Ohio-Style Apportionment (temporal variation).

[jimrtex]

Typically apportionment methods are flawed because they limit themselves to whole numbers, this results in the use of various rounding methods some of which are mathematically flawed and others which seek to minimize rounding errors. The argument for the use of Huntington-Hill is not that it is fair at all, or even the most fair, but that it is the least unfair.

The simplest way to avoid this is to permit fractional representational. One could use rational numbers. The amount of representation for a state (or county, etc.) is

R_i = R_USA * P_i / P_USA

That is the share of a State's representation is proportional to its share of the total population.

One could also used mixed decimal fractions. For example, rounding to the nearest 0.001 (the maximum relative error is around 0.05%).

If such a scheme were used, no one would object to territories such as the Virgin Islands or Guam having representation in the House of Representatives. Such a representative would have a small fractional vote.

Montana would have 1.4.. representatives. It is inane to argue that Montana should have one OR two representatives, particularly when one has to use square roots to determine the number or have the SCOTUS involved Department of Commerce v Montana


Alas, the US Constitution requires apportionment of whole numbers of representatives. But the US Constitution does not require that the number of representatives be constant throughout an apportionment decade.

Instead of apportioning 1 OR 2 representatives to Montana, we could apportion 1 representative some of the time, and two at other times. For convenience, this could be tied to the five Congresses in each decade. Since Montana is entitled to about 1.4 representatives, it could have 2 representatives for two sessions, and 1 representative for three sessions, for an average of 1.4 representatives over the decade.

Apportionment is simple. Multiply 435 (or 692 based on the cube root rule) by five and apportion that number of units, with a minimum guarantee of 5 units. I would use Webster's  method, using the arithmetic mean for the divisors, rather than the geometric mean.

This method does not work real well with single-member districts. It would be OK for Montana, but might be more difficult for California to devise 52-district and 53-district plans that varied from election to election. While the variation in districts in San Diego and Imperial would be small, in the middle of the state would be huge. An incumbent would find their district chopped in half with about a 50% chance of being paired. Proportional representation would be better. A multi-member district might have 3.4 representatives, 3 representatives some elections and four for other. For that district, there might be some bias, with a 2:1 delegation some elections and 2:2 delegation others, but these would tend to even out over the entire country.


Ohio used a system like this from 1851 until 1966. The details are in the original 1851 Ohio Constitution (PDF). See Article XI.

The original apportionment is in a Schedule to the Constitution, and is still technically part of the constitution.

Generally, a county would be apportioned as many representatives as they had full quotas, and would have an additional representative in one session for each additional 1/5 of a quota. If I am interpreting the text correctly, a county entitled to 1.39 representatives would be truncated down to 1.2.

There is also a guarantee that a county entitled to 0.5 or more representatives would be entitled to a whole representative, while a county entitled to 1.75 or more representatives to 2 representatives would be given two whole representatives.

Smaller counties were paired (see Schedule). It was likely felt that the smaller counties on the northern frontier would eventually fill in with farmers so that more counties would have full representation, but there were provisions providing for placing counties in multi-county district or separating them out.

Around 1900, Mark (Boss) Hannah pushed through a constitutional amendment guaranteeing every county one representative. Remarkably, the amendment was approved by a 99% vote. It was either a well-received change, or Hannah's appellation of "Boss" was well deserved.

The sessions that extra representatives were allocated was based on the size of the fraction.

n.2, the extra representative would be elected in the 5th session.
n.4  the extra representative would be elected in the 3rd and 4th session.
n.6  the extra representative would be elected in the 1st, 2nd, and 3rd session.
n.8  the extra representative would be elected in the 1st, 2nd, 3rd, and 4th session.

This provision as well as the provision for at-large elections would likely be considered OMOV violations. It appears that the legislature/governor were also concerned that the fractional apportionment would be found illegal, so the constitution was revised in 1966 to provide for single-member districts. The provisions in the Constitution are quite reasonable, but it appears that they were not actually followed.

[jimrtex]

This method does not work real well with single-member districts. It would be OK for Montana, but might be more difficult for California to devise 52-district and 53-district plans that varied from election to election. While the variation in districts in San Diego and Imperial would be small, in the middle of the state would be huge. An incumbent would find their district chopped in half with about a 50% chance of being paired. Proportional representation would be better. A multi-member district might have 3.4 representatives, 3 representatives some elections and four for other. For that district, there might be some bias, with a 2:1 delegation some elections and 2:2 delegation others, but these would tend to even out over the entire country.

I just realized that this could be easily handled by electing the extra representative at large. This might be permissive. A smaller state such as West Virginia might want to vary between 2 and 3 districts, while California might prefer to have 52 districts, with the 53rd representative elected at large.

In a typical Congress there would be around 27 at-large representatives (this includes those from VT, DE, ND, SD, MT, WY, and AK).

[jimrtex]

Will Minnesota receive 7.6 or 7.4 representatives, and will Alabama receive 6.6 or 6.4 representatives?

This will be answered at 3:00 EDT as the Census Bureau announce the 2020 Apportionment numbers.

It is certain New York will have 25 district representatives, but it is not known how many Congresses it will elect a 36th representative.

[Boss_Rahm]

This is a fascinating idea, here is the 2020 apportionment using this method:

AL   6.6
AK   1.0
AZ   9.4
AR   4.0
CA   52.0
CO   7.6
CT   4.8
DE   1.4
FL   28.4
GA   14.0
HI   2.0
ID   2.4
IL   16.8
IN   9.0
IA   4.2
KS   3.8
KY   6.0
LA   6.2
ME   1.8
MD   8.2
MA   9.2
MI   13.2
MN   7.6
MS   3.8
MO   8.0
MT   1.4
NE   2.6
NV   4.0
NH   1.8
NJ   12.2
NM   2.8
NY   26.6
NC   13.8
ND   1.0
OH   15.6
OK   5.2
OR   5.6
PA   17.0
RI   1.4
SC   6.8
SD   1.2
TN   9.0
TX   38.4
UT   4.4
VT   1.0
VA   11.4
WA   10.2
WV   2.4
WI   7.8
WY   1.0

I suppose a side effect would be that the Electoral College map would change every single election.

[jimrtex]

This is a fascinating idea, here is the 2020 apportionment using this method:

AL 6.6
AK 1.0
AZ 9.4
AR 4.0
CA 52.0
CO 7.6
CT 4.8
DE 1.4
FL 28.4
GA 14.0
HI 2.0
ID 2.4
IL 16.8
IN 9.0
IA 4.2
KS 3.8
KY 6.0
LA 6.2
ME 1.8
MD 8.2
MA 9.2
MI 13.2
MN 7.6
MS 3.8
MO 8.0
MT 1.4
NE 2.6
NV 4.0
NH 1.8
NJ 12.2
NM 2.8
NY 26.6
NC 13.8
ND 1.0
OH 15.6
OK 5.2
OR 5.6
PA 17.0
RI 1.4
SC 6.8
SD 1.2
TN 9.0
TX 38.4
UT 4.4
VT 1.0
VA 11.4
WA 10.2
WV 2.4
WI 7.8
WY 1.0

I suppose a side effect would be that the Electoral College map would change every single election.

You need to make sure that the total is 435.0 (or 2175 units).

