Paradoxes of Apportionment
The Alabama Paradox, Population
Paradox, and the New State’s
Paradox
What is a Paradox?
•
•
•
•
A seemingly contradictory statement that may
nonetheless be true: the paradox that standing is more
tiring than walking.
Something that exhibits inexplicable or contradictory
aspects: “The silence of midnight … rung in my ears” (Mary Shelley).
An assertion that is essentially self-contradictory,
though based on a valid deduction from acceptable
premises.
A statement contrary to received opinion.
- from www.dictionary.com
What is a Paradox?
•
We identify paradoxes of apportionment when we follow logical
steps and discover a result that may seem surprising because in
some way the result violates what we expect to be logical.
•
That is, the paradoxes arise when we follow logical steps and
derive a result that doesn’t seem logical.
•
Of course, the results we derive are perfectly logical in the sense
that they are the correct results of our deductions and it’s
important to recognize that in higher mathematics paradoxes are
often found in many areas and resolving these paradoxes opens
doors to new understanding.
Paradoxes of Apportionment
• Alabama Paradox – An increase in the number of
available items causes a group to lose an item (even
though populations remain the same).
• Population Paradox – Group A can lose an item to
group B even when the rate of growth of the population
of group A is greater than in group B.
• New States Paradox – The addition of a new group,
with a corresponding increase in the number of available
items, can cause a change in the apportionment of items
of among the other groups.
The Alabama Paradox
• Alabama Paradox – An increase in the number of
available items causes a group to lose an item (even
though populations remain the same).
• The Alabama Paradox first surfaced after the 1870 census. With
270 members in the House of Representatives, Rhode Island had 2
representatives but when the House size was increased to 280,
Rhode Island lost a seat. After the 1880 census, C. W. Seaton (chief
clerk of U. S. Census Office) computed apportionments for all House
sizes between 275 and 350 members. He then wrote a letter to
Congress pointing out that if the House of Representatives had 299
seats, Alabama would get 8 seats but if the House of
Representatives had 300 seats, Alabama would only get 7 seats.
The Population Paradox
• Population Paradox – Group A can lose an item to
group B even when the rate of growth of the population
of group A is greater than in group B.
• The Population Paradox was discovered around 1900, when it was
shown that a state could lose seats in the House of Representatives
in spite of a rapidly growing population. (Virginia was growing much
faster than Maine--about 60% faster--but Virginia lost a seat in the
House while Maine gained a seat.)
The New State’s Paradox
• New States Paradox – The addition of a new group,
with a corresponding increase in the number of available
items, can cause a change in the apportionment of items
among the other groups.
• The New States Paradox was discovered in 1907 when Oklahoma
became a state. Before Oklahoma became a state, the House of
Representatives had 386 seats. Comparing Oklahoma's population
to other states, it was clear that Oklahoma should have 5 seats so
the House size was increased by five to 391 seats. The intent was to
leave the number of seats unchanged for the other states. However,
when the apportionment was recalculated, Maine gained a seat (4
instead of 3) and New York lost a seat (from 38 to 37).
Example #1 – the Alabama Paradox
• Suppose Miami Dade College has 3 campuses and
enrollments at each campus are as given in the table
below.
Campus
North
Enrollment 10,170
Kendall
Wolfson
Total
9,150
680
20,000
• Suppose there are 40 full-time faculty members to be
distributed among these campuses according to their
enrollment.
Example #1 – the Alabama Paradox
Campus
Enrollment
North
10,170
Kendall
9,150
There are 3 groups (the campuses) and
40 items (the full-time faculty positions)
to be divided among them.
We will use Hamilton’s method to divide
the items so that each group gets a
proportional integer number of faculty.
(Faculty are less effective when cut into
pieces!)
Wolfson
Total
680
20,000
The standard divisor is
(total enrollment)/(number of items) =
20,000/40 = 500.
Next, we calculate the quotas for each
campus…
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Quota
North
10,170
10,170/500
20.34
Kendall
9,150
9150/500
18.3
Wolfson
680
680/500
1.36
Total
20,000
40
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Quota
Lower
Quota
North
10,170
10,170/50
0
20.34
20
Kendall
9,150
9150/500
18.3
18
Wolfson
680
680/500
1.36
1
Total
20,000
40
39
Add
remaining
seat
+1
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Lower
Quota
Add
remaining
seat
North
10,170
20.34
20
20
Kendall
9,150
18.3
18
18
Wolfson
680
1.36
1
Total
20,000
40
39
+1
Final
Apportionment
2
40
Example #1 – the Alabama Paradox
Campus
Enrollment
North
10,170
We conclude that, with 40 full-time
faculty, the College should make the
following apportionment:
North – 20 full-time faculty
Kendall
9,150
Wolfson
680
Total
20,000
Kendall – 18 full-time faculty
Wolfson – 2 full-time faculty
Example #1 – the Alabama Paradox
Campus
Enrollment
North
10,170
Kendall
9,150
Wolfson
680
Total
20,000
We concluded that, with 40 full-time
faculty, the College should make the
following apportionment:
North – 20 full-time faculty
Kendall – 18 full-time faculty
Wolfson – 2 full-time faculty
Now, suppose the College opens a new
line and decides to hire a new full-time
faculty member.
If the number of full-time faculty is
increased to 41, where will the new
faculty member be assigned? At which
campus will he or she teach?
Example #1 – the Alabama Paradox
Campus
Enrollment
The enrollments will remain the same.
North
10,170
The only change is to increase the
number of available full-time faculty.
Kendall
9,150
Now the standard divisor becomes
(total population)/(number of items) =
Wolfson
680
20,000/41 = 487.8
Total
20,000
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Quota
North
10,170
10170/487.8
20.85
Kendall
9,150
9150/487.8
18.76
Wolfson
680
680/487.8
1.39
Total
20,000
41
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Quota
Lower
Quota
Add
remaining
seats
North
10,170
10170/487.8
20.85
20
+1
Kendall
9,150
9150/487.8
18.76
18
+1
Wolfson
680
680/487.8
1.39
1
Total
20,000
41
39
Example #1 – the Alabama Paradox
Campus
Enrollment
Quota
Lower
Quota
Add
remaining
seats
Final
Apportionment
North
10,170
20.85
20
+1
21
Kendall
9,150
18.76
18
+1
19
Wolfson
680
1.39
1
1
Total
20,000
41
39
41
Example #1 – the Alabama Paradox
Campus
Enrollment
North
10,170
Kendall
9,150
Wolfson
680
Total
20,000
We concluded that, with 40 full-time
faculty, the College should make the
following apportionment:
North – 20 full-time faculty
Kendall – 18 full-time faculty
Wolfson – 2 full-time faculty
Now, with 41 full-time faculty, we
conclude that the College should make
the following apportionment:
North – 21 full-time faculty
Kendall – 19 full-time faculty
Wolfson – 1 full-time faculty
Example #1 – the Alabama Paradox
With 40 full-time faculty:
Campus
Enrollment
North
10,170
Kendall
9,150
Wolfson
680
Total
20,000
North – 20 full-time faculty
Kendall – 18 full-time faculty
Wolfson – 2 full-time faculty
With 41 full-time faculty:
North – 21 full-time faculty
Kendall – 19 full-time faculty
Wolfson – 1 full-time faculty
The Alabama Paradox has occurred! With
the addition of a new faculty member,
Wolfson loses a faculty member while North
and Kendall both gain one – all this in spite
of the fact that there were no changes in the
enrollments at any campus.