Talk:Apportionment paradox

Link to discussion pages
The three pages I merged into "Apportionment paradox" were "Alabama paradox", " New States paradox", and "Population paradox". Here are the existing talk pages for reference: Alabama,  New States,  Population. (Yes, as of now, New States Paradox has no discussion page.) -- Tklalmighty (talk) 05:02, 22 August 2008 (UTC)

smallest case
I got curious and went looking for the simplest possible example of the Alabama paradox; here it is:

Excluding the possibility of a zero (since the US Constitution specifies that each State shall have at least one Representative), the simplest case is this:

—Tamfang (talk) 02:01, 6 November 2008 (UTC)

Why is this "paradoxical"? The only solution to the problem is to increase the number of seats by one more. That will normally cure any Alabama paradox so long as the Hill method is used. The number of seats added to stay out of paradox city may be more than one. The House of Representatives needs to expand every 10 years as the population increases. Using Hill it is very easy to avoid the Alabama Paradox so long as you don't try to do the impossible. The approximate House size can be found as the cube root of the population as a guide and then the number of seats can be minimally adjusted upward to avoid any "paradox". The insanity of trying to keep the number of representatives to too small a number is what causes the problems.--The Trucker (talk) 07:07, 11 April 2009 (UTC)


 * Hm, the Alabama paradox does become less likely as House size increases, but I haven't tried to determine whether there is a number above which it is impossible; nor whether there can be a case where another increase to 'cure' Alabama causes a similar fluke elsewhere. — The Huntington-Hill rule is immune to the Alabama paradox regardless of house size. —Tamfang (talk) 04:05, 12 April 2009 (UTC)


 * I appreciate the comments. I did not think the Hill method was subject to the "Alabama Paradox" but I was not certain of it. This leaves me, however, with a problem in logic.  Either the illustration is not an example of "Alabama Paradox", or Hill is subject to the paradox.  I say this because when I apply what I think is the Hill method to the above example, it generates the same apportionment.  BUt my own rendition of the Hill method is just rounding to geometric mean as:


 * Find a beginning divisor as total population divided by seats (in the case of 7 seats that would be 1.85). Find the quotients as this common divisor divided into the population of the states (in this case we have 2.70, 2.70, and 1.62).  Then get the geometric mean for each case as (2.44, 2.44, and 1.41).  Then for each state if the quotient is greater than the geometric mean we round up, and if not we round down.  That gives us 3, 3, 2 but that's too many and the only way to get this rig to work is to increase the divisor until we generate the correct total number of seats. That is exactly what I do in my spread sheets that calculate the apportionments and it generates the correct stuff every time.  When I generate the target number (i.e. 435) the apportionments match what the book says they should be for Hill. Using Hill (my version) does not cure the problem (assuming there is a problem).  The only cure is to increase the size of the House to 8 instead of 7 or to leave it as it is.  But if you leave it as it is then individuals in the the two larger states have a lot less political power then the individuals in the smaller state.  Is such malapportionment "justiciable" on the basis of equal protection? Or on the basis of "Shall insure a republican form of government"?  —Preceding unsigned comment added by Mikcob (talk • contribs) 07:18, 12 April 2009 (UTC)


 * Either the illustration is not an example of "Alabama Paradox", or Hill is subject to the paradox. Or your version of Hill is flawed.  If I had more energy I'd work it out ... —Tamfang (talk) 06:19, 13 April 2009 (UTC)


 * I like my version of Hill because it can easily be entered in a spread sheet. But that is irrelevant. The loss of a seat to the least populous state is not a "paradox" unless you use the very loose definition of "paradox" which is merely "a result that appears counterintuitive".  And if you use that definition then a solution of 3, 2, 2 is equally paradoxical.  Any distribution of the seats will produce a malapportionment of voting power of .2666666666 or .26666667 and to have a malaportionment between identically populous states (parties) seems more "paradoxical" than to stick it to the small group that enjoyed the per capita voting power before the one seat increase. The small state is now on the other side of the power see-saw and that's why it "looks" so ugly.  If you are looking for the least malappotioinment of voting power the result of 3, 3, 1 is as correct a solution as any other.  No matter which of the major "methods" you use you will get the same result or a result that awards a different number of representatives to two equally populous states (parties). The only sane solution is to remove the cap and allow 8 members.  That was the solution to the "Alabama Paradox" at the time.  The "anomaly" occurred at 300 seats and was removed by raising the number to 325 where the methods of Webster and Hamilton produced identical apportionments.  Between 1870 and 1913 reapportionments were accomplished in such a way as to insure that no state lost a seat in the House.  —Preceding unsigned comment added by Mikcob (talk • contribs) 22:05, 13 April 2009 (UTC)