I have

TX 38.4
OH 15.4
MN 7.4
UT 4.2
DE 1.2

I think that is how I would handle the Electoral College. I have a scheme that would assign the extra seats on a regional basis, so that there is not a large scale shift between Congresses from West to East or North to South. But, for example Illinois would have an extra representative for four elections, and Iowa the remaining election. While they are neighbors, they are not politically linked. So the Republican candidate would likely pick up an elector if that was a presidential election when they also elected a 5th representative.

There are a total of 17 floating seats, but not all would move every presidential election.

[President Punxsutawney Phil]

So what would redistricting the Ohio House using the same methods and district size as pre-1966 result in?

[Boss_Rahm]

You need to make sure that the total is 435.0 (or 2175 units).

When I read about the Webster method, it called for setting the quota based on the target number of seats, but allowing the actual total to differ depending on how the rounding works out. But in any event I prefer your method of keeping the total number of seats fixed.

[jimrtex]

So what would redistricting the Ohio House using the same methods and district size as pre-1966 result in?

Depends on what method one used.

The 1851 Constitution gives a whole representative to any county with more than half a quota. The quota was 1/100 of the population, so that is the same as 0.5%.

Counties smaller than 0.5% were combined. There were 15 such counties, forming 7 districts of 6 pairs and one triplet.

The remaining 72 counties formed single county districts electing one or more representatives (counties were not divided into subdistricts). Note: Ohio had 87 counties in 1851. The 88th county (current number) was added later.

35 counties had one representative (population between 0.5% and 1.2%)
12 counties had 1.2 representatives (population between 1.2% and 1.4%)
14 counties had 1.4 representatives (population between 1.4% and 1.6%)
4 counties had 1.6 representatives (population between 1.6% and 1.75%)
4 counties had 2.0 representatives (population between 1.75% and 2.2%)
1 county (Muskingum) had 2.2 representatives (population between 2.2% and 2.4%)
1 county (Cuyahoga) had 2.4 representatives (population between 2.4% and 2.6%)
1 county (Hamilton) had 7.8 representative (population between 7.8% and 8.0%)

Cincinnati was truly a dominant city. The convention in 1851 met in Cincinnati even though Columbus had been the capital for around 30 years.

The average number of representatives was 102.8, with 91 elected to every session, and an average of 11.8 additional representatives. But this number varied greatly by session:

5 (fraction of 0.6 and 0.8)
5 (fraction of 0.6 and 0.8)
20 (fraction of 0.4, 0.6, and 0.8)
16 (fraction of 0.4 and 0.8)
13 (fraction of 0.2)

As you can see that having one representative per county was normative and quite typical for a western state legislature.

There is somewhat of a small county bias by awarding a whole representative to counties with 50% of a quota, and truncating fractions of larger counties to the lower 0.2%.

As time went and industrialization resulted in concentration of population in Cleveland, Akron, Columbus, Dayton, etc., agricultural counties increasingly fell below 1%, and in some cases required more counties to be combined into multi-county districts when they fell below 0.5%.


Mark Hanna's amendment eliminated the threshold of 0.5% for whole county representation. In 2019, Vinton with the smallest population (0.11%) of the total, and entitled to 1/9 of a representative would have one whole district.

In 1850, 47 counties had more than 1% of the population. This declined to 16 in 1960. It has rebounded somewhat as suburban counties, or satellite to suburban counties have gained in population (e.g. Butler, Clermont, Warren, Delaware, Licking, Fairfield, Medina, Lake, Lorain, Summit, Greene, Portage have been grown.

Most big Ohio counties other than Franklin have been declining in population since 1970.


If I were devising the plan, I would round to the nearest 1/5; expand the legislature based on the cube root rule; eliminate the small county guarantee (90% of the quota, or you are placed in a multi-county district), and ensure that the size of the legislature did not vary by session. I'd also figure out a way to avoid at-large elections in larger counties.

[jimrtex]

You need to make sure that the total is 435.0 (or 2175 units).

When I read about the Webster method, it called for setting the quota based on the target number of seats, but allowing the actual total to differ depending on how the rounding works out. But in any event I prefer your method of keeping the total number of seats fixed.

Under traditional Webster's method, you would choose the target size of the House, calculate the quota (USA population / Target size). For each state, divide its population by the quota, and round to the nearest integer.

This may produce more or less than the ideal House size, for the same reason that polls rounded to the nearest 1% might not total to 100%.

This led to the use of Hamilton's method, which chooses the state with the largest fraction getting the additional seats. In some decades, a fraction of 0.4 might be enough; in others a fraction of 0.6 might be barely enough. If the whole numbers are large enough, the fractions will be uniformly distributed, with a mean and median, around 0.5).

If the whole numbers are smaller, there will be a bias towards smaller fractions. If there were 100 representatives, we would see more states around 1.2 than around 1.8. The population distribution is highly skewed with California, Texas, Florida extreme outliers. There is one state with 52 representatives, the next with 38 representatives, but 6 with 2, and 7 with 1.

Hamilton's method has an anomaly known as the Alabama Paradox. If you increase the size of the House by one, a state can actually lose a representative. The paradox was named for an instance when the victim state was Alabama.

While we now think of the number of representatives as fixed, this was not true until 1930. Instead, the Census report to Congress would show the apportionment for various target sizes. Representatives would peruse the table to choose a size that was more favorable (e.g. Schumer and Nadler would argue that the House should be expanded to 437, to accommodate DC and keep the number of representatives odd (it was mere happenstance that this would benefit NY. There was a serious proposal to give DC a representative as a territory, and give Utah the representative that had been given to NC).

In 1871, The House was expanded enough to ensure that no State would lose representation. The elimination of the 3/5 rule meant that states with large former slave populations (aka as the States that lost the War Between the States), would gain representation at the expense of states without large Black populations (aka as the States that won the War Between the States).

This expansion was repeated through 1911, when no state lost representation compared to 1901. The number 435 was chosen because it had that property.

In 1921 no apportionment was done, in major part because they were concerned about increasing the House by 10% or more every decade (at that rate we would now have a House of 1240 (an increase from 1128 in 2010).

In 1929 President Hoover pushed through a law that fixed the number of representative (unless Congress changed it) and would automatically calculate the apportionment (unless Congress changed it). This permitted Congress to do what it does best: Do Nothing.

The version of Webster's method that you used was:

R_i = max (round (R_USA * P_i / P_USA), 5)

We can define a quota, Q, = P_USA / R_USA

and rewrite the expression above.

R_i = max (round (P_i / Q), 5)

We are simply saying that a state should be apportioned one unit for every 1/(435*5) of the total population. The guarantee of 5 units applies to Alaska, Vermont, and Wyoming.

But we can adjust the Quota. For this particular census and target number of seats the quota is 152,233.76. But that results in too many seats. We can adjust this upward by trial and error. Or we can assume that every State will get at least n seats where

n_i= floor ( max (P_i / Q), 5) )

That is, a state that is entitled to n.fraction seats should either get n seats or n+1 seats.

Using the above calculation we sum the minimum apportionment for every state, which comes out to 2152 seats when our goal is to apportion 435*5 or 2175 seats. We want to decide which 23 states are most deserving.

For each state we calculate the quotient:

quotienti = P_i / ( n_i + 0.5 )

If we used that quotient in place of the quota, then the apportionment for that state will round to n_i + 1

If we used a larger value, then it would round down. So we want to find the 23rd largest quotient. States with a quotient greater than equal to that will be awarded the extra seat.

The 23rd largest quotient is 152,441.89.

