what about if the quotient has a fraction.  If a state has a population of 230,000 persons, and it is entitled to 7²/₃ representatives, maybe it should be given 8 representatives.  So you might round fractions greater than 0.5.  Rounding this way, while proposed, has never been used.  What has been used is to drop the fraction entirely, or to apportion another representative regardless of the fraction.

A variation of this approach is to first determine the size of the House, and then calculate the number of persons per representative for the country, and then divide this into each State's population.  If fractions are ignored, then this won't produce enough seats to meet your target size (in general you will be about Nstates/2 seats short).  If you round fractions, you may get close to your target.  So instead, states with the largest fractions receive an extra seat.

In practice, Congress would experiment with various target sizes.  They might first use the current size of the House, and then compute the result (this would be done longhand by quill pen in the light of an oil lamp or candle, in some drafty boarding house in Washington).  If a State would lose a seat or two, its representatives might recommend that the size of the House be augments to reflect the growth of America.  And others would check to see if a particular size would benefit the Mugwumps or Know-Nothings or other parties.  And so they would pick a size.  The legislation would just say how many representatives each State was apportioned.   After the 1870 census, there were actually two apportionments as they decided they didn't like the results of the first apportionment which resulted in a few states losing representatives.

In 1880, the census bureau calculated the number of seats that each state would be apportioned based on various sizes of the House ranging from 275 to 350.  This would let Congress immediately see the effect of different sizes.  It was discovered that if the number of representatives was increased from 299 to 300, Alabama would lose a seat, and Texas and Illinois would gain a seat.  This is known as the Alabama paradox, where adding seats for the country can result in states losing seats.

When Oklahoma became a state in 1907 and given its fair share of 5 seats, it was discovered that recalculating the number of seats based on 391 instead of 386, would result in Oklahoma still getting its fair share, but also cause Maine to gain a seat and New York lose one.  And there were cases where a faster growing state could lose a seat to a slower growing state.

Congress, generally ignored the problem, by choosing a size that didn't have a problem, and trying to balance political considerations.  Following the 1920 census, they were not able to do this.  The previous decade had brought about an abrupt stop to the ever increasing farm population as the limits of lands in the West with enough rain for non-mechanized family farming were reached, and growth of manufacturing in the East meant a reversal of the long-term trend where western states always gained seats, and the House was increased in size so that eastern states didn't lose too many seats.

So the 1920 census was ignored, and the existing 1910 apportionment continued for the next decade.  As the 1930 census approached, President Hoover insisted that something be done about the problem.  After consulting mathematicians, they agreed to keep the House fixed at 435, and have the number of seats calculated automatically.  After the 1930 census, apportionment based on Equal Proportions and Major Fractions were computed.   They were the same result, so it can't be definitively said which was used.

After the 1940 census, the same procedure was used, and it was found that using Equal Proportions would shift a seat from Michigan to Arkansas, and vice versa.  It was a political decision, along with some support from mathematicians that resulted in "permanent" adoption of equal proportions.

A Brief History of Apportionment

History of apportionment.

A Little History

A brief history of apportionment methods, bullet version.

The House Apportionment Formula in Theory and Practice

This was a report prepared for Congress in 2000 (before the 2000 apportionment).  It mainly compares the method of equal proportions (the current method which uses the geometric mean; and major fractions, which uses the arithmetic mean for the divisors).  It appears to be somewhat biased towards use of major fractions, possibly indicating who had requested the study.  It has a priority list for the 1990 apportionment, which explains why Massachusetts sued to prevent the inclusion of the military overseas population.  It also doesn't question the fixed size of the House - treating what is just a common practice that was codified into law, as immutable.

A New Method for Congressional Apportionment

This isn't about history, but rather mathematics.  One desirable attribute of an apportionment method is that if a state is entitled to between n and n+1 seats, based on its relative share of the population, then it should be apportioned either n or n+1 seats (alternatively expressed as that the difference between the exact apportionment and the actual apportionment should be less than 1).

