General discussion about Congressional Apportionment
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Antonio the Sixth
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« Reply #25 on: February 22, 2010, 12:44:24 PM »

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

Or, Antonio lives in a country which apportions one representative in its Congress per 100,000 people. 

The funny thing is that this part is true (107,000 actually). Grin
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True Federalist (진정한 연방 주의자)
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« Reply #26 on: February 22, 2010, 05:58:34 PM »

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.
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jimrtex
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« Reply #27 on: February 24, 2010, 07:03:31 PM »

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.
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« Reply #28 on: February 25, 2010, 01:08:38 AM »

Which method was used before 1941 ? I'm extremely interested by these issues. Wink
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.Cool.  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.
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Antonio the Sixth
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« Reply #29 on: February 25, 2010, 02:01:06 PM »

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

Well, also if I recapitulate, the methods used were :
- 1791-1842 : Jefferson method (rounding down)
- 1842-1852 : Webster method (arithmetic mean)
- 1852-1911 : Hamilton method (strongest remainder)
- 1911-1941 : Webster method (arithmetic mean)
- 1941- : Huntington-Hill mthod (geometric mean)

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. Cheesy

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. Tongue Sad

Re-BTW, I'd really like to have such exercises as math homework. Grin
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jimrtex
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« Reply #30 on: February 26, 2010, 03:34:38 AM »

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

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. Cheesy

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. Tongue Sad

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

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.
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Antonio the Sixth
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« Reply #31 on: February 26, 2010, 04:33:05 AM »
« Edited: February 26, 2010, 04:58:42 AM by Antonio V »

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.

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

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.

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30,000 people per representative ?!? Shocked This would make a 10133-members House... Tongue
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.
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« Reply #32 on: February 26, 2010, 09:01:48 PM »

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".

So long as the total number of Representatives apportioned as a result of the 1790 census was 120 or less, a method that exceeded that limit for a particular state would have been Constitutional.

Incidentally, if any of Webster, Hamilton, or Huntington-Hill been applied to the 1790 Census, Delaware would have had 2 Representatives instead of 1 and Virginia would have had 18 instead of 19.  Under Huntington-Hill an additional change is that Vermont would have had 3 instead of 2 while Pennsylvania would have had 12 instead of 13.
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« Reply #33 on: February 27, 2010, 03:09:51 AM »

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 ?!? Shocked This would make a 10133-members House... Tongue
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.
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« Reply #34 on: February 27, 2010, 03:13:23 AM »

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%?
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« Reply #35 on: February 27, 2010, 03:29:29 AM »

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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« Reply #36 on: February 27, 2010, 05:04:24 AM »

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.

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.

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.

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 ?

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".

So long as the total number of Representatives apportioned as a result of the 1790 census was 120 or less, a method that exceeded that limit for a particular state would have been Constitutional.

Well, it's indeed quite ambiguous. But I still think that Washington's veto made sense.
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« Reply #37 on: February 27, 2010, 04:20:33 PM »

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.
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« Reply #38 on: February 27, 2010, 04:54:43 PM »

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?

Never said the contrary. Still, it does make some sense, in the perspective of Electoral College.


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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.[/quote]

Well, I hope it will result in something effective, even though there's a long and complicated path before the passage of a bill increasing the House's size.
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« Reply #39 on: February 27, 2010, 07:16:18 PM »

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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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.
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« Reply #40 on: February 28, 2010, 08:32:03 PM »

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.
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« Reply #41 on: March 01, 2010, 02:11:41 AM »

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.
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« Reply #42 on: March 04, 2010, 10:24:09 AM »
« Edited: March 04, 2010, 10:54:49 AM by Antonio V »

Also, I spent the past few days to look at the results of different methods for the 2000 Apportionment. Here are them.

StateHunt.-HillAdamsHamiltonWebsterJefferson
Maine22222
New Hampshire22222
Vermont11111
Massachusetts1010101010
Rhode Island22221
Connecticut56555
New York2928292930
Pennsylvania1919191919
New Jersey1313131313
Delaware12111
Maryland88888
West Virginia33332
Virginia1111111111
North Carolina1312131313
South Carolina66666
Georgia1313131313
Florida2524252526
Kentucky66666
Tennesse99999
Alabama77777
Mississipi45444
Louisiana77777
Arkansas44444
Texas3231323233
Ohio1817181818
Indiana99999
Illinois1919191920
Michigan1515151516
Wisconsin88888
Minnesota88887
Iowa55554
Missouri99999
North Dakota11111
South Dakota12111
Nebraska33332
Kansas44444
Oklahoma56555
Montana12111
Wyoming11111
Idaho22222
Colorado77777
Utah34433
New Mexico33332
Arizona88888
Nevada33333
California5350525355
Oregon56555
Washington99999
Alaska11111
Hawaii22221
Total435435435435435
Theor. Seats #435.69412.70N/A436.04456.87

