General discussion about Congressional Apportionment

[Kevinstat]

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)) ?

No, it's (2*n*(n+1))/(n+(n+1)), or 2n(n+1)/(n+(n+1)), or (2n^2+2n)/(2n+1)) as mathematicians might write it, but I prefer the middle one.

The harmonic mean of two numbers (or any number of numbers) is the reciprocal of the arithmetic mean of the reciprocal of those numbers.  Following order of operations, with multiplication and division ahead of addition in the absence of parentheses (exponents, which trump multiplication and division, and subtraction which is equal in priority to addition are not included here), ...

1/(((1/n+1/(n+1)))/2) = 2/((1/n+1/(n+1))) =  2/(((n+1)+n)/(n(n+1))) = 2n(n+1)/(n+(n+1)) .  This is for the "n+1th" seat.  The harmonic mean formula for a priority value the nth seat would be 2n(n-1)/(n+(n-1)) .

[Kevinstat]

BTW, the reason I oppose statehood for the Virgin Islands, American Samoa, Guam and the Northern Mariana Islands is that they don't have enough population to warrant a Representative of their own without the floor.  (Even if Guam and the NMI were to unite as the Mariana Islands, they'd be too small population-wise.)

You oppose statehood for Wyoming also? It'd be a lot fairer (though also completely absurd) if we could gerrymander a couple hundred thousand Montana voters into Wyoming's district.

No.  If we added one more seat to the HoR and doubled Wyoming's population in 2000, it would have legitimately earned 2 seats under the current apportionment formula (its 2nd seat would be the 387th seat under that scenario), so there's no reason for Wyoming to not be a State.  For the 2000 Census the cutoff point for Statehood under my criteria would be 456,567 (half of the minimum population needed to get a second Representative in a House with 436 members). Wyoming was above  that with an apportionment population 495,304.

The population of all four minor territories combined in 2000 was only 389,929 so even together they didn't meet the target. Possibly the combined Marianas (NMI + Guam) might be able to grow into Statehood, especially if we ever increased the size of the House to use the cube root formula as both Guam and NMI are growing at a faster rate than the U.S. With 676 seats in 2010, the target number would be approximately 322 thousand and the combined Marianas will have approximately 270 thousand.  (Kept at 435 seats and the target number for 2010 will be approximately 500 thousand, and Wyoming will again be comfortably over that number.)

Well, under the equal proportions (geometric mean), harmonic mean or smaller divisors methods, the divisor for a state to get its first seat, susing the same formula as for all additional seats under the respective methods, would be 0.  The limit as n approaches 0 from the positive side of any positive number/n is (positive) infinity, so under either of those three methods (including the current equal proportions method), the priority value for the 1st seat of any state with at least one resident or overseas person attibuted to that state would be infinity and the largest s states would have at least one seat up to the number s of seats, with any remaining seats being awarded on the basis of the non-infinite priority values.  That's the one thing I like about the equal proportions method as opposted to major fractions, besides the fact that my state could potentially benefit from it in a couple decades or so.  The constitutional requirement that each state, regardless of population, have at least one seat goes naturally with it.

Your doubling of a states population to see if it would get two seats seems disjoint to me given how the equal proportions method works.  The divisor for a state's second seat is less than half the divisor for a state's forth seat in any of the main divisor methods detailed by jimrtex except for the greatest divisors method, so why should we assume the divisor for a state's first seat if there was no "floor" or 1-seat guarantee would be half the divisor for a state's second seat under any of those same methods?

In Wyoming's case it would probably get a seat under major fractions with no 1-seat guarantee (the divisor for a state's first seat in that method would be .5, so you would multiply a state's apportionment population by 2 to get its priority value for its first seat), but probably not under greatest divisors with no such guarantee (where the divisor for a state's first seat would be 1), which is the only one of the five main divisor methods (with or without a 1-seat guarantee) where the divisor for a state's nth seat is exactly half the divisor for a state's 2nth seat.

Wow, True Federalist beat me to the point I was trying to make, not as well stated as he did, about the equal proportions (geometric mean) method not having a natural cutoff between 0 seats and having 1 seat, but I see that as an advantage to the equal proportions method as far as seat allocation among U.S. states is concerned because it goes with the U.S. Constitution's guarantee of at least 1 seat in the House of Representatives for each state.

[Kevinstat]

Also, Nevada wouldn't have had a single representative in the U.S. House under the methods used at the time with no floor for much of its time as a state, stretching through 1930s reapportionment and I believe through the 1940s reaportionment if the Congressional Dems hadn't switched from major fractions to equal proportions in 1941 to prevent then-solidly Democratic Arkansas from losing a seat to then-Republican-leaning Michigan.

[Kevinstat]

The House of Representatives is intended to represent the people, not the States.  The d'Hondt seat allocation method (equivilent to Jefferson's method) badly under-represents the people of the very smallest states those who vote for smaller parties.  If you have one state party with 121,000 22,355 votes and another with 59,000 11,087 votes, or barely over a 2:1 vote ratio in population, it produces a 4:1 ratio in representation.