On page 4605, there are a couple of examples using simulated population distributions: 1984A and 1984B.  Using population distribution 1984A, several large states receive extra seats beyond their quota.  If you look at the fractional apportionments, you can see why.   A large number of states have small fractions (37 have a fraction less than 0.4, 28 less than 0.3, when you would expect 20 and 15, assuming uniform distribution).  And a few large states have very large fractions CA, NY, PA, and TX have fractions over 0.9.

If you have 5 states with a fraction of 0.2, collectively they could be apportioned an additional seat (5 x 0.2 = 1.0), but individually none can claim the seat.  With the large number of states with small fractions, there is a surfeit of unclaimed seats.  Larger states are able to claim the extra seats, pushing them beyond their fair share of the quota.

1984B is the inverse phenomena.  Here there a large number of small states with a fraction above 0.5.  Two states each entitled to 1.55 seats, and 3.1 collectively might each be apportioned 2 seats.  There are also a relatively large number of states with very large fractions (16 have fractions greater than 0..  Since these seats are claiming a fraction of a seat beyond their ideal share, too many seats are being apportioned.  These come at the expense of the large states, which in this case have very small fractions.

Apportionment: Introduction
Apportionment II: Apportionment Systems

Some more math and overview.  An interesting tidbit is that the authors of the paper referenced above, ultimately rejected it as a valid method.

Congressional Apportionment and Population

The Datasets is interesting since it includes historical population data in spreadsheet form.

History, Myth Making and Statistics: A Short Story about the Reapportionment of Congress and the 1990 Census

An urban legend about apportionment and the census.

Discrete Mathematics Apportionment Methods

Some homework problems.

[jimrtex]

Damn, all this is so interesting... I spent the entire day reading the links you posted !

Well, also if I recapitulate, the methods used were :
- 1791-1842 : Jefferson method (rounding down)

What is particularly interesting is that Congress had originally passed Hamilton's method, but it was vetoed by George Washington, on the advice of his Secretary of State, Thomas Jefferson.  Because Hamilton's method can result in an average number of persons per representative than the divisor (because of the rounding up), it resulted in some states having representatives representing less than 30,000 persons - which violates the constitution.  This was the first legislation that was ever vetoed by a US President.

Jefferson's method is effectively the same as D'Hondt, other than all states are guaranteed a single representative.

It was probably a bit of inertia, and continuing respect for Washington that resulted in Jefferson's method continuing to be used.  As the population grew, and the divisor grew, it was possible to use Webster's or Hamilton's method, without violating the constitution.

Incidentally, there is a pending constitutional amendment that could have an effect on apportionment.  In 1789, Congress proposed 12 amendments to the Constitution, and specified that the States could approve or disapprove individual amendments.  Ten of them were ratified quickly, and are now known as the 'Bill of Rights', and numbered 1st to 10th.  In the proposal by Congress they were numbered 3 to 12.

One article regarding congressional pay was ratified about 200 years later, and is now known as the 27th Amendment.  There was no time limit on ratification.

The 12th proposed amendment set the number of persons per representative.  There was concern at the time of the ratification of the Constitution that the House of Representatives would not grow to accommodate an increasing population.  There may be a typo in the amendment that was actually proposed.  If it were ratified, it would simply increase the minimum number of persons per representative from 30,000 to 50,000.

- 1842-1852 : Webster method (arithmetic mean)
- 1852-1911 : Hamilton method (strongest remainder)

Though technically that of 1872 is not based on any mathematical formula.   This was the first reapportionment following the Civil War, and the passage of the 13th, 14th, and 15th amendments.

The original Constitution provided that slaves be counted as 3/5 of a person for apportionment purposes.  So if half the population was slaves, then a state would get about 90% of the representation it normally would have, though those 90% would be elected by the half of the population that could vote.

The 13th Amendment made the 3/5 clause moot.  So there was a concern that southern states would have increased representation in Congress.  Even if former slaves were not systematically disenfranchised based on race, there was a concern that property and literacy qualifications could be used to have the same effect  Such qualifications might also have the effect of disenfranchising poor whites, so that large landowners would have enormous political power.