[President Punxsutawney Phil]

Looking at what keeping a seat for every county would do. Assumed size of 137, as that was how large the Ohio House was in the 1960s pre-1966. Used 2019 population estimates.

Ashtabula 97,830 1
Trumbull 200,367 2 2.355
Mahoning 229,961 3 2.703
Columbiana 103,190 1
Lake 229,954 3 2.703
Geauga 93,843 1
Portage 162,511 2 1.910
Cuyahoga 1,247,452 15 14.662
Summit 541,334 6 6.363
Stark 372,404 4 4.377
Carroll 27,332 1
Jefferson 66,371 1
Harrison 15,211 1
Tuscarawas 92,335 1
Guernsey 39,111 1
Belmont 68,024 1
Monroe 13,942 1
Noble 14,416 1
Washington 60,426 1
Medina 177,980 2 2.092
Wayne 116,099 1
Holmes 43,901 1
Coshocton 36,585 1
Muskingum 86,131 1
Morgan 14,640 1
Athens 65,917 1
Meigs 23,078 1
Lorain 307,670 4 3.616
Ashland 53,536 1
Perry 36,022 1
Gallia 30,088 1
Erie 74,780 1
Huron 58,339 1
Richland 121,100 1
Knox 61,481 1
Licking 173,750 2 2.042
Fairfield 154,457 2 1.815
Hocking 28,390 1
Vinton 13,083 1
Jackson 32,450 1
Lawrence 60,184 1
Crawford 41,821 1
Morrow 35,043 1
Ottawa 40,632 1
Sandusky 59,029 1
Seneca 55,351 1
Wyandot 22,000 1
Marion 65,299 1
Delaware 201,135 2 2.364
Franklin 1,290,361 15 15.167
Pickaway 57,762 1
Ross 76,948 1
Pike 28,000 1
Scioto 76,040 1
Union 56,707 1
Madison 44,135 1
Fayette 28,620 1
Adams 27,776 1
Lucas 431,102 5 5.067
Wood 130,150 2 1.530
Hancock 75,837 1
Hardin 31,425 1
Logan 45,316 1
Champaign 38,845 1
Clark 134,726 2 1.584
Greene 166,502 2 1.957
Clinton 41,957 1
Highland 43,016 1
Brown 43,557 1
Fulton 42,253 1
Henry 27,208 1
Putnam 33,911 1
Allen 103,175 1
Auglaize 45,729 1
Shelby 48,749 1
Miami 105,371 1
Montgomery 531,609 6 6.249
Warren 229,177 3 2.694
Clermont 204,290 2 2.401
Williams 36,816 1
Defiance 38,160 1
Paulding 18,809 1
Van Wert 28,261 1
Mercer 40,884 1
Darke 51,513 1
Preble 41,093 1
Butler 380,019 4 4.467
Hamilton 813,589 10 9.563
165

Simplest way to combat this issue - that a quota for 137 combined with one district minimum for each county - and without ditching the guaranteed one district for each county - would be to strip seats from counties with multiple districts, with those with the least population per district coming first. Repeat over and over until you are at 137. Only 22 counties have more than one district.

This approach would have very detrimental impacts on the representation of Cuyahoga and Franklin counties.

[jimrtex]

Looking at what keeping a seat for every county would do. Assumed size of 137, as that was how large the Ohio House was in the 1960s pre-1966. Used 2019 population estimates.

Ashtabula 97,830 1
Trumbull 200,367 2 2.355
Mahoning 229,961 3 2.703
Columbiana 103,190 1
Lake 229,954 3 2.703
Geauga 93,843 1
Portage 162,511 2 1.910
Cuyahoga 1,247,452 15 14.662
Summit 541,334 6 6.363
Stark 372,404 4 4.377
Carroll 27,332 1
Jefferson 66,371 1
Harrison 15,211 1
Tuscarawas 92,335 1
Guernsey 39,111 1
Belmont 68,024 1
Monroe 13,942 1
Noble 14,416 1
Washington 60,426 1
Medina 177,980 2 2.092
Wayne 116,099 1
Holmes 43,901 1
Coshocton 36,585 1
Muskingum 86,131 1
Morgan 14,640 1
Athens 65,917 1
Meigs 23,078 1
Lorain 307,670 4 3.616
Ashland 53,536 1
Perry 36,022 1
Gallia 30,088 1
Erie 74,780 1
Huron 58,339 1
Richland 121,100 1
Knox 61,481 1
Licking 173,750 2 2.042
Fairfield 154,457 2 1.815
Hocking 28,390 1
Vinton 13,083 1
Jackson 32,450 1
Lawrence 60,184 1
Crawford 41,821 1
Morrow 35,043 1
Ottawa 40,632 1
Sandusky 59,029 1
Seneca 55,351 1
Wyandot 22,000 1
Marion 65,299 1
Delaware 201,135 2 2.364
Franklin 1,290,361 15 15.167
Pickaway 57,762 1
Ross 76,948 1
Pike 28,000 1
Scioto 76,040 1
Union 56,707 1
Madison 44,135 1
Fayette 28,620 1
Adams 27,776 1
Lucas 431,102 5 5.067
Wood 130,150 2 1.530
Hancock 75,837 1
Hardin 31,425 1
Logan 45,316 1
Champaign 38,845 1
Clark 134,726 2 1.584
Greene 166,502 2 1.957
Clinton 41,957 1
Highland 43,016 1
Brown 43,557 1
Fulton 42,253 1
Henry 27,208 1
Putnam 33,911 1
Allen 103,175 1
Auglaize 45,729 1
Shelby 48,749 1
Miami 105,371 1
Montgomery 531,609 6 6.249
Warren 229,177 3 2.694
Clermont 204,290 2 2.401
Williams 36,816 1
Defiance 38,160 1
Paulding 18,809 1
Van Wert 28,261 1
Mercer 40,884 1
Darke 51,513 1
Preble 41,093 1
Butler 380,019 4 4.467
Hamilton 813,589 10 9.563
165

Simplest way to combat this issue - that a quota for 137 combined with one district minimum for each county - and without ditching the guaranteed one district for each county - would be to strip seats from counties with multiple districts, with those with the least population per district coming first. Repeat over and over until you are at 137. Only 22 counties have more than one district.

This approach would have very detrimental impacts on the representation of Cuyahoga and Franklin counties.

If you apply a quota for 100, using the Hanna rules, you get an average of 136 seats (it varies from session to session: 135, 135, 140, 136, and 134.

That is the number 137 before 1966 comes from using a quota based on 100.

If you want to balance the sessions, map out the number of extra representative-sessions units.

Cincinatti (5): Hamilton 4, Butler 1

Dayton (4): Montgomery 2, Greene 2

Columbus (4): Licking 2, Franklin 1, Fairfield 1

Toledo (3): Lucas 3.

Cleveland-Northeast: Lorain 3, Cuyahoga 2, Medina 2, Summit 3, Portage 1, Trumbull 1.

You want groups of 5, so that there can be extra representatives every session.

So you have

Group 1: Butler 4, Hamilton 1
Group 2: Montgomery 2, Greene 2, Lucas 1
Group 3: Licking 2, Franklin 1, Fairfield 1, Lucas 1
Group 4: Lorain 3, Medina 2
Group 5: Cuyahoga 2, Summit 3
Group 6: Trumbull 3, Portage 1, Lucas 1

Groups 4 and 5 have been arranged for greater economic demographic cohesion, putting Cleveland and Akron in one group, and the suburban counties in the other.

