Chapter 4, Math 116, Notes
The Mathematics of Apportionment
1. Apportionment: To divide the assign in due and proper proportion or according to
some plan.
2. The States: This is the term we used to describe the "players" of the apportionment.
We can let
A , A , A ,... A
1
2
3
N
denote the number of players or states if you will.
3. The Seats: This term describes the set of M identical, indivisible, objects that are
being divided among the N states. For convenience, we will assume that M  N , (i.e.
that there are more seats then states.) This insures that each state can get at least one seat,
but doesn't mean that each state does get a seat!
4. The Populations: This is a set N positive numbers (for simplicity we will assume them
to be whole numbers) which are used as the basis for the apportionment of the seats to the
p , p , p ..., pN to denote the state's respective populations, and
states. We will use 1 2 3
"P" to denote the total population.
5. The Standard Divisor (SD): This is the ratio of population to seats. It gives us a unit
of measurement (SD people = 1 seat) for our apportionment calculations. Formally,
P
SD 
M .
6. The Standard Quotas: The standard quota of a state is the exact fractional number of
seats that the state would get if fractional seats were allowed. We will use the notation
q1 , q2 , q3 ,..., qN to denote the standard quota's of the respective states. To find a state's
p
qi  i
SD .
standard quota, we divide the state's population by the standard divisor:
7. Upper and Lower Quota: Associated with each of the standard quota's are two other
important numbers-the lower quota (the quota rounded down), and the upper quota (the
quota rounded up). In the unlikely event that the quota is a whole number, the lower and
upper quotas are the same. We can use "U's" to denote "Upper Quota's", and "L's" to
q  32.92
denote "Lower Quota's". For example i
, have a lower quota Li  32 , and an
upper quota u i  33 .
8. Hamilton's Method: Also known as Vinton's Method or the method of largest
remainders was used in the United States only between 1850 and 1900.
A. Calculate each states standard quota.
B. Give to each state (for the time being) its lower quota.
C. Given the surplus seats (one at a time) to the states with the largest fractional parts
until there are no more surplus seats.
Example: Hamilton's Method:
State
Population
A
B
C
D
E
F
Total
1, 646, 000
6, 936,000
154,000
2, 091,000
685,000
988,000
12,500,000
SD 
Step 1
Quota
32.92
138.72
3.08
41.82
13.70
19.76
250.0
Step 2 lower
quota
32
138
3
41
13
19
246
Fractional
parts
.92
.72
.08
.82
.70
.76
4.00
Step 3
Surplus
First
Last
Second
Third
4
Hamilton
Apportionment
33
139
3
42
13
20
250
1,646,000
12,500,000
 32.92
 50,000 q A 
50
,
000
250
.
9. The Quota Rule: Any State should not be apportioned a number of seats smaller than
its lower quota or largest than its upper quota. This is known as the quota rule!
10. The Alabama Paradox: The most serious flaw of the Hamilton's Method. This
paradox occurs when an increase in the total number of seats is apportioned, in and of
itself, forces a state to lose one of its seats. More seats for one population and less for
some other?
For example: An increase of a seat can cause some havocs.
State
Population
Bama
Tecos
Ilnos
Total
940
9030
10,030
20,000
SD 
Standard
Quota
9.4
90.3
100.3
200.0
Lower
Quota
9
90
100
199
Surplus
Apportionment
1
0
0
1
10
90
100
200.0
20,000
940
 100 q B 
 9.4
200
100
,
.
If now we increase the total seats "M" from 200 to 201 the next morning without any
increase in Population then the Standard Quota needs to be recalculated!
SD 
20,000
940
 99.5 q B 
 9.45
201
99.5
,
, producing a new table...
State
Population
Bama
Tecos
Ilnos
Total
940
9030
10,030
20,000
Standard
Quota
9.45
90.75
100.80
201
Lower
Quota
9
90
100
199
Surplus
Apportionment
0
1
1
2
9
91
101
201.0
Hamilton's Method also can fall victim to two other paradox's.
11. The Population Paradox: A state looses seats because it's population gets too big.
When state A loses a seat to state B even though the population of A grew at a higher rate
than the population of B.
12. The New-State Paradox: The addition of a new state with its fair share of seats can,
in and of itself, affect the apportionments of other states.
13. Jefferson's Method:
A. Find a "suitable divisor, D", ( A suitable divisor is a divisor that produces an
apportionment of exactly "M" seats when the quotas (populations divided by D) are
rounded down.
B. Each state is apportioned its lower quota.
For example: Jefferson's Method:
The first apportionment method used in the United States House of Representatives.
The hardest part of Jefferson's Method is step # 1, finding a suitable divisor D. So
approaches are very sophisticated see pg. 143 fig 4-1. We will use "Trial-and-Error"
Method for choosing "D".
Our target is a set of lower quotas whose sum is "M". For the sum of lower quotas to
equal M, we need to make the quotas somewhat bigger than the standard quotas. This
can only be accomplished by choosing a divisor somewhat smaller than the SD.