The 14th Amendment was intended to prevent that from happening.  Originally it was going to provide that apportionment would be based on the adult male population that was able to vote.  So if adults were prevented from voting due to property or literacy qualifications, it would simply mean greater political power for the north.  But there was concern that New England states might lose political power.  They had a relatively larger share of females and children.  As males reached adulthood, they might well strike out for the frontier, since they might not inherit the family farm for a couple of decades.  So instead, there is the version of the 14th amendment that was enacted, which provides for a proportionate reduction of the apportionment population based on disenfranchisement.  It had the bonus of permitting the restriction of the franchise to former rebels without a penalty.

The 15th Amendment greatly reduced the possibility that discriminatory property or other franchise qualifications would be enacted.  Under Reconstruction, the federal government could ensure that legislatures and constitutional conventions would not adopt discriminatory legislation.

Before the 1870 Census, Congress considered questions that would enumerate the number of adult males citizens over the age of 21 who could not vote for reason other than being a felon or a rebel, but never actually passed a law to that effect.  The head of the census bureau on his own had those overseeing the census in each state to make inquiries as to the number, but there was no formal effort, and the quality varies quite a bit.

In 1872, when Congress was considering the apportionment bill, as expected there was a shift in representation to the South and West.  Congress eventually passed a large increase in the size of the House so that most states would maintain their representation.  They discussed adjusting the apportionment population based on adults not franchised, but since they really didn't have quality numbers, and those formally disenfranchised were few in number (typically imbeciles or idiots) it wouldn't have made a difference.  States that excluded illiterates were likely to have few.  Western states were less likely to have literacy tests because they wanted to encourage local self government, and hadn't developed an elite that wanted to exclude the riff raff.  Eastern states ensured that children could read and write and cipher.

Eventually, Congress simply stuck the apportionment clause of the 14th Amendment into the US Code, where it still resides, but has never made a serious attempt to apply its provisions.

The 1872 apportionment bill had a number of other interesting provisions:

(1) Required representatives to be elected from districts.
(2) Required congressional elections to be held on the first Tuesday after the first Monday in November.

Both these provisions were later relaxed, and it wasn't until after 1960 that they were universally applied.

(3) Required congressional elections to be conducted by paper ballot.  This was not a government printed (Australian) ballot, but simply a piece of paper that a voter might write their preferred candidate's name.  Parties would print there own ballots as a "convenience" for voters.  At the time, some states elected by voice, where a voter would announce his vote.

A few months later, Congress passed a new apportionment bill, which apportioned another 9 representatives.  Because some state legislatures had already provided districts on the first apportionment, there was an option for states that gained a representative from the second apportionment, to elect him at large.

- 1911-1941 : Webster method (arithmetic mean)
- 1941- : Huntington-Hill method (geometric mean)

There was no apportionment following the 1920 census.

Congress required that the apportionment under the geometric mean and arithmetic mean be computed based on the 1930 census.  They happened to produce the same result, so it can't definitely which was used.  The 1930s reapportionment was the first that used a priority ranking method.

I'm really passionated by reapportionment methods and all these mathematical stuff... But I really wonder how you manage to know so many things. I'm really impressed.

And BTW, I spent weeks to search throughout the web for the population of each State by census, and then filling my own spreadsheet... All this to finally discover that I didn't found the right results (no overseas). And then you come with an already done spreadsheet with all the results ! Damn it.

Re-BTW, I'd really like to have such exercises as math homework.

Apportionment US

This is the website of a group that argues that the current apportionment is unconstitutional since it results in congressional districts that vary too much in population (Montana's single district is almost twice as populous as Wyoming's).  They have filed a lawsuit that would compel Congress to remedy this by increasing the size of the House.  Over on the right side, there are links to the latest briefs by the group and the US government.

Return the House of Representatives to the People
 435 Representatives Can Not Faithfully Represent 300 Million Americans!

This group advocates greatly increasing the size of the House of Representatives, so that congressional districts would have about 30,000 people, as they did in 1789.  This would increase the likelihood that representatives would be citizen-representatives rather than career politicians.

[jimrtex]

Washington's veto seems quite legitimate, since the Constitution did indeed set a minimun population per representative. Plus, I tend to think that Jefferson's method is actually quite fair and adapted to the House of Representatives. Indeed, small States are extremely overrepresented in the Senate (logical, that's its role). So, in some way, the House should compensate for this overrepresentation by favoring larger States, so that the Electoral College is closer to the real repartition of the population. Obviously, it's not a mathematical argument, but rather a political one.

