Adams method US House

[President Punxsutawney Phil]

This is what the US House would look like under the Adams method.
The first column is 2020 populations.
The second column is 2020 populations divided by the population of the 50 states divided by 435.
The third column is 2020 populations divided by the divisor of 801,000, which is yields 435 seats.
The fourth column is the seat apportionments that are yielded accordingly.
The fifth column is 2020 populations divided by said seat apportionments.
The sixth column is RL seat apportionments for comparison.

[President Punxsutawney Phil]

Practical impact would be:
-2 CA
-1 TX, FL, IL
+1 UT, ID, WV, DE, SD

Surprisingly less than I expected.

[President Punxsutawney Phil]

https://davesredistricting.org/join/24ea1b1c-8d96-467e-9071-80f666af110b
2 district SD

[President Punxsutawney Phil]

https://davesredistricting.org/join/d9fae3bb-e155-46e2-8ee1-06c2d57f9a0e
2 district Delaware

(I don't think DE Ds would allow the creation of a tilt R seat)

[President Punxsutawney Phil]

https://davesredistricting.org/join/429d36ab-4797-4b8e-bcc0-7e1e3f2ba058
WV under the 2020s, following least change principles.
WV-01 sheds Gilmer County and takes Wirt County.
WV-02 takes Gilmer, Nicholas, and Webster counties and loses Wirt and Putnam counties.
WV-03 loses Nicholas and Webster and gains Putnam.

[President Punxsutawney Phil]

https://davesredistricting.org/join/ce80b66e-d3ef-42f6-96ba-e8bdaeed0d56
ID has a commission, so I doubt Rs could go full gerrymander.
I assigned the district number of 1 to the Panhandle, 2 to Eastern Idaho, and 3 to Boise. Since Ada+the city of Nampa is basically perfect for a seat and still R-leaning overall, I figure that is what would occur.

[President Punxsutawney Phil]

https://davesredistricting.org/join/e4b38dff-9624-4095-ba4c-7ea9dabfcc65
UT-01 shrinks, and Salt Lake City remains united but in a safe R seat. Three seats are shared between Utah and Salt Lake counties, and Salt Lake County remains divided like a pizza. Lehi is the only city with a macro chop.

[jimrtex]

Practical impact would be:
-2 CA
-1 TX, FL, IL
+1 UT, ID, WV, DE, SD

Surprisingly less than I expected.

For other methods you need to calculate a different quota. The only way I know to do that is recursively.

S₀ = 435

For Jefferson's method begin with the quota of 801,000 (or Q = P_USA/S₀ = 435, and determine an apportionment entitlement for each state Ei = Pi/Q - 1 these should sum to S₁ = 385.

Calculate a new quota Q_i+1 = P_USA/( S_i / S_i-1 )

This will produce a smaller quota, which when calculating entitlement will sum closer to 435. Iterate until this converges.

Compare the entitlement between the two methods.

[Kevinstat]

Practical impact would be:
-2 CA
-1 TX, FL, IL
+1 UT, ID, WV, DE, SD

Surprisingly less than I expected.

For other methods you need to calculate a different quota. The only way I know to do that is recursively.

S₀ = 435

For Jefferson's method begin with the quota of 801,000 (or Q = P_USA/S₀ = 435, and determine an apportionment entitlement for each state Ei = Pi/Q - 1 these should sum to S₁ = 385.

Calculate a new quota Q_i+1 = P_USA/( S_i / S_i-1 )

This will produce a smaller quota, which when calculating entitlement will sum closer to 435. Iterate until this converges.

Compare the entitlement between the two methods.

You can also use "priority values", having a multiplier for the nth seat a state would be awarded as a function of n (for the Adams method, that multiplier would be 1/(n − 1), or n − 1 would be the divisor), multiplying each of those multipliers (or more than enough for each state) by each state's population and then starting with 1 seat for each state and awarding seats to the top 435 − 50 = 385 entries (or you could think of the multipliers for the first seat (where you would have to divide by 1 − 1 = 1 0) being (positive) infinity (∞), so you'll have 50 infinites, then California's whole population (which earns it a second seat as the 51st seat in the U.S. House), then Texas's, then Florida's, then New York's, then California's population divided by 2 (so seat #55 is California's third seat), and so on).

I'm explaining this somewhat lazily here.  But this heuristic can be useful if you're making a spreadsheet to calculate multiple potential apportionments (either changing the apportionment method or changing the number of seats) and not wanting to have to adjust the divisor each time.

