apportionment schemes

Apportionment Schemes

Dan VillarrealMATH 490-02Tuesday, Sept. 15, 2009

Sept. 15, 2009 Apportionment Schemes

But first…a quick PSA


What is Apportionment?

The apportionment problem is to round a set of fractions so that their sum is maintained at original value. The rounding procedure must not be an arbitrary one, but one that can be applied constantly. Any such rounding procedure is called an apportionment method.


Example

In the 1974-75 NHL season, the Stanley Cup Champion Philadelphia Flyers won 51 games, lost 18 games, and tied 11 games.

Won: 51 63.75% 64%Lost: 18 22.5% 23%Tied: 11 13.75% 14%

But this adds up to 101%, an impossibility!


Dramatis Personae

George Washington Alexander Hamilton Thomas Jefferson Daniel Webster

Delaware Virginia


The Constitution

Amendment 14, Section 2:

Article I, Section 2:“The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative”

“Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State”


The First Apportionment

For the third session of Congress (1793-1795) House of Representatives set at 105 15 states U.S. Population: 3,615,920

3,615,920

105= 34,437 people/district


Standard Divisors

34,437 is our standard divisor for 1790. More generally,

SDt =

Where HSt is the size of the House of Representatives (or whatever overall body) for year t.

Poptotal

HSt


Quotas

The number of Congressional districts a state should get is its quota:

Qi =

Take Delaware, for example…

Popi

SDt


If only it were that easy…

THERE’S NO SUCH THING AS .613

CONGRESSPERSONS.Hence the need for apportionment schemes, a way to map the quotas in R onto apportionments in Z.


If only it were that easy…State Population Quota

Virginia 630,560 18.310

Massachusetts 475,327 13.803

Pennsylvania 432,879 12.570

North Carolina 353,523 10.266

New York 331,589 9.629

Maryland 278,514 8.088

Connecticut 236,841 6.878

South Carolina 206,236 5.989

New Jersey 179,570 5.214

New Hampshire 141,822 4.118

Vermont 85,533 2.484

Georgia 70,835 2.057

Kentucky 68,705 1.995

Rhode Island 68,446 1.988

Delaware 55,540 1.613

Totals 3,615,920 105


More on quotas

The lower quota is the quota rounded down (or the integer part of the quota):

LQi = ⌊Qi⌋

The upper quota is the quota rounded up:

UQi = ⌈Qi⌉ = ⌊Qi + 1⌋


If only it were that easy…State Population Quota LQ UQ

Virginia 630,560 18.310 18 19

Massachusetts 475,327 13.803 13 14

Pennsylvania 432,879 12.570 12 13

North Carolina 353,523 10.266 10 11

New York 331,589 9.629 9 10

Maryland 278,514 8.088 8 9

Connecticut 236,841 6.878 6 7

South Carolina 206,236 5.989 5 6

New Jersey 179,570 5.214 5 6

New Hampshire 141,822 4.118 4 5

Vermont 85,533 2.484 2 3

Georgia 70,835 2.057 2 3

Kentucky 68,705 1.995 1 2

Rhode Island 68,446 1.988 1 2

Delaware 55,540 1.613 1 2

Totals 3,615,920 105 97 112


Alexander Hamilton

1755-1804 One author of Federalist Papers First Secretary of the Treasury

Most importantly for our purposes, devised the Hamilton Method for apportioning Congressional districts to states


The Hamilton Method

State i receives either its lower quota or upper quota in districts; those states that receive their upper quota are those with the greatest fractional parts


