Transcript Apportionment Schemes - William & Mary Mathematics

Apportionment Schemes
Dan Villarreal
MATH 490-02
Tuesday, Sept. 15, 2009
But first…a quick PSA
Sept. 15, 2009
Apportionment Schemes
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.
Sept. 15, 2009
Apportionment Schemes
Example
In the 1974-75 NHL season, the Stanley Cup
Champion Philadelphia Flyers won 51 games, lost
18 games, and tied 11 games.
Won:
Lost:
Tied:
51
18
11
 63.75%
 22.5%
 13.75%



64%
23%
14%
But this adds up to 101%, an impossibility!
Sept. 15, 2009
Apportionment Schemes
Dramatis Personae






George Washington
Alexander Hamilton
Thomas Jefferson
Daniel Webster
Delaware
Virginia
Sept. 15, 2009
Apportionment Schemes
The Constitution
Amendment
14, Section 2:
“Representatives shall be apportioned among the several States according to
their respective numbers, counting the whole number of persons in each State”
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”
Sept. 15, 2009
Apportionment Schemes
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

Sept. 15, 2009
3,615,920
105
= 34,437 people/district
Apportionment Schemes
Standard Divisors


34,437 is our standard divisor for 1790.
More generally,
SDt =
Poptotal
HSt
Where HSt is the size of the House of
Representatives (or whatever overall body) for
year t.
Sept. 15, 2009
Apportionment Schemes
Quotas

The number of Congressional districts a state
should get is its quota:
Qi =

Popi
SDt
Take Delaware, for example…
Sept. 15, 2009
Apportionment Schemes
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.
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009Totals
3,615,920
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009Totals
3,615,920
105
97Schemes 112
Apportionment
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
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
Back to 1790
State
Population Quota
Virginia
630,560
Massachusetts
State
Frac. Part
18.310 .310
Kentucky
.995
475,327
13.803 .803
South Carolina
.989
Pennsylvania
432,879
12.570 .570
Rhode Island
.988
North Carolina
353,523
10.266 .266
Connecticut
.878
New York
331,589
9.629
.629
Massachusetts
.803
Maryland
278,514
8.088
.088
New York
.629
Connecticut
236,841
6.878
.878
Delaware
.613
South Carolina
206,236
5.989
.989
Pennsylvania
.570
New Jersey
179,570
5.214
.214
Vermont
.484
New Hampshire
141,822
4.118
.118
Virginia
.310
Vermont
85,533
2.484
.484
North Carolina
.266
Georgia
70,835
2.057
.057
New Jersey
.214
Kentucky
68,705
1.995
.995
New Hampshire
.118
Rhode Island
68,446
1.988
.988
Maryland
.088
Delaware
55,540
1.613
.613
Georgia
.057
Sept. 15, 2009
Frac. Part
Apportionment Schemes
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
Sept. 15, 2009Totals
3,615,920
105
97Schemes 112
Apportionment
105
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
Sept. 15, 2009Totals
3,615,920
105
97Schemes 112
Apportionment
105
Sept. 15, 2009
Apportionment Schemes
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
PopDE
Sept. 15, 2009
2
=
55,540
Apportionment Schemes
1
<
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…
Sept. 15, 2009
Apportionment Schemes
The Alabama Paradox

US population in 1880 was 49,369,595
Sept. 15, 2009
Apportionment Schemes
The Alabama Paradox


US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
Sept. 15, 2009
Apportionment Schemes
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
Sept. 15, 2009
Apportionment Schemes
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
Population
Alabama 1,262,794
Illinois
3,078,769
Texas
1,592,574
Sept. 15, 2009
Apportionment Schemes
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
A
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
A
8
18
9
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
7
19
10
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
7
19
10
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
7
19
10
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
7
19
10
The Alabama Paradox



US population in 1880 was 49,369,595
For HS = 299, SD = 165,116
For HS = 300, SD = 164,565
HS = 299
Population LQ Frac.
Alabama 1,262,794
7 .647
Illinois
3,078,769
18 .646
Texas
1,592,574
9 .645
Sept. 15, 2009
Apportionment Schemes
HS = 300
A LQ Frac.
8
7 .674
18 18 .708
9
9 .677
A
7
19
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.
Sept. 15, 2009
Apportionment Schemes
Thomas Jefferson