Note : the Theoretical Number of Seats is the quotient of the total apportionment population by the Priority Value of the last seat granted. It corresponds to the number of seats which should theoretically be apportioned in order to actually apportion 435 seats under the system's criteria. The Hamilton method being based on a different aproach, there is no T#S.
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« Reply #43 on: March 04, 2010, 01:39:46 PM »

Also, the Adams method infringes the Fairness Rule for New York, Texas (one seat less for both) and California (two seats less). The Jefferson method brokes it for Florida (one seat more) and California (two seats more). So there are 4 seats uncorrectly awarded in Adams method and 3 in Jefferson.

And that led me to some considerations...

In fact, there are only two objective criteria that can lead to choose between one apportionment method or another. Every other criterion is merely subjective. The first objective criterion is the fairness rule : since the theoretical number of seats a State "deserves" is comprised between two integers, the actual number of seats the State will get should be one of those two integers. The second criterion is the "non-paradoxicality" : raising the number of the Hous should not result in one State losing a seat.
Well, I've come to the conclusion that no system can meet those two criteria at the same time. Because the simple fact of ensuring the respect of one criterion implies to broke the other one. Moreover, every proportional system necessarily respects one of these two criterions. Therefore, all the apportionment method can be classified in two categories.

The first category, which is the most used for Apportionments in the United States history, is the category of "divisors series", encomprising the Adams, Huntington-Hill, Webster and Jefferson methods. As was explained in one of the link you sent, these method consist in setting a "threshold" a State needs to reach in order to get a nth seat (this threshold being respectively n-1, ((n-1)*n)^0.5, n-0.5 and n for each of these method), and then to raise or lower the quota until the number of seats awarded corresponds to the number of seats you want to apportion). These category of systems necessarily results in ranking each seat in a "priority list", so if you add one seat to the House you don't need to compute again every seat : you just look at who is the next in the priority list. As a result, this method makes an "Alabama paradox" impossible to occur. Any seat granted to a State with n total seats will remain to this State with n+1, n+2 or n+100 total seats. However, the drawback of any system in this category is that there is absolutely no way to prevent an infringement of the fairness rule. We saw that for the Adams and Jefferson systems, but theoretically, the same may happen for Webster or Huntington method.
A random example for Webster's method (I made it myself) : there are four States and 851 seats to apportion. The first deserves 8.4 seats, the second 840.9, the third 1.4 and the fourth 0.3. To apportion 851 seats, we need one state to break the .5 threshold. So, we raise the quota in order to have 852 seats theoretically apportioned. The first State now "deserves" 8.41 seats, the second 841.9, the third 1.4 and the fourth 0.3. The second State has broken the 841.5 threshold, thus gets 842 seats. The final repartition is 8 seats for the first, 842 for the second and 1 for the third (the fourth gets no one, even though it's normally impossible, it doesn't change anything in this case). Normally, the second State should have got necessarily 840 or 841 because of the fairness rule.
The reason why it happens is simple : when apportioning the exact number of seats leads to too much or not enough States reaching the threshold, you're forced to lower or raise the quota, thus apportioning more or less seats than normally should be. In such situation, the big States are far more affected by a raise or a lowering, so that they could lose more than one seat. Actually, the fainess rule is respected. But in the case of Adams' method, it's the fairness rule that would apply to a 412.7 seats House, and for Jefferson's, it would apply to a 456.87 members House. And since the fairness rule is different for each seat number, it creates huge discrepacies. The reason why Webster and Huntington-Hill method are far less likely to break such rule is because they produce apportionments where the theoretical number of seats apportioned is far closer to the actual number of seats apportioned. However, the fairness rule can still be broken in an extreme situation such as the previous example.

The other category of system consists in those which imply a scrupulous respect of the fairness rule. Such systems are done in two steps. First of all, you calculate the number of seats theoretically deserved by each State, and you immediately grant each State a number of seats corresponding to the rounding down. Then, different methods exist to apportion the remaining seats, but no State can get more than one seat more. The most common method is the strongest remainder method, which corresponds to the Hamilton method previously used for the congressional apportionment. In this method, you just give the remaining seats to the States whose remainder is higher.
With a more elaborate method, you can divide the population of a State by the number of seats it would get if it got one more (D'Hond't strongest average method), by the arithmetic mean of the number of seats it owns and the numbers of seats it would get (Sainte Lagüe strongest average method), or by the number of seats it currently gets. These systems are a kind of combination between the precedent category and the SR method, but contrary to those, the Fairness Rule would necessarily prevail.
However, any of these method necessarily results in the possibility of an Alabama paradox. There's no need there for an example, there have been several in the history. The reason of these paradoxes is the same that makes the fairness rule unbrokable : such system creates an apportionment for a determined number of seats, and its particuar properties (importance of remainders) may totally change when the quota is raised or lowered.