Back to 1790State Population Quota Frac. Part

Virginia 630,560 18.310 .310

Massachusetts 475,327 13.803 .803

Pennsylvania 432,879 12.570 .570

North Carolina 353,523 10.266 .266

New York 331,589 9.629 .629

Maryland 278,514 8.088 .088

Connecticut 236,841 6.878 .878

South Carolina 206,236 5.989 .989

New Jersey 179,570 5.214 .214

New Hampshire 141,822 4.118 .118

Vermont 85,533 2.484 .484

Georgia 70,835 2.057 .057

Kentucky 68,705 1.995 .995

Rhode Island 68,446 1.988 .988

Delaware 55,540 1.613 .613

State Frac. Part

Kentucky .995

South Carolina .989

Rhode Island .988

Connecticut .878

Massachusetts .803

New York .629

Delaware .613

Pennsylvania .570

Vermont .484

Virginia .310

North Carolina .266

New Jersey .214

New Hampshire .118

Maryland .088

Georgia .057


If only it were that easy…State Population Quota LQ UQ Apportionment

Virginia 630,560 18.310 18 19 18

Massachusetts 475,327 13.803 13 14 14

Pennsylvania 432,879 12.570 12 13 13

North Carolina 353,523 10.266 10 11 10

New York 331,589 9.629 9 10 10

Maryland 278,514 8.088 8 9 8

Connecticut 236,841 6.878 6 7 7

South Carolina 206,236 5.989 5 6 6

New Jersey 179,570 5.214 5 6 5

New Hampshire 141,822 4.118 4 5 4

Vermont 85,533 2.484 2 3 2

Georgia 70,835 2.057 2 3 2

Kentucky 68,705 1.995 1 2 2

Rhode Island 68,446 1.988 1 2 2

Delaware 55,540 1.613 1 2 2

Totals 3,615,920 105 97 112 105


Back to Square One

President Washington vetoed the Apportionment Bill because he believed, following the counsel of Edmund Randolph and Thomas Jefferson, that it was unconstitutional:

ADE 2 1

PopDE 55,540 30,000= <


The Alabama Paradox

The Hamilton Method was Congress’s preferred method of apportionment from 1850 to 1900.

In 1881, the Alabama Paradox was first discovered.

The Census Bureau, as a matter of course, calculated apportionments for a range of House sizes; in this case, 275-350

Something interesting and very weird happened between the tables for HS = 299 and 300…


The Alabama Paradox

US population in 1880 was 49,369,595


The Alabama Paradox

US population in 1880 was 49,369,595 For HS = 299, SD = 165,116


The Alabama Paradox

US population in 1880 was 49,369,595 For HS = 299, SD = 165,116 For HS = 300, SD = 164,565


The Alabama Paradox


Population

Alabama 1,262,794

Illinois 3,078,769

Texas 1,592,574


The Alabama Paradox


HS = 299Population LQ Frac. A

Alabama 1,262,794 7 .647

Illinois 3,078,769 18 .646

Texas 1,592,574 9 .645


The Alabama Paradox


HS = 299Population LQ Frac. A

Alabama 1,262,794 7 .647 8

Illinois 3,078,769 18 .646 18

Texas 1,592,574 9 .645 9


The Alabama Paradox


HS = 299 HS = 300Population LQ Frac. A LQ Frac. A

Alabama 1,262,794 7 .647 8 7 .674

Illinois 3,078,769 18 .646 18 18 .708

Texas 1,592,574 9 .645 9 9 .677


The Alabama Paradox


HS = 299 HS = 300Population LQ Frac. A LQ Frac. A

Alabama 1,262,794 7 .647 8 7 .674 7

Illinois 3,078,769 18 .646 18 18 .708 19

Texas 1,592,574 9 .645 9 9 .677 10


Back to 1793…

This particular issue with the Hamilton Method was not discovered until 1881, but the Constitutional constraints meant that it could not be used in 1793.

A new method was proposed by Thomas Jefferson: the Jefferson Method.


Thomas Jefferson

Biographical Information:

You know this all already…

Had the good fortune never to take a class in Morton Hall


The Jefferson Method Rather than use the standard divisor SD, the

Jefferson Method uses the population of the smallest district, d.

Each state receives an adjusted quota; this will need to be rounded down the actual apportionment:

Ai = ⌊ ⌋Popi

d


The Jefferson Method In 1793, Jefferson used d = 33,000, so

AVA = 630,560 / 33,000 = 19.108 = 19⌊ ⌋ ⌊ ⌋

ADE = 55,540 / 33,000 = 1.683 = 1⌊ ⌋ ⌊ ⌋

But how do we determine d in the first place?