Lucas has been split up. It will get three extra sessions, but they will be coordinated with groups 2, 3, and 6.

Groups 2, 3, and 6 will have a representative in each session, with four of five in the home areas of Dayton, Columbus, and NE Ohio.

Groups 1, 4, and 5 will have a representative in each session.

There are five ways to arrange 4 extra sessions:

XXXXO
OXXXX
XOXXX
XXOXX
XXXOX

The one extra session complements it. It will help to see the patterns as rotations.

We throw a 5-sided dice, and it comes up 2.

Butler will get Session 2.
Hamilton will get Session 1, 3, 4, and 5.

There are five ways to arrange 3 extra sessions, while avoid making them consecutive, even in a circular fashion.

XXOXO
OXXOX
XOXXO
OXOXX
XOXOX

We roll our five-sided die for Lorain and again throw a 2.

So we have OXXOX or Medina, Lorain, Lorain, Medina, Lorain

We roll our five-sided die for Summit and throw a 4.

Our pattern is OXOXX or Cuyahoga, Summit, Cuyahoga, Summit, Summit.

We now roll to decide the three sessions for Lucas, and roll a 1.

So it will be Lucas, Lucas, ..., Lucas, ...

We need to determine which groups the three Lucas sessions are assigned to. We have a vase with balls containing "Dayton", "Columbus", and "NE Ohio".

We draw the first ball and it contains "Columbus". So we want to complete the pattern:

Lucas, Columbus, Columbus, Columbus, Columbus

There are three ways to assign the two sessions to Licking, without them being back to back:

LXOXO
LXOOX
LOXOX

We throw a 3-sided die and roll a 1. We then flip a coin, Franklin calls heads, and it comes up heads.

So the final pattern is: Lucas, Licking, Franklin, Licking, Fairfield

We pull the second ball out of our vase,and it says: NE Ohio

So we need to complete NEO, NEO, Lucas, NEO, NEO

There are two ways to place Portage, that also avoid putting Trumbull in three consecutive session:

PTLTT
TTLTP

Flip a coin, it comes up tails, and Portage is placed in the last session.

Trumbull, Trumbull, Lucas, Trumbull, Portage

The final ball contains "Dayton". There are two ways to avoid Montgomery and Greene being represented in consecutive sessions.

MGMGL
GMGML

Flip a coin, it comes up tails, so our final group:

Greene, Montgomery, Greene, Montgomery, Lucas

[Damocles]

Hmmm. So how would this work in the context of a proportional representation system, operating with multiple-member districts?

In the context of the United States House of Representatives, for example, could you not simply take the fractional quotas and transfer those seats to a “federal” constituency? It would smooth out these rough edges and provide each state with its own discrete representation, while the results in the “federal” seats would be determined by votes across all the states.

[jimrtex]

Hmmm. So how would this work in the context of a proportional representation system, operating with multiple-member districts?

In the context of the United States House of Representatives, for example, could you not simply take the fractional quotas and transfer those seats to a “federal” constituency? It would smooth out these rough edges and provide each state with its own discrete representation, while the results in the “federal” seats would be determined by votes across all the states.

The neatest thing about this method is that when I "invented" it, it might have been considered "creative", "ingenious", "whimsical", "crackpot", or "unworkable". It was only later that I discovered that Ohio had actually used such a system, had implemented it 160 years ago, and had used it successfully for 110 years.

It was like I had invented heavier than air flight or walked on the moon, and been told that some folks from Ohio had already done that.

Back to your original question. Using the Cube Root Rule, the new House should have 692 representatives. We apportion 692 * 5 units among the states, with a guarantee that every state will have at least 5 units (with that size house it is not a problem since Wyoming, Vermont, and Alaska will be apportioned 6, 7 and 8 units, respectively.

There are 15 states with less than 36 units, or more conventionally less than 6.0 representatives. They will elect the representatives statewide.

New Mexico is the largest with 4.4 representatives. In two elections during the decade they will elect 5 representatives, and in the other three they will elect four. The election could either be by STV or PAV.

Wyoming is the smallest with 1.2 representatives. It would elect two representative during one election, and one the other four.

Kansas is the next larger state with 6.2 representatives. It will be divided into two districts. Districts must have a minimum magnitude of 3.0, except for the smallest states. Districts have a maximum magnitude of 5.0, unless necessary to avoid districts with less than 3.0 magnitude, or division of counties, current districts, or other communities of interest, which may have a magnitude of up to 6.0.

There is no requirement that the two districts in Kansas have equal population, or that the two districts have population in a 32:30 proportion. It is more important that the two districts represent natural communities of interest. It would be desirable to have relatively stable districts.

Kansas is likely to have had two districts for several decades, with their magnitude decreasing over time. After each census, they would receive their share of the state's apportionment. The only adjustment that would have been needed is if the western district had dropped below 3.0.

The two districts in Kansas would have 3.2 and 3.0 magnitude. The district with a magnitude of 3.2 would elect four representatives in one election.

Now take the case of Utah with a magnitude of 3.8. The two districts might be 3.8 and 3.0 or 3.6 and 3.2 or 3.4 and 3.4. An unbalanced split might be preferred if it avoids chopping Salt Lake County. Or perhaps a doughnut could be drawn. Let's say that the split ends up a 3.2 and 3.6 split.

One district would elect four representative in one election, while the other would elect four representatives in 3 elections. The elections would be chosen so that only one district has an extra representative in any election. Utah would have a total of seven representatives in four of five elections, and six in the other.

California would have 82.8 representatives. This would require a minimum of 17 districts and permit a maximum of 27 districts. It is quite possible that no redistricting would be needed, just a calculation of the allocation of the representatives among the districts. Over the past decades, California might only have needed to split larger districts as they surpassed a magnitude of five or six.

Elections would be arrange so that extra representatives would float around Southern California and Northern California so a region would always range within one representative.

Having federal seats would violate the Constitution. My scheme is consistent with the Constitution. The apportionment is based on the census. There is nothing that says the apportionment has to be fixed during a decade. And representatives would be elected by the people of the states. The districts would likely be drawn by a federal commission.

[Boss_Rahm]

There is no requirement that the two districts in Kansas have equal population, or that the two districts have population in a 32:30 proportion.

If I'm understanding this system correctly, wouldn't a 32:30 ratio be required by the courts to satisfy OMOV?

[jimrtex]

There is no requirement that the two districts in Kansas have equal population, or that the two districts have population in a 32:30 proportion.

If I'm understanding this system correctly, wouldn't a 32:30 ratio be required by the courts to satisfy OMOV?

It depends.

If the two districts were 31:31 the error would about 3.1%. Under current precedent for legislative districts, anything other than 5% is presumed to be legal. A plaintiff would have to demonstrate that there was a some illegitimate intent.

Take a look at the current Kansas map. If I were going to divide Kansas into two parts, I would simply eliminate the split county at the northern end.

An alternative might be to move perhaps nine southeastern counties to the western district and extend the (northeastern) district out Manhattan (though splitting KSU and KU might more accurately reflect an urban-rural divide.

Kansas has an interest in not dividing counties and reasonably reflecting communities of interest.

I did think of a different way of conducting elections. so it would matter even less.

Elections would be conducted statewide by STV, but voters would be restricted to ranking candidates in their district.

