2020 House Seat Distribution -- Did New York and Ohio Get Screwed?

While this is supposed to be a forum about gambling and Vegas, it welcomes all intellectual discussion of a mathematical nature. In the spirit of the lively threads on Easy Math Puzzles and Wordle, I believe this is consistent with the spirit of the forum. Anyone wishing to press charges on the nature of this thread may do so through any other moderator than myself.

That said, I have been looking into the "Alabama Paradox," which says basically that if more seats were added to the House of Representatives, with all other things being equal, some states would get less seats.

Before going further, let me explain how the Constitution appropriates seats.

1. After each Census (which happen on years evenly divisible by 10), a count of the population is taken of all 50 states.
2. Each state is given as a minimum number of seats the greater of 1 and it's fair share of the total seats, which is currently 435, rounded DOWN.
3. Doing so will probably result in less than 435 seats apportioned, because of the rounding down rule. A count of extra seats needed will be taken to get to the total.
4. The number of states from step 3 that were rounded down the most shall each get an extra seat.

Following this method and 2020 Census data, I calculate how many seats each state should get, as follows. Note the table is sorted in "rounded down" order. This is the portion of a seat the state lost due to being rounded down.

Rank	State	Population	Fair Share of seats	Rounded down w/ 1 minimum	Rounded Down	Extra vote	Total votes	Actual votes	Difference
1	California	39,576,757	51.995	51	0.995	1	52	52	0
33	Arkansas	3,013,756	3.959	3	0.959	1	4	4	0
26	Kentucky	4,509,342	5.924	5	0.924	1	6	6	0
17	Indiana	6,790,280	8.921	8	0.921	1	9	9	0
40	Hawaii	1,460,137	1.918	1	0.918	1	2	2	0
34	Mississippi	2,963,914	3.894	3	0.894	1	4	4	0
35	Kansas	2,940,865	3.864	3	0.864	1	4	4	0
6	Illinois	12,822,739	16.846	16	0.846	1	17	17	0
41	New Hampshire	1,379,089	1.812	1	0.812	1	2	2	0
42	Maine	1,363,582	1.791	1	0.791	1	2	2	0
36	New Mexico	2,120,220	2.785	2	0.785	1	3	3	0
20	Wisconsin	5,897,473	7.748	7	0.748	1	8	8	0
29	Connecticut	3,608,298	4.740	4	0.740	1	5	5	0
9	North Carolina	10,453,948	13.734	13	0.734	1	14	14	0
23	South Carolina	5,124,712	6.733	6	0.733	1	7	7	0
24	Alabama	5,030,053	6.608	6	0.608	1	7	7	0
21	Colorado	5,782,171	7.596	7	0.596	1	8	8	0
37	Nebraska	1,963,333	2.579	2	0.579	1	3	3	0
27	Oregon	4,241,500	5.572	5	0.572	1	6	6	0
4	New York	20,215,751	26.559	26	0.559	1	27	26	-1
7	Ohio	11,808,848	15.514	15	0.514	1	16	15	-1
22	Minnesota	5,709,752	7.501	7	0.501	1	8	8	0
43	Rhode Island	1,098,163	1.443	1	0.443	0	1	2	1
44	Montana	1,085,407	1.426	1	0.426	0	1	2	1
38	Idaho	1,841,377	2.419	2	0.419	0	2	2	0
14	Arizona	7,158,923	9.405	9	0.405	0	9	9	0
12	Virginia	8,654,542	11.370	11	0.370	0	11	11	0
39	West Virginia	1,795,045	2.358	2	0.358	0	2	2	0
2	Texas	29,183,290	38.340	38	0.340	0	38	38	0
3	Florida	21,570,527	28.339	28	0.339	0	28	28	0
30	Utah	3,275,252	4.303	4	0.303	0	4	4	0
45	Delaware	990,837	1.302	1	0.302	0	1	1	0
10	Michigan	10,084,442	13.249	13	0.249	0	13	13	0
15	Massachusetts	7,033,469	9.240	9	0.240	0	9	9	0
11	New Jersey	9,294,493	12.211	12	0.211	0	12	12	0
28	Oklahoma	3,963,516	5.207	5	0.207	0	5	5	0
31	Iowa	3,192,406	4.194	4	0.194	0	4	4	0
46	South Dakota	887,770	1.166	1	0.166	0	1	1	0
13	Washington	7,715,946	10.137	10	0.137	0	10	10	0
18	Maryland	6,185,278	8.126	8	0.126	0	8	8	0
25	Louisiana	4,661,468	6.124	6	0.124	0	6	6	0
5	Pennsylvania	13,011,844	17.095	17	0.095	0	17	17	0
19	Missouri	6,160,281	8.093	8	0.093	0	8	8	0
8	Georgia	10,725,274	14.091	14	0.091	0	14	14	0
16	Tennessee	6,916,897	9.087	9	0.087	0	9	9	0
32	Nevada	3,108,462	4.084	4	0.084	0	4	4	0
47	North Dakota	779,702	1.024	1	0.024	0	1	1	0
48	Alaska	736,081	0.967	1	-0.033	0	1	1	0
49	Vermont	643,503	0.845	1	-0.155	0	1	1	0
50	Wyoming	577,719	0.759	1	-0.241	0	1	1	0
	United States	331,108,434	435.000	413	22.000	22	435	435	0