Finding the Critical Divisor

Start with the lower quota of each state; this is its tentative apportionment, ni.Next, find the critical divisor for each state:

Popi

ni + 1

For example, dVA = 630,560 / (18 + 1) = 33,187

dDE = 55,540 / (1 + 1) = 27,770

di =


The Critical Divisor

The critical divisor for each state is the divisor for which the state will be entitled to ni + 1 seats.

For example, if d > 27,770, Delaware gets only 1 seat, but for d ≤ 27,770, Delaware gets 2.

But then Virginia gets 630,560 / 27,770 = ⌊ ⌋22.707 = 22 seats. This will surely result in an ⌊ ⌋

overfull House Thus, d will need to be greater than 27,770


The Jefferson Method

Step 1: Assume a tentative apportionment of the lower quota for each state: ni = LQi

Step 2: Determine the critical divisor di for each state and rank by di

Step 3: If any seats remain to be filled, grant one to the state with the highest di; recompute di for this state since its ni has now increased by 1.

Step 4: Iterate Step 3 until the House is filled.


The Jefferson Method

This method actually was used for the 1793 apportionment, and it resulted in Virginia receiving 19 seats to Delaware’s one.

Used until about 1840 Not subject to the Alabama paradox But fails to satisfy the quota condition…


The Quota Condition

The quota condition is twofold: 1. No state may receive fewer seats than its

lower quota 2. No state may receive more seats than its

upper quota The Jefferson Method does just fine with 1, but

not 2


Example U.S. population in 1820 was 8,969,878, with a

House size of 213, so SD = 8,969,878 / 213 = 42,112

New York had a population of 1,368,775:

QNY = 1,368,775 / 42,112 = 32.503 So if the quota condition was satisfied, New

York’s delegation should be either 32 or 33 Using the Jefferson Method and d = 39,900, we

actually get 34 seats for New York


What’s the Problem? The Jefferson Method always skews in favor of

the large states. Let ui = pi / d be the state’s adjusted quota. Then

Ai = ⌊ui . Now compare ⌋ ui with the state’s quota:

M = = / = × =

Then ui = M * Qi => ai = ⌊M * Qi⌋ The rich only get richer…

ui Popi Popi Popi SD SD

Qi d SD d Popi d


The Webster Method

Daniel Webster devised an apportionment method that was similar in nature to Jefferson’s, but that did not unconditionally favor large states.

Used for 1840-1850 reapportionments, then 1900-1930


The Webster Method Step 1: Determine SD, and find the quota Qi for

each state i. Step 2: Round each quota up or down and let this

be the tentative apportionment ni for each state. Step 3: Determine the total apportionment at this

point. 3 cases: 1. The total apportionment equals HS 2. The total apportionment is greater than HS 3. The total apportionment is less than HS


Adjusting the Apportionment

If we have an overfill, at least one or more seats needs to be pared off. Let the critical divisor be di

- = pi / (ni - 1/2). The state with the smallest di-

will be the next to lose a seat. Conversely, if we have an underfill, we need to

add more seats. Let the critical divisor be di

+ = pi / (ni + 1/2). The state with the smallest di+

will be the next to gain a seat. Iterate either process until done.


Large State Bias

How does the Webster Method avoid susceptibility to the large-state bias exhibited by the Jefferson Method?

We get a similar expression for M: M = SD/d


Large State Bias

M > 1 when there is an underfill, thus in this circumstance, the larger states are more likely to receive another seat

But when there is an overfill and we must subtract, M < 1, and the larger states are more likely to get a seat subtracted

Equally likely to get an overfill or underfill Thus, equally likely that the Webster Method will

favor neither large nor small


Timeline

1790 1840 1850 1900

1900 1940 Present

Jefferson

MethodWebster

Method

Hamilton

Method

Webster

MethodHill-Huntington

Method


Hill-Huntington Method Step 1: Start with assumption that each state gets

1 seat (i.e., set ni = 1 for all i) Step 2: Calculate the priority value for each state

Popi

(ni(ni + 1))1/2

Step 3: The state with the greatest PVi is granted the next seat, increasing its tentative apportionment ni by 1; recalculate this state’s PVi

Step 4: Iterate Step 3 until the House is filled.