In addition, each candidate would be required to file a ranking of all candidates statewide. A voter will be presumed to have ranked all candidates that he did not personally rank in the order preferred by the candidate.

You could simply attach the candidate's ranking to the bottom of the voter's ranking, and then cross out all candidates ranked by the voter from this bottom part of the ballot.

Generally, there will be sufficient votes in each district to elect most of the representatives from the district. The cross-over votes would choose the final representative or two.

This method would also eliminate the need to balance district populations based on the electorate size, which may differ significantly from the total population.

[jimrtex]

This map is based on the rules in the 1851 Constitution prior to the 1902 Hanna amendment that guaranteed every county a district.

The 1851 Constitution gave every county with above 0.5 of quota its own representative. Smaller counties were grouped into multi-county districts with a single representative.

On this map, the multi-county districts are shown in various green colors (Europe if you play Risk). The small numbers are populations relative to the quota.

The non-green districts are single member districts. The large numbers show the number of representatives. The fraction in fifths, is the number of sessions the county would elect another representative (for example, Franklin, Cuyahoga, and Hamilton would have an extra representative in 1, 2, and 4 sessions, respectively).



The smaller counties are all contiguous to each other, with the exception of Ottawa. It was attached to Sandusky which has a population of 0.501. The Constitution reads that a county that was part of a multi-county district could have its own district so long as the remainder of the district continued to have its own district. So if Sandusky had qualified for its own district, if it had been paired with Ottawa, the pair would not have been broken.

This is somewhat the inverse, where a pairing was forced (at some time in the past, Ottawa presumably would have had its own district).

The small counties form a large loop around Columbus, with a long tail along the western border and smaller tails elsewhere. This tends to results in districts formed from strings of counties end-to-end rather than more compact districts. It also reduces flexibility in grouping counties, such that they tend to relatively underpopulated.

Collectively, the 44 smallest counties would be entitled to 13.317 representatives, but they are drawn into 16 districts with an average population of 0.832.

Overall, the map creates 108 districts, from a quota based on 100 districts. The average district is 92.6% of the ideal district. But this varies considerably with the largest district 15.8% overpopulated, to the smallest -50.0% underpopulated.

The large counties, entitled from 1.0 to 1.75 representatives, and 2.0 and greater are entitled to 65.002 representatives but only have 63. This is due to the truncation of representation to the next lower 1/5. This results in an error of -3.1%. If simple rounding were used, this would mostly disappear.

The four counties with a population between 1.75 and 2.00, Lake, Mahoning, Delaware, and Clermont are each given 2 whole representatives. Collectively entitled to 7.481 representative, they are given 8. This is overrepresentation of 6.9%.

The 21 counties between 0.5 and 1.0 are collectively entitled to 14.201 representatives, but given 21. This is an average 47.9% error.

The 44 counties below 0.5 are collectively entitled to 13.317 representatives, but form 16 districts. This is an average 20.2%.

The counties whose representation is truncated, are entitled to 65.002 representatives in a 100-member body, but given 63 representatives in an 108 member body. This is an average -10.3% underrepresentation.

The counties whose representation is raised to a higher level are entitled to 34.998 representatives in an 100-member body, but are given 45 members in an 108-member body. This is an average 19.1% overrepresentation.

Districts with 41.0% of the population have 50.4% of the representation; districts with 59.0% of the population have 49.6% of the representation.

This variance along with a systemic bias towards smaller counties would likely render a OMOV ruling against the plan.

[Boss_Rahm]

I was curious how this system would have played out at the national level over the 2012-2020 cycle. Based on the cube root rule, I apportioned 677 seats by Huntington-Hill:

AL   10.6
AK   1.6
AZ   14.0
AR   6.4
CA   81.8
CO   11.0
CT   7.8
DE   2.0
FL   41.4
GA   21.2
HI   3.0
ID   3.4
IL   28.2
IN   14.2
IA   6.6
KS   6.2
KY   9.6
LA   10.0
ME   3.0
MD   12.6
MA   14.4
MI   21.8
MN   11.6
MS   6.6
MO   13.2
MT   2.2
NE   4.0
NV   6.0
NH   2.8
NJ   19.2
NM   4.6
NY   42.6
NC   21.0
ND   1.4
OH   25.4
OK   8.2
OR   8.4
PA   27.8
RI   2.4
SC   10.2
SD   1.8
TN   14.0
TX   55.4
UT   6.0
VT   1.4
VA   17.6
WA   14.8
WV   4.0
WI   12.4
WY   1.2

12 states have n.0 representatives, and will elect the same number to each Congress. 7 states have n.8 representatives, and will elect an extra representative in 2012, 2014, 2016, and 2018. 10 states have n.6 representatives, and will elect an extra representative in 2012, 2014, and 2016. 11 states have n.4 representatives, and will elect an extra representative in 2018 and 2020. Finally, 10 states have n.2 representatives. 7 of those states will elect an extra representative in 2020, and one each will elect an extra representative in 2012, 2014, and 2016. By drawing lots, I picked Oklahoma for 2012, Missouri for 2014, and Kansas for 2016.

I will assume that each state uses some form of proportional representation (either mixed-member or party list). The results that follow are based on the real-life House popular vote in each state.

2012: 340 Democrats, 337 Republicans

https://www.yapms.com/app/?m=9ayo

This is such a close election that the temporal variation affects the outcome. Under the 2014 and 2016 maps, Democrats win 339-338, but under the 2018 and 2020 maps, Republicans win 340-337. Additionally, Republicans could have won a majority by flipping 560 votes in West Virginia and 12,495 votes in Missouri.

2014: 360 Republicans, 317 Democrats

https://www.yapms.com/app/?m=9azq

2016: 343 Republicans, 331 Democrats, 3 Libertarians

https://www.yapms.com/app/?m=9b6d

The Libertarians include 1 from Arkansas and 2 from Texas.

2018: 369 Democrats, 308 Republicans

https://www.yapms.com/app/?m=9b75

2020: 347 Democrats, 330 Republicans

https://www.yapms.com/app/?m=9b7t

[jimrtex]

I was curious how this system would have played out at the national level over the 2012-2020 cycle. Based on the cube root rule, I apportioned 677 seats by Huntington-Hill:

AL 10.6
AK 1.6
AZ 14.0
AR 6.4
CA 81.8
CO 11.0
CT 7.8
DE 2.0
FL 41.4
GA 21.2
HI 3.0
ID 3.4
IL 28.2
IN 14.2
IA 6.6
KS 6.2
KY 9.6
LA 10.0
ME 3.0
MD 12.6
MA 14.4
MI 21.8
MN 11.6
MS 6.6
MO 13.2
MT 2.2
NE 4.0
NV 6.0
NH 2.8
NJ 19.2
NM 4.6
NY 42.6
NC 21.0
ND 1.4
OH 25.4
OK 8.2
OR 8.4
PA 27.8
RI 2.4
SC 10.2
SD 1.8
TN 14.0
TX 55.4
UT 6.0
VT 1.4
VA 17.6
WA 14.8
WV 4.0
WI 12.4
WY 1.2

12 states have n.0 representatives, and will elect the same number to each Congress. 7 states have n.8 representatives, and will elect an extra representative in 2012, 2014, 2016, and 2018. 10 states have n.6 representatives, and will elect an extra representative in 2012, 2014, and 2016. 11 states have n.4 representatives, and will elect an extra representative in 2018 and 2020. Finally, 10 states have n.2 representatives. 7 of those states will elect an extra representative in 2020, and one each will elect an extra representative in 2012, 2014, and 2016. By drawing lots, I picked Oklahoma for 2012, Missouri for 2014, and Kansas for 2016.