* This column also reflects the rule that each state shall have a minimum of one Representative.

Note the 413 seats apportioned after rounding down. That means the 22 states who were rounded down the most should get an extra seat.

Also note the column for actual number of seats. This shows that New York and Ohio seem to be missing a seat. Meanwhile, Rhode Island and Montana have one too many. It's probably not a coincidence these states are all close to 22 down the list.

My question is why the disparity between my calculations and the actual apportionment?

Quote: Wizard

While this is supposed to be a forum about gambling and Vegas, it welcomes all intellectual discussion of a mathematical nature. In the spirit of the lively threads on Easy Math Puzzles and Wordle, I believe this is consistent with the spirit of the forum. Anyone wishing to press charges on the nature of this thread may do so through any other moderator than myself.

That said, I have been looking into the "Alabama Paradox," which says basically that if more seats were added to the House of Representatives, with all other things being equal, some states would get less seats.

Before going further, let me explain how the Constitution appropriates seats.

1. After each Census (which happen on years evenly divisible by 10), a count of the population is taken of all 50 states.
2. Each state is given as a minimum number of seats the greater of 1 and it's fair share of the total seats, which is currently 435, rounded DOWN.
3. Doing so will probably result in less than 435 seats apportioned, because of the rounding down rule. A count of extra seats needed will be taken to get to the total.
4. The number of states from step 3 that were rounded down the most shall each get an extra seat.

Following this method and 2020 Census data, I calculate how many seats each state should get, as follows. Note the table is sorted in "rounded down" order. This is the portion of a seat the state lost due to being rounded down.