There's a continuum of methods where the cutoff or "signpost" (in the heuristic where a state has to get to or past a certain real number (often not an integer) "signpost" to get n seats and the divisor is adjusted until you get the right total number of seats) or the divisor (in the heuristic assign a "priority value" for each state getting an nth seat for each n) is a generalized or power mean of the number of seats the state would already have at this point and the seat it is "going for".  It's a complex looking formula, which errors out at 0 and at positive and negative infinities, but using the mathematical concept of limits and substituting the limits where the function would error out you can get values for all real numbers, including ∞ and −∞.  The Adams method, sometimes called the method of "Smallest Divisors" (although in the "adjust the divisor by trial and error" method the divisor needed to get a certain number of seats would be the highest of any of the methods on this continuum; in the "priority value" method the divisors are indeed the smallest), corresponds to p = −∞.  The first method used, Jefferson's or "Greatest Divisors", corresponds to p = ∞.  Our current method of Equal Proportions or Huntington-Hill, which has been used from 1941 on, corresponds to p = 0 (the geometric mean).  Major Fractions or Webster's method, used in the apportionments following the 1840, 1910 and 1930 censuses, uses the arithmetic mean (the mean you probably learned in grade school) of the two adjacent seat numbers, which is the variant of the power mean where p = 1.

[Edited to correct a math error and one instance where I left off one side of an expression.]

[Kevinstat]

2020 Census (Apportionment Population)
Seat changes with ascending p in the power mean for seat divisors (the ⁺s and ⁻s indicate that the actual number is very slightly above or below, respectively, the number listed which is rounded to four decimal places.  I'm weird, I know.

[p → −∞: Smallest Divisors (a.k.a. Adams)]
p = (−28.8700)⁺ or −(28.8700⁻): Florida (27 → 28) gains from North Carolina (14 → 13)
p = (−15.3660)⁻ or −(15.3660⁺): California (50 → 51) gains from Utah (5 → 4)
p = (−11.9354)⁻ or −(11.9354⁺): Illinois (16 → 17) gains from Minnesota (8 → 7)
p = (−5.0593)⁺ or −(5.0593⁻): Texas (37 → 38) gains from South Dakota (2 → 1)
p = (−2.9816)⁺ or −(2.9816⁻): North Carolina (13 → 14) gains from West Virginia (3 → 2)
p = (−1.6198)⁻ or −(1.6198⁺): California (51 → 52) gains from Delaware (2 → 1)
[p = −1: Harmonic Mean (a.k.a. Dean)]
p = (−0.8155)⁺ or −(0.8155⁻): Minnesota (7 → 8) gains from Idaho (3 → 2)
[p → 0 (the geometric mean): Equal Proportions (a.k.a. Huntington–Hill)]
p = 0.0021⁺: New York (26 → 27) gains from Minnesota (8 → 7)
p = 0.1017⁺: Minnesota (7 → 8) gains from Montana (2 → 1)
p = 0.3119⁺: Ohio (15 → 16) gains from Rhode Island (2 → 1)
[p = 1 (the arithmetic mean): Major Fractions (a.k.a. Webster or Sainte-Laguë)]
[p = 2.5533⁻: Wyoming starts relying on 1-seat minimum rule (see below)]
p = 2.8734⁺: Texas (38 → 39) gains from Nebraska (3 → 2)
p = 3.8630⁺: Florida (28 → 29) gains from Minnesota (8 → 7)
[p = 4.3841⁺: Vermont starts relying on 1-seat minimum rule (see below)]
p = 6.8306⁻: California (52 → 53) gains from Maine (2 → 1)
p = 8.2404⁻: Minnesota (7 → 8) gains from New Hampshire (2 → 1)
p = 9.9984⁻: Virginia (11 → 12) gains from Oregon (6 → 5)
p = 14.7320⁻: Arizona (9 → 10) gains from New Mexico (3 → 2)
p = 15.7148⁺: Michigan (13 → 14) gains from Arizona (10 → 9)
p = 16.8906⁻: California (53 → 54) gains from Minnesota (8 → 7)
p = 20.2718⁺: Pennsylvania (17 → 18) gains from Michigan (14 → 13)
p = 21.9385⁺: Texas (39 → 40) gains from Pennsylvania (18 → 17)
p = 28.2563⁺: Pennsylvania (17 → 18) gains from Alabama (7 → 6)
p = 69.3347⁺: New York (27 → 28) gains from Virginia (12 → 11)
[p → ∞: Greatest Divisors (a.k.a. Jefferson or D’Hondt)]

[Assuming no automatic 1st seat for each state:]
p = 2.5533⁻: Texas (38 → 39) gains from Wyoming (1 → 0)
p = 2.9686⁻: Florida (28 → 29) gains from Nebraska (3 → 2)
p = 4.3841⁺: California (52 → 53) gains from Vermont (1 → 0)
p = 7.4037⁻: Virginia (11 → 12) gains from Maine (2 → 1)
p = 8.8174⁺: Arizona (9 → 10) gains from New Hampshire (2 → 1)
p = 12.5609⁻: Michigan (13 → 14) gains from Oregon (6 → 5)
p = 15.3280⁻: Pennsylvania (17 → 18) gains from New Mexico (3 → 2)
p = 15.5148⁺: California (53 → 54) gains from Pennsylvania (18 → 17)
p = 17.1518⁻: Pennsylvania (17 → 18) gains from Arizona (10 → 9)
p = 17.7507−: Arizona (9 → 10) gains from Minnesota (8 → 7)
p = 18.6646−: Texas (39 → 40) gains from Arizona (10 → 9)
p = 41.6333−: New York (27 → 28) gains from Alabama (7 → 6)
[p → ∞: Greatest Divisors (a.k.a. D’Hondt or Jefferson)]
(Virginia (12) would get the 434th seat; Michigan (14) would get the 435th seat)