I would use Webster's method. The goal is that each state gets its fair share of representation. It might not matter since you are apportioning 3385 units, and each state gets at least 5. As the number of units increase, the methods converge.

I did not care for the method that Ohio used for setting the extra sessions. Because there were so many counties entitled to 1.2 representative, they ended up dominating the fifth session. Since these were mid-sized counties, there was a systemic bias.

For the US, there may be less systemic bias, since states with a 0.2 are more mixed, and include IL, GA, and NJ; MO and IN; SC, OK, and KS; ID and WY. But there is also a bit of geographic clustering.

For states that had 2 or 3 extra representatives, I think it would be better to not have the extra seat in consecutive sessions.

What I would do is geographically cluster the extra seats in groups of five or ten. Five is preferable, though this is not always possible while keeping the clusters together.

My clusters for 2010:

New England (10): NH 4, VT 2, MA 2, RI 2.
NY Suburban (5): CT 4, NJ 1
Mid-Atlantic (10): PA 4, NY 3, MD 3.

South Atlantic (5): VA 3, NC 1, SC 1.
Redneck Riviera (5): AL 3, FL 2.
Gulf Coast (5): MS 3, TX 2.
Mid-South (5): KY 3, AR 2.

Great Lakes (10): MI 4, OH 2, WI 2, IN 1, IL 1.
Norse Belt (5): MN 3, ND 2.
Midwest (5): IA 3, MO 1, KS 1.

Badlands(5): SD 4, MT 1.
Mountains(5): NM 3, OK 1, WY 1.
Pacific (10): CA 4, WA 4, OR 2.
Great Northwest (5): AK 3, ID 2

There are 18 floating seats:
Northeast 5
South 4
North Central 4
West 5

There will be geographic balance, which will tend to provide political balance, demographic and economic balance. Contrast with Senate Classes, where one class is extremely Southern.

I will assume that each state uses some form of proportional representation (either mixed-member or party list). The results that follow are based on the real-life House popular vote in each state.

I would use STV within districts but with carry forward to the state level. The partisan result will be similar to party list, but there will be better personal and geographical representation.

District offices could be shared, with shared clerical staff. Each representative from the district could have a personal office at one of the district offices, with personal staff.

Districts would have a magnitude of 3 to 5. The smallest states would elect from a single at-large district. Larger seats up to 6 (or perhaps more) could be used to avoid splitting states or counties.

The nominal maximum number of districts would be

   floor(total/3)

The nominal minimum number of districts would be:

   ceil(total/5)

Though this might vary a bit when there are larger districts.

Candidates would run in a district and voters would be limited to those candidates, but results would be tabulated state wide. Each candidate would provide a ranking of all candidates in the state. The rankings of a voter's first choice would be appended to the voter's rankings, and any candidates that the voter had ranked struck off.

Thus if a voter's rankings were exhausted they would be transferred statewide.

This method would also eliminate discrepancies between eligible or actual voters, and population. A district with a lot of non-citizens with a nominal magnitude of 3 based on population might only have enough votes to elect two local candidates.

If a district was 60D:40R, and nominally elected three representatives, it might elect one Republican with a small surplus for statewide candidates. There might be one Democrat, and a second with enough votes to survive to accumulate enough transfers from other areas to be elected.

2012: 340 Democrats, 337 Republicans

https://www.yapms.com/app/?m=9ayo

This is such a close election that the temporal variation affects the outcome. Under the 2014 and 2016 maps, Democrats win 339-338, but under the 2018 and 2020 maps, Republicans win 340-337. Additionally, Republicans could have won a majority by flipping 560 votes in West Virginia and 12,495 votes in Missouri.

2014: 360 Republicans, 317 Democrats

https://www.yapms.com/app/?m=9azq

2016: 343 Republicans, 331 Democrats, 3 Libertarians

https://www.yapms.com/app/?m=9b6d

The Libertarians include 1 from Arkansas and 2 from Texas.

2018: 369 Democrats, 308 Republicans

https://www.yapms.com/app/?m=9b75

2020: 347 Democrats, 330 Republicans

https://www.yapms.com/app/?m=9b7t

Thanks for these maps.

[Boss_Rahm]

I don't love Ohio's method of assigning fractional seats either. It produces basically the same map for the first 3 cycles, and then major changes the last 2. However, it does have the advantage of being prescriptive - there are no arbitrary decisions that can be gamed for partisan advantage. Another strength is that it avoids as much as possible multiple changes in a state's number of seats. Your geographic method has advantages too, but I'd worry that whoever has the power to decide the groupings would find a way to favor one party over the other. I might prefer a purely random system to remove the element of choice altogether while avoiding bias.

[jimrtex]

I don't love Ohio's method of assigning fractional seats either. It produces basically the same map for the first 3 cycles, and then major changes the last 2. However, it does have the advantage of being prescriptive - there are no arbitrary decisions that can be gamed for partisan advantage. Another strength is that it avoids as much as possible multiple changes in a state's number of seats. Your geographic method has advantages too, but I'd worry that whoever has the power to decide the groupings would find a way to favor one party over the other. I might prefer a purely random system to remove the element of choice altogether while avoiding bias.

If you wanted a prescriptive method. Consider the case of state entitled to 1.8 representatives per session.

1st Session: 1.8
1st and 2nd Session: 3.6
1st, 2nd, 3rd Session: 5.4
1st, 2nd, 3rd, 4th: 7.2
1st, 2nd, 3rd, 4th, 5th: 9.0

If we round these two to the nearest integer, then the best arrangement for four extra representatives is:

4: + + 0 + +

Similarly for 3, 2, and 1 extra representatives:

3: + 0 + 0 +
2: 0 + 0 + 0
1: 0 0 + 0 0

Not unsurprisingly, the patterns for 4 and 1 extra are complementary, as are those for 3 and 2 extra.

If there were the same number of states with 4 and 1 extra representative, we could assign use the above pattern for of them. The same is true for states with 3 and 2 extra representatives.

We could also permit the size of the House vary per session. This was the case in Ohio, which I found highly undesirable. If you look at the schedule in the Ohio Constitution, there  is a large number electing 1.2 representatives, resulting in the 5th session of the decade having more representatives.

But varying the size of the House each session would seem to be highly undesirable.

Let's look at your data for 2010:

n4 = 7
n3 = 10
n2 = 11
n1 = 10
n0 = 12

Total: 7*4 + 10*3 + 11*2 + 10*1 + 12*0 = 90 = 18*5

We need to assign 18 representatives to each class. It seems reasonable to me that those states with the fewest extra representatives should have the most flexibility - which is the pattern you chose by randomly assigning the extra representative for KS, MO, and OK to a different session.

In general. Let t = the total number of extra representatives per session. In the 2010 example: t = 18.

We can always assign the n4 states to their standard pattern of ++0++.

The above is false, so we will need to handle this case.

So we have n4,n4,0,n4,n4 assigned sessions.

We then try to assign the n3 states. The standard pattern for these states coincides with the n4 pattern in the 1st and 5th sessions. There is no problem giving these states the 3rd session, and helps fill up the hole left by the n4 states.