 * then a solution of 3, 2, 2 is equally paradoxical. Well, not equally so; no one gets less as a result of increasing the total. —Tamfang (talk) 05:24, 14 April 2009 (UTC)

Can the Huntington-Hill method, sometimes called "method of equal proportions" be clarified or justified?
After this recent Judd Gregg and US Census flap, I've been considering the hot potato that apportionment of US Representatives will be after 2010. One problem is that of counting (who to count and who not to), but that's a different problem than what is concerning me at the moment. So, assuming we have undisputed census figures for each state, the (hopefully blind and objective) mathematical method for determining how many Representatives each state gets sure seems different than what is depicted at United_States_Congressional_Apportionment.

The constraints applied to this problem is that the total number of Representatives is fixed and determined in advance by law; 435, and that each state, even the least populous, must get at least one Representative.

Let
 * $$ K \ $$ be the number of states (currently 50).


 * $$ 1 \le k \le K \ $$ be the index of kth state. It doesn't matter how they're ordered.


 * $$ P_k \ $$ be the agreed census population for the kth state.


 * $$ P = \sum_{k=1}^{K} P_k \ $$ is the total population of all K states (excluding DC and the territories).


 * $$ N_k \ $$ is the number of Representatives for the kth state that we are trying to determine.


 * $$ N = \sum_{k=1}^{K} N_k \ $$ is the total number of Representatives in the House for all K states (excluding DC and the territories) which is currently 435.


 * $$ q > 0 \ $$ is the nationwide constant of proportionality or quota ratio for proportionately allocating Representatives or a state as a function of its population.

So, if we could actually have fractional numbers of persons as Representatives,


 * $$ N_k \ = \ q \ P_k \quad \quad \forall \ 1 \le k \le K $$


 * $$ \sum_{k=1}^{K} N_k \ = \sum_{k=1}^{K} \ q \ P_k = q \sum_{k=1}^{K} P_k $$

or


 * $$\ N = q P \quad $$


 * $$\ q = \frac{P}{N} \quad $$

But, of course, we cannot divide Congressional Representatives into fractions even if we might like to tear them apart on occasion. Each states House delegation must be an integer number of people at least as big as one. Wouldn't this mean:


 * $$ N_k \ = \lceil q \ P_k \rceil \quad \quad \forall \ 1 \le k \le K $$ ?

where


 * $$ \lceil x \rceil=\min\,\{n\in\mathbb{Z}\mid n\ge x\}$$ is the ceiling function (which means always round up).

Now if q>0 was arbitrarily small (but positive), then each state would get 1 Representative. It wouldn't be particularly well apportioned and it wouldn't add up to N=435. We want


 * $$ N = \sum_{k=1}^{K} N_k = \sum_{k=1}^{K} \lceil q \ P_k \rceil \ $$

Then couldn't q be increased monotonically, thus increasing some of the Nk and $$ \sum_{k=1}^{K} N_k = \sum_{k=1}^{K} \lceil q \ P_k \rceil \ $$ until it reaches the legislated N=435 value? Would that not be the meaning of proportional representation with the constraints that Nk must be an integer at least as large as 1? How is there any paradox in this method (assuming that, as q increases we don't have two states simultaneously increasing their integer Nk and the total jumping from 434 to 436) and where the heck does that Huntington-Hill method that is depicted at United_States_Congressional_Apportionment come from? How does that possibly have anything to do with true proportional allocation of a fixed number of seats in the House?

Can someone explain this? 96.237.148.44 (talk) 01:49, 14 February 2009 (UTC)


 * This question might be better raised in Talk:Apportionment (politics) or Talk:United States congressional apportionment.
 * Big states would object to a 'ceiling' rule because it gives some small states much more weight than in some other rules. (This bias is symmetric to the D'Hondt method's bias against small states.)  The Huntington-Hill rule "guarantees that no additional transfer of a seat (from one state to another) will reduce the ratio between the numbers of persons per Representative in any two states" (from which it follows that Nk is never zero); what's wrong with that as a criterion? —Tamfang (talk) 23:04, 16 February 2009 (UTC)


 * Answered at Talk: Huntington–Hill method . --84.151.17.20 (talk) 23:10, 17 November 2009 (UTC)

what is the Population Paradox?
One of the three properties proven incompatible in 1982 is given as this:
 * It does not have the population paradox: If one party gets more votes, whereas the other parties retain the same number of votes, that party does not get fewer seats.