Rank	State	Population	Fair Share of seats	Rounded down w/ 1 minimum	Rounded Down	Extra vote	Total votes	Actual votes	Difference
1	California	39,576,757	51.995	51	0.995	1	52	52	0
33	Arkansas	3,013,756	3.959	3	0.959	1	4	4	0
26	Kentucky	4,509,342	5.924	5	0.924	1	6	6	0
17	Indiana	6,790,280	8.921	8	0.921	1	9	9	0
40	Hawaii	1,460,137	1.918	1	0.918	1	2	2	0
34	Mississippi	2,963,914	3.894	3	0.894	1	4	4	0
35	Kansas	2,940,865	3.864	3	0.864	1	4	4	0
6	Illinois	12,822,739	16.846	16	0.846	1	17	17	0
41	New Hampshire	1,379,089	1.812	1	0.812	1	2	2	0
42	Maine	1,363,582	1.791	1	0.791	1	2	2	0
36	New Mexico	2,120,220	2.785	2	0.785	1	3	3	0
20	Wisconsin	5,897,473	7.748	7	0.748	1	8	8	0
29	Connecticut	3,608,298	4.740	4	0.740	1	5	5	0
9	North Carolina	10,453,948	13.734	13	0.734	1	14	14	0
23	South Carolina	5,124,712	6.733	6	0.733	1	7	7	0
24	Alabama	5,030,053	6.608	6	0.608	1	7	7	0
21	Colorado	5,782,171	7.596	7	0.596	1	8	8	0
37	Nebraska	1,963,333	2.579	2	0.579	1	3	3	0
27	Oregon	4,241,500	5.572	5	0.572	1	6	6	0
4	New York	20,215,751	26.559	26	0.559	1	27	26	-1
7	Ohio	11,808,848	15.514	15	0.514	1	16	15	-1
22	Minnesota	5,709,752	7.501	7	0.501	1	8	8	0
43	Rhode Island	1,098,163	1.443	1	0.443	0	1	2	1
44	Montana	1,085,407	1.426	1	0.426	0	1	2	1
38	Idaho	1,841,377	2.419	2	0.419	0	2	2	0
14	Arizona	7,158,923	9.405	9	0.405	0	9	9	0
12	Virginia	8,654,542	11.370	11	0.370	0	11	11	0
39	West Virginia	1,795,045	2.358	2	0.358	0	2	2	0
2	Texas	29,183,290	38.340	38	0.340	0	38	38	0
3	Florida	21,570,527	28.339	28	0.339	0	28	28	0
30	Utah	3,275,252	4.303	4	0.303	0	4	4	0
45	Delaware	990,837	1.302	1	0.302	0	1	1	0
10	Michigan	10,084,442	13.249	13	0.249	0	13	13	0
15	Massachusetts	7,033,469	9.240	9	0.240	0	9	9	0
11	New Jersey	9,294,493	12.211	12	0.211	0	12	12	0
28	Oklahoma	3,963,516	5.207	5	0.207	0	5	5	0
31	Iowa	3,192,406	4.194	4	0.194	0	4	4	0
46	South Dakota	887,770	1.166	1	0.166	0	1	1	0
13	Washington	7,715,946	10.137	10	0.137	0	10	10	0
18	Maryland	6,185,278	8.126	8	0.126	0	8	8	0
25	Louisiana	4,661,468	6.124	6	0.124	0	6	6	0
5	Pennsylvania	13,011,844	17.095	17	0.095	0	17	17	0
19	Missouri	6,160,281	8.093	8	0.093	0	8	8	0
8	Georgia	10,725,274	14.091	14	0.091	0	14	14	0
16	Tennessee	6,916,897	9.087	9	0.087	0	9	9	0
32	Nevada	3,108,462	4.084	4	0.084	0	4	4	0
47	North Dakota	779,702	1.024	1	0.024	0	1	1	0
48	Alaska	736,081	0.967	1	-0.033	0	1	1	0
49	Vermont	643,503	0.845	1	-0.155	0	1	1	0
50	Wyoming	577,719	0.759	1	-0.241	0	1	1	0
	United States	331,108,434	435.000	413	22.000	22	435	435	0

* This column also reflects the rule that each state shall have a minimum of one Representative.

Note the 413 seats apportioned after rounding down. That means the 22 states who were rounded down the most should get an extra seat.

Also note the column for actual number of seats. This shows that New York and Ohio seem to be missing a seat. Meanwhile, Rhode Island and Montana have one too many. It's probably not a coincidence these states are all close to 22 down the list.

My question is why the disparity between my calculations and the actual apportionment?
link to original post

Trump.





(Teasing. Please no suspension!)

To answer my own question, it would seem that the method I described in my original post has become antiquated. Since 1941, the way it has been done is:

1. After each Census (which happen on years evenly divisible by 10), a count of the population is taken of all 50 states.
2. Each state is given as a minimum number of seats the greater of 1 and it's fair share of the total seats, which is currently 435, rounded DOWN.
3. Doing so will probably result in less than 435 seats apportioned, because of the rounding down rule. A count of extra seats needed will be taken to get to the target total.
4. What I will call the "shortfall factor" is calculated for each state p/sqrt(n*(n+1)), where:
   p = population of the state
   n = votes after step 2 above
5. An extra given to the states with the highest "shortfall" factors, such that the number of states getting an extra votes equals the shortfall from step 3.

