Apportionment, Continued
Apportionment II
1/26
Last Time
Last time we discussed the issue of apportioning the U.S. House of
Representatives, and we discussed the method Alexander Hamilton
invented. We’ll review it with the example of the 1791
apportionment. We will also discuss a method invented by Thomas
Jefferson, which was the method used in 1791.
We will look at some problems with Hamilton’s method and will
introduce other methods of apportionment.
Apportionment II
2/26
The quota is the exact amount that would be allocated to a state if a
whole number was not required.
The quota for a state is
Population of a State
· Number of House Members
Population of the U.S.
The lower quota of a state is the whole number part of the quota,
which is obtained by rounding the quota down to the next whole
number. The upper quota is the whole number you get by rounding
the quota up to the next whole number.
Apportionment II
3/26
Last time we saw that Hamilton’s method allows a paradox, called the
Alabama Paradox: It is possible to increase the number of seats in the
House and have a state get fewer seats because of this. In 1880 this
was discovered because this would have happened to Alabama in
going from 299 to 300 seats.
A second paradox can occur with Hamilton’s method. It is the
Population Paradox: It is possible for a state’s population to increase
and its apportionment to decrease, while another state’s population
decreases and its apportionment increases.
Apportionment II
4/26
Here is an example to show the population paradox. Suppose that a
country has four states, A, B, C, and D, and that it has a Parliament
of 100 members. Suppose that Hamilton’s method of apportionment
is used. We will apportion Parliament witH 1990 census data, and
reapportion it with 2000 census data.
State
A
B
C
D
Totals
1990 Population
5,525,381
3,470,152
3,864,226
201,203
13,060,962
2000 Population
5,657,564
3,507,464
3,885,693
201,049
13,251,770
Note that State A, B, and C’s population increased and State D’s
population decreased.
Apportionment II
5/26
State
A
B
C
D
Totals
1990 Population
5,525,381
3,470,152
3,864,226
201,203
13,060,962
Lower Quota
42
26
29
1
98
Fractional Part
0.305
0.569
0.586
0.540
State
A
B
C
D
Totals
1990 Population
5,525,381
3,470,152
3,864,226
201,203
13,060,962
Lower Quota
42
26
29
1
98
Apportionment
42
27
30
1
100
Apportionment II
6/26
Clicker Question
Given the 2000 population data, which two states will receive an extra
seat?
State 2000 Population Lower Quota Fractional Part
A
5,657,564
42
0.692
B
3,507,464
26
0.468
C
3,885,693
29
0.322
D
201,049
1
0.517
Totals
13,251,770
98
A A and B
D B and C
B A and C
E B and D
C A and D
Apportionment II
7/26
The correct answer is C, because A and D should get an extra seat.
State 2000 Population Lower Quota Fractional Part
A
5,657,564
42
0.692
B
3,507,464
26
0.468
C
3,885,693
29
0.322
D
201,049
1
0.517
Totals
13,251,770
98
State
A
B
C
D
Totals
Apportionment II
2000 Population
5,657,564
3,507,464
3,885,693
201,049
13,251,770
Lower Quota
42
26
29
1
98
Apportionment
43
26
29
2
100
8/26
Comparison of the 1990 and 2000 Apportionment
State
A
B
C
D
Totals
1990 Pop.
5,525,381
3,470,152
3,864,226
201,203
13,060,962
2000 Pop.
5,657,564
3,507,464
3,885,693
201,049
13,251,770
1990 App.
42
27
30
1
100
2000 App
43
26
29
2
100
While State B’s population increased, its apportionment decreased.
At the same time, State D’s population decreased but its
apportionment increased.
Apportionment II
9/26
Thomas Jefferson’s Method
Apportionment II
10/26
Because President Washington vetoed the bill to make Hamilton’s
method of apportionment law, another method was needed. Thomas
Jefferson, the third president of the U.S., created a method, and this
method was used in the first apportionment in 1791, and was used
until 1842.
Jefferson’s method requires the choice of a divisor, a whole number d.
Jefferson interpreted d as the minimum population of a congressional
district.
Jefferson computes the quota of a state by dividing its population by
d. He then rounds the quota down to get the apportionment.
Apportionment II
11/26
Clicker Question
Q Suppose we have the following population data and we are going to
to try to use d = 50.
State Population
A
105
B
200
C
50
What would State A’s apportionment be? You get it by dividing the
state’s population by d and round down.
A We calculate 105/50 = 2.1. The apportionment is then 2.