But this isn't the same paradox described below: two parties grow at different rates and the small one with rapid (proportional) growth loses to a bigger one with slower (proportional) growth. I'd need to see an example of a scheme in which a party that grows while all others are static loses a seat. —Tamfang (talk) 04:19, 25 July 2009 (UTC)


 * It's surely possible to give such an example (not with Hamilton's method, but with some other method for which the quota rule holds), but such an example isn't necessary for the population paradox. The criterion for the paradox according to Balinski/Young is that state i gets fewer seats and j more, although i's population increases and j's decreases. --84.151.17.20 (talk) 01:59, 18 November 2009 (UTC)


 * Do you agree that the case you describe is not the case described in the bullet point quoted above? —Tamfang (talk) 02:27, 11 December 2009 (UTC)

paradoxes and the US Congress

 * The [population] paradox arises because of rounding in the procedure for dividing the seats. See the apportionment rules for the United States Congress for an example.

User:CRGreathouse added the last sentence; someone else deleted, saying the current rule is not subject to this paradox, and CRG reverted. Unfortunately neither United States congressional apportionment nor Huntington–Hill method (the current rule) says whether or not the p.p. applies; can CRG give more specific information? —Tamfang (talk) 02:25, 11 December 2009 (UTC)

Balinski–Young theorem
The class of non-deterministic algorithms (with the last seat awarded by proportional dice rolls) can satisfy all the stated requirements on an amortized basis. It used to be that the population at large was mortified to even contemplate a stochastic mechanism in an election context. But this was before blockchain ledgers entered the zeitgeist. It's relatively trivial to turn a blockchain into a distributed stochastic algorithm that's verifiably fair (to the extent that the blockchain itself is not compromised by a majority tenant). &mdash; MaxEnt 14:41, 30 September 2018 (UTC)

Question I removed from the article
Not my question, but it felt mean to remove it completely.

Using data from Wikipedia results in a New Jersey, not an Alabama, Paradox. Is there an alternate source of 1880 Census information?

All the best — Preceding unsigned comment added by 62.254.205.3 (talk) 11:28, 14 October 2020 (UTC)

This is misleading
The US House of Representatives is constitutionally required to allocate seats based on population counts, which are required every 10 years. The size of the House is set by statute.

The Alabama paradox does not happen if the size of the house is fixed. 24.8.46.212 (talk) 02:11, 12 March 2021 (UTC)dave@twinsprings.com

Example of quota rule violation with divisor method
Suppose you follow the procedure,

1) Find natural quota for each state (or party). Assume that no states have exactly the same populations.

2) Round each quota up or down to the nearest whole number.

3) If the result from step 2 doesn't match the desired number of seats, proportionally increase or decrease every quota until one state (or party) gains or loses a seat. Repeat until desired number of seats is reached.

* According to Why it’s mathematically impossible to share fair, rounding to the nearest integer like this is the same as Webster/Sainte-Laguë_method although that article describes a more confusing way of reaching this result.

Now consider the case where the natural quotas are 9000 from one very large state (with 90 million population), and approximately 1.6 each from 625 small states (with between 15600 and 16400 population each, no two exactly the same). We desire 10 000 seats. If we round, we end up with 9000+625*2 = 10250 seats, 250 more than we want.

So we proportionally decrease all quotas, until 250 seats have been eliminated. However, when we do this, the large state is the first to lose a seat; and it is also the second. The quota rule has already been violated. In fact, before we reduce the smallest state from 1.56 quota to 1.5, we have already reduced the largest state by more than 250 seats, so all 250 seats required to change the total to 10 000 are subtracted from the large state.

If you're like me, you might conclude that the "quota rule" is not very important here and an apportionment system should not be rejected for not satisfying it; I don't want to edit the article because some editors might revert said edits as "original research", but if you find this interesting you could try to add it to the article. 73.65.167.168 (talk) 05:57, 27 October 2022 (UTC)