The House of Representatives is intended to represent the people, not the States.  Jefferson's method badly under-represents the people of the very smallest states.  If you have one state with 121,000 and another with 59,000, or barely over a 2:1 ratio in population, it produces a 4:1 ratio in representation.

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Actually, on my count, it would make 80% instead of 90%. Wouldn't it ?

You are right, 80%.

And which criterion did they use for the final 1872 apportionment if it was neither Webster nor Hamilton method ? I hope they didn't dare distributing seats arbitrarily.

Most early apportionment legislation simply designated how many representatives each State would get, so it looks much like the original Constitution, which simply said how many representatives each State would have before the first census.

In 1872, the first apportionment law simply listed the number of seats for each state.  The supplemental apportionment law just listed the 9 states that would get an additional seat.

The size of the House had grown from its original 65 to 240 by 1830.  It was then held roughly constant for the next 3 censuses.  It was at 243 after the 1860 census, including the new states of Nevada, Kansas, and Nebraska.

Had the size remained constant, every New England state would have lost a representative, New York would have lost 3, Pennsylvania 2, and Ohio 2.  Kentucky would also have lost 1.

Most of the gains would have been in the West, with Illinois, Iowa, and Missouri gaining 2 each, and Michigan, Wisconsin, Minnesota, Kansas, and California one each.

None of the southern states would have lost any, and Texas and Georgia would have gained one each.  While it was a really a shift from East to West, in the post-Civil War era it had the appearance of the victors losing massively, while the losers were held harmless.

They found that if they increased the nominal size of the House to 280, and applied Webster's method it resulted in 283 members (in Webster's method you determine the average number of persons per representative for the US, and then divide that it into each state's population to determine the number of representatives for each State, rounding fractions greater than 1/2).  Moreover, this meant that every State except NH and VT would have the same or have a gain.  In the East, there was a small reversal, so that MA, NY, PA(+2), and OH gained rather than lost, and CT, RI, and ME would stay the same rather than losing.

A few months later, there was a supplemental apportionment.  They started out using a nominal size of 290, and applying Webster's method, it produced a House of 290 members, and more importantly restored Vermont's 3rd representative.

They then observed that NH and FL had fractions greater than 0.4, and were the only two such states.  They then decided that since NH and FL were closest to getting to 0.5 that they would be given the final two seats (they probably didn't care about Florida, but it happened to have the larger fraction (0.428 vs. 0.422).  So it was sort of a pseudo-Hamilton's method, awarding the two seats to NH and FL because they were among the 14 states with the largest fraction.  The only problem with this is that with a nominal size of 290, 278 whole seats would be awarded, and you would only need to award seats to the 12 states with the largest fraction, who in this case, coincidentally, all had a fraction greater than 1/2.

In the debate, they wouldn't use decimal fractions, but would rather use the remained after dividing by the average number of persons per representative.   For 290 representatives, this was 131,433 persons.

So for example, Alabama had a population of 996,982.  If you divide 996,982 by 131,433, you get 7.586, so Alabama would have 7 representatives and be awarded an 8th representative based on its fraction being greater than 1.2.

They would do the division and get a result of 7⁷⁶⁹⁶¹/₁₃₁₄₃₃.  The remainder of 76,961, would be called a fraction.  And they would say that since 76,961 was more than half of 131,433 that Alabama should be given an 8th seat for that fraction. Alabama was among the 9 states that gained a seat in the supplemental apportionment.

If they would have applied Hamilton's method correctly for 292 seats, then New York and Illinois would have been awarded the 291st and 292nd seats.

If the goal had been simply to give New Hampshire a 3rd seat, they would have had to apportioned 299 seats using Webster's method (Florida would have required 308 in order to get its 2nd seat, but this wasn't really the goal).

If Huntington-Hill had been invented, Florida's 2nd would have been the 289th overall and New Hampshire 295th.

Now go forward to the Hayes-Tilden election of 1876.