Case I if n4+n3 <= t, we can use the regular pattern for all n3 states.

See below.

s3 = n3
e34 = 0

Case II if n4+n3 > t we must assign some of the n3 states a pattern that is different than the standard pattern.

It seems reasonable to either assign these states:

0++0+ or +0++0

The other possible choice of 0+++0 seems less desirable since it would result in the extra representatives being assigned in consecutive sessions, even though we would need to assign fewer states to that pattern.

let e34 = n4 + n3 - t
and s3 = n3 - 2 * e34

s3 could be negative which would also have to compensated for.

e34 is the number of states that will be assigned 0++0+ AND +0++0 (they will be randomly chosen in pairs).

After assigning the sessions for n4 and n3 we would have:

n4+s3,n4+e34,n3,n4+e34,n4+s3

We would then try to assign

n2 instances of 0+0+0

But this might not be possible, so we would have to fit in some:

+00+0 and 0+00+ or perhaps +000+

But there might be cases where we were having to assign all the extra sessions in the third  session, even if it meant assigning 2 or 3 extra representatives into a single session.

It might be possible to handle all cases in a prescriptive manner, but it would require a fairly complex analysis.

Even if we could assign all the extra sessions in a straightforward manner, as is the case in the 2010 data set, it could still result in multiple states being in phase together, which could result in a geographic bias, just like there are among senate classes.

If you look at the four census regions (but move DE and MD to Northeast), your distribution is fairly balanced, but that may be by luck, depending on how OK, KS, and MO were chosen. If you had randomly selected MT, WY, and NJ the distribution would be worse.

More thoughts to come ...

[Boss_Rahm]

I am genuinely impressed at the level of thought you've put into this system.

The more I think about it, the more I favor a purely random system. So for the 2010 example, you would put 90 balls in a lottery machine, and then draw them into the 90 available positions. Then you would reshuffle at random if a state ends up with multiple extra seats in the same Congress. I'll try to explain why I think that would be preferable to a more predictable approach.

My prior is that over the long run, state populations will follow a roughly Zipfian distribution; i.e., ratios of 1:1/2:1/3:...:1/50. This means there will be more states with smaller numbers of seats than there will be states with larger numbers of seats.

I'm not positive, but my intuition is that this will bias the set of states with remainders of .2 to be smaller states on average than states with remainders of .8. We are apportioning enough seats that this effect will not be very announced, but I think it will still show up in the long run. As a sanity check, I ranked the states by population 1-50, and then calculated the average rank for each set of states with a common remainder. Here are the results:

1 extra seat: 23.7
2 extra seats: 25.8
3 extra seats: 24.7
4  extra seats: 20.7

It's not a perfect correlation, but this tracks with the intuition that larger states (i.e., ones with smaller ranks) will tend to have larger remainders. If that is indeed the case, then following your scheme would mean that the third Congress of each decade would tend to systematically overrepresent smaller states and systematically underrepresent larger states. Again, the effect will be small, and in some decades may not be present at all. But I think it's worth the effort to try to avoid systematic bias of this kind.

Randomness by no means guarantees that the resulting map won't be biased towards a particular size of state, or a particular region, or a particular demographic group. But it does remove systematic bias.

[jimrtex]

I am genuinely impressed at the level of thought you've put into this system.

The more I think about it, the more I favor a purely random system. So for the 2010 example, you would put 90 balls in a lottery machine, and then draw them into the 90 available positions. Then you would reshuffle at random if a state ends up with multiple extra seats in the same Congress. I'll try to explain why I think that would be preferable to a more predictable approach.

My prior is that over the long run, state populations will follow a roughly Zipfian distribution; i.e., ratios of 1:1/2:1/3:...:1/50. This means there will be more states with smaller numbers of seats than there will be states with larger numbers of seats.

I'm not positive, but my intuition is that this will bias the set of states with remainders of .2 to be smaller states on average than states with remainders of .8. We are apportioning enough seats that this effect will not be very announced, but I think it will still show up in the long run. As a sanity check, I ranked the states by population 1-50, and then calculated the average rank for each set of states with a common remainder. Here are the results:

1 extra seat: 23.7
2 extra seats: 25.8
3 extra seats: 24.7
4  extra seats: 20.7

It's not a perfect correlation, but this tracks with the intuition that larger states (i.e., ones with smaller ranks) will tend to have larger remainders. If that is indeed the case, then following your scheme would mean that the third Congress of each decade would tend to systematically overrepresent smaller states and systematically underrepresent larger states. Again, the effect will be small, and in some decades may not be present at all. But I think it's worth the effort to try to avoid systematic bias of this kind.

Randomness by no means guarantees that the resulting map won't be biased towards a particular size of state, or a particular region, or a particular demographic group. But it does remove systematic bias.

A concern that I had with random was that there might be some clustering. That is, a state with two extra representatives might have them in consecutive sessions. If you consider the five sessions to form a circular pattern, then there are five ways to arrange two extra representatives:

+0+00
0+0+0
00+0+
+00+0
0+00+

These not only ensure that the extra representatives don't occur in consecutive sessions, but that the there are not three consecutive sessions without an extra representative. That is, while +000+ keeps the extra sessions separate, it also means three consecutive sessions without representation.

There are five patterns of three extra representatives, each of which is a complement of a pattern for two extra representatives.

0+0++
+0+0+
++0+0
0++0+
+0++0

We can pair a state with two extra representatives with a state with three extra representatives and assign them any pair of complementary patterns. If there are multiple pairs of +2 and +3 states we assign them different pairs of pattern so that the states aren't in phase.

There are also five patterns to arrange four extra representatives:

++++0
0++++
+0+++
++0++
+++0+

And five complementary patterns to arrange one extra representative:

0000+
+0000
0+000
00+00
0000+

We will have identified

n4 states with 4 extra representatives.
n3 with 3 extra
n2 with 2 extra
n1 with 1 extra
n0 with no extra.

Because of the way we apportioned the extra seats, we know that:

n4*4 + n3*3 + n2*2 + n1*1 + n0*0 is a multiple of five.

It is also possible that there are zero in any sets of states with a particular number of extra representatives.

For example, we could have 90 extra representatives with n4=15, n2=15, n0=20, and n1=n3=0. Or we could have n4=22, n2=1, n0=27, n1=n3=0.

While these are possible, they are quite improbable. Presumably the distribution will be fairly uniform, with a small residual effect due to the roughly Zipfian distribution of populations.

We can pair +4 and +1 states.

p14 = min(n4,n1)

We choose the +4 and +1 states at random. We randomly assign the patterns so that among five pairs of states that each pattern is used once.

So we would have one vase with n4 balls with a state name inside; a second vase with n1 balls with a state name inside; and a third vase with five balls with the numbers 1 to 5 inside.

If the third vase is emptied, we replace it with another vase with the numbers 1 to 5 inside.

In drawing the +4 and +1 states, we will exhaust all the +4 states, or all the +1 states or all of both. That is

n4 = p14 < n1
n1 = p14 < n4
n1 = n4 = p14

This will leave a residual

r1 = n1 - p14; r4 = 0
r4 = n4 - p14; r1 = 0
r1 = r4 = 0

The same process can be used to +3 and +2 states.