Apportionment II
12/26
For the 1791 apportionment, Jefferson used d = 33,000.
For example, using the 1790 census data, New York’s population was
331,589. We then calculate
331,589
331,589
=
= 10.048
d
33,000
Rounding down, New York then receives 10 seats.
Apportionment II
13/26
Vermont’s population in 1790 was 85,533. Its quota was then
calculated as
85,533
= 2.59.
33,000
Vermont then received 2 seats, since that is what we get by rounding
2.59 down to the nearest whole number.
The following table shows how the 1791 House was apportioned with
Jefferson’s method.
Apportionment II
14/26
Apportionment in 1791 via Jefferson’s Method
Apportionment II
15/26
The value of d must be chosen in such a way that using Jefferson’s
method apportions the house into the correct number of seats.
This value of d can by found by trial and error. You can experiment
with the spreadsheet Apportionment.xlsx to find the correct value of
d.
Apportionment II
16/26
For the 1990 population, we would need to choose a different value of
d to come out with the correct house size of 435 members.
With the appropriate value of d, taking data from the 1990 census,
several states would be apportioned according to the following table.
Apportionment of the house with Jefferson’s Method
State
1990
Population / d Apportionment
Population
California
29,839,250
54.65
54
Montana
803,655
1.47
1
New Mexico
1,521,779
2.79
2
New York
18,044,505
33.05
33
Texas
17,059,805
31.25
31
Apportionment II
17/26
As we see below, Jefferson’s method apportions the house differently
than the method currently used.
Apportionment of the house, Jefferson’s Method
versus the current method
State
Apportionment via
Actual
Jefferson’s Method Apportionment
California
54
52
Montana
1
1
New Mexico
2
3
New York
33
31
Texas
31
30
Apportionment II
18/26
Jefferson’s method generally benefits larger states. This is possibly
the reason other methods were introduced into Congress. John
Quincy Adams, after stepping down as President and joining Congress
as a representative of Massachusetts, introduced a method, which
never was accepted.
In the same year, Webster, a senator from Massachusetts, introduced
his method. It took 10 years for Congress to finally approve it.
Following the 1840 census, an Ohio representative introduced
Hamilton’s method. It was approved, along with a bill to change the
house size to 234. This size results in both Hamilton and Webster’s
method giving the same apportionment.
Apportionment II
19/26
The Hill-Huntington Method
Webster’s method was used in the 1840’s before being replaced by
Hamilton’s method. It was reintroduced in 1900 and used until 1940.
Our current method of apportionment, credited to Joseph Hill and
Edward Huntington, has been used to apportion the U.S. House of
Representatives since 1940. Hill was a statistician in the Census
Bureau and Huntington was a mathematics professor at Harvard.
This method was developed around 1911 and was a competitor of
Webster’s method for nearly 30 years before it was approved. It, like
other methods of apportionment, comes down to how does one round
quotas.
Apportionment II
20/26
Comparison Between Three Methods
Apportionment II
21/26
Clicker Question
Which method do you think would be the most fair for apportioning the
1791 house?
A Jefferson’s Method
B Webster’s Method
C The Hill-Huntington Method
Apportionment II
22/26
One feature of Jefferson’s method is that it is possible for a state to
receive more than its upper quota. This is an indication that
Jefferson’s method favors large states.
It then violates the quota condition: Each state receives either its
lower quota or its upper quota.
Hamilton’s method satisfies the quota condition, since each state
receives either its lower or its upper quota.
Apportionment II
23/26
We saw that Hamilton’s method produces paradoxes, but Jefferson’s
method violates the quota condition.
It turns out that the Hill-Huntington method also violates the quota
condition.
Maybe there is an apportionment method that does not produce
paradoxes and satisfies the quota condition.
Apportionment II
24/26
Unfortunately, not.
Balinski and Young showed in 1982 that any apportionment method
that does not violate the quota rule must produce paradoxes, and any
apportionment method that does not produce paradoxes must violate
the quota rule.
Therefore, it is impossible for a method to satisfy the quota rule and
have no paradoxes.
Deciding on methods of apportionment will then remain a political
problem, ultimately.
Apportionment II
25/26
Next Time
On Friday we will watch the short movie Flatland. This movie is
based on the book Flatland, written by Edwin Abbott and published
in the 1880’s. A free copy of the book can be found at
www.geom.uiuc.edu/∼banchoff/Flatland
Apportionment II
26/26