Had the original apportionment been in effect, Tilden would have won 180 to 177.  But Colorado had entered the Union in 1876 and its 3 electors would have tied it at 180 to 180.

If 290 representatives had been apportioned, and Colorardo entered the Union with the 291st Representative, Tilden would have won 184 to 183.  But the extra(neous) representatives for New Hampshire and Florida gave Hayes his 185 to 184 margin.

Had 299 representatives been apportioned in a proper manner so that New Hampshire would have retained 3 representatives, and Colorado been given the 300th representative, then the electoral vote would have been a tie of 188 to 188.

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30,000 people per representative ?!? This would make a 10133-members House...
I also favor a huge increase of the size of the House (1000 Representatives seems fair in a so big country), but not at this point.
In the other link you posted, they instead advocate for a 1761-members House. Still, it seems quite irrealistic to me, even though I strongly agree with what they say.

When the US was a largely agrarian country, legislatures did a pretty good job keeping up with population growth and drawing congressional and legislative districts.  Farm areas typically have comparable densities,so that two districts with about the same area will have similar populations.  So when drawing districts it is almost like creating districts of equal area.  In addition, legislatures might be growing in size, so that new legislators could be assigned to the growth areas.

By the early 1900s, the limits of arable land were reached, and manufacturing and cities were rapidly developing.  With mechanization, rural populations were often declining.  Redistricting should have mean creation of many small districts in urban areas, and combining districts in rural areas.  Since the people in the two areas were often politically antagonistic, legislators often were reluctant to redistrict.  Sometimes, they would analogize a legislature to Congress, and say that if each State has 2 senators, shouldn't each county have a state senator.

By the 1960s, districts had become severely malapportioned, and the US Supreme Court finally intervened.  States would make small adjustments, and the courts would overturn the new plan.  Eventually the courts let plans with legislative districts that were within 5% of the ideal stand, if they were trying to meet some other restrictions, such as respecting county or city boundaries.  It is not so much that the courts have said 5% is good enough, they have never said that 5% was not acceptable.

[jimrtex]

continuing....

The Apportionment US people are arguing that the same sort of standard should apply to Congress with respect to apportionment of the House of Representatives.  If Congress apportions one representative to each of Montana and Wyoming, a person will have much more influence over Congress than his cousin who lives across the border in Montana.

If a State is forced to adjust its apportionment for legislative districts to meet the 10% standard, why not Congress?  The US government will argue that the current apportionment is the best that can be done with 435 representatives, and Congress's choice of apportionment method.  And the Supreme Court has already ruled that the choice of apportionment method is a political decision.  The Constitution doesn't define "apportion among the States".

The Apportion US people will counter that use of 435 representatives is an arbitrary decision, and that Congress can't deny the people of Montana equal representation simply because Congress might have to buy a few more desks.

If you were to determine the the number of persons per representative for each State, and calculate the ratio between the largest and smallest values (for N=435, Montana and Wyoming) for each value of N, you would find that the ratio generally declines as N increases.  But it doesn't do so monotonically.  If N = 500, it might be worse than for N - 435.   But if you go to N = 1761, the ratio will finally go below 110%, it won't stay there but 1761 members would be sufficient.

But if you use a different population distribution, say that following the 2010 census, you may find that 1761 isn't such a good number.  As it happens it appears that a number so small as 1761 is an anomaly.   And that for most censuses, the minimum number of representatives needed to meet a 110% standard is significantly higher.

That is, if Congress were to lose the lawsuit and reapportion to 1761 representative in time for the 2010 elections (Primaries have already been held in Illinois, and are being held in Texas.  But this is no legal obstacle, the primary results can be discarded if they are based on an unconstitutional procedure), they probably will find that 1761 is not sufficient after the 2010 census.  The Apportionment US people might simply file a new law suit requiring that the new number be found to meet the 110% standard.

But what is so sacrosanct about the 110% standard?  States generally have other constraints.  The size of the Texas House is limited by the Constitution to 150 members, and there are other standards.  The current Texas districts do not fully comply with the Texas Constitution, but they largely do, particularly the size of the House, and respect for county boundaries, and they are able to do so while meeting a 110% standard.  If the standard were much tighter, Texas would simply be forced to ignore its respect for county boundaries.