PVi,n =


Hill-Huntington in 2000

US population in 2000: 281,421,906 435 seats in the House California population: 33,871,648 Texas population: 20,851,820 PVCA,1 = 33,871,648 / (2)1/2 = 23,992,697 PVTX,1 = 20,851,820 / (2)1/2 = 14,781,356 PVCA,2 = 33,871,648 / (6)1/2 = 13,852,190 http://www.census.gov/population/censusdata/

apportionment/00pvalues.txt


The Population Paradox

The Hill-Huntington Method is immune to the Alabama paradox, but may violate the quota condition.

In the 1970s, two mathematicians attempted to devise a method that was immune to both violations, and they did…but another paradox popped up: the population paradox.

This paradox occurs when the population of one state increases at a greater rate than others, but fails to gain a seat.



Exercise 10, COMAP page 535:

State Old Census

Q A

A 5,525,381

B 3,470,152

C 3,864,226

D 201,203

Tot 13,060,962



Exercise 10, COMAP page 535: House size set at 100

State Old Census

Q A

A 5,525,381

B 3,470,152

C 3,864,226

D 201,203

Tot 13,060,962




SD = 13,060,962 / 100 = 130,610

State Old Census

Q A

A 5,525,381

B 3,470,152

C 3,864,226

D 201,203

Tot 13,060,962




SD = 13,060,962 / 100 = 130,610

State Old Census

Q A

A 5,525,381 42.304

B 3,470,152 26.569

C 3,864,226 29.586

D 201,203 1.540

Tot 13,060,962




SD = 13,060,962 / 100 = 130,610

State Old Census

Q A

A 5,525,381 42.304 42

B 3,470,152 26.569 27

C 3,864,226 29.586 30

D 201,203 1.540 1

Tot 13,060,962




State Old Census

Q A New Census Q A

A 5,525,381 42.304 42 5,657,564

B 3,470,152 26.569 27 3,507,464

C 3,864,226 29.586 30 3,885,693

D 201,203 1.540 1 201,049

Tot 13,060,962 13,251,770




SD = 13,251,770 / 100 = 132,518

State Old Census

Q A New Census Q A

A 5,525,381 42.304 42 5,657,564

B 3,470,152 26.569 27 3,507,464

C 3,864,226 29.586 30 3,885,693

D 201,203 1.540 1 201,049

Tot 13,060,962 13,251,770




SD = 13,251,770 / 100 = 132,518

State Old Census

Q A New Census Q A

A 5,525,381 42.304 42 5,657,564 42.693

B 3,470,152 26.569 27 3,507,464 26.468

C 3,864,226 29.586 30 3,885,693 29.322

D 201,203 1.540 1 201,049 1.517

Tot 13,060,962 13,251,770




SD = 13,251,770 / 100 = 132,518

State Old Census

Q A New Census Q A

A 5,525,381 42.304 42 5,657,564 42.693 43

B 3,470,152 26.569 27 3,507,464 26.468 26

C 3,864,226 29.586 30 3,885,693 29.322 30

D 201,203 1.540 1 201,049 1.517 2

Tot 13,060,962 13,251,770




State D lost population, yet gained a seat!

State Old Census

Q A New Census Q A

A 5,525,381 42.304 42 5,657,564 42.693 43

B 3,470,152 26.569 27 3,507,464 26.468 26

C 3,864,226 29.586 30 3,885,693 29.322 30

D 201,203 1.540 1 201,049 1.517 2

Tot 13,060,962 13,251,770


SO MANY PARADOXES!!!

The apportionment methods that Congress has used have either violated the quota condition (Jefferson, Webster, Hill-Huntington) or the Alabama and population paradoxes (Hamilton)

The quota method (never used by Congress) violates the population paradox

Is this just another instance of that old joke?