Using this method, I get the following corrected table, which agrees with exactly with the current votes per state. Note is it sorted in Shortfall Factor order.

Rank	State	Population	Fair Share of seats	Rounded down w/ 1 minimum	Shortfall Factor	Extra vote	Total votes
40	Hawaii	1,460,137	1.918	1	1,032,473	1	2
41	New Hampshire	1,379,089	1.812	1	975,163	1	2
42	Maine	1,363,582	1.791	1	964,198	1	2
33	Arkansas	3,013,756	3.959	3	869,996	1	4
36	New Mexico	2,120,220	2.785	2	865,576	1	3
34	Mississippi	2,963,914	3.894	3	855,608	1	4
35	Kansas	2,940,865	3.864	3	848,955	1	4
26	Kentucky	4,509,342	5.924	5	823,289	1	6
29	Connecticut	3,608,298	4.740	4	806,840	1	5
37	Nebraska	1,963,333	2.579	2	801,527	1	3
17	Indiana	6,790,280	8.921	8	800,242	1	9
23	South Carolina	5,124,712	6.733	6	790,760	1	7
20	Wisconsin	5,897,473	7.748	7	788,083	1	8
6	Illinois	12,822,739	16.846	16	777,493	1	17
43	Rhode Island	1,098,163	1.443	1	776,519	1	2
24	Alabama	5,030,053	6.608	6	776,154	1	7
9	North Carolina	10,453,948	13.734	13	774,898	1	14
27	Oregon	4,241,500	5.572	5	774,388	1	6
21	Colorado	5,782,171	7.596	7	772,675	1	8
1	California	39,576,757	51.995	51	768,517	1	52
44	Montana	1,085,407	1.426	1	767,499	1	2
22	Minnesota	5,709,752	7.501	7	762,998	1	8
4	New York	20,215,751	26.559	26	762,994	0	26
7	Ohio	11,808,848	15.514	15	762,258	0	15
2	Texas	29,183,290	38.340	38	758,071	0	38
3	Florida	21,570,527	28.339	28	756,977	0	28
14	Arizona	7,158,923	9.405	9	754,617	0	9
12	Virginia	8,654,542	11.370	11	753,281	0	11
38	Idaho	1,841,377	2.419	2	751,739	0	2
10	Michigan	10,084,442	13.249	13	747,509	0	13
11	New Jersey	9,294,493	12.211	12	744,155	0	12
5	Pennsylvania	13,011,844	17.095	17	743,838	0	17
15	Massachusetts	7,033,469	9.240	9	741,393	0	9
8	Georgia	10,725,274	14.091	14	740,114	0	14
13	Washington	7,715,946	10.137	10	735,687	0	10
39	West Virginia	1,795,045	2.358	2	732,824	0	2
30	Utah	3,275,252	4.303	4	732,369	0	4
16	Tennessee	6,916,897	9.087	9	729,105	0	9
18	Maryland	6,185,278	8.126	8	728,942	0	8
19	Missouri	6,160,281	8.093	8	725,996	0	8
28	Oklahoma	3,963,516	5.207	5	723,636	0	5
25	Louisiana	4,661,468	6.124	6	719,280	0	6
31	Iowa	3,192,406	4.194	4	713,844	0	4
45	Delaware	990,837	1.302	1	700,628	0	1
32	Nevada	3,108,462	4.084	4	695,073	0	4
46	South Dakota	887,770	1.166	1	627,748	0	1
47	North Dakota	779,702	1.024	1	551,333	0	1
48	Alaska	736,081	0.967	1	520,488	0	1
49	Vermont	643,503	0.845	1	455,025	0	1
50	Wyoming	577,719	0.759	1	408,509	0	1
	United States	331,108,434	435.000	413	37,691,283	22	435

Source: Census.gov

I would like to thank Matt Parker who explains the issue in his video Why it’s mathematically impossible to share fair. Jump to the 31:50 point to get to the issue at hand.