But Congress is under no limit other than the minimum of 30,000 persons/representative.  Congress proposed a constitutional amendment over 220 years ago that would have set a higher limit, and the States to date have not seen a need to ratify it.  So what reason is there to stop at 110%?

[jimrtex]

Washington's veto seems quite legitimate, since the Constitution did indeed set a minimum population per representative

The Constitutional limit is "The Number of Representatives shall not exceed one for every thirty Thousand" not "The Number of Representatives for each State shall not exceed one for every thirty Thousand".

Washington's veto message says:

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[jimrtex]

The House of Representatives is intended to represent the people, not the States.  Jefferson's method badly under-represents the people of the very smallest states.  If you have one state with 121,000 and another with 59,000, or barely over a 2:1 ratio in population, it produces a 4:1 ratio in representation.

I also find it dramatically unfair for any representation system. But what I was saying isd that it could actually be quite good for the Electoral College. To take your example, the 4-1 ratio in the House would become a 6-3 ratio in the Senate, therefore corresponding to the correct ratio between the two States. And note also that this example is the most extreme you could find : for example 241,000 to 119,000 produces a 8-3 ratio (still unfair, but fairer than the preceding), and 1,201,000 to 599,000 just produces a 40-19 one.

In 1790, Virginia had 630,560 persons, so with a ratio of 30,000 it would get 21 seats.  Delaware had 55,540 persons, so it would only get one seat.  So with 11.35 times as many people, Virginia would get 21 times as much representation.

Washington and Jefferson were from Virginia.  Maybe it was politically motivated?

So in 1872 they first used the Webster method and then added two more seats to the strongest remainders ? Well, of course combining methods that way is extremely dangerous, and obviously serves political purposes.

Yes.

Well, the 110% ratio seems to make sense, but it's also necessary to face reality, and understand that nobody would accept a 1761-members House, neither the people nor the legislators. It's however very important to push for a strong increase of the House size. Do you know if a bill has already been proposed in the past to this purpose ? And how did it end up ?

The People approved the Constitution, including the 30,000 ratio; and the equal protection clause of the 14th Amendment.

HR 3972 To establish a commission to make recommendations on the appropriate size of membership of the House of Representatives and the method by which Members are elected.

[jimrtex]

The Constitutional limit is "The Number of Representatives shall not exceed one for every thirty Thousand" not "The Number of Representatives for each State shall not exceed one for every thirty Thousand".

Washington's veto message says:

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Since 8 States were affected, they must have originally tried to apportion the maximum 120 possible Representatives in the bill Washington vetoed, but doing so would run afoul of Washington's stated objection no matter what method was used.  With Washington's stated objection the maximum number of Representatives could be 112 for the 1790 census.

So any objection to using Hamilton's method on Constitutional grounds is a sham as by appropriately reducing the size of the House, any method could meet the stated reason, and without a reduction, none could.

The maximum size for the House apportioned using the 1790 Census and Hamilton's method while keeping the Ratio in each State above 30K per Representative would have been 100, which would have given Virginia only 17 of 100 Representatives under Hamilton (or under Webster or Huntington-Hill, but the House would need to be even smaller under those two methods, at most 99 for Webster and 92 for Huntington-Hill).

Yet rather than increase the quota to be used for Hamilton's method, they switched to a different method that produced a bias for large States such as Virginia.  It could have been worse.  Using a quota that would have produced a House of 112 Jefferson's method would have given Virginia 21 of 112 seats for an even larger bias.

Note: All of the stated maximums above come from selecting a quota that produces as large a House as possible using the 1790 Census data, but does not necessarily work a priori, assuming the Representation limit applies per State without knowing what the numbers were.  Using a quota of 30,000 in Jefferson's method works with any set of data but results in there being a House of 105 members using the 1790 data. Webster's quota needs to be 40,000 (or more) which results in a House of 91 members. Huntington-Hill needs a priority value of at least 42,427 (30,000 * sqrt(2)) which results in a House of 86 members. The quota needed for Hamilton to work under all data sets is 60,000 which results in a House of 60 members.