[Kevinstat]

There has been some academic study done into using what are sometimes called "stationary signposts" in apportionment (where the cutoff values or divisors, depending on the heuristic used, all have the same fractional component (say 0.4, 1.4, 2.4, ... or 0.7, 1.7, 2.7, ...)).  This corresponds to using a weighted arithmetic mean of the number of seats a state (or party, as a lot of this study deals with representation in an elected body under a party list-type system) "already has" and the number its "going for".  q, which can range for 0 to 1 (or beyond if you want but you could get not even remotely equal representation there, and it would no longer be any kind of mean), is sometimes used as the variable here, where the signpost or divisor for a state or party to get an nth seat is n − 1 + q.  q = 0 corresponds to using the smaller number (as in the Adams method), while q = 1 corresponds to using the larger number (as in Jefferson's method).

Seat changes with ascending q in weighted arithmetic mean for seat divisors, a.k.a. “stationary signposts”:

[q = 0: Smallest Divisors (a.k.a. Adams)]
q = 0.1158⁻: Illinois (16 → 17) gains from South Dakota (2 → 1)
q = 0.1175⁺: Florida (27 → 28) gains from Utah (5 → 4)
q = 0.2495⁺: California (50 → 51) gains from Minnesota (8 → 7)
q = 0.2598⁺: Minnesota (7 → 8) gains from Delaware (2 → 1)
q = 0.2939⁺: Texas (37 → 38) gains from West Virginia (3 → 2)
q = 0.3911⁻: California (51 → 52) gains from Idaho (3 → 2)
q = 0.4171⁻: Ohio (15 → 16) gains from Montana (2 → 1)
q = 0.4361⁻: New York (26 → 27) gains from Rhode Island (2 → 1)
[q = 0.5: Major Fractions (a.k.a. Webster or Sainte-Laguë)]
q = 0.5405⁺ Texas (38 → 39) gains from Minnesota (8 → 7)
q = 0.6035⁻ Florida (28 → 29) gains from Nebraska (3 → 2)
q = 0.6417⁻ California (52 → 53) gains from Oregon (6 → 5)
[q = 0.7851⁺: Wyoming starts relying on 1-seat minimum rule (see below)]
q = 0.8433⁻: California (53 → 54) gains from Alabama (7 → 6)
q = 0.8595⁺: Alabama (6 → 7) gains from Maine (2 → 1)
q = 0.8724⁺: Texas (39 → 40) gains from Alabama (7 → 6)
[q = 0.8836⁻: Vermont starts relying on 1-seat minimum rule (see below)]
q = 0.8887⁻: Alabama (6 → 7) gains from New Hampshire (2 → 1)
q = 0.9200⁻: Pennsylvania (17 → 18) gains from New Mexico (3 → 2)
q = 0.9390⁻: Virginia (11 → 12) gains from Alabama (7 → 6)
q = 0.9774⁻: New York (27 → 28) gains from Virginia (12 → 11)
[q = 1: Greatest Divisors (a.k.a. Jefferson or D’Hondt)]

[Assuming no automatic 1st seat for each state:]
q = 0.7851⁺: California (53 → 54) gains from Wyoming (1 → 0)
q = 0.8626⁻: Texas (39 → 40) gains from Maine (2 → 1)
q = 0.8836⁻: Virginia (11 → 12) gains from Vermont (1 → 0)
q = 0.8968⁺: Pennsylvania (17 → 18) gains from New Hampshire (2 → 1)
q = 0.9284⁻: Michigan (13 → 14) gains from New Mexico (3 → 2)
q = 0.9352⁺: New York (27 → 28) gains from Michigan (14 → 13)
q = 0.9663⁻: Michigan (13 → 14) gains from Alabama (7 → 6)
[q = 1: Greatest Divisors (a.k.a. D’Hondt or Jefferson)]
(Virginia (12) would get the 434th seat; Michigan (14) would get the 435th seat)

The apportionment in the interval between 0.3911⁻ and 0.4171⁻ matches the current apportionment under Equal Proportions, but there won’t necessarily be such an interval on this continuum for any given population figures.  If Minnesota's 2020 apportionment population had been 26 people less and/or New York's 89 people greater, then New York would have kept its 27th U.S. House seat and Minnesota would have dropped to 7 seats under Equal Proportions, but the apportionment in this theoretical interval wouldn't have changed and there would have been no q in a weighted arithmetic mean that would have corresponded to what would have been the apportionment under Equal Proportions.  Of course, one could also do a weighted geometric mean (or a weighted power mean with variables p and q), and the continuum of apportionments of the weighted geometric mean would include the actual apportionment under Equal Proportions (in a range including q = 0.5).