In 2010 data set:

n4 = 7
n3 = 10
n2 = 11
n1 = 10
n0 = 12

p14 = 7
p23 = 10

r4 = 0
r3 = 0
r2 = 1
r1 = 3
r0 = 0

r4*4 + r3*3 + r2*2 + r1*1 + r0*0 will be a multiple of five, since each pair of +4 and +1, or +3 and +2 states removed 5 extra representatives of the total.

At most, two of r4, r3, r2, and r1 can be non-zero; and at least one of r4 and r1 will be zero; and at least one of r2 and r3 will be zero.

If any of r4, r3, r2, r1 is five or greater we can randomly pick five states and assign each one of the five patterns. We can repeat until r4, r3, r2, r1 is 4 or less.

If only one of r4, r3, r2, r1 is non-zero, then it is a multiple of 5 and all remaining unassigned states can be handled that way.

Case I: 0 < r4 < 5 and 0 < r3 < 5

That is, n4 > n1 and n3 > n2, and we have removed any excess groups of five.

Case IA If r4 = 1, r3 = 2 (remember that r4*4 + r3*3 = a multiple of 10)

We can assign a pattern to the +4 state, and then choose patterns for the +3 states such that they both have an extra representative in the session where the +4 state did not have a representative (there are three such +3 patterns, so we pick 2).

Case 1C if r4 = 3, r3 = 1

We can assign the +3 pattern randomly, and then assign the +4 patterns which don't have a representative in sessions where the +3 pattern does.

Case 1B if r4 = 2, r3 = 4

We can divide this in to two sets of +4,+3,+3 and +4,+3,+3 and use the assignment of Case IA above. We would use different +4 patterns to minimize the number of +4 states that have the same pattern.

Case 1D if r4 = 4, r3 = 9

We can divide this in two sets of +4,+3,+3 and +4,+4,+4,+3 and use the assignments of IA and IC above.

We can repeat the same analysis

Case II: 0 < r4 < 5 and 0 < r2 < 5
Case III: 0 < r3 < 5 and 0 < r1 < 5
Case IV: 0 < r2 < 5 and 0 < r1 < 5

For the 2010 data set:

r2 = 1
r3 = 3

This is an instance of Case IV, and we assign the one remaining +2 state to one of the +2 patterns, and each of the three remaining +1 states to sessions where the +2 state was not represented.

This method assures that we minimize +4 and +3 states being in phase, and keeps +2 and +3 states having too many consecutive sessions. It avoids the instances where we would have to reject a drawing and pick again. It also avoids the possibility of political corruption  in assigning states to pairs geographically.

This is my draw for the 2010 data set. The first 7 rows are for pairs of +4 and +1 states. The session for the first five +1 states SC, OK, WY, MT, GA were drawn randomly avoiding repeats. The sessions for the next two +1 states MO, NJ were also drawn randomly.

The next 10 rows are for pairs of +3 and +2 states, the sessions for the first five rows and the next five rows were drawn randomly, such that each pattern is used twice.

The last row is for the residual patterns r2 = 1, r1 = 3. The session for the +1 states IL, IN, and KS use sessions that the other +1 states that had been paired with +4 states had not used.

2012	2014	2016	2018	2020
WA	WA	WA	WA	SC
OK	SD	SD	SD	SD
NH	NH	WY	NH	NH
PA	PA	PA	MT	PA
CA	GA	CA	CA	CA
MI	MI	MI	MO	MI
NJ	CT	CT	CT	CT
MN	TX	MN	MN	TX
WI	MD	WI	MD	MD
AR	AL	AL	AR	AL
MS	OR	MS	OR	MS
KY	KY	MA	KY	MA
VA	VA	FL	VA	FL
NY	OH	NY	NY	OH
NM	VT	NM	VT	NM
ID	AK	ID	AK	AK
ND	IA	IA	ND	IA
RI	IL	IN	RI	KS

There is pretty good distribution of the larger states, except 2018 only has 2 large states (NJ or larger).

The regional distribution is pretty even:

West: 4,3,5,5,3
South: 5,5,3,3,5
Midwest: 4,5,6,4,5
Northeast: 5,5,4,6,5

This is particularly true if we remember that 654 of the total 672 representatives are static.

If I understand your simulations of the results, that the extra representative for a given election in a given state is not necessarily assigned to the larger party. That is the representative share is the vote share times the number of total representatives then rounded. Which party gets the favorable rounding for the final seat is somewhat random.

[Boss_Rahm]

I'm satisfied with the procedure you've laid out here. As far as I can tell it will be unbiased.

If I understand your simulations of the results, that the extra representative for a given election in a given state is not necessarily assigned to the larger party. That is the representative share is the vote share times the number of total representatives then rounded. Which party gets the favorable rounding for the final seat is somewhat random.

That's correct. I use the Droop Quota for allocating seats (mostly because it's easy to do in Excel).

So, 659 of the 677 are fixed, and the other 18 float. 38 states will have delegations that fluctuate in size.

In 2014, 2018, and 2020, the 659 fixed seats give the winning party a majority on their own (in the case of 2020, with no seats to spare).

In 2016, Republicans are guaranteed 336 seats, Democrats are guaranteed 320, and Libertarians are guaranteed 3. Of the 38 states with floating seats, the marginal seat would go to a Democrat in 19 and a Republican in the other 19. But even if the stars aligned to deny Republicans a majority, I imagine they would govern in coalition with the Libertarians. So I won't analyze this election further.

In 2012, Republicans are guaranteed 332 seats, and Democrats are guaranteed 327 seats. However, 23 states would give Democrats a marginal seat, and 15 would give it to Republicans. In your example below, Democrats win 12 marginal seats and Republicans win 6, giving Democrats a 1-seat majority. A natural question would be, what are the odds of a draw that produces a Democrat majority, or a draw that produces a Republican majority? I worked out all the combinatorics, and here are the probabilities of each possible number of seats for the Democrats:

333   0.008%
334   0.191%
335   1.623%
336   7.022%
337   17.489%
338   26.563%
(majority)
339   25.244%
340   15.050%
341   5.510%
342   1.169%
343   0.126%
344   0.005%

In sum, Democrats win a majority 47.103% of the time.

[jimrtex]

I am genuinely impressed at the level of thought you've put into this system.

This is a thread where I had previous explored the idea. It extends the idea to multi-seat congressional districts in each state.
   
Temporal Weighted Apportionment

This is where I first started developing the idea. I had created some maps, but they aren't uploaded to atlas.
   
Cube Root Rule Legislative Districts

This is the message where I discovered that I had re-invented the wheel.

Nothing New Under The Sun

[jimrtex]

I'm satisfied with the procedure you've laid out here. As far as I can tell it will be unbiased.

In 2012, Republicans are guaranteed 332 seats, and Democrats are guaranteed 327 seats. However, 23 states would give Democrats a marginal seat, and 15 would give it to Republicans. In your example below, Democrats win 12 marginal seats and Republicans win 6, giving Democrats a 1-seat majority. A natural question would be, what are the odds of a draw that produces a Democrat majority, or a draw that produces a Republican majority? I worked out all the combinatorics, and here are the probabilities of each possible number of seats for the Democrats:

333 0.008%
334 0.191%
335 1.623%
336 7.022%
337 17.489%
338 26.563%
(majority)
339 25.244%
340 15.050%
341 5.510%
342 1.169%
343 0.126%
344 0.005%

In sum, Democrats win a majority 47.103% of the time.

Do you have lists of the 23 states where Democrats would be likely to win the extra seat, and 15 states were Republicans would?