I've been reading through the debates, and there were conflicting views of how to interpret the Constitution.   Fisher Ames (MA) was very forceful in first determining the total number of seats, and then apportioning them.  He pointed out that a proper apportionment was simply calculated by the Rule of Three, which is based on the relationship:

state_rep / USA_rep = state_pop / USA_pop

or

state_rep = (state_pop * USA_rep) / USA_pop

I'm not sure if rounding is applied or not.  And he read into the record tables that showed with this method how close the population / representative would be to 30,000.  He also pointed out that would be how you would apportion taxes among the States.

James Madison argued the only way to do it was to decide a quota first, and then simply divide it into the population of each state, claiming that was the method specified by the Constitution.  It is somewhat suspicious, given how favorable the method was to Virginia with its population of 630,560 which is only 560 above 21 x 30,000 and Washington, Jefferson, and Madison all being from Virginia.  I think that the Senate switched to something like Hamilton's method, and then the House agreed and that was what was vetoed.  The Senate would of course be more favorable to the smaller states.  And there were those who argued that it was OK to have a large state bias, since that compensated for the senate and electoral college.

Some of those who argued against applying the apportionment to the entire country also argued that it would be like treating the country as simply being divided into districts, rather than being a Union of States who were given their own allotment based on their "respective numbers".

The bill that was vetoed by Washington did not include a reasoning, and the first part of his veto message was that the bill failed to include a "ratio", so I think it just included numbers of representative.  I haven't read any of the Senate debate yet, just that in the House.

There was very serious consideration to ordering a new census (in 1796) with the ratio pre-determined, so representatives could not choose the method of apportionment after seeing the effect.

There was also a suggestion that the apportionment should come into effect at an earlier date, such as October 1792.  If I understand it was actually a compromise that resulted in the current system where the new apportionment comes into effect for elections first, and then for Congress at the beginning of the new term on March 3, 1793.

At least for some of the debate, South Carolina had not completed its census.  I don't think anyone realized that could have a consequence on an apportionment based on relative shares of the population.

The vote to override the veto in the House was 23:33, and some of the earlier votes had been very close, so basically the House ended up having to go along with Jefferson and Washington.

[jimrtex]

I made it through the all the debate in Congress.  It was pretty confusing.  The Senate met in secret until 1794, so there is no record of debate, just the journal which shows things like motions and record votes, but that is all.

The bill started in the House, which is probably the right thing to.  The main issue was whether to use 30,000 vs a higher ratio.  Those arguing for a higher ratio wanted a smaller House.

A motion to set the ratio at 35,000 failed 21:38.  And then with 30,000 (Jefferson method) it passed by 43:12.

The Senate then voted down a proposal for Webster's method based on 30,000 (9:15); and to switch to a ratio of 33,000 (11:13); and another proposal for 106 members that appears to be based on Webster's method with 32,500 for the ratio (9:15).   Finally, the Senate approved the House version on a 15:11 vote.

The next day they switch to 33,000 on a 12:12 margin with Adams having the casting vote.  The main reason for 33,000 is because it does the best job of reducing fractions.

The House defeated a motion to switch to Webster's method based on 30,000 (23:37), and then defeated a motion to agree to the Senate amendments 29:31.

Neither House would give way, and they appeared to decide to try again later (this was late December 1791).

The House then created a committee of Madison, Gerry, and Benson to draft a bill, that would also include a provision for a 2nd census.

The House voted to use a divisor of 30,000 (30:21), and voted down a proposal to use Hamilton's method.  The bill was passed 34:16 with the same apportionment as the House had approved before, but with a provision for a 1796 census.

The Senate then voted down a 30,000 Webster's (13:14); as well using 33,000 (11:16) vote.

They later approved a 30,000 Hamilton's (14:13) but then voted down a motion explaining how the numbers were derived, instead simply listing each State's apportionment.

They went back and forth on the 2nd census: eliminating it, moving it to 1798, and finally eliminating it.

Neither house would give way, and a conference committee was appointed.  The conference committee could come to an agreement.

Finally the House gave way on a 31:29 vote, so the version sent to Washington was Hamilton's method based on a ratio of 30,000; and hence 120 members.