Now the main question is : which criterion should be privileged, avoiding paradoxes or guaranteeing an absolute fairness ? On a practical level, such question is in great part useless : the odds to break the fairness rule with the Webster or Hamilton method are extremely low, so are the odds to get an Alabama paradox with a strongest average method. However, theoretically it remains interesting. I tend to think that the fairness rule should be privileged, and thus I will always prefer Hamilton's method to Adams' or Jefferson's.
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« Reply #44 on: March 04, 2010, 07:13:40 PM »

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.

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« Reply #45 on: March 04, 2010, 08:18:09 PM »

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 

qn+1 = qn (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.
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« Reply #46 on: March 05, 2010, 06:15:44 AM »

Also, an interesting thing to do is to calculate the number of ill-assigned seats, i.e. the sum of the absolute values of the differences of each State's theoretical number of seats and the actual number of seats it gets. For example, Iowa "deserves" only 4.53 seats but it gets 5 with every method except Jefferson's, so it has 0.53 ill-assigned seats in Jefferson method, and 0.47 in others.

For Adams method, the number of IAS is 19.11
For Jefferson's, it's 18.93
For Hamilton's, it's 11.67
For Humtington-Hill and Webster's, it's 11.69

By definition, the Strongest Remainder Method (ie Hamilton's) minimizes the number of IAS, because the remaining seats are given to the States which have a larger "share of seats". That's why Utah, with 3.46 deserved seats, receives 4, while California receives only 52 because it would deserve 52.45. With Huntington's method, Utah has 0.46 IAS and California has 0.55. With Hamilton's, Utah has 0.54 and California 0.45. So the total is 1.01 for Huntington and 0.99 for Hamilton.
The high results for Jefferson nd Adams methods aren't surprising, since they are the most distorting from real proportionality. A State whose number of seats infringes the fairness rule is characterized by a number of IAS superior to 1.

The number of IAS can be considered as a criterion of fairness. That means that with Hamilton's, the least possible seats are assigned to States they don't belong to.

Also to note, the quotient of the number of IAS/total number of seats decreases when the total number of seats increase.
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« Reply #47 on: March 05, 2010, 04:27:50 PM »

Actually if you're going to talk fairness, one ought to consider the effect of removing the States that under any system will only get the mandated minimum of 1 from consideration.

If you do that and remove Alaska, North Dakota, Vermont, and Wyoming (the states with less than 1/435 of the apportionment population) you get slightly different values, tho no effect under your fairness criteria, which would still recommend shifting a seat from California to Utah.

However, in 1970 your method would recommend shifting a seat from South Dakota to Connecticut and a second from Montana to Oregon, whereas my refinement of only considering the 432 seats left after removing States that under any rational system get only 1 seat (North Dakota had more than 1/435 of the population in 1970) would have South Dakota's 2nd seat going to Oregon, and then Oregon handing the seat it just got from Oregon over to Connecticut.  (I had to go to four decimal places to see whether Montana or Oregon would send a seat to Connecticut.)
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« Reply #48 on: March 05, 2010, 06:04:23 PM »

Actually if you're going to talk fairness, one ought to consider the effect of removing the States that under any system will only get the mandated minimum of 1 from consideration.

If you do that and remove Alaska, North Dakota, Vermont, and Wyoming (the states with less than 1/435 of the apportionment population) you get slightly different values, tho no effect under your fairness criteria, which would still recommend shifting a seat from California to Utah.

However, in 1970 your method would recommend shifting a seat from South Dakota to Connecticut and a second from Montana to Oregon, whereas my refinement of only considering the 432 seats left after removing States that under any rational system get only 1 seat (North Dakota had more than 1/435 of the population in 1970) would have South Dakota's 2nd seat going to Oregon, and then Oregon handing the seat it just got from Oregon over to Connecticut.  (I had to go to four decimal places to see whether Montana or Oregon would send a seat to Connecticut.)

Well, there is no reason to remove Alaska, North Dakota, Vermont and Wyoming since even though they "deserve" less than one seat, they still deserve more than 0.5. Thus Vermont, with 0.94 theoretical seats, deserves its first seat more than Alabama, with 6.90, deserves its seventh. No State in 2000 had less than 0.5 theoretical seats.
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Ernest
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« Reply #49 on: March 05, 2010, 07:34:28 PM »

But we're not considering how to allocate 435 seats, but rather how to allocate 385 seats.
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