Biographical Information:
You know this all already…

Had the good fortune never to take a class in
Morton Hall
Sept. 15, 2009
Apportionment Schemes
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:
Popi
Ai =
d
⌊ ⌋
Sept. 15, 2009
Apportionment Schemes
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?
Sept. 15, 2009
Apportionment Schemes
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
di =
ni + 1
For
example, dVA = 630,560 / (18 + 1) = 33,187
dDE = 55,540 / (1 + 1) = 27,770
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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.
Sept. 15, 2009
Apportionment Schemes
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…
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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:
ui
Popi SD
SD
M=
=
/
=
×
=
Qi
d
SD
d
Popi d


Popi
Popi
Then ui = M * Qi => ai = ⌊M * Qi⌋
The rich only get richer…
Sept. 15, 2009
Apportionment Schemes
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 19001930
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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 diwill 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.
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Apportionment Schemes
Timeline
1790
Jefferson
Method
Hamilton
1840 1850
1900
Method
Webster
Method
1900
Webster
1940
Hill-Huntington
Method
Method
Sept. 15, 2009
Apportionment Schemes
Present
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
PVi,n =
(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.
Sept. 15, 2009
Apportionment Schemes
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/app
ortionment/00pvalues.txt
Sept. 15, 2009
Apportionment Schemes
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.
Sept. 15, 2009
Apportionment Schemes
The Population Paradox

Exercise 10, COMAP page 535:
State Old Census Q
A
5,525,381
B
3,470,152
C
3,864,226
D
201,203
Tot
13,060,962
Sept. 15, 2009
A
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q
A
5,525,381
B
3,470,152
C
3,864,226
D
201,203
Tot
13,060,962
Sept. 15, 2009
A
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q

A
5,525,381
B
3,470,152
C
3,864,226
D
201,203
Tot
13,060,962
A
SD = 13,060,962 / 100 = 130,610
Sept. 15, 2009
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
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
Sept. 15, 2009
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
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
Sept. 15, 2009
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
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
Tot
13,060,962
1
SD = 13,060,962 / 100 = 130,610
Sept. 15, 2009
Apportionment Schemes
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q
A
New Census Q
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
201,049
Tot
13,060,962
Sept. 15, 2009
1
13,251,770
Apportionment Schemes
A
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q

A
New Census Q
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
201,049
Tot
13,060,962
1
13,251,770
SD = 13,251,770 / 100 = 132,518
Sept. 15, 2009
Apportionment Schemes
A
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q

A
New Census Q
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
201,049
1.517
Tot
13,060,962
1
13,251,770
SD = 13,251,770 / 100 = 132,518
Sept. 15, 2009
Apportionment Schemes
A
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
State Old Census Q

A
New Census Q
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
201,049
1.517
Tot
13,060,962
1
13,251,770
SD = 13,251,770 / 100 = 132,518
Sept. 15, 2009
Apportionment Schemes
A
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
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
201,049
1.517
Tot
13,060,962
1
13,251,770
SD = 13,251,770 / 100 = 132,518
Sept. 15, 2009
Apportionment Schemes
2
The Population Paradox


Exercise 10, COMAP page 535:
House size set at 100
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
201,049
1.517
Tot
13,060,962
1
2
13,251,770
State D lost population, yet gained a seat!
Sept. 15, 2009
Apportionment Schemes
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?
Sept. 15, 2009
Apportionment Schemes
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.
Sept. 15, 2009
Apportionment Schemes
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
Sept. 15, 2009
Totals 3,615,920
105
105
Apportionment
Schemes
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
Sept. 15, 2009
Totals 3,615,920
105
105
Apportionment
Schemes
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
105
105
Sept. 15, 2009
Totals 3,615,920
105
105
Apportionment
Schemes
The point of the story being…

Delaware should’ve gotten 2 seats.
Sept. 15, 2009
Apportionment Schemes
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/documen
ts/1880-01.pdf
http://www.ams.org/featurecolumn/archive/apport
ion2.html
Sept. 15, 2009
Apportionment Schemes
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_H
amilton_portrait_by_John_Trumbull_1806.jpg
 Jefferson:
http://www.usnews.com/dbimages/master/3165/FE_DA_080128moore_vert_2
0410.jpg
 Webster: http://en.wikipedia.org/wiki/Daniel_Webster
 Delaware: http://www.national5and10.com/images/grey%20Delawhere%20tshirt.JPG
 Virginia: http://wwp.greenwichmeantime.com/timezone/usa/virginia/images/state-flag-virginia.jpg
 Constitution: http://cache.boston.com/bonzaifba/Globe_Photo/2008/08/15/we__1218837534_8547.jpg
 Veto: http://kraigpaulsen.com/blog/wp-content/uploads/2009/05/veto.png
Sept. 15, 2009
Apportionment Schemes