## Congressional apportionment

For the discussion of math. Duh.

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Lazar
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### Congressional apportionment

On this election day, I have a question about the apportionment formula used for the US House of Representatives after each decennial census. A while ago I was doing some worldbuilding and was trying to apportion seats for a fictional United States, but I was intimidated by the description of the current Huntington-Hill method, which requires hundreds of calculations. So I came up with my own, simpler method. I divide each state's population by the combined population of the 50 states and then multiply it by 435, and I place the resulting numbers in two columns: the digits to the left of the decimal point in column A (which is the starting number of seats for each state), and those to the right in column B. If a state's A value is 0, I round it up to 1. Then I add up the A column and subtract it from 435, giving me x remaining seats to be assigned. Then I rank the states by their B values (excluding any states which have been rounded up to 1), and assign the x remaining seats to the top x states. At the time, I tested this with figures of the 2010 census and came up with results identical to those of our world. So am I missing something, or is my method guaranteed to yield the same results as the one that's really used?
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Nicias
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### Re: Congressional apportionment

Your method is the Largest Remainder Method.It is not the same as the H-H method. You can give this a look for more information:

http://www.ams.org/samplings/feature-co ... portionii1

Lazar
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### Re: Congressional apportionment

Oh okay, so mine would be subject to the Alabama paradox. That's unforch. But thanks.
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Derek
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### Re: Congressional apportionment

Lazar wrote:Oh okay, so mine would be subject to the Alabama paradox. That's unforch. But thanks.

I believe all (realistic) apportionment methods are subject to the Alabama paradox.

mike-l
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### Re: Congressional apportionment

I may be incorrect, but H-H would seem immune to the paradox as it is blind to the total number of seats
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Qaanol
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### Re: Congressional apportionment

It is worth noting that the problem of Congressional apportionment is identical to that of proportional representation, so any method for one works the same for the other. You might be interested to look into the various and sundry formulæ for proportional representation that exist.
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Derek
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### Re: Congressional apportionment

mike-l wrote:I may be incorrect, but H-H would seem immune to the paradox as it is blind to the total number of seats

Well according to a theorem, no apportionment method can satisfy all three of these "intuitive" criteria:
-If the fair share of seats for a state is between N and N+1 seats, then it gets either N or N+1 seats.
-It does not have the Alabama paradox.
-If the (proportional) population of state A grows and state B shrinks, no seat is transferred from A to B.

According to this source, the Huntington-Hill method violates the first of these criteria (in rare circumstances). (So my original statement was wrong)

mike-l
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### Re: Congressional apportionment

That's interesting (though having spent quite a bit of time thinking on election methods, not overly surprising. Impossibility theorems seem to be the name of the game). I'd definitely like to look closer at the claim that every divisor method violates quota, and I don't particularly trust rangevoting.org as a source, so I'd like to verify the initial claim, but definitely interesting (and believable, prior comments notwithstanding) nonetheless.
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DavCrav
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### Re: Congressional apportionment

mike-l wrote:That's interesting (though having spent quite a bit of time thinking on election methods, not overly surprising. Impossibility theorems seem to be the name of the game). I'd definitely like to look closer at the claim that every divisor method violates quota, and I don't particularly trust rangevoting.org as a source, so I'd like to verify the initial claim, but definitely interesting (and believable, prior comments notwithstanding) nonetheless.

One suggestion is, I have ten states, one with a million people in it and nine with 1 person each. If I distribute (say) 10 seats between then, they each get 1, but by quotas shouldn't the big state get between 9 and 10, having more than 99% of the population in it? Or have I misunderstood?

mike-l
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### Re: Congressional apportionment

Yeah I simply haven't sat and convinced myself that there is no way around it. Like picking one state at random to get one seat and giving 9 to the big one, can that be consistently done while maintaining the two monotonicity criteria?