SO MANY PARADOXES!!! It turns out that this is endemic to the situation Theorem “No apportionment method that satisfies

the quota condition is free of paradoxes” (COMAP, p. 519)

Proof The only methods that are free of paradoxes are the divisor methods (Jefferson, Webster, Hill-Huntington). But the divisor methods are all subject to violating the quota condition.

Thus, we are basically screwed.


Hill-Huntington…in 1790!State Population Quota J

Virginia 630,560 18.310 19

Massachusetts 475,327 13.803 14

Pennsylvania 432,879 12.570 13

North Carolina 353,523 10.266 10

New York 331,589 9.629 10

Maryland 278,514 8.088 8

Connecticut 236,841 6.878 7

South Carolina 206,236 5.989 6

New Jersey 179,570 5.214 5

New Hampshire 141,822 4.118 4

Vermont 85,533 2.484 2

Georgia 70,835 2.057 2

Kentucky 68,705 1.995 2

Rhode Island 68,446 1.988 2

Delaware 55,540 1.613 1

Totals 3,615,920 105 105


Hill-Huntington…in 1790!State Population Quota J H

Virginia 630,560 18.310 19 18

Massachusetts 475,327 13.803 14 14

Pennsylvania 432,879 12.570 13 13

North Carolina 353,523 10.266 10 10

New York 331,589 9.629 10 10

Maryland 278,514 8.088 8 8

Connecticut 236,841 6.878 7 7

South Carolina 206,236 5.989 6 6

New Jersey 179,570 5.214 5 5

New Hampshire 141,822 4.118 4 4

Vermont 85,533 2.484 2 2

Georgia 70,835 2.057 2 2

Kentucky 68,705 1.995 2 2

Rhode Island 68,446 1.988 2 2

Delaware 55,540 1.613 1 2

Totals 3,615,920 105 105 105


Hill-Huntington…in 1790!State Population Quota J H H-H

Virginia 630,560 18.310 19 18 18

Massachusetts 475,327 13.803 14 14 14

Pennsylvania 432,879 12.570 13 13 12

North Carolina 353,523 10.266 10 10 10

New York 331,589 9.629 10 10 10

Maryland 278,514 8.088 8 8 8

Connecticut 236,841 6.878 7 7 7

South Carolina 206,236 5.989 6 6 6

New Jersey 179,570 5.214 5 5 5

New Hampshire 141,822 4.118 4 4 4

Vermont 85,533 2.484 2 2 3

Georgia 70,835 2.057 2 2 2

Kentucky 68,705 1.995 2 2 2

Rhode Island 68,446 1.988 2 2 2

Delaware 55,540 1.613 1 2 2

Totals 3,615,920 105 105 105 105


The point of the story being… Delaware should’ve gotten 2 seats.


Selected Sources COMAP, For All Practical Purposes, 7th ed.

(2006), Chapter 14 Wikipedia (multiple pages) http://www.usconstitution.net/const.html http://www2.census.gov/prod2/statcomp/

documents/1880-01.pdf http://www.ams.org/featurecolumn/archive/

apportion2.html


Photo Credits

Images: George Washington: http://www.morallaw.org/images/George%20Washington

%20portrait.gif Alexander Hamilton:

http://igs.berkeley.edu/library/hot_topics/2008/Dec.2008/Images/Alexander_Hamilton_portrait_by_John_Trumbull_1806.jpg

Jefferson: http://www.usnews.com/dbimages/master/3165/FE_DA_080128moore_vert_20410.jpg

Webster: http://en.wikipedia.org/wiki/Daniel_Webster Delaware: http://www.national5and10.com/images/grey%20Delawhere%20t-

shirt.JPG Virginia: http://wwp.greenwichmeantime.com/time-zone/usa/virginia/images/state-

flag-virginia.jpg Constitution:

http://cache.boston.com/bonzai-fba/Globe_Photo/2008/08/15/we__1218837534_8547.jpg

Veto: http://kraigpaulsen.com/blog/wp-content/uploads/2009/05/veto.png

apportionment schemes

Documents

apportionment schemesif

apportionment method

apportionment schemesmore

apportionment problem

apportionment schemeswhat

apportionment schemesexamplein

stateapportionment schemessept

apportionm schemessept