Washington vetoed the bill, because it didn't specify a ratio; and 8 states had less than 30,000 per representative (NH, MA, CT, NJ, DE, NC, SC, and VT)

The House failed to override the veto 28:33.

A committee produced a new version which specified Jefferson's method using 30,000.  The House changed this to 33,000 (matching the original Senate version from 4 months earlier) on a 34:30 margin.  The Senate then passed the final bill, with all 3 readings occurring on the same day.

[jimrtex]

Actually the problem with the fairness rule that the divisor methods have results from choosing to have a fixed number of seats.  If one instead chooses a fixed priority value, then the divisor methods will always meet the fairness rule, but there might be extra or fewer seats compared to the ideal.  For example, in 2000, using the Hamilton-Hill method with a fixed priority value of 646,952 (which is 281,424,177 / 435) one ends up with a 433 seat House, with California and North Carolina both not getting a seat that they received due to the requirement for a fixed size of the house., but in 1990, one ends up with a 438 seat House, with Massachusetts, New Jersey, and New York getting the 436th, 437th, and 438th seats respectively.

There is no requirement that the size of the House be determined a priori.

Using Tot_Pop/Target_Size will in general produce a larger number of seats than the target, for Huntington-Hill, though it would not in 2000.

A better approach is to start with 645,952 as a quota, q, and calculate a normalized share of the population, x, and assume that it is geometry mean of y+1/2 and y-1/2, where y is the seat entitlement for that state.

x = sqrt( (y+1/2)(y-1/2) )

squaring:

x² = y² - 1/4

and solving for y

y = sqrt ( x² + 1/4 )

If we sum the values of y, we will get a number slightly larger than 435.  Adjust the quota according to this amount 

q_n+1 = q_n (sum(y) / 435)

And repeat.  This converges very quickly.  Similar approaches can be used for the other divisor methods.

Either approach - not using a fixed House size generally avoid another paradox that may be exhibited in 2010.  California's relative share of the population may increase, while it's relative share of the House decreases.  California's estimated growth rate is very close to the national estimated growth rate, so it may end up that its share of the population decreases, while it loses a seat; its share of the population decreases, while it keeps 53 seats; its share of the population increases, while its loses a seat; or its share of the population increases, while it keeps 53 seats.

[jimrtex]

Sure, every method have advantages and drawbacks. But in such case, wouldn't the best solution be the one situated right in the middle of the two...
The Adams method is undoubtedly the silliest, and the one that creates the biggest distortion. A party could theoretically ge a seat with 1 vote ! On the other hand, the d'Hondt divisors method is quite excessive too, since, as Jim pointed out, a 2/1 difference in votes may create a 4/1 distortion... Not to mention the fact that the State which deserves 0.99 seats gets no one while one deserving 1.01 gets one. Both method have great odds to violate the fairness rule.
The Sainte Laguë method avoids both extremities. It is situated "right in the middle" between Adams' and Jefferson"s (while the geometric and harmonic means are closer to Adams method). Also, the number of ill-assigned seats is extremely close to the minimum reached with the Strongest Reminders. So, I view it as the best possible compromise.

So for 2010 we find that the average workload for a Representative is 708,000 persons, and that a representative can do an adequate job working an 8-hour day.    But if you use Ste. Lague to assign the representation tasks, you can end up with some representatives having to work 12 hours.  They may be tired, so they actually don't represent any of their constituents as well, despite spending 50% more time each day.  And for why?  So some big state representative can knock off a few minutes early.  Hardly fair at all.

But let consider Adams.  We set a maximum representation load of 748,000.  So some representative might have to work 8 hours and 27 minutes.  Hardly overworked.  And if the workload for some state runs over 748,000 per representative, you hire another representative.    Sure a few representatives might get a cushy job and not have to work 8 hours.  But isn't that acceptable considering the alternative.

St.Lague is only the middle ground if we consider D'Hondt at all acceptable.  But if we would use D'Hondt, we might as well bring back child labor with 8-years olds working from dawn to dusk picking clinkers out at the coal mine, sleeping on a hard board and being fed thin gruel twice a day.