The actual method is this:
1. Each state starts with 1 seat.
2. One seat at a time, give the next seat to the state with the highest value of population / (seats already awarded x (seats already awarded + 1)).

I calculated this with the 2020 numbers, mainly to see who would lose a seat if DC got one (I think it was Oregon), and noticed that had there been a 436th seat, it would have gone to New York, which, IIRC, would have gotten the 435th seat if its population was something like 100 larger.

The "quick version" of the Alabama Paradox: if you use the method of awarding seats = population / total population x total seats, rounded down, with any left over going to the states with the largest fractional values in the calculation, then it's possible to have a distribution where, if you add a seat, instead of one state gaining a seat, one state actually loses one of its seats while two others gain one.

Quote: ThatDonGuy

The actual method is this:
1. Each state starts with 1 seat.
2. One seat at a time, give the next seat to the state with the highest value of population / (seats already awarded x (seats already awarded + 1)).
link to original post

That's the same procedure as I described.

Interesting - I did play around (on a spreadsheet) using the p/SQRT(N N+1) method, and although the rankings of the states did change (e.g. 21st and 22nd), but it didn't result in the -1 +1 +1 (or similar). I'm guessing under some situations it could do, but I haven't analysed it at all.

I did play with needing a number of people to get a seat, in other words for each 721300 population you get a seat, and this gave California 54 (cf. 51+1).while Maine would only get 1. I think the p.. method favours smaller states, so it may be where the numbers allow a bigger state to get/lose an extra seat, that causes the problem.

btw the population numbers you quote can be found https://www2.census.gov/programs-surveys/decennial/2020/data/apportionment/apportionment-2020-tableA.pdf

A trivia factoid- Although Congress set the number of congressmen at 435, there were a few years when it expanded to 437.
When Alaska and Hawaii entered in 1959, we already had 435 congressmen so the option was to deny the two states a seat in Congress, somehow remove two elected congressmen to replace them with new reps or expand the house by two until Congress was reapportioned in 1963.

Quote: charliepatrick

Interesting - I did play around (on a spreadsheet) using the p/SQRT(N N+1) method, and although the rankings of the states did change (e.g. 21st and 22nd), but it didn't result in the -1 +1 +1 (or similar). I'm guessing under some situations it could do, but I haven't analysed it at all.

Here is an example of the Alabama Paradox in action:

A town of 10,000 people is dividing its 9 council seats among 4 districts.
District 1 has 4820 people; District 2 has 3780; District 3 has 810; District 4 has 590.
Multiplying the fraction of people in each district by 9, you get:
District 1 - 4.338
District 2 - 3.402
District 3 - 0.729
District 4 - 0.531
At first, District 1 gets 4 seats, and District 2 gets 3; the other two go to the districts with the largest fractions, which are 3 (0.729) and 4 (0.531), so the final apportionment is 4, 3, 1, 1.
However, it is then decided that there should be 10 members of the council. Multiplying the fraction of people in each district by 10, you get:
District 1 - 4.82
District 2 - 3.78
District 3 - 0.81
District 4 - 0.59
Again, it starts with District 1 getting 4 and District 2 getting 3; the remaining three go to 1 (0.82), 3 (0.81), and 2 (0.78), resulting in an apportionment of 5, 4, 1, 0.
Adding a seat means that District 4 loses one, at the expense of Districts 1 and 2.

^ Thanks - the p/SQRT(N (N-1)) doesn't work when N=0! Thus I think the US system needs a guaranteed 1 seat to work. I can see where it goes wrong where the front two start on a low remainder, so the last district gets a seat; but if you increase the number of seats the front two land up with a high remainder and overtake the last district; so they each gain a seat and the last one loses one. Here's a simple example where D1=52%, D2=38%, D3=10% and there are 14 or 15 seats.
NOTE: The US system lands up giving 8 5 2 as the numbers are 6.949, 6.938, 7.071; but suggests that there might be values where even the US system goes wrong!