Fairness in Apportionment
How do you decide whether a
method for apportioning
representatives is fair?
Absolute Fairness
• If a state has a certain percentage of the
total population, it should have that same
percentage of seats in the House.
But, this usually results in a fractional
number of seats, called the ideal quota or
standard quota for the state. Thus, in
most cases we have to go up or down to
get an integer number or representatives.
Upper and Lower Quota
Suppose a state has an ideal quota of 8.87
representatives.
• Its upper quota will be 9.
• Its lower quota will be 8.
Satisfying Quota
• Any apportionment system which gives
each state no fewer representatives than
its lower quota and no more
representatives than its upper quota is
said to satisfy quota.
• An apportionment system which fails to do
this for even one state is said to violate
quota.
An example
A state has an ideal quota of 23.71.
If, under a particular apportionment system,
the state receives less than 23
representatives or more than 24
representatives, the system will have
violated quota.
What about our methods?
• Hamilton’s method never violates quota.
• Jefferson, Adams and Webster (and, in
fact, any divisor method) can violate
quota, although they do not always do so.
We will see an historical example of this
for Jefferson’s method on the next slide.
Jefferson ♥ NY!
• New York’s population, 1820 census, was
1,368,775 while the total population of the US
was 8,969,878.
• The house size was 213 so the standard divisor
was 8,969,878 ÷ 213 = 42,112. Thus, NY’s ideal
quota was 1,368,775 ÷ 42,112 = 32.503.
• Jefferson’s method, using a divisor of 39,900,
gave NY int[1,368,775 ÷ 39,900] = 34 seats!
This violated upper quota.
Jefferson’s woes continued
• If Jefferson’s method had been used (it
wasn’t) since 1850, it would have violated
quota in every apportionment!
Paradoxes
• Consider the following
Population
data for the Dale
North High 9061
County schools.
South High 7179
Valley High 5259
Meadow H. 3319
Ridge High 1182
Total
26,000
Getting the Ideal Quotas
• We want to consider how to apportion Councils
of size 26, 27 and 40 using Hamilton’s method.
26
27 40
Seat seat seat
Standard
Divisor
1000 963 650
Table of Ideal Quotas
26 seats
27 seats
40 seats
North
9.061
9.4095
13.94
South
7.179
7.455
+1
11.045
Valley
5.259
5.4613 +1
8.0908
Meadow
3.319 +1
3.4467
5.1062
Ridge
1.182
1.2275
1.8185 +1
+1
The Hamilton Apportionments
26 seats
27 seats
40 seats
North
9
9
14
South
7
8
11
Valley
5
6
8
Meadow
4
3
5
Ridge
1
1
2
26
27
40
Alabama Paradox
• Notice what happened to Meadow High when
Council size went from 26 to 27 seats! There
was no change in total population or in any
school population and yet Meadow lost a seat
when the Council size increased by 1!!
• This is called the Alabama paradox. (Alabama
would have lost a seat going from a 299
member house to a 300 member house after the
1880 census.) At this point Congress ditched the
Hamilton method!
Population Paradox
• This occurs when state X loses a seat to
state Y even though X’s population grew at
a higher rate than Y’s population!
• From 1900 to 1910 Virginia’s population
grew by 19,767 while Maine grew by only
4648. Also, Virginia’s population grew at a
60% faster rate, yet, under Hamilton’s
method, Virginia lost a seat to Maine!
Population Paradox: A simple
example
• We have 5 states: A,B,C,D,E with a total
population of 900 and a 50 seat assembly, making
the standard divisor d = 18. Using Hamilton, this
gives:
Population Paradox (cont.)
State
Population
Ideal Quota Apportion
A
150
8.33
8
B
78
4.33
4
C
173
9.61 +1
10
D
204
11.33
11
E
295
16.39 +1
17
Total
900
50
Population grows:
Only C and E grow
State
Population
Ideal Quota Apportion
A
150
8.25
8
B
78
4.29 +1
5
C
181
9.96 +1
10
D
204
11.22
11
E
296
16.28
16
Total
909
50
Look what happened!
• State E, which gained population, lost a
seat to state B, which did not gain
population!!
New States Paradox
• The addition of a new state, with its fair
share of representatives can change the
apportionments of other states.
• In 1907 Oklahoma joined the Union with 5
seats. Using Hamilton’s method to
reapportion the House would have NY
giving a seat to Maine, even though
neither state (or any other) had a
population change!
Are there any apportionment
methods that do not violate quota
and avoid all paradoxes??
• (Balinsky and Young, 1980) There is no
perfect apportionment method! Any
method which does not violate quota
must produce paradoxes, and any
method that does not produce
paradoxes can violate quota.
• No apportionment method is
completely fair!
What about our methods?
• Hamilton satisfies quota but produces
paradoxes (Alabama, population, new
states)
• Jefferson, Adams, Webster (and all divisor
methods) avoid all paradoxes, but can
violate quota.