See the West Belfast 1996 Forum election for a real example of this.  I crossed off the badly in jimrtex's post because I actually like d'Hondt as a seat-allocation method for parties if a list method is going to be used, since, to use the West Belfast 1996 example, if (Provisional) Sinn Féin in West Belfast had divided into two parties with half of the voters who voted for Sinn Féin in real life voting for each party, and all else being equal, each of those two parties would have bet the Social Democratic and Labour Party out for a second seat, and those voters, as crazy as they were for voting for SF, shouldn't be penalized for their consolidation.

Interestingly enough, the one successful SDLP candidate in that election, Joe Hendron, was the MP for West Belfast at the time, although redistricting had occured since his election as an MP and he had only won narrowly and because of Unionist tactical anti-SF voting.  Sinn Féin leader Gerry Adams, whom Hendron had unseated in 1992, defeated Hendron in 1997 and has been elected as West Belfast's MP (although he has never taken his seat in Westminster as that would require taking an oath to the Queen) in every election since.

[Kevinstat]

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.

Well, the Jefferson or d'Hondt method (I think I goofed by capitalizing the first letter d earlier; I've fixed that now) minimizes the largest (best) level of representation per capita or per voter, respectively, of any two or more states or parties, respectively, given the total number of seats awarded to those states or parties, except where overriden in the Jefferson apportionment method, as it would have to be practiced in the U.S., by every state having to have at least one seat in the governing body for which the seats are apportioned.  Said another way, the Jefferson or d'Hondt method maximizes the smallest (best) number of people or votes, respectively, per seat of any two or more states or parties, respectively, given the total number of seats awarded to those states or parties, except where overriden in the Jefferson apportionment method, as it would have to be practiced in the U.S., by every state having to have at least one seat in the governing body for which the seats are apportioned.

The d'Hondt method could thus be said to minimize the greatest "fat rat effect" benefitting the voters of any party among two or more parties given the total number of seats awarded to those parties, as long as that "fat rat effect" is measured in either absolute terms (seats per vote or votes per seat, not compared to any other party or parties (or slate or slates of candidates, including "slates" of only one candidate)) OR is measured in comparison to all parties or slates of candidates combined (i.e. how a the largest number of seats per vote or the smallest number of votes per seat, repsectively, among a given set of two or more parties compares to the total number of seats divided by the total number of votes, or vice versa, for the governing body being elected).

If you're more concerned about parties making out like fat rats than about parties being greatly underrepresented (and there will likely be slates of candidates receiving no seats anyway, and comparing the underrepresentation of voters for any slate receiveing no seats is like comparing the underrepresentation of voters who voted in 2008 for John McCain to that of those who voted for Chuck Baldwin - both were 100% unrepresented in terms of the number of Presidential and Vice Presidential seats awarded), than d'Hondt is your method.  Practically every elected official in a non-proportional election will talk about how he or she represents all his or her constituents, not only those who voted for him or her.  While a lot of that is just talk, I think it's reasonable for someone who to be more concerned about minimizing underepresentation in terms of voters having someone representing them who they get to vote for or against, while being more concerned about limiting the "fat rat effect" when it comes to the partisan or ideological makeup of the governing body in question.

It just so happens that, everything else being equal, larger parties will fare the same or better, while smaller parties or states will fare the same or worse, under a method (d'Hondt or Jefferson) that minimizes the "fat rat effect" than under one which minimizes the worst case of underrepresentation (the smallest divisors or Adams method), absolute difference in votes per representative (harmonic mean) or representation per voter (major fractions, Webster or Sainte-Laguë) between any two parties given the number of seats between them, relative (percentage) difference either in votes per representative or representation per capita/voter (equal proportions) between any two parties given the number of seats between them, or one which minimizes the number of ill-assigned seats (largest remainders or Hamilton).

[Kevinstat]

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.

Just to be clear, I do not support D'Hondt for a seat apportionment method for a regional unit of voters (such as a state).  My favorite method there would probably be major fractions (Sainte Laguë), but with the current apportionment method (geometric mean or equal proportions) a close second, harmonic mean third, the paradoxical strongest remainders/Hamilton method fourth and even smallest divisors/Adams ahead of greatest divisors/Jefferson, but not by much.  But when you consider that one of congresswoman Allison Shwartz (D-PA-13)'s constituents is or has been Keystone Phil and one of congressman Geoff Davis (R-KY-4) constituents is Bandit3, you can see how representation by someone you get to vote for or against is different from representation by someone you actually voted for.  Currently, if Keystone Phil still lives in PA-13 and Bandit3 still lives in KY-4, I doubt either one has any representative in the U.S. House of Representatives (and Bandit3 in the U.S. Congress period; I'm not sure if Keystone Phile voted for Specter in the 2004 general election) whom they voted for.  That's democracy (or the democratic aspects of our republic) for you.  If that's okay than having D'Hondt (which can't be any less proportional than winner-take all) for multi-winner elections should be okay too.