The claim is believable, and Id bet on it being correct, but I've been burned believing rangevoting.org blindly before so I just want to convince myself.
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Xanthir
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### Re: Congressional apportionment

DavCrav wrote:
mike-l wrote:That's interesting (though having spent quite a bit of time thinking on eating contest methods, not overly surprising. Impossibility theorems seem to be the name of the game). I'd definitely like to look closer at the claim that every divisor method violates quota, and I don't particularly trust rangevoting.org as a source, so I'd like to verify the initial claim, but definitely interesting (and believable, prior comments notwithstanding) nonetheless.

One suggestion is, I have ten states, one with a million people in it and nine with 1 person each. If I distribute (say) 10 seats between then, they each get 1, but by quotas shouldn't the big state get between 9 and 10, having more than 99% of the population in it? Or have I misunderstood?

That's different - minimum rep requirements cause their own problems. They're not related to the "can't satisfy all three" result.
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Derek
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### Re: Congressional apportionment

DavCrav wrote:One suggestion is, I have ten states, one with a million people in it and nine with 1 person each. If I distribute (say) 10 seats between then, they each get 1, but by quotas shouldn't the big state get between 9 and 10, having more than 99% of the population in it? Or have I misunderstood?

You can give 0 seats to each of the small states and still meet the quota requirement. Of course in the US House of Representatives this is not allowed, but the mathematical theorem isn't concerned with that.

Yeah I simply haven't sat and convinced myself that there is no way around it. Like picking one state at random to get one seat and giving 9 to the big one, can that be consistently done while maintaining the two monotonicity criteria?

The claim is believable, and Id bet on it being correct, but I've been burned believing rangevoting.org blindly before so I just want to convince myself.

Well I got the impossibility theorem from Wikipedia, so I would suggest starting there and checking their sources.

mike-l
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### Re: Congressional apportionment

Yeah I tracked down the original paper, so I'm happy now to accept it, though still working through the details to understand it.

In any case, it looks like the impossibility result is nonetheless a very rare occurrence for H-H, (and indeed has never ocurred in practice), despite arbitrarily large violations of quota being possible, but while the remainder method suffers the Alabama paradox much more often (orders of magnitude more often). So, despite an ideal system being impossible, there is still a very quantifiable difference and advantage between systems.
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mathmannix
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### Re: Congressional apportionment

Derek wrote:
mike-l wrote:I may be incorrect, but H-H would seem immune to the paradox as it is blind to the total number of seats

Well according to a theorem, no apportionment method can satisfy all three of these "intuitive" criteria:
-If the fair share of seats for a state is between N and N+1 seats, then it gets either N or N+1 seats.
-It does not have the Alabama paradox.
-If the (proportional) population of state A grows and state B shrinks, no seat is transferred from A to B.

According to this source, the Huntington-Hill method violates the first of these criteria (in rare circumstances). (So my original statement was wrong)

Under the current method, the two most populous states, California and Texas, have 53 and 36 seats in the House, respectively. There are seven states with the minimum of one representative, and the total House size has been set at 435 since 1911 - but this is merely a statute, not from the Constitution, which merely states that the seats in the House of Representative are apportioned by population (according to the decennial Census), with a minimum of one for the least populous state(s).

Wouldn't a simple method of having one representative for every X people in a state satisfy all these methods? (From the time it was implemented, of course.) It wouldn't be practical to keep the number X constant, as the size of the House would theoretically grow without bound. But it would be fair.

If we set X at 500,000 people (somewhat arbitrary, as it is not too far below the current population of the smallest state - Wyoming, 563,626 in the 2010 census) then California would have 74 seats, Texas 50, and the total House 592, not tremendously more than the current 435.

But what if X were defined as the population of the least populous state? This has been Wyoming since the 1990 census, previously Alaska (since it was a state) and before that Nevada. Currently (2010 census), if X were 563,626, then California would have 66 seats, Texas 44, and the total House 524. While this could also theoretically grow without bound (especially if the population of Wyoming or Vermont were to decrease until they were essentially empty, like rotten boroughs), it should be slower. Does this violate any of the criteria above?
Last edited by mathmannix on Thu Nov 13, 2014 1:20 pm UTC, edited 1 time in total.
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mike-l
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### Re: Congressional apportionment

That method seems to work. The impossibility theorem is for fixed house sizes.
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