General discussion about Congressional Apportionment

[Antonio the Sixth]

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 ?

[Antonio the Sixth]

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

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.

[Antonio the Sixth]

Yes, I see, it always comes to the opposition between those who think "fairness" means the smaller differences in terms of people per seat and those who think it means the  smaller differences between the theorical number of seats and the actual number. I guess it's gonna be an endless debate.

Anyways, the best solution would be indeed to dramatically raise the number of seats in the House. The US should have at least 650, and even then they would be far from other democracies in terms of representativity.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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.

Oh, that's weird but interesting.
So if I well understand the HM's formula is (n^2*(n+1))/(n+(n+1)) ?

[Antonio the Sixth]

Oh yeah, I see now.
(1/3+1/2)/2=0.416667
1/0.416667=2.4

Interestingy, the geomentric mean between two numbers is also the geometric mean between their arithmetic mean and their geometric mean...

[Antonio the Sixth]

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

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).

[Antonio the Sixth]

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)
- 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.

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.

[Antonio the Sixth]

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 ?!? 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.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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

State	Hunt.-Hill	Adams	Hamilton	Webster	Jefferson
Maine	2	2	2	2	2
New Hampshire	2	2	2	2	2
Vermont	1	1	1	1	1
Massachusetts	10	10	10	10	10
Rhode Island	2	2	2	2	1
Connecticut	5	6	5	5	5
New York	29	28	29	29	30
Pennsylvania	19	19	19	19	19
New Jersey	13	13	13	13	13
Delaware	1	2	1	1	1
Maryland	8	8	8	8	8
West Virginia	3	3	3	3	2
Virginia	11	11	11	11	11
North Carolina	13	12	13	13	13
South Carolina	6	6	6	6	6
Georgia	13	13	13	13	13
Florida	25	24	25	25	26
Kentucky	6	6	6	6	6
Tennesse	9	9	9	9	9
Alabama	7	7	7	7	7
Mississipi	4	5	4	4	4
Louisiana	7	7	7	7	7
Arkansas	4	4	4	4	4
Texas	32	31	32	32	33
Ohio	18	17	18	18	18
Indiana	9	9	9	9	9
Illinois	19	19	19	19	20
Michigan	15	15	15	15	16
Wisconsin	8	8	8	8	8
Minnesota	8	8	8	8	7
Iowa	5	5	5	5	4
Missouri	9	9	9	9	9
North Dakota	1	1	1	1	1
South Dakota	1	2	1	1	1
Nebraska	3	3	3	3	2
Kansas	4	4	4	4	4
Oklahoma	5	6	5	5	5
Montana	1	2	1	1	1
Wyoming	1	1	1	1	1
Idaho	2	2	2	2	2
Colorado	7	7	7	7	7
Utah	3	4	4	3	3
New Mexico	3	3	3	3	2
Arizona	8	8	8	8	8
Nevada	3	3	3	3	3
California	53	50	52	53	55
Oregon	5	6	5	5	5
Washington	9	9	9	9	9
Alaska	1	1	1	1	1
Hawaii	2	2	2	2	1
Total	435	435	435	435	435
Theor. Seats #	435.69	412.70	N/A	436.04	456.87

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.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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

Even so, the least populous States (Wyoming) deserves 0.67 seats, enough to secure it.

[Antonio the Sixth]

I've always wondered where the idea that "it's fair and normal to under-represent small party in proportional elections". The voice of a person who votes for a small party has to be heard as well as the one those who chose large parties. The D'Hondt method is hardly proportional, just as the Jefferson method of Apportioning Congress would be blatantly unfair, hadn't the Senate existed to compensate it.

[Antonio the Sixth]

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.

[Antonio the Sixth]

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.

I think this argument is really too pragmatic to be considered for apportionment. We don't have to forget that it's about apportioning Representatives to States according to which deserves a Rep the most.
And Jim, remind that differences between all those members remain marginal enough to avoid too overworked Representatives. I'm not worried about that.

[Antonio the Sixth]

Well, my ranking is :

Webster/Sainte-Laguë method, corrected with the fairness rule
Hamilton/Strongest remainder method
Huntington-Hill method, corrected with fairness rule
Webster/Sainte-Laguë method
Huntington-Hill method
Hamronic mean method, corrected with the fairness rule
Hamronic mean method
Jefferson method, corrected with the fairness rule
Adams method, corrected with the fairness rule
Jefferson method
Adams method

This ranking is a generic one. In the case of US Congress apportionment, I'd rank Jefferson method higher (because of the Senate). In the case of a voting system, I'd exclude Adams, Huntington-Hill and harmonic mean, because this would give 1 seat to every party.

[Antonio the Sixth]

I'm bumping this so that we can use this thread to comment the recent apportionment numbers.